William Fulton
Intersection
Theory
Second Edition
Springer
Berlin
Heidelberg
New York
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara trgSk^i o
Singapore Vw Springer
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William Fulton
Department of Mathematics
University of Chicago
Chicago, IL 60637
USA
e-mail: fulton@math.uchicago.edu
Preface to the Second Edition
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Fulton, William: Intersection theory / William Fulton. - 2. ed. - Berlin; Heidelberg; New York;
Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo:
Springer, 1998
(Ergebnisse der Mathematik und ihrer Grenzgebiete; Folge 3, Vol. 2)
ISBN3-540-62046-X
We thank P. Aluffi, P. Belorousski, G. Ellingsrud, L. van Gastel, H. Gillet,
B. Gross, G. Kennedy, S. Kimura, S. Kleiman, K. Kurano, K. F. Lai, F. Oort,
D. Perkinson, D. Ramakrishnan, W. Raskind, N. Ring, M. Saito, C. Soule,
H. Tamvakis, A. Vistoli, W. Vogel, S. Xambo, and some anonymous critics for
supplying corrections.
No attempt has been made to survey the many developments in intersection
theory since 1983, other than adding some references which appeared not long
after first edition. A few indications to more recent literature, as well as an
informal introduction to the main ideas of this book, can be found in the 1996
edition of the author's Introduction to Intersection Theory in Algebraic Geometry,
CBMS 57, Amer. Math. Soc, 1984, 1996.
October, 1997
William Fulton
Mathematics Subject Classification A991): 14C17,14-02,14C10,14C15,
14C25,14C40,14E10, i4Mi2,14M15,14N10,55N45
ISSN 0071-1136
ISBN 3-540-62046-X Springer-Verlag Berlin Heidelberg New York
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Preface to the First Edition
From the ancient origins of algebraic geometry in the solution of polynomial
equations, through the triumphs of algebraic geometry during the last two cen-
centuries, intersection theory has played a central role. Since its role in founda-
tional crises has been no less prominent, the lack of a complete modern treatise
on intersection theory has been something of an embarrassment. The aim of
this book is to develop the foundations of intersection theory, and to indicate
the range of classical and modern applications. Although a comprehensive his-
history of this vast subject is not attempted, we have tried to point out some of the
striking early appearances of the ideas of intersection theory.
Recent improvements in our understanding not only yield a stronger and
more useful theory than previously available, but also make it possible to devel-
develop the subject from the beginning with fewer prerequisites from algebra and
algebraic geometry. It is hoped that the basic text can be read by one equipped
with a first course in algebraic geometry, with occasional use of the two appen-
appendices. Some of the examples, and a few of the later sections, require more spe-
specialized knowledge. The text is designed so that one who understands the con-
constructions and grants the main theorems of the first six chapters can read other
chapters separately. Frequent parenthetical references to previous sections are
included for such readers. The summaries which begin each chapter should fa-
facilitate use as a reference.
Several theorems are new or stronger than those which have appeared be-
before, and some proofs are significantly simpler. Among the former are a new
blow-up formula, a stronger residual intersection formula, and the removal of a
projective hypotheses from intersection theory and Riemann-Roch theorems;
the latter includes the proof of the Grothendieck-Riemann-Roch theorem.
Some formulas from classical enumerative geometry receive a first modern or
rigorous proof here.
Acknowledgements. The intersection theory described here was developed
together with R. MacPherson. The author whose name appears on the cover is
responsible for the presentation of details, and many of the applications and
examples, but the extent to which it forms a coherent theory derives from collab-
collaboration with MacPherson. Previously unpublished results of R. Lazarsfeld,
and joint work with Lazarsfeld, and with H. Gillet, is also included. During the
course of the work, many helpful suggestions were made by A. Collino, P. De-
ligne, S. Diaz, J. Harris, B. Iversen, S. L. Kleiman, A. Landman, Lazarsfeld, and
J-P. Serre. Although other contributions and historical precedents are acknowl-
acknowledged in the text, many others, such as those of students and others who have
responded to talks on these subjects, must be silently, but gratefully, cited.

VIII
Preface to the First Edition
This undertaking was made possible by the support of several foundations
and institutions. The Guggenheim Foundation provided a fellowship in
1980—81, the Sloan Foundation provided support in 1981-82, and grants have
been received from the National Science Foundation during six years of re-
research and writing on this subject. The support and hospitality of several institu-
institutions and their staffs has been equally vital: Mathematisk Institut of the Univer-
University of Axhus, Denmark A976—77); Institute des Hautes Etudes Scientifiques,
Bures-sur-Yvette, France A981); Institute for Advanced Study, Princeton
A981—82); and Brown University. A summer course in Cortona, Italy in 1980
provided a chance to test a preliminary version of the first portion of the book.
Thanks are due to the staffs at the IAS and Brown, especially to K. Jacques, for
expert typing, and to the publishers for their cooperation.
Contents
Introduction 1
Chapter 1. Rational Equivalence 6
Summary 6
1.1 Notation and Conventions 6
1.2 Orders of Zeros and Poles 8
1.3 Cycles and Rational Equivalence 10
1.4 Push-forward of Cycles 11
1.5 Cycles of Subschemes 15
1.6 Alternate Definition of Rational Equivalence 15
1.7 Flat Pull-back of Cycles 18
1.8 An Exact Sequence 21
1.9 Affine Bundles 22
1.10 Exterior Products 24
Notes and References 25
Chapter 2. Divisors 28
Summary 28
2.1 Cartier Divisors and Weil Divisors 29
2.2 Line Bundles and Pseudo-divisors 31
2.3 Intersecting with Divisors • . . . 33
2.4 Commutativity of Intersection Classes 35
2.5 Chern Class of a Line Bundle 41
2.6 Gysin Map for Divisors 43
Notes and References 45
Chapter 3. Vector Bundles and Chern Classes 47
Summary 47
3.1 Segre Classes of Vector Bundles 47
3.2 Chern Classes . . • 50
3.3 Rational Equivalence on Bundles 64
Notes and References 68
Chapter 4. Cones and Segre Classes 70
Summary 70
4.1 Segre Class of a Cone 70

X Contents
4.2 Segre Class of a Subscheme 73
4.3 Multiplicity Along a Subvariety 79
4.4 Linear Systems 82
Notes and References 85
Chapter 5. Deformation to the Normal Cone 86
Summary 86
5.1 The Deformation 86
5.2 Specialization to the Normal Cone 89
Notes and References 90
Chapter 6. Intersection Products 92
Summary 92
6.1 The Basic Construction 93
6.2 Refined Gysin Homomorphisms 97
6.3 Excess Intersection Formula 102
6.4 Commutativity 106
6.5 Functoriality 108
6.6 Local Complete Intersection Morphisms 112
6.7 Monoidal Transforms 114
Notes and References 117
Chapter 7. Intersection Multiplicities 119
Summary 119
7.1 Proper Intersections 119
7.2 Criterion for Multiplicity One 126
Notes and References 127
Chapter 8. Intersections on Non-singular Varieties 130
Summary 130
8.1 Refined Intersections 130
8.2 Intersection Multiplicities 137
8.3 Intersection Ring 140
8.4 Bezout's Theorem (Classical Version) 144
Notes and References 151
Chapter 9. Excess and Residual Intersections 153
Summary 153
9.1 Equivalence of a Connected Component 153
9.2 Residual Intersection Theorem 160
9.3 Double Point Formula 165
Notes and References 171
Contents XI
Chapter 10. Families of Algebraic Cycles 175
Summary 175
10.1 Families of Cycle Classes 176
10.2 Conservation of Number 180
10.3 Algebraic Equivalence 185
10.4 An Enumerative Problem 187
Notes and References 193
Chapter 11. Dynamic Intersections 195
Summary 195
11.1 Limits of Intersection Classes 196
11.2 Infinitesimal Intersection Classes 198
11.3 Limits and Distinguished Varieties 200
11.4 Moving Lemmas 205
Notes and References 209
Chapter 12. Positivity 210
Summary 210
12.1 Positive Vector Bundles 211
12.2 Positive Intersections 218
12.3 Refined Bezout Theorem 223
12.4 Intersection Multiplicities 227
Notes and References 234
Chapter 13. Rationality 235
Summary 235
Notes and References 241
Chapter 14. Degeneracy Loci and Grassmannians 242
Summary 242
14.1 Localized Top Chern Class 244
14.2 Gysin Formulas 247
14.3 Determinantal Formula 249
14.4 Thom-Porteous Formula 254
14.5 Schur Polynomials 263
14.6 Grassmann Bundles 266
14.7 Schubert Calculus 271
Notes and References 278
Chapter 15. Riemann-Roch for Non-singular Varieties 280
Summary 280
15.1 Preliminaries 280
15.2 Grothendieck-Riemann-Roch Theorem 286
15.3 Riemann-Roch Without Denominators 296

XII
Contents
15.4 Blowing up Chern Classes 298
Notes and References 302
Chapter 16. Correspondences 305
Summary 305
16.1 Algebra of Correspondences 305
16.2 Irregular Fixed Points 315
Notes and References 318
Chapter 17. Bivariant Intersection Theory 319
Summary 319
17.1 Bivariant Rational Equivalence Classes 320
17.2 Operations and Properties 322
17.3 Homology and Cohomology 324
17.4 Orientations 326
17.5 Monoidal Transforms 332
17.6 Residual Intersection Theorem . . .' 333
Notes and References 337
Chapter 18. Riemann-Roch for Singular Varieties 339
Summary 339
18.1 Graph Construction 340
18.2 Riemann-Roch for Quasi-projective Schemes 348
18.3 Riemann-Roch for Algebraic Schemes 353
Notes and References 368
Chapter 19. Algebraic, Homological and Numerical Equivalence . . . 370
Summary 370
19.1 Cycle Map 371
19.2 Algebraic and Topological Intersections " . . . 378
19.3 Equivalence on Non-singular Varieties 385
Notes and References 391
Chapter 20. Generalizations 393
Summary 393
20.1 Schemes Over a Regular Base Scheme 393
20.2 Schemes Over a Dedekind Domain 397
20.3 Specialization 398
20.4 Tor and Intersection Products 401
20.5 Higher ^-theory 403
Notes and References 404
Appendix A. Algebra 406
Summary 406
Al Length 406
Contents
XIII
A2 Herbrand Quotients 407
A3 Order Functions 411
A4 Flatness . 413
A.5 Koszul Complexes 414
A.6 Regular Sequences 416
A.7 Depth 418
A.8 Normal Domains 419
A.9 Determinantal Identities 419
Notes and References 425
Appendix B. Algebraic Geometry (Glossary) 426
B. 1 Algebraic Schemes 426
B.2 Morphisms 427
B.3 Vector Bundles 430
B.4 Cartier Divisors 431
B.5 Projective Cones and Bundles 432
B.6 Normal Cones and Blowing Up 435
R7 Regular Imbeddings and l.c.i. Morphisms 437
B.8 Bundles on Imbeddable Schemes 439
B.9 General Position 440
Bibliography 442
Notation 462
Index 464

Introduction
A useful intersection theory requires more than the construction of rings of
cycle classes on non-singular varieties. For example, if A and B are sub-
varieties of a non-singular variety X, the intersection product A ¦ B should be
an equivalence class of algebraic cycles closely related to the geometry of how
A OB, A and B are situated in X. Two extreme cases have been most familiar.
If the intersection is proper, i.e., dim {A 0 B) — dim A + dim B — dim X, then
A • B is a linear combination of the irreducible components of Af]B, with
coefficients the intersection multiplicities. At the other extreme, if A = B is a
non-singular subvariety, the self-intersection formula says that A ¦ B is re-
represented by the top Chern class of the normal bundle of A in X. In each case
A ¦ B is represented by a cycle on A 0 B, well-defined up to rational
equivalence on AC]B. One consequence of the theory developed here is a
construction of, and formulas for, the intersection product A ¦ B as a rational
equivalence class of cycles on Af]B, regardless of the dimensions of the
components of Af]B. We call such classes refined intersection products.
Similarly other intersection formulas such as the Giambelli-Thom-Porteous
formulas for the degeneracy loci of a vector bundle homomorphism, are
constructed on and related to the geometry of these loci, including the cases
where the loci have excess dimensions.
To give an idea of the main thrust of the text, we sketch what we call the
basic construction, from which such refined classes are derived. To a closed
regular imbedding i:X^> Y of codimension d, and a morphism/: V-* Y, with
V a fc-dimensional variety (or any purely fc-dimensional scheme), this con-
construction produces a rational equivalence class of (k — d)-cycles on W=f~l (X).
This intersection class, denoted XYV, can be formed as follows. Since i is a
regular imbedding, the normal cone to X in Y is a vector bundle; let TV denote
the pull-back of this bundle to W. The normal cone C to Win Kis a fc-dimen-
sional closed subscheme of N. Using the lengths of local rings of C along its
irreducible components as coefficients, C determines an algebraic k-cyde,
denoted [C], on N. One may construct X-YV by intersecting [C] with the zero
section of N. Thus a (k — d)-cycle Sw^Zj] on W represents XYV if
Yjmi[Nz] is rationally equivalent to [C] on N, where NZt is the restriction of
NtoZj.
Three situations show the utility of this construction: A) If X is a d-
dimensional non-singular variety, the diagonal imbedding of I in Ixl is
regular. With Y=XxX, V=AxB the Cartesian product of subvarieties A, B
of X, the construction determines the intersection class A ¦ B on AC\B. In
particular, this determines the ring structure on the rational equivalence

2 Introduction
classes on X. B) If //,,..., Hd are effective Cartier divisors on a variety X, and
V is a subvariety of X, the product imbedding of H\ x ... x Hd in X x ... x X is
regular. If / is the diagonal imbedding of V in X x ... x X, the construction
determines a class on Hl(\...[\HiC\V. This is useful in enumerative
geometry, where X parametrizes geometric figures, and the hypersurfaces //,
represent "simple conditions" on the figures. C) If E is a vector bundle of rank
d on a variety X, any section i of E is a regular embedding of X in E. Applying
the construction to Y=E, V=X, /the zero section, this produces a class on the
zero-scheme of i which represents the top Chern class cd(E) n [X] on X. This
is used on Grassmann and flag bundles to represent determinantal formulas
by cycles on degeneracy loci.
To use the basic construction one needs to verify several properties which
are not obvious from this description. For example, if /: K-> Y is also a
regular imbedding, then there is a commutativity property: V ¦ YX = X • YV.
This assures, that intersection products which could be formed in different
ways, e.g. by A) or B) above, lead to the same classes. One also needs to
know that the construction passes to rational equivalence. Precisely, with
i.X^Y a. regular imbedding of codimension d, f:Y'-*Y any morphism, and
a = ? «i[^j] any algebraic /c-cycle on Y', define a class i'a of (k — d)-cycles on
X' = f~i(X) by setting i'a = ? ni(X-rK)- W a' IS rationally equivalent to a on
Y', it must be shown that i a' is equal to r a; the analogous statement with
rational equivalence replaced by algebraic equivalence refines the "principle
of continuity". A third important property is functoriality (or associativity) of
this construction: if i: X -> Y, j : Y -* Z are regular imbeddings, then j ° i is
also, and (J ° i)! = t ° f. The homomorphisms r refine the Gysin homo-
morphisms i* :AkY^> Ak-d X which had been constructed by Verdier.
In addition, one needs formulas for these classes. Any cone C on a scheme
W determines a Segre class s(C) in the group of rational equivalence classes
on W. In case C = E is a vector bundle, this Segre class is dual to the inverse
total Chern class of E: s(E) = c(E)~l n [W]. In the situation of the basic
construction, one then has the formula
X-YV = {c(N)ns(C)}k_d.
where {}m denotes the m-dimensional component of the term in braces. When the
imbedding of W in V is also regular, this gives an excess intersection formula for
XrV. By replacing V by the blow-up of V along W, one may always reduce to
this situation. Another basic property expresses the compatibility of these classes
with push-forward by proper morphisms. The value of allowing V-* Y to be a
morphism which is not an imbedding is evident in this reduction, as is the need
to allow V to have arbitrary singularities.
The basic construction, properties, and formulas are based on joint work
with R. MacPherson, and work of J.-L. Verdier. Originally these sources
depended on the previously constructed "Chow rings" of cycle classes on non-
singular quasi-projective varieties, as developed by Severi, B. Segre, Todd,
Chevalley, Chow, Samuel, Weil, Grothendieck, et al. It was indicated in
Fulton-MacPherson A), however, how one could use the basic construction to
develop intersection theory from scratch. This is the program carried out in
Introduction 3
Chapters 1 — 8 here. Note that in this program one also has no need for a
preliminary study of intersection multiplicities. In the case of proper inter-
intersections, the intersection class is automatically a well-defined cycle, whose
coefficients are then the intersection multiplicities; indeed, one sees readily
that in this case the construction agrees with that of Samuel A). Although the
refined intersection classes respect variation in families of cycles, no moving
lemma, or quasi-projective hypotheses, are needed. In a sense the approach is
rather close to that of B. Segre D), and relies on explicit deformations, and
blowing up, rather than an abstract moving lemma. Ideas related to this point
of view have also been published by H. Gillet, J. P. Jtfuanolou, J. King, A. T.
Lascu, D. Mumford, J. P. Murre, and D. B. Scott. Work in intersection theory
by S. L. Kleiman, D. Laksov, and R. Piene has particularly influenced this
book.
Outline. The first chapter contains the definition of the group Ak X of
rational equivalence classes of algebraic fc-cycles on an algebraic scheme X,
and the verification that the natural definition of push-forward of cycles makes
Ak a covariant functor for proper morphisms. In addition, a flat morphism of
relative dimension n determines pull-back homomorphisms, raising dimen-
dimensions by n. In the second chapter the basic construction is studied in the case of
codimension one. This includes a construction of a first Chern class for line
bundles. In Chapter 3 Chern classes Cj(E) are constructed as homomorphisms
a -> Cj(E) n a from AkX to Ak-tX, for E a vector bundle on an algebraic
scheme X. The expected formulas are proved for these Chern classes; they are
used to prove that the pull-back from AkX to Ak+rE, r = rank(?), is an
isomorphism. The first Chern class is also used to construct the Segre class of a
cone, studied in the next chapter; this includes the notion of the multiplicity of
a scheme along a subvariety. In Chapter 5 the deformation to the normal cone
is constructed. This is a rational family of closed imbeddings containing a
given imbedding W -> V and the zero-section imbedding W -> C in the
normal cone C to W in V. The existence of such a deformation, together with
the "principle of continuity", helps explain the key role of normal cones in the
construction of intersection products. Chapter 6 contains the general construc-
construction and basic properties of the intersection products XrV and classes i'a.
This chapter also contains a new general blow-up formula for the pull-back of
cycles by a monoidal transformation.
The rest of the book consists of largely independent applications of the first
six chapters. The next two chapters consider the special cases of proper
intersections (intersection multiplicities) and intersections on non-singular
varieties. In Chapter 9 we prove a residual intersection theorem, more general
than those previously available. With notation as in the basic construction, and
Z a given closed subscheme of the intersection scheme W, this formula writes
the intersection class X rV as a sum of a class on Z and a class on a residual
set R with ZUR = W. Following Laksov C), this is applied to deduce a general
double point formula.
Chapter 10 considers the variation of intersection classes in families,
including a strong form of the principle of continuity. The decomposition of
the normal cone into its irreducible components determines a decomposition

4 Introduction
of the intersection product. Chapter 11 includes R. Lazarsfeld's infinitesimal
construction of intersection classes, and his proof that the decomposition
constructed by decomposing the normal cone agrees with that obtained by a
dynamic method, along lines suggested by Severi.
From our construction of intersection classes, various positivity or ample-
ness hypotheses on the normal bundle to X in Y can be expected to force
corresponding positivity of intersection classes XyV. Such theorems, and
applications to a refined Bezout's theorem, and inequalities for intersection
multiplicities, are discussed in Chapter 12; this is joint work with Lazarsfeld.
Similarly, since the construction is valid over an arbitrary field, and produces
classes on the loci of interest, it can be used to prove the existence of rational
solutions to algebraic equations (Chapter 13).
Formulas for degeneracy loci are among the most important applications of
intersection theory. Our method, combined with ideas from Kempf-Laksov
A), gives refinements of the usual formulas, producing classes on the degener-
degeneracy loci in question, and valid on possibly singular varieties (Chapter 14). The
classical Schubert calculus is deduced from these formulas. In Chapter 15 the
geometry of the deformation to the normal bundle is used to give a short
conceptual proof of the Grothendieck-Riemann-Roch theorem, as well as the
formula for blowing up Chern classes. The basic algebra of correspondences is
included in Chapter 16; our intersection formulas yield classical formulas of
Fieri and Severi for the virtual number of fixed points of a correspondence,
when the fixed point locus is infinite.
The bivariant language of Fulton-MacPherson C) is introduced in Chapter
17. This codifies and strengthens the machinery of Chapters 1-8. Indeed, the
book could be considerably shortened by an uncompromising use of this
formalism throughout, but at the cost of its usefulness as a reference for those
unfamiliar with it. In Chapter 18 it plays a key role in the analysis of
MacPherson's graph construction, which is used to extend Riemann-Roch to
singular quasi-projective varieties, as in Baum-Fulton-MacPherson A) and
Verdier E), In addition, recent joint work with H. Gillet is included, which
removes all quasi-projective hypotheses from these Riemann-Roch theorems.
Chapter 19 shows that, for complex varieties, the cycle map, from
algebraic cycle classes to homology classes, is compatible with (refined) inter-
intersection products; included is a brief survey comparing rational, algebraic,
homological and numerical equivalence of cycles on non-singular complex
projective varieties. The final chapter sketches generalizations of the preceding
chapters to schemes over Dedekind domains and other non-algebraic base
schemes. Serre's intersection multiplicity, and Bloch's formula relating rational
equivalence to higher A^-theory, are also mentioned.
Appendix A contains the commutative algebra needed for Chapters 1-6,
together with references for a few facts used later; this appendix can be
consulted as required, and need not be read as prerequisite. Appendix B is a
glossary of basic concepts and constructions needed from algebraic geometry;
it is hoped that occasional use of Appendix B will help bridge gaps between
the language of various introductory treatments of algebraic geometry and that
used here. In addition a few special conventions adopted here are pointed out.
Introduction 5
Among these are the following, to be understood otherwise indicated: schemes
are algebraic schemes over an arbitrary field K; varieties are irreducible and
reduced schemes; points are closed points; subvarieties and imbeddings are
assumed to be closed; cycles are algebraic cycles, i.e., integral linear combina-
combinations of subvarieties; non-singular varieties are smooth over K; flat morphisms
are assumed to have some relative dimension; P(E) denotes the projective
bundle of lines in a bundle E.
A substantial portion of the book consists of Examples at the end of
sections. As one would expect, these include illustrations and special cases of
the theorems, and classical and modern applications. In addition, there are
generalizations of the theorems, or counterexamples to possible generaliza-
generalizations. Some examples, such as a series on intersections of plane curves in the
first chapter, are included primarily to motivate later developments. Unless
otherwise indicated, the proofs of assertions in examples should be reasonably
straight-forward from the preceding text. Hints are included in parentheses.
References preceded by "cf." indicate where similar results may be found,
although often with a different approach. References without "cf", or with the
more direct "see", indicate a closer relation of the example to the reference,
which may be consulted for details. An unspoken assumption in computational
examples is that the ground field is algebraically closed of characteristic zero;
the interested reader may make the necessary modifications in positive
characteristics.
At the end of each chapter a Notes and References section contains some
historical remarks on material related to the chapter, and an attempt to
attribute sources for the main ideas. Many other references may be found in
the examples. Although it is hoped that an impression of the interesting
history of intersection theory emerges from these notes and the examples, a
thorough historical analysis is beyond the capacity of this book or its author.
For similar reasons, only rarely do we discuss to what extent classical
references meet modern standards of rigor. Other surveys are referred to for
much of the closely related history of enumerative geometry, but we have tried
to point out important contributions which are likely to be unfamiliar to
modern readers. Both in the notes and the examples, emphasis is placed on
classical topics, such as excess intersection formulas, which are closely related
to the basic view-point of the text presented here. References are given by
referring to the author, followed by a number in parentheses; however,
Grothendieck-Dieudonne A) is referred to by the familiar [EGA] and
Berthelot-Grothendieck-Illusie et al. A) by [SGA 6].
The bibliography is similarly only a sampling of the vast literature in
intersection theory. Omission of topics or references which should be included
may be attributed to lack of space, as usual, but is more likely due to incom-
incompetence of writer.

Chapter 1. Rational Equivalence
Summary
A cycle on an arbitrary algebraic variety (or scheme) X is a finite formal sum
X nv\V\ of (irreducible) subvarieties of X, with integer coefficients. A rational
function r on any subvariety of X determines a cycle [div (/•)]. Cycles differing
by a sum of such cycles are defined to be rationally equivalent. Alternatively,
rational equivalence is generated by cycles of the form [V@)] - [K(oo)] for
subvarieties VofXx IP1 which project dominantly to IP1. The group of rational
equivalence classes on X is denoted An X.
For a proper morphism f:X-> Y, there is an induced push-forward of
cycles. The fundamental theorem of this chapter states that rational equiva-
equivalence pushes forward, so there is an induced homomorphism /, from A^X to
A* Y, making A* a covariant functor for proper morphisms.
For flat morphisms /: X -* Y (of constant relative dimension) there are
contravariant pull-back homomorphisms/* from A%Y to AtX. There is a
useful exact sequence
AtY^AtX-+At(X- Y) -»0
for a closed subscheme Y of X, and exterior products
AtX®AtY^At(Xx Y).
The groups AtX will play a role analogous to homology groups in
topology. In succeeding chapters it will be shown how geometric objects
(vector bundles, regularly imbedded subschemes,...) give rise to operations on
these groups (Chern classes, intersection products,...). Eventually correspond-
corresponding contravariant, ring-valued functors A* will be constructed, with cap-
products from A*X®AtX to A%X and other properties familiar from
topology. When X is non-singular, A*X^AifX\ in the non-singular case, but
not in general, A* X will have a ring structure. The actual relation of these
groups to homology groups is discussed in Chapter 19.
1.1 Notation and Conventions
Until Chapter 20, by a scheme we shall mean an algebraic scheme over a field '
(see Appendix B.I). A variety will be a reduced and irreducible scheme, and a
1 Except where exterior products (§ 1.10) occur, the ground field could be replaced by any
local Artinian ring with no significant changes.
1.1 Notation and Conventions 7
subvariety of a scheme will be a closed subscheme which is a variety. A point
on a scheme will always be a closed point.
Affine n-space is denoted A"; projective n-space is P".
The local ring of a scheme X along a subvariety V is denoted 0V<X, its
maximal ideal ^#v,x- The field of rational functions on a variety X is denoted
R(X); the non-zero elements of this field form the multiplicative group
R(X)*.
Little will be lost if a reader wishes to assume that all ambient schemes are
varieties over algebraically closed fields in the sense of Serre A). It is
important, however, that arbitrary closed subschemes be allowed. In other
words, the defining ideals must be remembered (cf Example 1.1). For
rationality questions (Chapter 13) it is useful, and no more difficult, to work
over an arbitrary ground field.
It is particularly important that ambient varieties are allowed to have
singularities: our constructions of intersection cycles, even on non-singular
varieties, will involve blowing up along singular subschemes.
The role of subschemes can be seen already in the modern definition of
intersection numbers for plane curves. Although the situation is considerably
more complicated in higher dimensions, several important features of inter-
intersection theory can be seen in the plane curve examples in this chapter.
Example 1.1.1. If f(x,y) and g(x,y) are polynomials defining affine plane
curves F and G over an algebraically closed field K, the intersection scheme Z
is the subscheme of A2 defined by the ideal (/ g) in K [x, y] generated by /
and g. If P = (a, b) is a point in the plane, the intersection multiplicity of F and
G at P is defined to be
i(P,F- G) = dimKJP,z=dimKJP,li,/(f, g).
This intersection number satisfies the following properties:
1) i(P,GF) = i(P,FG).
2) i(P, (F{ + F2) ¦ G) = i{P,Fx ¦ G) + i(P,F2 ¦ G),
where F\ + Fi is the curve defined by/, f2, with/ defining F,.
3) i(P,F'G) = i(P,FG),
if F' is defined by/+ g h, some h e K [x, y].
4) i(P,FG) = 0 if P$F[}G,
and i(P, F- G) = oo if F and G have a common component through P.
Otherwise i(P,F• G) is finite and positive.
5) i{P,FG) = \ if f=x-a, g=y-b,
or more generally if the Jacobian d(f, g)/d(x, y) is not zero at P.
6) i(P,G- HK?min(i(P,FG),i(P,F- H))
if P is a simple point on F, and F has no common component with G or H
through P (see Namba A) 2.3.2).

8 Chapter 1. Rational Equivalence
For fixed P, properties l)-5) characterize the intersection number (see
Fulton AK.3). A similar definition, with analogous properties, is valid when
the plane is replaced by an arbitrary non-singular surface. These intersection
multiplicities agree with the general definitions to be given later (cf.
Example 7.1.10). In general, however, the multiplicities will not be determined
by the intersection scheme alone.
1.2 Orders of Zeros and Poles
Let A' be a variety, V a subvariety of X of codimension one. The local ring
A —&v,x is a one-dimensional local domain. Let r e R{X)*. We will define the
order of vanishing of r along V, ord v(r), which will be a homomorphism, i.e.,
(*) ord v(r s) = ord y(r) + ord v(s)
for/-, seR(X)*.
Any r 6 R(X)* may be written as a ratio r = a/b, for a, b, e A. By (*) we
must define
ord^(r) = ordy(a) — ord v(b).
Thus ord v will be determined if ord v(f) is defined for r e A.
In case X is non-singular along V, i.e., A is a discrete valuation ring, then
r = utm, for u a unit in A, t a generator for the maximal ideal of A, and m an
integer. In this case one may set ord v(r) = m. When A' is a curve over an
algebraically closed field K, this is the same as setting
or&v(r) = dimKA/(r).
The latter definition extends to singular curves, but not to higher dimensions,
for then A/(r) is not finite-dimensional over the ground field. The correct
general definition for r e A is
where lA denotes the length of the /1-module in parentheses. That this
determines a well-defined homomorphism ordt- from R{X)* to 1 is proved in
Appendix A3.
For a fixed reR(X)*, there are only finitely many codimension one
subvarieties Kof X with ordv(r) 4= 0 (Appendix B.4.3).
Example 1.2.1. Let/and g be polynomials defining plane curves F and G,
with/irreducible. Let g be the rational function on the curve F defined by the
residue class of g in K[x, y]/(f). If P e F, then
i(P,F- G) = ordP(g).
Example 1.2.2. Let/(x, y) and g (x, y) define plane curves F and G, over an
algebraically closed field K. Let r(x) e K[x] be the resultant of/and g with
1.2 Orders of Zeros and Poles 9
respect to the variable y. Write f(x, y) = 2.1 =o /•(*) /• If f*(a) * °>then
In case the intersections of F and G have distinct x-coordinates, the intersection
numbers are given by the order of vanishing of the resultant. This is one of the
classical definitions of intersection number for plane curves (cf. Walker A)).
(The equality follows from Example A.2.1 and the fact that K [x, y]/(f, g),
being Artinian, is the direct product of its localizations <^/>A./(/ g).)
Example 1.2.3. Let I be a variety, X -» X the normalization of X in its
function field. If r 6 R(X)* = R(X)*, then
ordr(r) = E ordp(r) [R(V):R(V)],
where the sum is over all subvarieties V of X which map onto V, and
[R(V):R(V)] denotes the degree of the field extension. (This follows from
Example A.3.1.). The more familiar order function on normal varieties there-
therefore determines the order function on arbitrary varieties.
Example 1.2.4. If r e 0KX, then
ordK(r) ^ max{n|r 6 Jl"Y x}.
This inequality is an equality if X is non-singular along V, but strict if r e ~^v,x
and X is singular along V.
Example 1.2.5. Let/ g define plane curves F, G in the affine plane over an
algebraically closed field K, and let P = @, 0).
(a) i (P. F-G) = dim^tx, yj/(f, g)) ,
where K^x, y\ is the ring of formal power series (see Zariski-Samuel A) VII,
VIII for properties of formal and convergent power series).
(b) If/has only one branch at P, and (x(t),y(t)) is a power series para-
metrization of this branch, then i(P,FG) is the order of vanishing of
g{x(t),y(t)) at t = 0. (K([x, yj/(f, g) is imbedded in K^tJ, with finite
dimensional cokernel; apply Lemmas A.2.1 and A.2.4.)
(c) If K = <C, then convergent power series may be used in place of formal
power series in (a) and (b).
(d) The definition and properties of intersection multiplicities of Example
1.1 extend to (germs of) analytic curves, and to the formal case, i.e., to
arbitrary/ g in K^x, y\
(e) If K= <E, and F and G have no intersections but P inside an e-neighbor-
hood of P, then for sufficiently small rj + 0, i(P,FG) is the number of
intersections of the curves F= r\ and G = 0 inside an e/2-neighborhood of P.
(Factoring g in <C {x, y), one may assume g has one branch; then use (b).)
(f) (Zeuthen's rule). Let u\,...,um (resp. V],...,vn) be the roots of/(x, Y)
resp. (g (x, Y)) in some extension of <C I*]]. Then

10
Chapter 1. Rational Equivalence
Here «,-, vt e <C §_x[/NJ for some N, ord^*1'") = \/N. If axes are chosen so that
F and G have no common points on the j>-axis except at P, the right side is the
order of the resultant of/and g.
For more on intersection multiplicities for plane curves, see Example 12.4.2,
C. Segre A), Zeuthen C), Walker A), and Fulton A).
1.3 Cycles and Rational Equivalence
Let X be an algebraic scheme. A k-cycle on A" is a finite formal sum
I n,[V,]
where the Vt are ^-dimensional subvarieties of X, and the «, are integers. The
group of ^-cycles on X, denoted Zk X, is the free abelian group on the k-
dimensional subvarieties of A"; to a subvariety V of X corresponds [V\ in ZkX.
For any (k+ l)-dimensional subvariety Wof X, and any r e R(W)*, define
a k-cyc\e [div (/-)] on X by
the sum over all codimension one subvarieties V of W; here ord v is the order
function on R ( W)* defined by the local ring 0Vi w.
A ?-cycle a is rationally equivalent to zero, written a ~ 0, if there are a finite
number of (k + l)-dimensional subvarieties Wt of A", and r, 6 R(Wi)*, such that
<x=?[div(r,)].
Since [div (r)] = — [div (r)], the cycles rationally equivalent to zero form a
subgroup Ka\kX of Zk X. The group of k-cycles modulo rational equivalence on
X is the factor group
AkX=ZkX/KsXkX.
Define Z*A" (resp. A^X) to be the direct sum of the Z^A" (resp. AkX) for
k = 0, 1,..., dim (A1). A cycle (resp. cycle class) on X is an element of Z^A"
(resp. A*X). A more classical definition of A* X will be given in § 1.5.
If a is a class inA^X, and k is an integer, we denote by
the component of a in AkX. Thus a = Y.k>o{a}k-
A cycle is positive if it is not zero, and each of its coefficients is a positive
integer. A cycle class is positive if it can be represented by a positive cycle.
Example 1.3.1. (a) A scheme and its underlying reduced scheme have the
same subvarieties, and therefore the groups of cycles and rational equivalence
classes are canonically isomorphic:
1.4 Push-forward of Cycles 11
(b) If A" is a disjoint union of schemes A",,..., X,, then Z* X = © Z* A", and
AkX=®At{X,).
(c) If X] and X2 are closed subschemes of A", then there are exact sequences
Ak(XlC\X1)-*AkXl ® AkX2 -^A^X^XJ-^O.
(See Example 1.8.1 for a generalization.)
Example 1.3.2. If X is ^-dimensional, An X= Zn X is the free abelian group
on the n-dimensional irreducible components of X. More generally, any two
rationally equivalent cycles on X contain any irreducible component V of X
with the same coefficient. (Indeed, a cycle of the form [div(/-)], re R(W)*,
cannot include an irreducible component of X.) For any a e A^X, and any
irreducible component V of X, we define the coefficient of V in a to be the
coefficient of [F] in any cycle which represents a.
1.4 Push-forward of Cycles
Let /: X -» Y be a proper morphism. For any subvariety V of X, the image
W=f(V) is then a (closed) subvariety of Y. There is an induced imbedding of
R(W) in R(V), which is a finite field extension if W has the same dimension
as ^(Appendix B.2.2). Set
\[R{V):R{W)] if
I 0 if dim(^<dim(F)
where [R (V): R (W)] denotes the degree of the field extension. Define
This extends linearly to a homomorphism
These homomorphisms are functorial: if g is a proper morphism from Y to Z,
then (gf)*= g*f*, as follows from the multiplicativity of degrees of field
extensions. In the complex case, if dim W = dim V, V is generically a covering
of W with deg(K/H/) sheets, and the push-forward agrees with the push-
forward in topology (cf. § 19.1).
Theorem 1.4. If f:X —> Y is a proper morphism, and a is a k-cycle on X
which is rationally equivalent to zero, then /* a is rationally equivalent to zero
on Y.

12
Chapter 1. Rational Equivalence
There is therefore an induced homomorphism
so that A* is a covariant functor for proper morphisms.
Proof. We may assume a = [div (r)], where r is a rational function on a
subvariety of X. We may replace X by this subvariety, and we may replace Y
by f{X), so we may assume Y is a variety and /is surjective. The theorem then
follows from the following more explicit proposition.
Proposition 1.4. Let f: X
and let r e R{X)*. Then
(a) /,[div (r)]
(b)/* [div (r)]
Y be a proper, surjective morphism of varieties,
0 // dim ( Y) < dim (JT).
[div (iV (/¦))] ;/
In (b), /?(X) is a finite extension of R(Y), and iV(r) is the norm of r, i.e.,
the determinant of the /?(F)-linear endomorphism of R(X) given by multipli-
multiplication by r.
Proof. Casel: y=Spec(*:), if a field, X=P|. Then R(X) = K(t), with
? = X|/x0. Since the order functions are homomorphisms, we may assume r is
an irreducible polynomial, of degree d, in K[t\ Then r generates a prime ideal
p in K[t] corresponding to a point P in X with ord/>(r) = 1. The only other
point along which r has non-zero order is the point />oo=@:l) at infinity,
where s=\/t is a uniformizing parameter. Then sdr is a unit at Px, so
ord/.^ (r) = - d. Therefore
Now /?(?) = A:[?]//> is an extension of K of degree d, while
Therefore
Cose 2: /is finite. Let K= R(Y), L = /?(X). Let W be a subvariety of F of
codimension one, ^ = 0WiY,Jf =-jfWtY. There is a domain B, finite over .4, with
quotient field L, B®AK= L, so that the subvarieties V, of A' mapping onto W
correspond to the maximal ideals^, of B, with B., = ffVi,x- (To see this, one
may assume Y and X are affine, with coordinate rings F and A respectively;
then A is the localization of F at the prime ideal corresponding to W, and
B = A ®rA.) To prove part (b) we must show that
Since N and the order functions are homomorphisms, it is enough to prove
this when r e B. By Lemmas A.2.3 and A.2.2 of Appendix A, the left side of
this equation is lA(Coker((p)), where <p is the endomorphism of B induced by
multiplication by r. The norm N(r) is by definition det(<pK), where <pK is the
induced endomorphism of L. The required equality
= ord,y(det(<pK))
is a special case of Lemma A. 3.
1.4 Push-forward of Cycles
13
The general case of (b) can be proved in the same way, since there is
always a B as in Case 2 (Appendix B. 2.4). For a more elementary proof, one
may let X -» X, Y -> Y be the normalizations of X and Y in their function
fields; the morphism / induces a morphism /: X ->?. By functoriality and the
case proved for the (finite) normalization maps, we may assume X and Y are
normal. If A is the local ring of Won Y, A is a discrete valuation ring. Let 5 be
the integral closure of A in L = R (X). By the valuative criterion for
properness, for each maximal ideal pt in B, Bp. dominates a local ring (9Vi x; the Vt
thus obtained are distinct since proper maps are assumed to be separated.
Since fiVi%x is one-dimensional and normal, 0Vi%x = BPl, and one concludes as in
Case 2.
For the general case of (a), we may assume dim (Y) = dim (X) - 1. Let
K = R (Y). The coefficient of Y in/* [div (r)] is
Zordy(r)[R(V):K],
the sum over all codimension one subvarieties V of X which map onto Y. We
may replace Yby Spec(/Q, and Zby the base extension XK= Xx KSpec(iO- so
we may assume A' is a curve over Y = Spec(K). Let h:X-^X be the
normalization of X, and choose a finite morphism g: X -> P^. Let p be the
projection from Pi to Y,sofoh=p°g, and let f e R{X) be the image of r by
the isomorphism /? (X) ^ /?(A0- By functoriality, and Case 2 for /*,
The latter is zero by an application of Case 2 to g and Case 1 to p. ?
Definition 1.4. If X is a complete scheme, i.e., X is proper over S = Spec(K),
K the ground field, and o. = Y.pnp[P] is a zero-cycle on X, the degree of a,
denoted deg(a), or \x a, is defined by
Equivalently, deg (a) = /> * (a), where /> is the structure morphism from X to S,
and /40S=Z[5] is identified with Z. By the theorem, rationally equivalent
cycles have the same degree. We extend the degree homomorphism to all of
by defining \x a = 0 if a e AkX, k>0. For any morphism /: X -> Y of complete
schemes, and any <xe A^X,
a special case of functoriality. We often write j in place of \x.
Convention 1.4. Let Yly..., Yr be closed subschemes of a scheme X. Let Y
be a closed subscheme of X which contains all the Yh Given a, 6 A* F,,

14
Chapter 1. Rational Equivalence
i = 1,..., r, and ft e A^Y, we will usually write "/S = ?[=i a; in A^Y" in place
of the precise equation /? = X/=i <Pi*(a,) where <pf is the inclusion of Yt in 7.
Example 1.4.1. Theorem 1.4 implies Bezout's theorem for plane curves
over an algebraically closed field (see Chapters 8 and 12 for generalizations):
if F and G are projective plane curves of degrees m, n, with no common
components, then
Y, i(P,F-G) = mn.
PeP'
(One may assume F is irreducible. If G and G' both have degree n, then GIG'
defines a rational function r on the curve F, and
i(P, F ¦ G) - Z i(P,
= Z ordp(r) = 0 ,
the last equation by Theorem 1.4. Taking G' = L" for L linear, one is reduced
to the case where G is linear. Similarly, one reduces to the case where F is also
linear, where it is obvious.) B6zout's theorem may also be proved via
resultants, using Example 1.2.2.
Example 1.4.2. The fact that proper morphisms are separated is crucial for
the truth of the theorem. If X is constructed by identifying two copies of P^
except at A:0), and/is the projection from X to Spec(A~), and r = x,Ax0, then
Example 1.4.3. Let A' be a non-singular projective curve of genus g over an
algebraically closed field. Then A0X is the Picard group Pic (A") of divisor
classes on X. The kernel of the degree homomorphism is the Jacobian variety
of A", an abelian variety of dimension g. If g > 0, A0X is not finitely generated.
Example 1.4.4. Let X be an abelian variety over an algebraically closed
field. If a zero-cycle X "/[-PJ on -^is rationally equivalent to zero, then the sum
X "/jP/ is zero in X. This determines a homomorphism
S:A0X-4 X.
More generally, if X is any non-singular variety, and <p : X -» Alb (X) is the
universal mapping to the Albanese variety of X, then there is a homomorphism
S:A0X-> Alb(X)
taking X «/ [P.] to ? n, <p(Pi). (Let a = ? n, [PJ. By the definition of rational
equivalence, there are non-singular curves Cj, finite morphisms fj: Cj ~* X
mapping C, birationally onto their images, and r,-e/?(C,-)*, so that a =
?.//*[div(r;-)]. Each fj induces a morphism from Alb(Cy) to Alb (A"), com-
compatible with S. The conclusion follows by the identification of Alb(C,) with
the Jacobian of Cj. See Roitman A) or Murthy-Swan A) for details.)
1.6 Alternate Definition of Rational Equivalence
1.5 Cycles of Subschemes
15
Let X be any scheme, and let Xlt ...,X, be the irreducible components of X.
The local rings fix,,x are all zero-dimensional (Artinian). The geometric
multiplicity mi of A",- in X is defined to be the length
The (fundamental) cycle [X] of X is the cycle
i
[X] = E m,[X,].
;-i
This is regarded as an element of Z*A". By abuse of notation we also write [X]
for its image in A*X. If X is purely ^-dimensional, i.e., dim A",= k for all /,
then [X]eZkX. In this case ZkX = AkX is the free abelian group on
[A-,],..., [A",].
If A" is a closed subscheme of a scheme Y, then Z*A"c Z* Y, and we write
also [X] for the image of [X] in Z* Y, and for its image in A* Y.
Example 1.5.1. Let V be a variety of dimension k+ 1, and /: K-» P1 a
dominant morphism. Let 0= A:0), oo =@:1) be the zero and infinite points
of P1. The inverse image schemes (cf. Appendix B.2.3)/~'@) and/~'(oo) are
purely ^-dimensional subschemes of V, and the cycle
is the cycle [div(/)] defined in § 1.2, where/also denotes the rational function
in R (V) determined by the morphism /. (This follows immediately from the
definitions given in § 1.2, § 1.3, and § 1.5.)
1.6 Alternate Definition of Rational Equivalence
Let A" be a scheme, and let Xx P1 be the Cartesian product of A" with P1. Letp
be the projection from Xx P1 to X. Let Kbe a (k + l)-dimensional subvariety
of A"xP' such that the projection to the second factor induces a dominant
morphism / from V to P1. For any point P in P1 which is rational over the
ground field, the scheme-theoretic fibre /"' (P) is a subscheme of A"x{P},
which p maps isomorphically onto a subscheme of X; we denote this
subscheme by V(P). Note in particular that/>* [/"' (P)] = [V(P)] in Zk X.
The morphism/: K-» P1 determines a rational function/6 R(V)*. From
Example 1.5.1 it follows that
[/-'@)]-[/"'(oo)] = [d
where 0 = A:0) and oo=@:l) are the usual zero and infinity points of P'.

16
Therefore
Chapter 1. Rational Equivalence
[K@)]-[K(oo)]=p,[div(/)],
which is rationally equivalent to zero on X by Theorem 1.4.
Proposition 1.6. A cycle a in ZkX is rationally equivalent to zero if and only
if there are (k + \)-dimensional subvarieties Vu..., V, of XxW1, such that the
projections from Vt to P1 are dominant, with
in ZkX. ;¦
Proof. Let a = [div(/-)], r e R(W)*, W a (k + l)-dimensional subvariety of
X. Then r defines a rational mapping from W to P1, i.e., a morphism from
some open U <= Wto P1. Let Kbe the closure of the graph of this morphism in
A"x P1. The projection/; maps Kbirationally and properly onto W. Let /be the
induced morphism from Kto P'. Then
[div (/¦)] = />* [div (/)] by Proposition 1.4 (b)
The proposition follows from this and the preceding remarks. D
With more intersection theory at our disposal, we will see that two cycles
are rationally equivalent if they are members of a family of cycles parametrized
by any rational or unirational variety (cf. Example 10.1.7).
Example 1.6.1. A k-cyc\e a on a scheme X is rationally equivalent to zero if
and only if there are a finite number of normal varieties Vh with rational
functions / on V-, determined by morphisms from V{ to P1, and proper
morphisms p{ from V, to X, with a = ?/>,*([div (/¦)]). (Replace the Vt of
Proposition 1.6 by their normalizations.)
Example 1.6.2. Say that a cycle Z = ? n,-[K,] on Xx P1 projects dominantly
to P1 if each variety Vj which appears with non-zero coefficient in Z projects
dominantly to P1; in this case set Z@) = ? «/[K@)], Z(oo) = ? «, [^(oo)].
Two ^-cycles a, a' on a scheme X are rationally equivalent if and only if
there is a positive {k + l)-cycle ZonlxP1 projecting dominantly to P1, and a
positive A>cycle /? on X, with
Z@) = a + y? and Z(oo) = a'+ $.
(If a — a' = Z'@) — Z'(oo) for some positive Z', choose a positive cycle /? so
that the cycle y = a - Z' @) + /? is positive. Write y = ? [ K], and set
Z = Z' + E[^xP'].)
Example 1.6.3. Let A' be a projective scheme over an algebraically closed
field. Let S"X be the n* symmetric product of X, whose points are identified
with positive 0-cycles of degree n on X. Two 0-cycles a, a.' are rationally
equivalent if and only if there is a morphism/: P1 -» S"X, for some n, and a
positive 0-cycle /? on X with
/@) = a + jff and /(oo) = a' + 0.
1.6 Alternate Definition of Rational Equivalence
17
(If X is a smooth curve, this follows from the existence of the universal 0-cycle
on XxS"X; in general,/:P' -» S"X factors through S"C for some smooth
(possibly disconnected) curve C which maps finitely to X.)
This result may be generalized to ^-cycles, k > 0, with the varieties S"X
replaced by Chow varieties parametrizing positive ^-cycles on X (cf. Samuel
C) Thm. 3).
Even if a and a' are positive, the criterion fails if /? is omitted. (Let X be
the blow-up of C x P1 at P x 0, C a non-singular, non-rational curve, P e C; let
a = [P x oo], and let a' be any point on the exceptional divisor except the one
determined by the line P x P1.)
Example 1.6.4. Let € be a category of algebraic schemes over a fixed field.
Assume that all morphisms in 4 are proper, and that for any projective
morphism from X' to X, if X e <?, then A" e <?. Let H be a covariant functor
from iS to the category of abelian groups. Assume that every variety V in € has
a class cl{V)'u\H(V). Assume
(i) if/: K-» W is a surjective morphism of varieties in4, then
This determines a natural transformation c /: Z* -> // of covariant functors.
Assume also
(ii) if A" is a normal variety \w4 and/: X -> P1 is a dominant morphism,
then
c /(I/ @)]) = c /([/-' (oo)]).
Then c I passes to rational equivalence, inducing a natural transformation
cl:A*-*H of covariant functors. Thus A* is the finest theory satisfying (i)
and (ii).
Example 1.6.5. The Grothendiek group KOX of coherent sheaves on X has
a filtration F^K^, with FkKoX generated by sheaves whose support has
dimension at most k. If / is a proper morphism, the higher direct image
functors induce a map from KOX to KOY which preserves this filtration
(cf. Example 15.1.5). The associated graded groups Gr^KoX therefore become
covariant for proper morphisms. A reader familiar with this machinery may
prefer to use Gr^Ko in place of A^. The homomorphism from Z^X to Gr^KeX
which takes [V] to the class of the structure sheaf (Sv satisfies the conditions of
Example 1.6.4. This gives a natural transformation A^-* Gr^A^. In Chapter 18
we will see that this becomes an isomorphism after tensoring with (Q.
Example 1.6.6. Let A' be a complete scheme, and let
If the ground field is algebraically closed, and X is irreducible, the Ao (X) is a
divisible group. (Aa(X) is generated by zero-cycles of the form/* ([P] - [g]),
/: C -» X, C a non-singular projective curve, P, Q e C. A0(Q is the Jacobian
variety of C, and any abelian variety over an algebraically closed field is
divisible (cf. Mumford D) p. 62).)

18
Chapter 1. Rational Equivalence
1.7 Flat Pull-back of Cycles
Let/: X -» Y be a flat morphism of relative dimension « (cf. Appendix B.2.5).
The examples of primary importance for us will be:
(i) an open imbedding (n = 0).
(ii) the projection of a vector bundle or A "-bundle (cf. § 1.9), or a projective
bundle, to its base.
(iii) the projection from a Cartesian product X =Y xZ to the first factor,
where Z is a purely n-dimensional scheme.
(iv) any dominant morphism from an (n + l)-dimensional variety to a non-
singular curve.
Convention. In this book, a. flat morphism is always assumed to have relative
dimension n for some integer n.
For such/: X -> Y, and any subvariety Kof Y, set
Here/ (V) is the inverse image scheme (cf. Appendix B.2.3), a subscheme of
X of pure dimension dim (V) + n, and [f~'(V)] is its cycle (§1.5). This extends
by linearity to pull-back homomorphisms
f*:ZkY-^Zk+nX.
Lemma 1.7.1. Iff:X-> Y is flat, then for any subscheme Z of Y,
Proof. Let W be an irreducible component of/~'(Z), let Kbe the closure
off(W). The first assertion of Lemma A.4.1 implies that Kis an irreducible
component of Z. The second assertion of Lemma A.4.1, applied to A=ffvz,
B =&wj-'z, implies the required equality of multiplicities. ?
It follows from this lemma that flat pull-backs are functorial: if/:!"-* Y
and g:Y^Z are flat, then gf is flat, and (gf)*=f*g*. For if V is a
subvariety of Z, then
Proposition 1.7. Let
X'^X
n 1/
Y'7Y
be a fibre square, with g flat and f proper. Then g' is flat, f is proper, and for all
a e Z,X,
fig'*cc = g*f*a
in Z* Y'.
1.7 Flat Pull-back of Cycles
19
Proof. Since flatness and properness are preserved by base change, we may
assume X and Y are varieties,/is surjective, and a = [X]. Let/* [X] = d [ Y]. We
must show that/* [X'\ = d[Y'\. This is a local calculation involving local rings
of irreducible components, so we may assume X = Spec(L), F = Spec (AT),
with K, L fields, F' = Spec(^), with A local Artinian, and X' = Spec (B),
B = A ®KL. Then the result follows from Lemma A.1.3. ?
Theorem 1.7. Let f: X—> Ybe aflat morphism of relative dimension n, and a
a k-cycle on Y which is rationally equivalent to zero. Then f* a is rationally
equivalent to zero in Zk+nX.
There are therefore induced homomorphisms, the flat pull-backs,
f*:AkY-^Ak+nX,
so that A* becomes a contravariant functor for flat morphisms.
Proof. By Proposition 1.6 we may assume a = [F@)]-[K(oo)], where Kis
a subvariety of FxP' and the projection g from Kto P1 is dominant, hence
flat. Let W = (fx 1)"'(K), a closed subscheme of Xx P1, and let h : W-» IP1 be
the morphism induced by the projection to P1.
Let/> :XxWx -^ X,q: Yx P1 -> Fbe the projections. Then
= />*(/* 1)* (
by Proposition 1.7, and this equals
by Lemma 1.7.1. Let WX,...,W, be the irreducible components of W, hi the
restriction of h to Wh Let [ W\ = ? m\ W,]. Since
-'@)]-[</-'(oo)])
preserves cycles rationally equivalent to zero, it suffices to verify that
for P = 0 and P = oo. This is a special case of the following general lemma.
Lemma 1.7.2. Let X be a purely n-dimensional scheme, with irreducible com-
components X\,..., Xr, and geometric multiplicities mx,..., mr. Let D be an effective
Cartier divisor on X, i.e., a closed subscheme of X whose ideal sheaf is locally gen-
generated by one non-zero-divisor. Let Dt = DC]Xt be the restriction of D to Xt. Then
[?] = g>. [A]
inZn-x(X).
Proof. It must be checked that each codimension one subvariety V of X
appears with the same multiplicity in both sides of the equation. Let A be the
local ring of X along V, a a local equation for D in A. The minimal prime
ideals pt in A correspond to the irreducible components A', of X which contain

20
Chapter 1. Rational Equivalence
V. The multiplicity m( of [Xi] in [X] is the length lApi (Ap). The multiplicity of
[V\ in [D] is lA(A/aA). The multiplicity of [V\ in [?>,] is lA,pl(A/p,¦+ a A). The
required equality
lA (A/a A) = X ™, lA/pt (A/pt + aA)
is given by Lemma A.2.7; the fact that a is a non-zero-divisor is used to know
that lA (AlaA) is the multiplicity eA (a, A) of § A.2. D
Example 1.7.1. Theorem 1.7 may fail if/is not assumed to have constant
relative dimension. Lemma 1.7.2 may fail if X is not pure-dimensional. (For
example, let X be the subscheme of A3 defined by the ideal (zx, zy), and E the
Cartier divisor defined by the function z — x.) In Fulton B) § 1.5, Prop. 3, this
assumption was mistakenly omitted.
Example 1.7.2. Lemma 1.7.2 is also valid for Cartier divisors which are not
effective.
Example 1.7.3. Let D be an effective Cartier divisor on an n-dimensional
scheme X. Let [X]n = ?J=i m^X^ denote the n-dimensional component of the
cycle [X], and let ?, = Df]Xi. Then (by the proof of Lemma 1.7.2)
Example 1.7.4. Let /: X' -> X be a finite and flat morphism; each point of
X has an affine neighborhood U such that the coordinate ring of /"'([/) is a
finitely generated free module over the coordinate ring of U. One says that/
has degree d if the rank of this module is d, for all such U. Then for all
subvarieties Kof XJtf* [V\ = d[V] in Z*(X). The composite
SA.X
is multiplication by d.
Example 1.7.5. If a subgroup R+X of Z*X is specified for all algebraic
schemes X over a given field, which is preserved under proper push-forward
and flat pull-back, and /?*P' contains [0] — [oo], then R*X contains all cycles
rationally equivalent to zero, for any X.
Example 1.7.6. Suppose a finite group G acts on a variety Y, with quotient
variety X= Y/G. (Such X exists if Y is quasi-projective, cf. Mumford D)
p. 111.) Then there is a canonical isomorphism
Here A^X^ denotes A*X®zQ; G acts on A%Y by covariance (§ 1.4), and
04* Fq)c denotes the G-invariant subgroup of A+Yq. (Let n: Y -» X be the
quotient map. For any subvariety W of Y, let
lw= {g 6 G
1.8 An Exact Sequence
be the inertia group, and let
ew= card(I
21
where V=n(W) and deg^HVK) is the degree of inseparability of R(W) over
). For a subvariety K of X set
the sum over all irreducible components W of n~](V). This determines an
isomorphism Z*ArQ = (Z*F(Ij)c, and (A, FQ)C is the quotient of (Z* yQ)c
modulo the subspace generated by
Note that the composite
A^X^
is multiplication by card (G).
^ .4*
1.8 An Exact Sequence
Proposition 1.8. Le? F be a closed subscheme of a scheme X, and let U = X - Y.
Let i: Y —> X, j: U —> X be the inclusions. Then the sequence
is exact for all k.
Proof. Since any subvariety V of U extends to a subvariety V of X, the
sequence
ZkY-^ZkX-^ ZkU^0
is exact. If a 6 ZkX andj* a ~ 0, then
for r, 6 R(Wi)*, W, subvarieties of U. Since R(W) = R(W), r, corresponds to
a rational function F, on Wt, and
j * (a -S[div (,-)]) = 0
in Zt G. Therefore
for some ft e ZkY, which implies the proposition. D

22
Chapter 1. Rational Equivalence
Example 1.8.1. Let
'I V
be a fibre square, with i a closed imbedding, p proper, such that p induces an
isomorphism of X' — Y' onto A'— Y. Then the sequence
Ak Y' -i Ak Y © Ak X' -^ Ak X -> 0
is exact, where a(a) =(?»a, —./*«), b(a, /?) = i%<x + />*/?. (Use the definition of
§ 1.3 to show that Ker(/*) surjects onto Ker (/*).)
1.9 Affine Bundles
A scheme E, together with a morphism p : E -> X, is an affine bundle of rank n
over X\tX has an open covering by ?/*, and there are isomorphisms
such that p restricted to
to ?/,.
^) corresponds to the projection from UaxA"
Proposition 1.9. Let p : E -^ X be an affine bundle of rank n. Then the flat
pull-back
Ak+nE
is surjective for all k.
Proof. Choose a closed subscheme Y of X so that U = X — Y is an affine
open set over which E is trivial. There is a commutative diagram
A* Y
1
A*X
G -> 0
F)
0
where the vertical maps are flat pull-backs, and the rows are exact by
Proposition 1.8. By a diagram chase it suffices to prove the assertion for the
restrictions of ? to U and to Y. By Noetherian induction, i.e., repeating the
process on Y, it suffices to prove it for X = U. Thus we may assume
E = Xx A". The projection factors
X,
so we may assume n = 1.
1.9 Affine Bundles
23
We must show that [V] is in p*AkX for any (k + l)-dimensional subvariety
Fof E. We may replace A'by the closure of p(V) (cf. Proposition 1.7), so we
may assume A" is a variety and p maps V dominantly to X. Let A be the
coordinate ring of X, K = R(X) the quotient field of A, and let q be the prime
ideal in^[r] corresponding to V. If dim X= k, then V= E, so V=p*[X]. So we
may assume dim X = k+ \. Since Vdominates X and K=# E, the prime ideal
q K [t] is non-trivial; let r 6 K [t] generate q K [t\ Then
for some (k + 1 )-dimensional subvarieties V: of E whose projections to X are
not dominant. Therefore Vt = p~](IV,), with Wt¦ = p(J^), so
[] [()] S1
as required. ?
In particular, Ak (A") is zero for k <n, while An (A") = 1.
When E is a vector bundle over X, we will see in Chap. 3 that p* is an
isomorphism.
Example 1.9.1 (cf. Chow B)). Let X be a scheme with a "cellular decomposi-
decomposition", i.e., X has a filtration X = I,3l,.1 =3... dJjdI., =0by closed sub-
schemes, with each Xt -I,.^ disjoint union of schemes Uu isomorphic to affine
spaces A'. Then A^X is (finitely) generated by {[^y]}, where VtJ is the closure
of Ut] in X. Using homology (cf. Example 19.1.11 (b)) one can show that the [Vtj]
form a basis for A^X. The Grassmann and flag varieties are examples (cf.
Chap. 14).
Example 1.9.2. The conclusion of Proposition 1.9 holds for any locally trivial
fibre bundle whose fibre is isomorphic to an open subscheme of an affine
space A" (cf. Grothendieck A) § 6).
Example 1.9.3. Let Lk be a ^-dimensional linear subspace of P", k = 0 ... n.
(a) Ak(P") is generated by [Lk\ (Apply Proposition 1.8 with X=P",
Y=L"-\U=A\)
(b) Ak(P") = I.[Lk] = I,. (For k=n-\ this follows from the form of
rational functions on P". For k <n - 1, if d[Lk] = ? n,-[div(/-,-)], r, 6 R{ K()*,
let Z be the union of the Vt, and let/: Z -> P*+l be projection from a linear
{n — k — 2)-dimensional subspace disjoint from Z; apply Theorem 1.4 to/.) A
more general theorem will be proved in Chap. 3.
Example 1.9.4. Let H be a (reduced) hypersurface of degree d in P". Then
[H] = d [L], for L a hyperplane, and
4,_,(P"-//) =
Example 1.9.5. (a) Let f:X'-*X be a finite, birational morphism of
^-dimensional varieties. For each codimension one subvariety Kof X, let d{V)
be the greatest common divisor of the degrees of all field extensions R (W) of
R{V), for all subvarieties W of X' such that/(W)= V. There is an exact
sequence
:->0.

24
Chapter 1. Rational Equivalence
(b) Let Xa A3 be the Whitney umbrella: x2 = yz2. Then AlX=T./21,
generated by the class of the j'-axis. (Apply (a) to/: A2 -* X,f(s, t) = (st,s2,t).)
(c) Let X be the quotient of A2 obtained by identifying (s, 0) and (p. s, 0)
for all n* roots of unity [i; equivalently,
X = Spec (K[sn, st,s2t,...,s"~] t, t]).
ThenAoX=O,AiX=Z/nZ,A2X=Z.
Example 1.9.6. (cf. Fulton B) p. 166). Let H(X) = A*(X)<Si = A*(X) <g> Q,
regarded as a covariant functor from the category of complete schemes over a
fixed field, to abelian groups. Suppose T: H -> H is a natural transformation
such that T[P"] = [P"] + /}„ for some class fin of dimension < n on P",
n = 0, 1, 2,.... Then Tmust be the identity. (Show first that /?„ = 0, by choosing
/:P"-^P" with/*[P"] = rf[P"], ft[Lk] = dk[Lk], Lk a fc-plane in P", k < n,
dk*d, comparing f*T[P"] to 7"/*[P"] shows that the coefficient of [Lk] in A,
must be zero. To show that T[X] = [X] for all varieties X, by Chow's lemma
([EGA] II.5.6) one may assume X is projective. Replacing A' by a variety that
maps finitely to X, one may assume a finite group G acts on X with X/G s P".
Let T[X] = [X] + ft. Applying the covariance with respect to the projection/
from X to P", and the preceding case for P", one derives /*/?=(). By the
covariance with respect to the automorphisms in G, /? e H(X)G. By Example
1.7.6, y?=0, as desired.)
Similarly on the category of all algebraic schemes over a fixed field, if
T: H -> H is covariant with respect to proper morphisms, and contravariant
with respect to open imbeddings of quasi-projective schemes, and T[W]
= [P"] + pn, dim(jffH) < n, then Tis the identity.
1.10 Exterior Products
For algebraic schemes X, Y over a field, Xx Y denotes the Cartesian (fibre)
product of X and Y over the ground field.
The exterior product
ZkX®Z,Y^Zk+I(Xx Y)
is defined by the formula
[V]x[W] = [Vx W]
for V, W subvarieties of X, Y, and extending bilinearly to general cycles. (If the
ground field is not algebraically closed, Vx Wmay not be irreducible; its cycle
[Vx W\ is defined by the prescription of § 1.5.)
Proposition 1.10. (a) If<x~0or/3~ 0, then a x fi ~ 0.
(b) Let f:X'-*X, g : Y' —> Y be morphisms, fx g the induced morphism
fromX'x Y'toXx Y.
Notes and References
(i) Iffandg are proper, so isfxg, and
25
for all cycles a on X', ft on Y'.
(ii) If f and g are flat of relative dimensions m and n, then fx g is flat of
relative dimension m + n, and
for all cycles a on X, /? on Y.
Proof. For (b), factoring fxg into (fxidy) ° (idx^xg) reduces one to the
easy cases where/or g is the identity; cf. Proposition 1.7 for (i). For (a), if a~0,
we may assume /?= [W], W a subvariety of Y, and then assume W = Y by (b)
(i). In this case ax ft = p* (a), where p: Xx W -> X is the projection, and the
assertion is a special case of Theorem 1.7. D
It follows that there are exterior products
AkX®A,Y^Ak+l(Xx Y)
satisfying the formulas of Proposition 1.10(b).
Example 1.10.1. The exterior product is associative:
(a x P) x y = a x (/? x y) for a e A*X, (I e A*Y, y e A*Z .
Example 1.10.2. If A'has a cellular decomposition as in Example 1.9.1, then,
for all schemes Y, the product maps
© AkX®A,Y-
k+l=m
Am(XxY)
are surjective.
Notes and References
For divisors on a non-singular variety, rational equivalence coincides with
linear equivalence, a subject which has long been central in algebraic
geometry. For zero-cycles on a curve this is the study of the Jacobian variety.
Much of the Italian school of algebraic geometry was devoted to the study of
linear systems of curves on a surface.
The notion of rational equivalence for cycles of codimension greater than
one was originated by Severi. The canonical divisor class had been a vital
classical tool. In his seminal 1932 paper (Severi F)), he revealed that — to use
modern language — the second Chern class of a surface was not just a number,
but was a well-defined rational equivalence of zero-cycles. Much of the
subsequent work of Severi, as well as B. Segre, Eger and Todd, was devoted to

26
Chapter 1. Rational Equivalence
developing the idea of rational equivalence for cycles of arbitrary dimension,
and constructing "canonical" classes in all dimensions.
Crucial to these developments was the construction of an intersection
product for two cycle classes on a non-singular variety. For example,
intersecting canonical classes would lead to numerical invariants, generalizing
the self-intersection number of a canonical divisor on a surface. These related
problems will be discussed in later chapters.
Several definitions of rational equivalence were proposed by Severi. One
difficulty at first was caused by the desire to phrase everything in terms of
rational families of positive cycles (cf. Examples 1.6.2, 1.6.3). Evidently
motivated by Lefschetz's ideas in topology, Todd B) explicitly introduced the
notion of the group of virtual cycles, and the subgroup of cycles rationally
equivalent to zero; this simple artifact led to considerable clarification.
Definitions generating rational equivalence by intersecting families of ratio-
rationally equivalent divisors were also proposed. Discussions of these ideas may
be found in Severi A4), A9), and Baldassarri A). Attempts to reconcile
competing definitions and develop a satisfactory theory of rational equivalence
with intersections led to considerable debate among Severi, Van der Waerden,
Samuel, and Weil (cf. Van der Waerden E), F)).
Weil E), Samuel C), and Chow A) began a systematic study of
equivalence relations on cycles, based on the new foundations of Weil B).
Many of Seven's intuitive geometric notions about families of cycles were
replaced by a precise algebraic language of specialization, and a more
axiomatic approach to equivalence relations on cycles was developed. The
paper of Chow gave an accepted proof that rational equivalence classes (on a
non-singular projective variety) determine a well-defined intersection class; his
proof used a "moving lemma" which was based on a construction of Severi's.
In 1958 Chevalley's seminar focused on rational equivalence. In the notes
(Chevalley B)) the theory is developed from first principles, with no reference
to, or even mention of, previous work on the subject; there the ring of rational
equivalence classes was named the "Chow ring". The fact that a general
foundational crisis was taking place in algebraic geometry helps to explain this
break with the past. Indeed, rational equivalence and intersection theory were
used as a major testing ground for the new foundations. It would be
unfortunate, however, if Severi's pioneering work in this area were forgotten;
and if incompleteness or the presence of errors are grounds for ignoring
Severi's work, few of the subsequent papers on rational equivalence would
survive.
Although most work on rational equivalence assumed a non-singular
ambient variety, Chevalley B) and Grothendieck A) pointed out that the
notion of rational equivalence, and several basic properties, can be extended to
singular varieties. These ideas were worked out in the first part of Fulton B),
on which the present Chap. 1 is based. The proof that rational equivalence
pushes forward follows Chevalley B); the exact sequence of § 1.8 and
Proposition 1.9 are found in Grothendieck A). The direct proof (Example
1.2.2) of the agreement of modern and resultant definitions of intersection
Notes and References
27
multiplicities for plane curves is apparently new. The alternative in Example
1.6.5 was proposed by S. Kleiman. R. Lazarsfeld suggested Example 1.9.5.
In spite of the formal analogy with homology groups, the groups A*X are
much more difficult to calculate. Some of the literature on this is discussed in
Chap. 19. On the other hand, an equation in A*X is much finer than the
corresponding equation in homology. For zero-cycles on a curve, this dif-
difference amounts to knowing a point on the Jacobian; for an application of this
principle, see Example 14.4.6. In addition, the exact sequence of § 1.8 may be
more useful than the analogous long exact homology sequence. For applica-
applications to affine and projective surfaces, see Murthy-Swan A). Collino A) has
calculated At(X) when X is a symmetric product of a non-singular curve, as a
module over the Chow ring of the Jacobian of the curve.

Chapter 2. Divisors
Summary
If D is a Cartier divisor on a scheme X, and a is a &-cycle on X, we construct
an intersection class
where \D\\ \a\ are the supports of D and a. For a=[F], V a. subvariety,
D[V] is defined by one of two procedures: (i) if V<t \D\, D restricts to a
Cartier divisor on F, and D • [F] is defined to be the associated Weil divisor of
this restriction; (ii) if Fez \D\, the restriction of the line bundle ^(Z)) to Fis
the line bundle of a well-defined linear equivalence class of Cartier divisors on
F, and D ¦ [ F] is represented by the associated Weil divisor of any such Cartier
divisor.
We prove that if a is rationally equivalent to zero on X, then D ¦ a is zero in
Ak-\ {\D\)\ there are therefore induced homomorphisms
AkX ^ Ak-\{\D\).
In the special but important case where D is the inverse image of a point for a
morphism from X to a smooth curve, D ¦ a is the specialization of a; in this
case (or whenever D is principal) D ¦ a can be well-defined as a cycle, setting
D[F] = 0 if Fez D. The above fact therefore includes the assertion that
rational equivalence is preserved under specialization.
If D and D' are Cartier divisors on a scheme X, and a is a &-cycle on X, a
crucial property is the commutative law
D(D'(X) =
- a)
in ^4fc_2(I-DIHj?>'| Pi|a|). Consider, for example, the case where /:I-»A!
is a morphism, and D and D' are the inverse images of the two axes. One may
specialize a cycle first to the part of X over the .x-axis, and then specialize the
resulting cycle to/ @); or one may first specialize over the y-axis, then over
the origin. The resulting cycles one arrives at by these two routes may well be
different2, but the above says they are rationally equivalent.
1 This shorthand for Supp(Z)) should not be confused with a notation for complete linear
systems, which do not occur in this chapter.
2 This corresponds to the fact that a family of cycles parametrized by a smooth parameter
variety has a unique limiting cycle over a missing point when the parameter variety is a
curve, but not when it has dimension two or more.
2.1 Cartier Divisors and Weil Divisors
Both of the above facts follow from the identity (Theorem 2.4):
29
D[D'] = jy-
in
for Cartier divisors D, D' on an n-dimensional variety X, with [D], [D'\ their
associated Weil divisors.
A Cartier divisor D on a scheme X determines a line bundle L=0X (&) and
a trivialization of L over X- | D |. Only the line bundle, the support, and the
trivialization are needed to carry out the above intersection construction3.
These concepts are formalized in the notation of a pseudo-divisor (§ 2.2); there
is the added advantage that a pseudo-divisor, unlike the stricter notion of a
Cartier divisor, pulls back under arbitrary morphisms.
Intersecting with divisors is used to construct homomorphisms
AkX -* Ak-\X, a -¦ Ci (L) n a ,
for a line bundle L on X, and to construct Gysin homomorphisms
when i is the inclusion of an effective Cartier divisor D in X. These operations
will be generalized to higher codimension in subsequent chapters.
2.1 Cartier Divisors and Weil Divisors
Let X be an n-dimensional variety. A Weil divisor on X is an (n — l)-cycle on
X. The Weil divisors form the group Zn_,Xof § 1.3.
A Cartier divisor on X is defined by data (U^,/,,), where the Ua form an
open covering of X and the/, are non-zero functions in /?(i/a) = R{X), subject
to the condition that/c,//^ is a unit (i.e., a regular, nowhere vanishing function)
on UaC\Up. The rational functions fa are called local equations of D; they are
determined up to multiplication by units on Ua (see Appendix B4).
If D is a Cartier divisor on X, and Fis a subvariety of X of codimension
one, write
ordyD = ordy(fa)
3 Over the complex numbers, a model for these constructions is available from topology. A
complex line bundle I on I has a first Chern class c, (Z.) e H2(X). If L =#X(D), L is
trivialized on X- \D\, so cx (L) comes from a class we may call c(D) in H2(X, X— \D\)
(cf. Example 19.2.6). For Y closed in X there are cap products (cf. § 19.1)
In particular, capping with c(D) gives homomorphisms from HIk Y to H2k-2{Yf)\D\). If
D,D' are Cartier divisors, then c(D) u c(D') = c(D')vc(D) in H*{X, X - {\D\\J |Z)'|)),
so
c(D) n (c(D') n a) = c(D') n (c(D) n a)
in H2k_i{YC\\D\ fll-CI) for <xsH2kY - which motivates the commutativity law. Other
properties such as the projection formula are also familiar in topology.

30
Chapter 2. Divisors
where ord^ is the order function on R(X) defined by V in §1.2, and fa is a
local equation for D on any affine open set Ua with UaOV + 0; this is well-
defined since/, is well-defined up to units. Define the associated Weil divisor
[D] of D by setting
[D]
the sum over all codimension one subvarieties V of X\ as in § 1.2, there are
only finitely many Fwith ordyD =t= 0 (Appendix B.4.3).
The Cartier divisors form an abelian group Div(A'): if D and E are given
by data (i/a,/a) and (Ua,ga), the sum D + E is given by (U^,faga). By the
additivity of the order functions, the mapping D -* [D] is a homomorphism
Any/in R(X)* determines a. principal Cartier divisor div(/), by taking all
local equations equal to/ Note that the Weil divisor associated to div(/) is
the cycle [div(/)] defined in § 1.3.
Two divisors D, D' are linearly equivalent if they differ by a principal
divisor: D' = D + div(/). From the definition of rational equivalence, it
follows that [D] and [Dr] are rationally equivalent cycles. If Pic(X) denotes the
group of linear equivalence classes of Cartier divisors, there is an induced
homomorphism
This homomorphism is in general neither injective nor surjective (see
Examples 2.1.1-2.1.3).
We shall see that Cartier divisors can be intersected with arbitrary cycles,
corresponding to the fact that elements of Pic (A1) define cohomology classes.
Weil divisors in general do not have this ability — they determine homology
classes (see Example 2.4.5).
The support of a Cartier divisor D, denoted by Supp(D), or \D\, is the
union of all subvarieties Z of X such that a local equation for D in the local
ring ^z,^ is not a unit. This is a closed algebraic subset of X.
On a general scheme X, an effective Cartier divisor is a subscheme which is
locally defined by one equation, which is required to be a non-zero-divisor.
The notion of Cartier divisor also extends to schemes which are not varieties
(Appendix B.4), but this is not required for present purposes.
Example 2.1.1 (cf. [EGA] IV.21.6). If X is normal (resp. locally factorial)
then DivpQ -* Zn_, (X) and Pic(A') -* An-\ (X) are injective (resp. iso-
isomorphisms). It follows for example that Pic(P") = Z, with generator 0(Y) (cf.
Example 1.9.3).
Example 2.1.2. Let X be the projective plane curve over C defined by the
homogeneous equation y2z = x3. Then AqX^TL, and the homomorphism
Pic(A) -^ A0(X) is surjective, with kernel the additive group C. In case X is
the curvey2z = x2z + x3, the kernel is C*.
Example 2.1.3. Let X be the surface in A3 defined by the equation z2 = x y.
The line V: x = z = 0 (a generator for the cone) defines a Weil divisor which is
not a Cartier divisor. In this case Pic(X) = 0, and A^X) = TLJ1TL.
2.2 Line Bundles and Pseudo-divisors
31
Example 2.1.4. Let X be a projective scheme, and let L be an ample line
bundle on X. For a (k + l)-dimensional sub variety V of X, and non-zero
sections st, s2 of the restriction of L9" to V, with divisors of zeros DUD2, [D]]
is rationally equivalent to [D2]. The group Rat^A' is generated by cycles
[/),] - [D2], as V, n, .?,, ^2 vary. (If r e R (V)*, there is, for large n, a section ¦$,
so that D] + div(r) is an effective divisor on V.)
2.2 Line Bundles and Pseudo-divisors
If D is a Cartier divisor on X and/: X' -* X is a morphism, a pull-back Cartier
divisor f*D is defined only under certain assumptions (cf. [EGA]IV.21.4). If
X' is a variety, for example,/*/) is defined by pulling back local equations for
D provided f(X') <t\D\, but no Cartier divisor pull-back is defined if
/(A") <= | D |. We will introduce a simple generalization of the notion of Cartier
divisor, which will not have this defect, but will still carry enough information
to determine intersection operations on cycle classes.
Definition 2.2.1. A pseudo-divisor on a scheme X is a triple (L, Z, s), where
L is a line bundle on X, Z is a closed subset of X, and s is a nowhere vanishing
section of L on X — Z (equivalently, s is a trivialization of the restriction of L
to X— Z). We call L the line bundle, Z the support, and s the section, of the
pseudo-divisor. Data (U, Z', s1) define the same pseudo-divisor if Z = Z' and
there is an isomorphism a of L with L' such that the restriction of a to X — Z
takes s to s'. Note that a pseudo-divisor with support X is simply an
isomorphism class of line bundles on X.
Any Cartier divisor D on a scheme X determines a pseudo-divisor4 (#x(D),
D\, sD) on X, where 0x(P) is the line bundle of D, \D\ is the support of D,
and sD is the canonical section of #x(D) (Appendix B.4.5). We say that a
Cartier divisor D represents a pseudo-divisor (L, Z, s) if | D \ c Z, and there is
an isomorphism from <?x(D) to L which, off Z, takes so to s. Note that we
allow Z to be larger than | D |; for example, if Z = X, all linearly equivalent
Cartier divisors represent the same pseudo-divisor.
A general pseudo-divisor will often be denoted by a single letter D, and we
write #x(D) for its line bundle, \D\ for its support, sD for its section. This
agrees with the notation for Cartier divisors, except that a Cartier divisor may
have smaller support than a pseudo-divisor it represents.
Lemma 2.2. If X is a variety, any pseudo-divisor (L, Z, s) on X is represented by
some Cartier divisor D on X. Moreover,
4 A Cartier divisor is a line bundle together with a "regular meromorphic" section, up to
isomorphism (cf. [EGA]IV.21.1.4). More generally, a line bundle together with any
"meromorphic" section ([EGA]IV.20.1) determines a pseudo-divisor in our sense. Both
these definitions require conditions on the section over the support, for which we have no
need.

32
Chapter 2. Divisors
(a) IfZ^X, D is uniquely determined.
(b) IfZ = X, D is determined up to linear equivalence.
Proof. Let g^ be transition functions for L, for some affine open covering
{?/a} of X. Fix one index a0, and set /„ = gxaLa. Then fa/fp= gxp, so the data
(?4,/a) define a Cartier divisor D with &x(P) = L. In case Z = X, this gives
the existence of D.
If Z + X, let U= X— Z. The section .? is given by a collection of regular
functions sa on i/Df/j such that sa = gxpSp. (The functions fa give the
canonical section sD.) Since ^//c = ^//^, there is a rational function r e R (X)*
with r = sjf(t for all a. Set Z)'= Z) + div(r). The local equations for D' are
f'*=f*' f = Sa, so the canonical section so' corresponds to s. This proves the
existence of D in case Z+I.
For the uniqueness, if D and D\ with local equations /, and /„, both
determine (L, Z, s), then there is an/e /?(A0* with/; =/./for all a. If ?/ * 0,
and 5C. = sc, /a' and fa must agree on i/fl Ua, so / = 1 on U, i.e., / = 1 and
Definition 2.2.2. If D is a pseudo-divisor on an n-dimensional variety X,
and | D | is its support, define the Weil divisor class
of D as follows. Take a Cartier divisor which represents D, and let [D] be the
class in An-i(\D\) of the associated Weil divisor. In case \D\ 4= X, this Cartier
divisor is unique (Lemma 2.1) and then [D] is a well-defined (n-l)-cycle
\\
on \D\, as reflected in the fact that Zn_,(|D|) is ^n-,(|D|). In case \D\ = X,
the Cartier divisor is only determined up to linear equivalence, but its
associated Weil divisor is well-defined \nAn-\ (X) (§ 2.1).
If D = (L, Z, s) and D' = (Ll, Z', s') are pseudo-divisors on X, the sum
D + D' is the pseudo-divisor
D + D' = (L ® L', Z [J Z', s ® s').
(This agrees with the sum for Cartier divisors, except that the sum of two
Cartier divisors may have smaller support than the union of their supports.)
Similarly define
-Z) =(/,-', Z, l/s).
For fixed Z <= X closed, the pseudo-divisors with support Z form an abelian
group.
If/: X' -* X is a morphism and D = (L, Z, s) is a pseudo-divisor on X, the
pull-back f*D is the pseudo-divisor (/*!,/"' (Z),/*j) on A". This pull-back is
functorial, and agrees with pull-backs of representing Cartier divisors when
those pull-backs are defined, and takes sums to sums.
Example 2.2.1. Let Xi\yzX be the group of pseudo-divisors on X with
support Z. If/: X' -* X is a morphism, pull-back determines a homomorphism
f* :DivzX ^Divz'X', Z'=f'](Z). If X is an n-dimensional variety, the
mapping D \—> [D] determines a homomorphism from DivzX to An_L Z.
2.3 Intersecting with Divisors
33
2.3 Intersecting with Divisors
Definition 2.3. Let D be a pseudo-divisor on a scheme X, and let V be a &-
dimensional subvariety of JT. Define a class, denoted D ¦ [ V\ or D ¦ V, in
^-,A1I0 K) as follows: Let j be the inclusion of V in X. The restriction
(pull-back) j*D is a pseudo-divisor on V whose support is J ?> | H K. Define
D ¦ [V] to be the Weil divisor class (Definition 2.2.2) of;*D:
Z>-[F] = [/*/>].
When D is a Cartier divisor, this may be rephrased as follows: if F<t \D\, D
restricts to a Cartier divisor j*D on V, and D ¦ [V] is its associated Weil
divisor; if F<= \D\, D ¦ [V] is the class in /!*_, (F) represented by [C], for any
Cartier divisor C on Fwhose line bundle ffv(C) is isomorphic toj*#x{D).
In line with the convention in § 1.4, we will write D ¦ [V\ also for the image
of the above class in ^4A._] (Y), for any closed subscheme Y of X which contains
\D\C\V.
For any &-cycle tx=^ny[V] on X, the support of a, written |a|, is the
union of the subvarieties V appearing with non-zero coefficient in a. For a
pseudo-divisor D on X, each Z) • [V] is a class in Ak^l(\D\V\\ol\), and we
define the intersection class D ¦ a in ^4fe_1(|Z)|n|a|) by setting
As above, we also regard D ¦ a in ,4^,G) for any |Z)|fl|a| <= Y <= X
These intersection classes will be used for two important constructions:
A) lfL=<e(D) is a line bundle on X, and |Z)| = JT, Z) • a will be c, (Z,) n a,
the action of the first Chern class of L on a (§ 2.5).
B) If D is an effective Cartier divisor on X, and i is the inclusion of D in X,
D ¦ a will be the Gysin pull-back i* (a) (§ 2.6).
Remark 2.3. In one important case, intersecting with D is defined on the
D\ is a trivial line bundle, then D
cycle level. If the restriction of #x(D) to
determines a homomorphism
also denoted a -> D • a. As before, set D ¦ [V\ = \J*D], withy the inclusion of V
in X, if V<t | D |, but set Z) ¦ [ V\ = 0 if V <= | D j.
This condition holds when D is a principal Cartier divisor on X, or on a
neighborhood of Z) in X. For example if a variety X is mapped dominantly to
a curve C, and P is a simple point of C, then the inverse image X(P) of /"
satisfies this condition. The resulting homomorphisms from Zk(X) to
are called specialization homomorphisms (cf. § 10.1).
Proposition 2.3. (a) If D is a pseudo-divisor on X, and a., a' are k-cycles on X,
then
D ¦ (a + a') = D ¦ a + D ¦ a'

34 Chapter 2. Divisors
(b) IfD, D' are pseudo-divisors on X, and a. is a k-cycle on X, then
(D + D') ¦ a = D ¦ a + D' ¦ a
(c) (Projection formula) Let D be a pseudo-divisor on X/:X'-»Ia proper
morphism, a a k-cycle on X', and g the morphism from f~l(\D\)P\\a\ to
|D|n/(|a|) inducedbyf. Then
<?,(/*/)• a) =
(a)
inAk^(\D\nf(\<x\)).
(d) Let D be a pseudo-divisor on X, f: X' -* X a flat morphism of relative
dimension n, a a k-cycle on X, and g the induced morphism from f~l (\D\f)\<x\)
to\D\r\\*\. Then
(e) If D is a pseudo-divisor on X whose line bundle #x(D) 's trivial, and a is a
k-cycle on X, then
Da = 0
inAk-,(\a\).
Proof, (a) follows directly from the definition. For (b)-(e), there is
therefore no loss of generality in assuming a = [V], Fa variety, (b) follows
from the fact that restricting to V and forming associated Weil divisor classes
is compatible with sums. For (c), by functoriality of pull-back and push-
forward we may assume a = [K], V= X' andf(V) = X; D is represented by
a Cartier divisor, which we also denote D, and the content of (c) is the identity
of cycles on X:
This identity is a local assertion on X, so we may assume D = div (r) for some
rational function on X. Then from Proposition 1.4, with d = deg (X7X), one
has
/¦[div(/*r)] = [div (N (/*/•))] = div(^) = d [div (/•)],
as required.
For assertion (d) we may also also assume V— X, so D is represented by a
Cartier divisor. The identity to prove is that
[f*D]=f*[D]
as cycles on X'. This too is local on X, so we may assume D is the difference
of two effective divisors. Since both sides are additive, it suffices to prove the
identity when D is effective. This case is a special case of Lemma 1.7.1.
For (e), we may assume a = [V\, V= X, and D is represented by a Cartier
divisor on X. The assertion is then that [D] — 0 in Ak-, (X) when D is principal,
which we have seen (§ 2.1). ?
2.4 Commutativity of Intersection Classes
35
Example 2.3.1. Let a be a &-cycle on X, fi and /-cycle on Y, let D be a
pseudo-divisor on X, and let p be the projection from Xx Y to X. Then
(/>*D) • (a x/?)=(/)• a) x/?
in i4t+,_,((|i)|n|a|) x |/?|). (Reduce to the case fl = [Y], Y a variety, and apply
Proposition 2.3 (d).)
2.4 Commutativity of Intersection Classes
If D and D' are Cartier divisors on a variety X, with associated Weil divisors
[D] and [?>'], one may form the intersection classes D ¦ [/)'] and D' ¦ [D], both of
which are well-defined classes in A^(\D\n\D'\). When D and D' intersect
properly, i.e., no codimension one subvariety of X is contained in | D \ 0 | D' \,
the equality of these classes is quite straightforward. The "classical" method of
moving D or D' to linearly equivalent divisors that intersect properly may be
used — when such moving is possible — to show that the two classes agree in
At,(\D'\) or A^(\D\), but this does not prove their equality in A^(\D\f] \D'\),
which we shall need. Instead we proceed by blowing up along subschemes of X
to achieve the situation where D and D' are sums of divisors, each pair of
which either intersect properly, or are equal (the other case where commuta-
commutativity is obvious!).
Theorem 2.4. Let D and D' be Cartier divisors on an n-dimensional variety X.
Then
D ¦ [D1] = D' ¦ [D]
Proof. Case I: D and D' are effective and intersect properly. Let W be any
codimension two subvariety of X, let A =<?w,x, and let a, a' be local equations
for D, D' in A. The subvarieties V of X of codimension one which contain W
correspond to height one primes p in A. The coefficient of [V\ in [?>'] is
lA,{Ap/a'Ap). The coefficient of [W] in D-[V\ is lA/p(A/p + a A). The coef-
coefficient of [W] in D ¦ [D'} is therefore
YdlA,(Ap/a'Ap)-lA,p(.A/p + aA).
p
By Lemma A.2.7 applied to the ring A/a'A, this coefficient is
eA(a,A/a'A).
By Lemma A.2.8.
eA (a, A/a'A) = eA{a', A/a A) ,
which by the same argument is the coefficient of [W] in D' ¦ [D]. (In short, the
coefficient of [W\ in both sides is the multiplicity eA(a,a',A) discussed in
Example A.5.2.) ?

36
Chapter 2. Divisors
Some preparation is needed before proving the general case of the theorem.
If D and D' are effective Cartier divisors on a variety X, define the excess
of intersection e(D, D') by the formula
e(D, D') = max {ovdv{D) • ordy(D') | codim (V, X) = 1},
the maximum over all codimension one subvarieties V of X. Thus D and D'
meet properly precisely when e(D, D') = 0.
Let DOD' be the intersection scheme of D and D'. This is the subscheme
of X which on an affine open set U is defined by the ideal {a, a'), where a and
a' are local equations for D and D' in U. Let
n:X-*X
be the blow-up of X along DC\D', and let E = n~l (D n D') be the
exceptional divisor. The local equations for n*D and n*D' are divisible by the
local equations for E, so
n*D = E+C, n*D' = E+C
for effective Cartier divisors C, C on X.
Lemma 2.4. With the above notation,
(a) C and C" are disjoint.
(b) If e(D,D')>0, then e(C,E) and e(C',E) are strictly smaller than
e(D, D').
Proof. The assertions are local on X, so we assume X = Spec(A), and
D = div(a), Z)' = div(a') are principal divisors. Then X = Proj (©/"), where
/ = (a, a'). The surjective graded homomorphism
A[S, T]-*®In
given by sending S to a, T to a', determines a closed imbedding of X in
Proj(A[S, T]) = XxV>:
In fact, X is contained in the subscheme of Xx P1 where a' S— aT vanishes.
Let 0(Y) be the pull-back of the standard line bundle on P1 to X, and let s, t be
the sections of 0(X) induced by S, T. Then C is the zero-scheme Z(s) of s, and
C* = Z(t). Indeed, the equation a' =(tls) a shows that n*D agrees with E on
the open set where s 4= 0, and the equation a = (s/t) a' shows that n*D agrees
with E + Z(s) on the open set where t 4= 0. Since Z(s)f)Z(i) = 0, this proves
that
as desired, and (a) follows.
We also see from this description that C <= Xx {0} and C <= Xx {oo} map
isomorphically by n into subschemes of D and D' respectively. Therefore if
2.4 Commutativity of Intersection Classes
37
V is any codimension one subvariety of X contained in Cf]E or COE, then
V = n(V) is a codimension one subvariety of X contained in DOD'. Since
[D] = Trji" + C] (by Proposition 2.3 (c)),
ord^Z) §
and similarly for D'. Suppose e(C, E) ^ e(D, D') > 0, and F<=cn? is chosen
with ordyC • ord^? = e(C, ?). Then
Z)') ^ ordyD ¦ ordyD'
a (ordpE + ord(?C) (ordpE + ordpC)
since ord pE > 0, this is a contradiction. ?
To prove Theorem 2.4 we will also need the following fact:
(*) If D, D' are Cartier divisors on X, n: X -* X is a proper birational
morphism of varieties, n* D = B ± C, n*D' = B' ± C", for Cartier divisors
B, C, B\ C on X with \B\\J\C\ <= n'l(\D\), \B'\\J\C\an-l(\D'\), and the
theorem holds for each of the pairs (B, B'), (B, C"), (C, B'), and (C, C) on X,
then the theorem holds for (D, D') on X.
Indeed, by Proposition 2.3, if g is the induced morphism from
'] = g*((B±Q[B'±C'])
= g*{B [B'}± B [C]± C [B']
= g* (B' ¦ [B] ± C ¦ [B] ±B'[C]
C
C- [C])
We return to the proof of Theorem 2.4. Case 2: D and D' are effective. This
is proved by induction on e(D, /)'), the start e = 0 being Case 1 proved above.
If e (D, D') > 0, blow up X along Df)D' as in the lemma. By assertion (b) of
the lemma, and the inductive hypothesis, the theorem holds for (?, C") and
(C, E). The theorem is trivially true for (E, E), and it is true for (C, C") since
CH C = 0. An application of (*) completes the proof.
Case 3: One of D, D' is effective. Suppose D' is effective. Let f be the ideal
sheaf of denominators for D: on an open affine U = Spec (A) where D has local
equation d,/^is determined by the ideal
{a 6 A | ade A}.
Let n: X -* X be the blowup of X along the subscheme defined by/", and let ?
be the exceptional divisor. Then
for an effective divisor C on X. Since |C|U|?| cr^'dDI), and Case 2 covers
the pairs (C, 7r*Z)') and (?, n*D'), an application of (*) completes the proof.

38
Chapter 2. Divisors
Case 4: D, D' arbitrary. Let n: X -* X be the blowup along the denomi-
denominators of D as in Case 3. Then the pairs (C, n*D') and (E, n* D') are covered
by Case 3, and we conclude by another application of (*). ?
Corollary 2.4.1. Let D be a pseudo-divisor on a scheme X, a a k-cycle on X
which is rationally equivalent to zero. Then
Z)<x = 0
Proof. If a = [div (/•)], r e R(V)*, Fa subvariety of X, we may replace X by
V, and D by a representing Cartier divisor. Then
Z)-[div(r)] = div(r)-[Z)] = O
in Aic-i (\D\), by Theorem 2.4 and Proposition 2.3 (e) respectively. ?
Definition 2.4.1. If D is a pseudo-divisor on a scheme X, and Y is a closed
subscheme of X, the mapping a -* D ¦ a determines homomorphisms
By Corollary 2.4.1 (applied to the restriction of D to Y), D ¦ a = 0 if a ~ 0 on
Y, so these homomorphisms pass to rational equivalence, defining homo-
homomorphisms
also denoted a -* D • a, and called intersecting with D.
Corollary 2.4.2. Let D and D' be pseudo-divisors on a scheme X. Then for
any &-cycle <x on X,
D(D'a) = D'(D- a)
Proof. Note that- D' ¦ ae Ak_l(\D'\f)\a\), and intersecting with D takes
4*-i(|?'|n|a|) to ^t.2(|Z)|n|Z)'|n|a|). Taking a = [V], and restricting
D and D' to F, one is reduced to Theorem 2.4. ?
Definition 2.4.2. Let D,,..., Dn be pseudo-divisors on a scheme X For any
a 6 ZkX, define D1 ¦... ¦ Dn-ain Ak.,(\D11 fl... fl |DJ fl |a|) by induction:
ZV...ZV<x=Z),(ZJ-...ZVa).
From Corollary 2.4.2 this is independent of the ordering of the Dh and from
Proposition 2.3 it is linear in each variable D\,..., Dn, and a. More generally,
for any homogeneous polynomial P(T,, ...,Tn) of degree d with integer coeffi-
coefficients, and any closed subscheme Z of X containing (|D, |U ... U|DJ)n|a|,
the class
P(Dl,...,Dn)aeAk-d(Z)
is defined by adding terms in the preceding definition. This is additive in P
and in a.
2.4 Commutativity of Intersection Classes
39
If n = k, and y=|D1|n...n|Z)jn|<x|is complete, we define an intersection
number (D, •... ¦ Dn • oi)x by
Similarly ifZ = (|D1|U...U|Z)B
as above, define
|a|is complete, k = d, and P is a polynomial
{P{Du...,Dn)-a)x=\P{Du...,Dn)-a.
z
If Vis a pure A:-dimensional subscheme of X, we may write P(DU ..., Dn) ¦ V
in place of P(D\,..., Z)n) • [V\. If A1 is pure dimensional, we may abbreviate
this further, writing simply P (D,,..., Dn) in place of P (D,,..., Dn) ¦ [X]. For
example, we may write D' e Ak-,(\D\) in place of D' ¦ [X], if X is purely k-di-
mensional. Similarly if X is purely ^-dimensional, and P is homogeneous of
degree k, we write (P (D,,..., Dn))x in place of (P (D,,..., Dn) ¦ [X])x.
Example 2.4.1. Let n: X -* A2 be the blow-up of A2 at the origin. Let D
and D' be the inverse images of the A:-axis and >>-axis. Then D and D' are
principal divisors on I, so fl- [D'] and D' ¦ [D] are well-defined cycles on
D f] D' = E, the exceptional divisor (cf. Remark 2.3). These cycles are not
equal, although they are rationally equivalent in E = P1.
Example 2.4.2. The operations of Definition 2.4.2 are compatible with
proper push-forward and flat pull-back, as in Proposition 2.3 (c), (d). In
particular, the construction is local, in the following sense. If U is an open
subscheme of X which contains |D,|n...n|Z)jn|a|, and Df, av denote re-
restrictions of Dj, a to U, then
ZV...-ZVa=Z)r..-Z)?-a"
Example 2.4.3. If/: X' -* X is a proper morphism, Z),, ...,/)„ are pseudo-
divisors on X, P a polynomial of degree d, and a is a d-cycle on X' with
|D, |n...n|Djn/(|a|) complete, then
Example 2.4.4. Let V be an irreducible surface, P a singular point of V,
and let n: X -* V be a proper morphism, E = n~x (P). Assume that X is
regular, that 7t maps X—E isomorphically onto V—P, and E is connected.
Then(cf. Mumford(l))
(D D)x<0
for any effective, non-zero, divisor D on X which is supported in E. (To see
this (cf. Deligne-Katz A) X), let E\,...,Er be the irreducible components of
E. Choose a rational function/on V, so that
= 'Z m,E,

40
Chapter 2. Divisors
with m,|> 0, Z a divisor not containing any ?,, (Z?,)xS0 for all i, some
(ZEi)x > 0. Set D; = m{ ?r. Replacing D by a positive multiple, one may assume
D = Y.Ui atDi- Then
(Z) • D)x =
a, (a, - a,) (Z), • />,)* -So? (A • Z)x
= - Z (fly - ad2 (D, ¦ Dj)x - E fl? (A • Z)* < 0 .
Corollary 2.4.1 was used for the second equality.)
Example2.4.5. Let (x:y:z:t) be homogeneous on P3, and let X be the
singular cone defined by the equation z2 = x y. Let D be the Cartier divisor on
X defined by the equation x = 0. Let / be the line x = z = 0, /' the line
y = z = 0, P the point @:0:0:1). Then [D] = 2[/] and D ¦ [/'] = [/>]. It follows from
Theorem 2.4 that there cannot be a Cartier divisor D' on X with [/)'] = [/'],
either as cycles or as classes in A\ (X).
Example 2.4.6. In the construction in the proof of Lemma 2.4, the sub-
scheme of Xx P1 defined by the equation a'S = aT may not be reduced; in
particular, it may not be equal to X. (Let X = V(x w —yz) <= A4, a = x, a' = y.)
Example 2.4.7. Let/: X -* C be a dominant morphism from a variety X to
a curve C, P a simple point on C, D =/"' (/>). Then D is an effective Cartier
divisor on X, and D2a = 0ini4fe_2(|D|n|a|)for any &-cycle a on X. (Shrink C,
so Z) becomes principal.)
Example2.4.8. Let Dlt...,Dn be effective divisors on an n-dimensional
variety X, which intersect in a finite set. Assume that for each point P in the
intersection, local equations for D,,...,Dn form a regular sequence in #PX-
Then
This is the case whenever X is Cohen-Macaulay (Lemma A7.1). For more on
this intersection number, see Example 12.4.8, and Lomadze A).
Example 2.4.9 (cf. Mumford B), Deligne-Katz A)X). Let X be a non-singular
surface, D, E Cartier divisors on X, with |Z)|Hi-fi11 complete, so that the inter-
intersection number (D ¦ E)x is defined.
(a) If D and E are effective and meet properly, then
(DE)x = deg[D0E].
(b) If D is effective and complete, then
(D-E)x=degp(E)\D).
(c) If D and E are effective, then
(D ¦ E)X=X(<
2.5 Chern Class of a Line Bundle
(d) If Xis complete, then
41
In (c) and (d), x denotes the Euler characteristic (cf. § 15.1). (The formulas
(b)-(d) are bilinear in D and E, so one is reduced to the case where D and E
are effective and either meet properly or are equal.)
2.5 Chern Class of a Line Bundle
Let L be a line bundle on a scheme X. For any ^-dimensional subvariety V of
X, the restriction L \ v of L to Fis isomorphic to ^(C) for some Cartier divisor
C on F, determined up to linear equivalence (§2.2). The Weil divisor [C]
determines a well-defined element in Ak_1{X\ which we denote by
Ci(L)n[V]:
cl(L)n[V] = [C].
This is extended by linearity to define a homomorphism aHc,(L)r;a from
Zk(X) to /4*-i (JT). If L = #x(D) for a pseudo-divisor D on X, it follows from
the definition of the intersection class (§ 2.3) that
n <x = D ¦ a
Proposition 2.5. (a) If a is rationally equivalent to zero on X, then C[ (L) na
= 0. There is therefore an induced homomorphism
(b) (Commutativity). If L, L' are line bundles on X, a a k-cycle on X, then
c, (L) n (c, (//) n a) = c, (/,') n (c, (L) n a)
inAk-2(X).
(c) (Projection formula). Iff: X' -* X is a proper morphism, L a line bundle
on X, a a k-cycle on A", then
(d) (Flat pull-back). Iff: X' -* X is flat of relative dimension n, L a line
bundle on X, a a k-cycle on X, then
c,(/*I)n/*a=/*(c,(Z,)noO
inAk+n-i(X').
(e) (Additivity). If L, L' are line bundles on X,aa k-cycle on X, then
and
inAk-\X.
] (L ® L') n a = c, (L)na + c, (Z/) n a
Ci (Z.v) na = -C| (Z,) n a

42
Chapter 2. Divisors
Proof. Since a line bundle on X determines a pseudo-divisor on X with
support X, the assertions follow from the corresponding facts for pseudo-
divisors, i.e., Corollaries 2.4.1, 2.4.2, and Proposition 2.3 (c), (d), and (b). ?
Definition 2.5. From (a), (b), and (e) of the Proposition, it follows that
arbitrary polynomials in Chern classes of line bundles act on A+X. If
L],...,Ln are line bundles on A', a e Ak X, and P (Tt,..., Tn) is a homogeneous
polynomial of degree rfwith integer coefficients, then
P (c, (L,),..., c, (/,„)) n a
is defined in Ak^d(X). In particular, for a line bundle L on X, a e Ak (X)
is the element in Ak-jXdefined inductively by C\ (L)dna=c1 (L) n(c\ (L^'na).
Example 2.5.1. Let Lk be a A>dimensional linear subspace of P", k = 0,..., n.
Then
for k=\,...,n. This gives another ' proof that AkP" = TL ¦ [Lk] = TL for
k = 0,..., n (cf. Example 1.9.3(b)).
The degree deg (a) of a &-cycle a on P" is defined to be the integer d such
that a ~ d[Lk\ Equivalently,
deg(a)= JCl@(l))*na.
Example 2.5.2. Let Ibea closed subscheme of P" of dimension ^ k, and
let F(X) be the homogeneous coordinate ring of X.
(a) For t sufficiently large, the dimension of the rth graded piece F(X), is a
polynomial in / of degree S k called the Hilbert polynomial (cf. Hartshome
EI.7.5). Define dk(X) to be the coefficient of tk/k\ in this polynomial.
(b) If JT|,..., Xr are the irreducible components of X, m, the multiplicity of
Xj inX, then dk(X) = Y. midk{X,).
(c) If X is an irreducible variety, and H is a hypersurface of degree m not
containing A", then dk_1(XC\H) = mdk{X). (Here Xfltf is the scheme-
theoretic intersection, so there are exact sequences
0 -» /-(*),_„ -» r(X), -» r(*D ff), ^ 0 for t » 0.)
(d) For any purely fe-dimensional subscheme X of P",
(For X a subvariety, Ci@(l)) n [A"] = [ATIH], where ff is a hyperplane not
containing X.)
Example 2.5.3. If a, /? are cycles on X, Y, L a line bundle on X, p the projec-
projection from Xx Y to X, then (cf. Example 2.3.1)
c,(/>*Z,)n(ax/?)=(c,(Z,)na)x/?.
2.6 Gysin Map for Divisors
43
Example 2.5.4. The operation C\ (L) n_ is uniquely determined by proper-
properties (c), (e) of Proposition 2.5, and the normalization that
c,(*(Z>))n [*] = [/>]
for D an effective Cartier divisor on a normal variety X. (Argue as in the proof
of Theorem 2.4, Case 3.)
Example 2.5.5. If D is an effective Cartier divisor on a variety X, the
restriction of^x(D) to D is the normal bundle NDX, and
If L is another line bundle on X whose restriction to D is NDX, however,
C] (L) n [X] need not be represented by [D]; cf. Proposition 2.6(c).
Example 2.5.6. Plitcker formulas. Let C be a non-singular projective curve
of genus g, and let C (/¦)<= Cx C be the subscheme defined by the ideal sheaf
¦S'+], where -f is the ideal sheaf of the diagonal; let p and q be the first and
second projections from C(r) to C. For a line bundle L on C, the bundle of
principal parts P'(L) is the sheaf on C defined by:
Then /""(I) = L, and for r > 0 there is an exact sequence
0 -> (f2?)®f ® Z, -> Pr(Z,) - /"-' (Z,) -> 0 .
Therefore Pr(L) is locally free of rank r + 1, and
/»(I)) =(/•+ 1) c, (?,) +
c,
c,
If F<= //°(C, ?) is a subspace, there are canonical homomorphisms of
vector bundles on C,
a:Cx V^ P"(L).
If dim F= r+ 1, i.e., the linear system determined by Fhas dimension r, then
det(tr) is a section of Ar+lPr(L), well-defined up to scalars. If det((r) #= 0, its
divisor of zeros, denoted 5^, measures the osculation of the linear system. Then
deg (.5,) = 0-+ 1) deg (I) + ('$') B <7 - 2).
For a sample of applications see Piene A), Laksov D), Eisenbud-Harris A).
2.6 Gysin Map for Divisors
Let D be an effective Cartier divisor on a scheme X, and let /: D
inclusion. There are Gysin homomorphisms
i*:ZkX -+ Ak-XD
X be the

44
Chapter 2. Divisors
defined by the formula
i* (a) = D ¦ a
where D ¦ a is the intersection class in Ak-t D defined in § 2.3.
Proposition 2.6. (a) If a is rationally equivalent to zero on X, then i*a = 0.
There are therefore induced homomorphisms
Ak-\D .
(b) If a is a k-cycle on X, then
(c) If a is a k-cycle on D, then
i*i*(a) = C) (N) n a ,
where N = i*tfx(D).
(d) If X is purely n-dimensional, then
i*[X] = [D]
inA^D.
(e) IfL is a line bundle on X, then
i*(ci (L) n a) = c, (i*L) n i*(a)
in Ak-2 {D)for any k-cycle a on X.
Proof, (a) and (e) are special cases of Corollary 2.4.1 and 2.4.2 respectively,
(b) and (c) follow from the definitions: in both cases, both sides are
represented by the intersection class D ¦ a. (d) says that [D] = D ¦ [X], which is
a restatement of Lemma 1.7.2. ?
Example 2.6.1. Let I be a line bundle on X, p: L -* X the projection,
i:X -* L the imbedding of Xin L by the zero section. Then i*(p*a) = a for all
a 6 AkX. One concludes from Proposition 1.9 that
p*:AkX->Ak+lL
is an isomorphism (see § 3.3 for generalizations).
Example 2.6.2. Let X be a closed subscheme of P", and let X' be a cone
over Xin P"+l. Then/H(A") = Z, and Ak(X') s Ak-i (X) for k > 0. (If P is the
vertex of the cone, the complement of P in X' is a line bundle over X. Use
Proposition 1.8 and Example 2.6.1.)
Example 2.6.3. (a) Let L be a line bundle on X, let L - {0} be the
complement of the zero section, and let n be the projection from L — {0} to X.
Then for all k ^ 0, the sequence
AkX X Ak+l (L - {0}) - 0
is exact.
45
Notes and References
(b) Let X be a closed subscheme of P", with canonical line bundle
Let Va A"+1 be the affine cone over X. Then Ao F= 0, and for k > 0 there is
an exact sequence
Example2.6.4. Pure-dimensionality is needed in Proposition 2.6(d). Let
X= V(z x, zy) <= P3, D the Cartier divisor on X defined by the equation
z — x = 0. Let i be the inclusion of D in X. Then (cf. Example 1.7.1)
D[X] = i* [X] + [D], c,
n
* [/)].
Example 2.6.5. Let D be an effective Cartier divisor on a scheme X, / the
inclusion of D in X. Let K be a closed subscheme of X. Assume that Y is
purely n-dimensional and D0Y has dimension n — 1. Let Vl,...,Vr be the
(reduced) irreducible components of Z) f] Y. Let ^4, be the local ring of Y
along Vt, and let a, be a local equation for D in AK. Then
where e^a^A^) is the multiplicity defined in Appendix A.2. More generally, if
one assumes only that dim Y^n and dim Df)Y ? n — 1, and l^,..., Vr are
the components of D0Y of dimension n— 1, then the right side of this
equation gives a formula for ;*([y]n), where [Y]n is the n-dimensional
component of the cycle of Y. (These formulas follow from Lemma A.2.7.)
Example2.6.6. Let Jbea scheme, and let a be a &-cycle onlx P1. Let i'o
and la, be the imbeddings of X in Xx P1 at 0 and oo respectively. Then
it ol = iS, a.
in Ak-\X. (If <x = [V], and the projection of Fto P1 is not dominant, both sides
are zero. If the projection is dominant, the equation says [V@)] = [V(oo)].)
More generally, for any / 6 P1, rational over the ground field, if i, is the
imbedding of X in X x P' at /, then i*, = i% = ij.
Notes and References
Generalities on Cartier divisors and Weil divisors, including the material of
§ 2.1, may be found in [EGAjIV.21 (Many otherwise standard sources discuss
one or both kinds of divisors only on normal or locally factorial varieties.)
The intersection class constructed in § 2.3 has not appeared before, either
for Cartier divisors or the pseudo-divisors introduced here; its construction
was implicit in Fulton-MacPherson A), however. The statement and proof of
the fundamental Theorem 2.4 follow Fulton F). A special case of Corollary
2.4.1 - that rational equivalence specializes - had been conjectured by
Grothendieck ([SGA6JX.7) in the non-singular case; this case was proved in

46
Chapter 2. Divisors
Fulton B), by a method similar to that used in Baum-Fulton-MacPherson A)
II.2.5, to prove a Riemann-Roch theorem. Conversely, having Theorem 2.4 will
simplify the proof of the Riemann-Roch theorem (cf. § 18.1).
It has long been understood (cf. Hirzebruch A), Grothendieck A), B)) that
a good knowledge of first Chern classes for line bundles determines higher
Chern classes for vector bundles. The development followed in the present text
may be seen as a corresponding principle in intersection theory: a good
knowledge of how divisors intersect determines intersections in arbitrary
codimension. An important case of this principle was given by Verdier E),
who constructed Gysin maps in arbitrary codimension from the case of
divisors.
Chapter 3. Vector Bundles and Chern Classes
Summary
We will construct, for any vector bundle ? on a scheme X, Chern class
operations
C;(E)n_:AkX^>Ak-iX,
satisfying properties expected from topology. From the special case of line
bundles done in § 2.5, we first construct inverse Chern classes, or Segre classes,
which are then inverted to produce Chern classes. The first Chern class
operations are also used to describe/I*? and A*P(E) in terms of A*X.
Chern classes will be used later for one of the constructions of general
intersection products. Although Chern classes are not absolutely needed for
intersection theory, they are used in most applications. For the quickest route
to intersection theory proper, the reader will need only Proposition 3.1 (a) and
Theorem 3.3.
For vector bundles, Chern classes and Segre classes determine each other;
Chern classes are preferred since they vanish beyond the rank of the bundle.
We will see in the next chapter that for cones — "singular vector bundles" -
there is a natural analogue of Segre classes, but not Chern classes. Segre classes
for normal cones have other remarkable properties not shared by Chern classes
(cf. § 4.2).
3.1 Segre Classes of Vector Bundles
Let Ebe a vector bundle of rank e + 1 on an algebraic scheme X. Let P = P(E)
be the projective bundle of lines in E, p = pE the projection from P to X, and
let 0{X) =0e(X) denote the canonical line bundle on P, i.e., its dual fi{— 1) is
the tautological subbundle ofp*E (see Appendix B.5).
Define homomorphisms a ->.?,(.?) na from AkX to Ak-iX by the formula
Herep* is the flat pull-back from 4* X to Ak+eP (§ 1.7), c,(<!?(l))e+/n _ is the
iterated first Chern class homomorphism from Ak+eP to Ak-iP (§ 2.5), and />*
is the push-forward immAk-iP toAn-iX(% 1.4).

48
Chapter 3. Vector Bundles and Chern Classes
Proposition 3.1. (a) For all a e Ak X,
(i) ii(?)na = 0 for i<0;
(ii) sQ(E) na = a.
(b) IfE, Fare vector bundles onX,a e Ak X, then, for all i,j,"
s, (E) n (sj (F) n a) = sj (F) n fa (E) n a).
(c) Iff: X' —> X is proper, E a vector bundle on X, a e A*X', then, for all i,
/, fa (/¦?) n a) = j,(?)n/¦(«).
(d) ///: JT' -> X is flat, E a vector bundle on X, a e A*X, then, for all i,
s,(f*E)nf*a=f*(s,(E)na).
(e) IfE is a line bundle onX,a e A* X, then
S] (E) n a = — C\ (E) n a.
Proof. We first prove (c) and (d). Given/: X' -* X, and a vector bundle E
on Jf, there is a fibre square
p'l
^/*f(l)- If/is proper, a e A*X', then
/» (*,(/¦?) n a) =/,/>; (c, (^/.
= />*
= P*
A))'+/ n /?'* a)
^(l)r+'n/7'* a) (§ 1.4)
t+I n/;/»'* a) (Proposition 2.5 (c))
*+' n /»*/*«) (Proposition 1.7)
The proof of (d) is similar, using the corresponding fact (Proposition 2.5 (d))
for line bundles, and is left to the reader.
To prove (a), we may assume a = [V\, with Ka fc-dimensional subvariety of
X. By (c), we may assume X= V. Then Ak-, X is zero for / < 0, which proves
(i). Also,
jo(?) n a =p* (c, (^A))' n [/>]) = m [JT]
for some integer w. To check that m = 1, by (d) we may restrict to an open set
of X, so we may assume E is trivial. Then P (E) = X x Pe, and ^A) has
sections whose zero scheme is X x J*. Then
by definition of the Chern class of a line bundle. Repeating this e times shows
that m must be 1.
3.1 Segre Classes of Vector Bundles
To prove (b), form the fibre square
49
P(E)
P{F)
4
wherep and q are the projections. Let/+ 1 be the rank of F. Then
s,{E) n(Sj(F) n a) =P% (c, (^
= /;» (c, (
; n /7*(9* (d
/ n ?;//¦ (ci
l))/+y n 9* a))
(Proposition 1.7)
= P* qi (ci (?'V?A))'+/ n (c, W*0f(\)y+> n /* ?•«))
Proposition 2.5 (c), (d))
= 9*^; (c, (p'*#F(l))f+J n (c, (?'*^A))'+/ n ?'V* a))
by Proposition 2.5 (b) and the functoriality of push-forward and pull-back. The
steps may now be reversed, to arrive at
Sj(F)n(Si(E)na).
For (e), note that P(E) = X and 0E{- 1) = E in this case, so ^?A) = Ev,
and
sx(?) n a = c1(
by Proposition 2.5 (e). D
n a = - cx(?) n a
Corollary 3.1. The flat pull-back
p*:AkX->Ak+e(P(E))
is a split monomorphism.
Proof. By (a) (ii), an inverse is fi -> />* (c, (^?(l))e n /?). ?
For a geometric appearance of Segre classes, see Example 3.2.22.
Example 3.1.1. Let ? be a vector bundle of rank e+ 1, L a line bundle.
Then
sp(E ® L) = t (~ ir'Gtt) Ji(?) c, W' ¦
(Identify />(?) with P(E ® L), with universal subbundle ^? (- 1) ® p*L. Then
)Y+" np* a).)

50 Chapter 3. Vector Bundles and Chern Classes
3.2 Chern Classes
Let ? be a vector bundle on a scheme X. Consider the formal power series
00
s,(E) = X Si(E) t'=l+ st (?) t + s2(E) t2 + ....
Define the Chern polynomial
00
c,(?) = X c,(E) t'=l+ c, (?) f + c2(?) t2 + ...
i-0
to be the inverse power series (which will be shown to be a polynomial):
c, (?) = *,(?)"'.
Explicitly,
co(?)=l, c, (?) = -*,(?),
c.(?) =-«!(?) cn.x{E) - S2(E) c._2 (?)-...- «„(?).
Here the j,-(?) are regarded as endomorphisms of A*X, with products
denoting composition; since all such endomorphisms commute (Proposi-
(Proposition 3.1 (b)), there is no ambiguity. We write, for a e At X,
c,(?) n a
for the element in Ak-, Jf obtained by applying the endomorphism c,-(?) to a.
The Jofa/ C/ie/Ti daw c (E) is the sum
c(?)=
r=rank(?).
In other words, c(?) n a = 2i=0 c,(?) n a for all a e /i^X Similarly, the
Segre class is
Although an infinite number of terms appear in this formal sum, when applied
to a e AJC, s(E) n a = Y.ao s,(E) n a, only a finite number of non-zero terms
appear.
Theorem 3.2. 77ie Chern classes satisfy the following properties:
(a) (Vanishing) For all vector bundles E on X, all i > rank (?),
c,(?) = 0.
(b) (Commutativity) For a// vector bundles E, F on X, integers i,j, and cycles a
onX,
c, (E) n (cj (F) n a) = c, (F) n (c, (?) n a).
(c) (Projection formula) Le? E be a vector bundle on X,f:X'^Xa proper
morphism. Then
/¦ (c, (/¦?) n a) = c,(?)n/„,(«)
for all cycles a on X', all i.
3.2 Chern Classes
51
(d) (Pull-back) Let E be a vector bundle on X, f: X' -» X a flat morphism.
Then
c,(/*?)n/*a=/*(c,(?)na)
for all cycles a on X, all i.
(e) (Whitney sum) For any exact sequence
0 ->?'->?->?"-> 0
of vector bundles on X, then
c,(?) = c,(?')-c,(?"),
i.e.,
= Z
@ (Normalization) If E is a line bundle on a variety X, D a Cartier divisor
on X with ^(D) S E, then
c,(E)n[X] = [D].
Note that from (c) and @ it follows that the first Chern class for a line
bundle defined here agrees with the definition given in § 2.5. Formula @ will
be generalized to bundles of arbitrary rank in Chap. 14.
Proof. Properties (b), (c), (d), and @ follow directly from corresponding
facts proved for Segre classes in Proposition 3.1. There are several proofs of (a)
and (e). The following proof is based on the splitting construction, which will
be useful later. Two shorter proofs are sketched in Examples 3.2.9 and 3.2.10.
Splitting construction. Given a finite collection y of vector bundles on a
scheme X, there is a flat morphism/: X' -> X such that
A)/* : A*X -> A*X' is injective, and
B) for each E in S,f*E has a filtration by subbundles
/*? = ?r=>?r_|3... =>?, z>?0 = 0
with line bundle quotients L, = ?,-/?,-_ i.
For one bundle E, f is constructed by induction n the rank of E. Let
P= P(E), p: P -» X the projection. Then p* is injective by Corollary 3.1, and
p*E has a subbundle fiE(- 1) of rank one. If ?' is the quotient bundle
p*E/fiE{- 1), inductively we construct q: X' -» P with q* injective and q*E'
filtered. Thenf-pq has/* injective, and an induced filtration on/¦?. The
process may be repeated for any finite collection S of bundles.
To prove (a), it suffices by the splitting construction to prove that
when E has a filtration by subbundles
? = ?,=>?,_, =>...=>?0 =

52
Chapter 3. Vector Bundles and Chern Classes
with quotient line bundles L,= ?,/?,_,. For then i(f:X'-*X is as in the
splitting construction,
/* (c, (?) n a) = c, (/* ?) n /* a = 0
for i > r = rank(?), and (a) follows since/* is injective.
Lemma 3.2. /feume that E is filtered as above, with line bundle quotients
Llt..., Lr. Let s be a section of E, and let Z be the closed subset of X where s
vanishes. Then for any k-cycle a on X, there is a (k — r)-cycle p on Z with
II c, (L,) n a = ?
/-i
in Ak_r(X). In particular, if s is nowhere zero, then I7'=i ci(^d — 0-
Proof. The section s determines a section J of the quotient bundle Lr. If Y
is the zero-scheme of J, then (L,, Y, s) determines a pseudo-divisor Dr on X
(§ 2.2). Intersecting with Dr gives a class D, ¦ a in /U-i (Y) such that
where / is the inclusion of Y in X. By the projection formula (Proposition
2-5(c)),
lr-\
IIc.(Z.,)na=yJnciO"*-
(Dr ¦ a) .
/
The bundle y*?f-i has a section, induced by s, whose zero set is Z. By
induction on r the term in parenthesis on the right side of the preceeding
formula is represented by a cycle on Z, which concludes the proof. ?
We return to the proof of (*). Let/;: P(E) -> Xbe the associated projective
bundle. The universal subbundle ^(—1) of p*E corresponds to a trivial line
subbundle of p*E®0{\), i.e., to a nowhere vanishing section of /?*
Since /?*?®^A) has a filtration with quotient line bundles /?*
Lemma 3.2 implies that
Let f = c, (^A)), and let cr, (resp. <?,•) be the i'* elementary symmetric function
of c,(Lx),...,cx{Lr) (resp. c,(/»*L,),..., c,(/»*Lr))- Then c{{p*L,®# {\))
= Cjfp* L,) + C by Proposition 2.5(e), so the above displayed equation may be
written
{'+a, {'-'+ ¦¦• + <?, = ().
Therefore, with e = r - 1,
for all; § 1. It follows that for all a e /J*Jf,
3.2 Chern Classes 53
From the definition of Segre classes, and the projection formula, this says
Si(E) na + ff, j,--,(?) na + ... + arsHr(E) n a = 0,
which means that
and this is equivalent to (*).
The Whitney sum formula (e) also follows readily from (*). Given an exact
sequence of vector bundles as in (e), apply the splitting construction to find
/: X -> X, with/* injective, so that/*?' filters with line bundle quotients L\,
and /*?" filters with line bundle quotients L". Then /*? has an induced
filtration with line bundle quotients L', and L". The formula
c,(/*?) = C,(/*?')c,(/*?")
follows from (*), and the required formula on X follows from the injectivity of
/*. ?
Remark 3.2.1. Uniqueness. The Chern classes are uniquely determined by
properties (d), (e) and (f) of the theorem, together with the property that
co(?) = 1. This follows from the splitting construction as in the preceding para-
paragraph.
Remark 3.2.2. Notation and conventions. By the commutative law (b), any
polynomial in the Chern classes of vector bundles on X operates oxvA+X. If
p = P(ck (?,),..., clm(Em))
for vector bundles Et,..., EmonX, and a e A*X, we write
for the result of applying this polynomial to a. If p is isobaric of weight d, i.e.,
homogeneous of degree d, where c,(?) has weight /, and a e AkX, then
p n a e Ak-dX.
Notation is often abused by writing simply p in place of/7 n [X]. When X is
non-singular, we will see in Chap. 8 that one may recover the action of p on
any a from the class p n [X], so there is no loss of information in this case. In
any case, we may write
in place of \x p n [X], the degree of the zero-dimensional component of p n [X].
Another shorthand is quite useful. If/? is a polynomial in Chern classes on
X, as above, and a morphism/: X' -> X is specified, then for any a e A+X', we
may write p n a for the result of pulling back the bundles to X' and operating
on a by the same polynomial in the pull-back bundles, i.e.,
p n a = />(?,,(/*?,),..., c.(/•?„)) n a .
If g : X" -> X' is a proper morphism, the projection formula (c) then reads
simply

54
Chapter 3. Vector Bundles and Chern Classes
for a e A*X". Similarly, if g is flat,
for a e A*X'.
In Chap. 17, we will define contravariant functors A*, with cap products
Then a polynomial in Chern classes of bundles on X will be an element of
A*X. The convention introduced in the previous paragraph will become part of
a general notation.
Remark 3.2.3. Splitting principle. Suppose one wants to prove a universal
formula involving Chern classes of a finite number of vector bundles in a
certain relation with each other. If the formula is true whenever the bundles
have nitrations with line bundle quotients, and the relation among the bundles
is preserved by flat pull-back, then the formula holds in general. This follows
from the splitting construction and the pull-back property (d).
If the Chern polynomial of a bundle ? of rank r is factored
then <Xi,..., ar are called Chern roots of E. This factorization may be regarded
as purely formal: the Chern classes of E are the elementary symmetric
functions of a\,...,a.r. Or one may use the splitting construction: if/*? is
filtered with line bundle quotients Lit then
with a, = Ci (Li). Any symmetric polynomial in the Chern roots of E has a
definite expression as a polynomial in the Chern classes of E. The Whitney
sum formula says that for an exact sequence as in (d), Chern roots for E' and
E" taken together give Chern roots for E. We give some other applications.
(a) Dual bundles. The Chern classes of the dual bundle Ev are given by the
formula
C/(?v)=(-l)'c,(?).
We have seen this when ? is a line bundle (Proposition 2.5 (e)). If E has a
filtration with quotients L,, then ?v has a filtration with quotients I^_,. Thus
if <X|,...,<xr are Chern roots for ?, — <Xi,..., —<xr are Chern roots for ?v,
which gives the displayed formula.
(b) Tensor products. The formula for Chern classes of a tensor product
? ® F has a simple expression in terms of Chern roots. If <Xi,..., a, are Chern
roots for ?, and /?i,..., /?s are Chern roots for F, then the sums
are Chern roots for E® F. This follows from the splitting principle and the
corresponding result for line bundles (Proposition 2.5 (e)). Thus <:*(? ® F) is
the kih elementary symmetric function of oti+ /?i,..., ar + /?,, which may be
3.2 Chern Classes
55
written as a universal polynomial in the symmetric functions of a,,..., ar and
/?i,...,/?5, i.e., as a polynomial in the Chern classes of E and F. In general
explicit formulas for these polynomials are rather complicated (see Example
14.5.2). Incase F= L is a line bundle, however, there is the useful formula
cr(E®L) = t cx(L)'cr^(E)
i-0
for the top Chern class. More generally,
which follows from the identity
where c, is the i* elementary symmetric function of a,,...,<xr. For a
reformulation see Example 3.2.2.
(c) Exterior powers. Similarly, if <X|,..., ar are the Chern roots of ?, then
c,(A'?)= El (l+K + ... + a,,)t).
n<... < iP
In particular
c1(A'E) = c1(E).
For this one uses the fact that if there is an exact sequence
0^L^?^?'^0
with L a line bundle, then there are exact sequences
0 -> A"~' ?' ® L -> A"? -> A"?' -> 0
(cf. Hi rzebruch AL.13).
Remark 3.2.4. In the course of the proof of Theorem 3.2 the identity
was proved. Here ? is a bundle of rank r on a scheme X, p is the projection
from P(E) to X, and ( = cl(#E(\)). Alternatively, this follows from the fact
that p*(E) ® 0e(\) has a nowhere vanishing section; then by the Whitney
formula, cr(p*(E) ®^?A)) = 0, which gives the displayed formula by Re-
Remark 3.2.3 (b).
Additional identities for Chern classes are given in Chap. 14, as well as in
the examples.
Example 3.2.1. Let/?: P(E) -> X be as in § 3.1, and let LE = 0E(- 1). Then
s(E) n <x = p*(s(LE) np*a)
for all a e A*X, and c(?) mx = s(E)~' n a.

56 Chapter 3. Vector Bundles and Chern Classes
Example 3.2.2. Let ? be a vector bundle of rank r, L a line bundle. Then,
(Equate coefficients of f in the formula for c,(E®L) in Remark 3.2.3.)
Another equivalent formulation is
c,(E® L)=(\+xt)'cT{E)
with x = c, (L) and x = t/(l + x t).
Example 3.2.3. The Chern character ch(?) of a bundle E is defined by the
formula
r
= ?exp(a,)
where exp(x) = ex = ?™=o x"/n\, and a.y,..., ocr are the Chern roots of E. The
first few terms are
ch(?) = r + Cl + ^(cl- 2c2) + g (c? - 3Cl c2 + 3c3)
+ — (cf - 4c? c2 + 4c, c3 + 2c\ - 4c4) + ...
where c;= c, (?). Then «th term is/?„/«!, where pn is determined inductively by
Newton's formula:
Pn~ C,pn-l + C2pn-2- ...+(- I)" Cn_,/»,+(- l)"«Cn = 0.
For any exact sequence of vector bundles as in Theorem 3.2 (e),
ch (?) = ch (?') + ch (?") ,
while for tensor products
ch(?®?') = ch(?)-ch(?')
Example 3.2.4. The Todd class td (?) of a vector bundle E is defined by the
formula r
where
E* the Bernoulli numbers), and a,,..., a, are the Chern roots of E. The first
few terms are
td(?) = 1 + y c, + -y (c? + c2) + y^- (c, a)
+
- cf + 4c? c2 + 2c22 + c, c3 - c4) + ... .
3.2 Chern Classes 57
For an exact sequence of vector bundles as in Theorem 3.2 (e),
td(?) = td(?')-td(?").
Example3.2.5 (cf. Borel-Serre A) Lemma 18). Let ? be a vector bundle of
rank r. Then
p-0
v) = cr(?) • td(?)-'.
(If a,,..., ar are Chern roots for E, the left side is
r r
X (- 1)" ? exp (- a,,.-...- cO = II 0 - exp (- a,))
0
= «,-...-
-«,¦))/«,),
which equals the right side.)
Example 3.2.6. If cti,..., <xr are Chern roots for E, then the sums
m\ <X| + ... + m,ar, for all /--tuples of non-negative integers mx, ...,mr adding
to m, are the Chern roots for the mih symmetric power Sym (?) of E. For
m = 2, explicit formulas are given in Example 14.5.1.
Example 3.2.7. (a) For bundles ?, F on X write
= 1 +(c, (F) - c, (?)) +(c2(F) - c, (F) c, (?) + c, (?J- c2(?)) + ...
and let c* (F — ?) be the kih term in this expansion. If rank E = r and
xavk.F = s, then
c,_r+I(F-?) na = Pif{cs{p*F®^E(\))np*a)
for all a e A*X, where /? projects P(?) to X. Generalizations of this formula
will be given in Chap. 14.
(b) For any element ?, in the Grothendieck group K°X of vector bundles
on X (cf. § 15.1), there is a well-defined Chern class c(?), and power series
X/}]-?[?,], then
It follows from the Whitney formula that these are well-defined on K°X.
Example 3.2.8. Let X, Y be schemes, p and q the projections from X x Y to
X and 7, ? and F vector bundles on X and 7, a e /J*Jlf, P e A*Y. Then (cf.
Example 2.5.3)
and
(c (?) ni)x(c(f)nffl = c(/>*? © ?*/=) nfixfl.
Example 3.2.9. The vanishing property (a) of Chern classes follows from
the Whitney sum formula (e) and the splitting principle.
Example 3.2.10. There are other simple proofs of the Whitney sum
formula (e):

58
Case I: rank E"=
fora = [K], K= JST. Let
Chapter 3. Vector Bundles and Chern Classes
1. It suffices to show that s(E') n a = c(E")s(E) n a,
LE = fiE{- 1). Consider the diagram
On />, the composite LE <= />*? -»p*E" determines a section of p*E" ®
whose zero-scheme is P'. So
i*[P']=(cl(p*E")-cl(LE))n[P].
c(E") s{E) na= p*(c{p*E") s{LE) n [/>])
Now
and
= P*{s{LE) c, (/?*?") n [/>]) -/>»(*
Therefore, when rank (?) > 1,
?") i (?) — 5 (?')) n a = p* (s (LE) c (LE) n j
n [/>]).
=/7*[P] = 0.
General case. Let 2 = P(E"V), q:Q-*X the projection, and let Z, be the
universal line bundle quotient of q*E". Construct a commutative diagram of
vector bundles on Q with exact rows and columns:
</*?'
1
9*?'
0
4
->¦ G -
4
->q*E->
4
L =
4
0
0
4
/
4
q*
4
?
4
0
Then c (q*E) = c (G) c(L), c (q*E") = c(F)c(L), and c(G) = c (q*Er) c (F), by
Case 1 and induction. Combining these gives
c(q*E)nq*a = c(q* ?') c(q* E") nq* a
for all a e A*X, which suffices since q* is one-to-one.
One may also deduce the general case from Case 1 by realizing the blow-up
of P(E) along P(E') as a projective bundle P(H) over /"(?"), where H fits
into an exact sequence
0 -> /?"*?' -> // -* L?» -> 0 .
3.2 Chern Classes
One has a commutative diagram
59
P(H)
x/-
x
with re*L? = LH. Then
j(?) n [Jf] = p*(s(LE) n [P(?)]) = p*K*(s(LH) n
= p'i(s(H) n [P(?")]) =/*'(^(/'*?') c
= j(?')n(j(?")n[JT]).
Example 3.2.11. From the exact sequence
on P" (Appendix B.5.8),
n [/>(?")])
where //= C|(^P»A)) is the class of a hyperplane. More generally, if ? is a
vector bundle on a non-singular variety X, p.P (?) -> X the projection, then
CG>(?)) = c,(/7*^) • c,(p*E ® ^A)) = c,(/7*7^) A + x tycx(p*
where x = cx @E(\)), r=rank?, t= t/(l + x t). (From Appendix B.2.7 and
B.5.8, there is an exact sequence
0
0 .)
Example 3.2.12. Adjunction formula. Let i:X -* Y be a closed imbedding
of codimension d of non-singular varieties with normal bundle iV. From the
exact sequence (Appendix B.7.2)
one has
If Jf is an intersection of divisors Du ..., Dj, then
d
From the projection formula one deduces
U(c(Tx) n [X]) = c{TY). JJ(A- D2, + D]-...).
For example, if Y = Pm, and deg /), = «,, then

60
Chapter 3. Vector Bundles and Chern Classes
h = c\ (i*0 A)). In particular,
/ d \
Ci(Tx)=[m+l-Z»i}h.
Example 3.2.13. Let X be a non-singular n-dimensional variety with tangent
bundle Tx. The total Chern class of X is c(Tx) n [X]. The Euler characteristic is
\x cn(Tx\ F°r example, if n = 1, the Euler characteristic is 2 — 2 g, where g is the
genus of the curve X (see § 15 for other definitions). The Todd class of X is td G^).
The Todd genus of X is {* td(Tf). When n = 1, the Todd genus is 1 - g.
Example 3.2.14. If C is an effective Cartier divisor on a complete surface
X, then
where iV is the normal bundle to C in X. If C and X are non-singular, and i is
the inclusion of C in X, then iV = i*Tx/Tc, so
Equivalently,
(C-(C+K))x=2g-2,
where K = — c\ (Tx) is the canonical divisor class of X, and g is the genus of C.
In case Jf is a non-singular surface of degree d in P3,
Example 3.2.15. (a) Let X be an n-dimensional abelian variety, i: X -> Pm a
closed imbedding, with normal bundle iV. Then
c(jV) = A + A)m+I
where h = cx(i*<9(\)).
It follows that there can be no such imbedding if m < 2 n. If m = 2 n, such an
imbedding is possible only if (cf. Corollary 6.3)
For example, if n = 2, m = 4, X must have degree 10; Horrocks and Mumford
A) have constructed such an example.
(b) Let / be the d-Md Veronese imbedding of P" in P, m ={ntd) - 1,
/V(l)^(rf)Th
c(A0=(l +^)m+7(l +/i)"+1,
where h = c, (^p.(l)). For the Veronese surface P2 ^ P5,
c(AT)= l+9/i + 30 h2,
= \-9h + 5lh2.
3.2 Chern Classes
61
(c) Let ; be the Segre imbedding of X = P"' x... x F"' in Pm,
™ = ri[=1(«i+ 1)- IThen
r
where A,¦ = c, (pr*^p-i A)), pr, the projection from X to the /* factor.
Example 3.2.16. Let ? be a vector bundle of rank r on X, s a section of E,
Z the zero-scheme of s.
(i) For any a e AkX, there is a class /? in Ak-r{Z) whose image in Ak-T{X)
is cr(?) n a. In particular, if Z = 0, cr(E) = 0. (Use the splitting principle and
Lemma 3.2.)
(ii) If X is purely H-dimensional, and ? is a regular section of E, then Z is
purely (n — /^-dimensional, and
These facts will be reproved and generalized in § 14.1. (cf. Grothendieck B)
§ 5 in the projective non-singular case).
Example 3.2.17. Let F be a subbundle of a vector bundle E, with quotient
bundle G. There is a regular section of p*G ® ^A) on P(E), whose zero-
scheme is P(F) (Appendix B.5.6). By the preceeding example,
r
[P(F)] = cr(p*G®#E(l)) n[P(E)]=?Jcl(#E(l)y np*(cr-,{G) ^[X])
;-0
in A*P(E), where r is the rank of G. In the non-singular projective case this
formula may be found in Ilori-Ingleton-Lascu A).
Example 3.2.18. Let Ym be a smooth subvariety of codimension d of a
smooth variety X, with normal bundle N. Applying the preceding example to
'1
X
one has
[Y] = cd(q*N ® * A)) n [PGV| y)] = ? {' n q*(cd-i(N) n[YJ)
in ^2m_, (PGV| y)), with C = c, (^ A)). Therefore,
d
in /42m_, (Jf), a formula of D. B. Scott A).
Example 3.2.19. Let X be a non-singular variety. Let X = P(TX) be the
projective bundle of tangent directions, p : X -> X the projection, L <= p*Tx the
universal line sub-bundle. Let/:Jf-> Pr be a morphism, i.e., a linear system

62
Chapter 3. Vector Bundles and Chern Classes
without base points, and let df: Tx ->/*7> be the induced differential. On X
there is a composite
L<=p*Tx-+p*f*Tr.
The locus where this composite is zero is the set of tangent lines at points
x e X which are tangent to every member of the linear system passing through
x. The vanishing of the composite corresponds to the vanishing of the
corresponding section of /7*/*7V® Lv, and the set is given a scheme structure
2 as the zero scheme of this section. If codim B, Jt) = r, then (cf. Example 3.2.16
and Lemma A.7.1)
] = cr (p*f* TP,®Lv)n[X]
*7» n[X\)
where ( = c, (Lv) = c, (^A)), and h = c, (/VPr(l)) is the divisor class of any
member of the linear system.
If a linear system has a base locus B, the above applies on X—B. If
codim(B, X) § /- + 1, then the restriction from AkX to Ak(X-p~' (B)) is an
isomorphism for fc = dim Z, so the above formula holds for the closure of [Z]
inA*(X) (cf. Vainsencher A)). Whenr= 1, this gives the formula
[Z] = (+2p*(h)
which is useful for giving a geometric interpretation of ( (cf. Example 16.2.4).
Example 3.2.20. Ramification. Let /: X" -» 7" be a morphism of non-
singular varieties. Let R(f) be the subset of points of X where the induced
map of tangent spaces is not an isomorphism. A scheme structure on R (/) is
given locally by the vanishing of the Jacobian determinant, i.e., R(f) is the
zero scheme of the map
/\"df: AnTx^> A"f*Tr,
or the zero scheme of a section of the line bundle A"f*Tr® A" T%. If
R(f)*X, then
[R(f)]=(cl(f*TY)-cl(Tx))n[X].
If n = 1, taking degrees of both sides yields the Riemann-Hurwitz formula
2gx- 2 = deg(/) BgY- 2)
where gx(resp. gY) is the genus of X (resp. Y). (See Examples9.3.12 and
14.4.8 for generalizations.)
Example 3.2.21. Dual varieties (cf. Deligne-Katz A) XVII).
(a) Let X" be a non-singular subvariety of P. Let IP be the projective
space of hyperplanes in P, and let
X = {(/>, //) e Xx Pm| 7> JTc: H)
3.2 Chern Classes
63
where 7> Jif is the tangent n-plane to X at P in P. The projection n:X-*X
identifies X with P(NV), where iV is the normal bundle to X in P. The image
of Jfby the projection/: X -> P is the dim/ vaner^ to Jf, denoted A^. Then
' v ' Jx(l+hJ
where h = c{ (#VA)). In particular, X* is a hypersurface if and only if the right
side is non-zero; in this case
deg(Xv) =
(-1)"
J
c(Tx)
" deg(l/Xv) i A + /zJ'
(The inclusion of X in Jf x P comes from the inclusion
so/V(l) =^-
(- 1).Therefore,
l
= Z
AndcG» = (l + /i)m+4(N) by Examples 3.2.11, 3.2.12.)"
(b) (cf. Kleiman (8) p. 364) Let X be a non-singular plane curve of degree
d. Imbed X in P, w = H2) - 1, by the /"-fold Veronese imbedding of P2 in P.
The dual variety to X in P is the variety of plane curves of degree r which are
tangent to X. Using (a), the degree of this variety is seen to be d(d + 2r - 3).
Example 3.2.22. Regard P3 as the Grassmannian of 2-planes in P3, with
universal rank 3 subbundle S and quotient line bundle Q. The variety Y of
conies in P3 may be identified with the projective bundle of Sym2Sv, or with
P(E), E = Sym2(Sv ® Qv). With this description, the variety of conies meeting a
given line is given by the vanishing of a section of 0E(i). (Since this concerns
divisors, it suffices to verify it on the complement of the set of planes containing
the line.) Thus the locus of conies meeting n given lines is represented by Cj (&B(\)f.
Computing the Segre class of E,
s(E)= l + Sh + 34h2 + 92hi
where h = C\ (Q), one sees in particular that
fCl@?(l))8= f *3(?) = 92,
the number of conies meeting 8 general lines. Similarly there are
JcatMl)O*-! s2(E)h = 34

64
Chapter 3. Vector Bundles and Chern Classes
conies meeting 7 general lines, whose planes pass through a given general
point, and §s1 (E) h2 = & conies meeting 6 general lines, whose planes contain a
given general line (cf. Schubert A) § 20).
3.3 Rational Equivalence on Bundles
Let ? be a vector bundle of rank r = e + 1 on a scheme X, with projection
n:E-*X. Let P(E) be the associated projective bundle,p the projection from
P{E) to X, and^ A) or^?(l) the canonical line bundle on P(E).
Theorem 3.3. (a) The flat pull-back
is an isomorphism for all k.
(b) Each element /? in Ak P (E) is uniquely expressible in the form
/?= I c,(^(l))'n/>*«,,
for a, 6 Ak_e+l(X). Thus there are canonical isomorphisms
Proof The surjectivity of n* was proved in § 1.9. Let 6E be the canonical
homomorphism from ©f=0 A^X to AifP(E) defined in (b):
To see that 0E is surjective, the same Noetherian induction argument used in
the proof of Proposition 1.9 reduces it to the case when E is trivial. By
induction on the rank, it suffices to prove that 8F is surjective when Be is
known to be surjective, where F= E® 1 is the direct sum of E and a trivial
line bundle.
Let P= P{E), Q= P(F) = P(E® 1), q:Q -> X the projection. We have a
commutative diagram
p ^qIe
identifying Q as the projective completion of E, P as the "hyperplane at
infinity" (see Appendix B.5). By Proposition 1.8, the row in the following
commutative diagram is exact:
i'
Ak-,X
3.3 Rational Equivalence on Bundles
Lemma 3.3. For all a e A*X,
65
Proof. It suffices to prove this for a = [V\, Va subvariety of X. Since#F(l)
has a section vanishing precisely on P (Appendix B.5), the fact that
c, @fW) n [q'1 V\ = [p-lV] follows from the definition of Chern classes. ?
Now if /? e A+Q, write./*/?= n*n for some a inA*X. Then /?— q*n is in the
kernel oij*. Since Ker(/*) = Im(/Kt), and inductively we are assuming 6E is
surjective,
i-0
for some a, e A*X. Since i*#r(\) = #e(\), the projection formula rewrites the
right side as
Now apply the lemma, obtaining
which shows that 9F is surjective.
To show that the expression in (b) is unique, suppose there is a nontrivial
relation e
Let / be the largest integer with a, =4= 0. Then
by Proposition 3.1 (a), a contradiction.
Finally, to see that n* is injective, let F=E®\,
notation as before. If ri*a = 0, a + 0, theny*g*<x = 0, so
= P{F), with other
c,
)' n /7*a,) = E c,
/ i-0
n
using Lemma 3.3 again. But this contradicts the uniqueness assertion of (b) for
the bundle E 0 1. ?
Definition 3.3. Let s = se denote the zero section of a vector bundle E; s is a
morphism from X to E with n° s = idx. The result of Theorem 3.3 (a) allows
us to define Gysin homomorphisms
r = rank E, by the formula
This Gysin homomorphism is an important intersection operation: given any
subvariety of E, or fc-cycle /? on E, no matter how it meets the zero section,

66
Chapter 3. Vector Bundles and Chern Classes
there is a well-defined cycle class ?*(/?) in Ak-rX. By the surjectivity of n*, s*
is determined by the fact that s*[iC[{V)] = [V] for all Vc X, and the fact that
s* preserves rational equivalence.
This ability to intersect with zero sections of vector bundles will be the
basis for the construction of general Gysin homomorphisms and intersection
products in later chapters.
Note that Theorem 3.3 was proved without using the higher Chern classes
constructed in § 3.2. Only Chern classes of line bundles, and Proposition 3.1 (a)
were used.
The following proposition gives a somewhat more constructive formula for
s*, using higher Chern classes.
Proposition 3.3. Let /? e AkE, and let /? be any element of Ak(P{E® 1))
which restricts to /? in AkE. Then
where q is the projection from P(E® 1) to X, and ? is the universal rank r
quotient bundle ofq*(E © 1).
If /?=X"f[^]' one may take P t0 be X«/[^]> Vi the closure of V, in
P{E © 1). Note that the composite q* 1 cr q*(E © 1) -> ?, gives a section of ?
which vanishes precisely on the zero section s (X).
Proof. Let F = E © 1, Q = P(F), i the inclusion of P = P(E) in Q, j the
inclusion of ? in Q. We must show that
rt*qAcr{S)nP)=j*fi
for all fie A^Q. By Theorem 3.3(b) and Lemma 3.3, we may write
for some classes y eA^X, 5 eA*P. Since j*q* = n* and/1";* = 0, the required
formula follows from the two formulas
(i) q*(cr{?)nq*y) = y
(ii) c,(f)ni^ = 0.
To prove (i), since ?, is a quotient of q*E © 1 by 0F(-1), the Whitney sum
formula gives
r
By Proposition 3.1 (a), (d),
?„ (cM) nq*y) = q* (c, @f(\))' n q*y) = y.
For (ii), since i*?, has a nowhere vanishing section, i.e., a trivial subbundle,
cr(i*i) = 0 by the Whitney formula. Therefore,
cr(Oni*5=i*(cr(i*Ond) = 0. D
Example 3.3.1. When L is a line bundle on X, the zero section 5 imbeds X
as a Cartier divisor on L. In this case the Gysin homomorphism of Definition
3.3 Rational Equivalence on Bundles
67
3.3 agrees with the Gysin homomorphism s* defined in § 2.6. (It suffices to
observe that the homomorphism of § 2.6 takes [n~[ (V)] to [ V\.)
Example 3.3.2. Us is the zero section of a bundle E of rank r on Jf, then
s*sif{a) = cr(E) n a
for all <x?,4*X (If /? = j^(a), take ]}=sif(a) in Proposition 3.3, where
s: Jf-> 2 is the section induced by s, i.e., s=j ° ?.) This is a special case of the
excess intersection formula (§ 6.3).
Example 3.3.3. If E has rank e + 1, and ?/ is the universal rank e quotient
bundle of/?* EonP (E), then for all aeA*X,
if i = e
if i=t=e.
(Argue as in the proof of (i) in the proof of Proposition 3.3, or as in the proof
of Proposition 3.1 (a).)
Example 3.3.4. Let X be a non-singular subvariety of a non-singular n-di-
mensional variety Y, and let N be the normal bundle. Let Y be the blow-up of
Y along X, X = P(N) the exceptional divisor, tf.X-*X the projection. Let Xk
denote the self-intersection of the divisor X with itself k times:
A — A. ... ^l [ I J ,
which by the construction of § 2.4 is a well-defined class in An..k(X). Since
#y (X) restricts to 0N (- 1) on X, we also have
It follows that
k?]
in /i* Jf, where s (N) is the total Segre class of N.
The left side of this formula was used by B. Segre D) to construct in-
invariants (or "covariants") of the imbedding of X in Y. For example, if
Y=XxX, and X is diagonally imbedded in Y, this class is the total inverse
Chern class of Tx. Segre inverted these, much as in § 3.2, to give a new and
intrinsic construction of the canonical classes of X, which had been constructed
earlier by Severi, Segre, Eger and Todd, and are now known as Chern classes
of X (up to sign).
From the results in Chapter 2, the classes (- I)* tj* (Xk) make sense for an
arbitrary closed subscheme X of an arbitrary variety Y, X + Y. If X is regularly
imbedded in Y, they are Segre classes of the normal bundle. In general they
live only inA*X, not as operators; they will play an important role in our con-
construction and calculation of intersection products.
Note also that if Segre's construction is applied to the case Y=E or
Y=P (?©1), with X imbedded by the zero-section, the Segre classes of an
arbitrary bundle E result.

68
Chapter 3. Vector Bundles and Chem Classes
Example 3.3.5. Let ? be a vector bundle of rank n on a scheme X, and let F
be the flag manifold of complete flags in E, with projection p:F->X, and
universal flag of bundles 0 <=?,<= E2 <=...<=?„ = /;*? on F; F may be con-
constructed by the splitting construction of § 3.2. Let xt = c, (?,-/?,_,). Then every
element of A+Fcan be written as a polynomial in X\, ...,xn, with coefficients
inA*(X); this expression is unique modulo the relations
(T,--/>*C;(?) = 0, 1^/^71,
where a, is the i'h elementary symmetric function of x,,..., xn. (Factor p into
F-+P(EV)-*X and induct; for details and a generalization to arbitrary flags,
see Grothendieck A) § 3. Example 3.2.17 also generalizes to flag bundles (cf.
Ilori-Ingleton-Lascu A).)
Example 3.3.6. The assertion of Example 2.6.6, that the Gysin map
is independent of t e P', is also a consequence of Theorem 3.3 (b), and the fact
that /'T^(l) is trivial.
Notes and References
Although the appearance of numerical invariants and canonical classes of
varieties in algebraic geometry often anticipated their discovery in topology,
the reverse is true for characteristic classes of vector bundles. Stiefel-Whitney
classes in topology were constructed in the 1930's. Until the new foundations of
Weil and Serre, which included flexible notions of abstract algebraic varieties,
general vector bundles were not a natural object of study in algebraic ge-
geometry.
With those new foundations, Grothendieck B) constructed Chem classes,
in the rational equivalence ring, for an algebraic vector bundle on a non-
singular quasi-projective variety. His method was to compute the intersection
ring of the associated projective bundle as an extension of the intersection ring
of the base, and to define the Chem classes by the formula of Remark 3.2.4.
For this, intersection theory on such varieties had to be developed first. In
Fulton B), Chem classes were defined for vector bundles on singular quasi-
projective varieties by using Grothendieck's theorem and the fact that such
bundles are restrictions of bundles on suitable non-singular varieties.
The present treatment is considerably simpler, and requires no quasi-pro-
quasi-projective hypotheses. No intersection theory beyond the facts proved in Chapter 2
for divisors is needed. In fact, Grothendieck's procedure will be quite reversed,
in that the results proved here about vector bundles will be used in later
chapters to construct general intersection products.
The formula giving inverse Chem classes of E in terms of pushing forward
powers of the first Chern class of the canonical line bundle on P (?) appears in
Notes and References
69
Washnitzer A); it has been rediscovered by almost everyone who has written
about Chern classes. Although Segre D) didn't work with bundles, he did
construct higher codimension classes by a similar construction (cf. Example
3.3.4) which explains the naming of Segre classes of bundles. We shall see that
Segre's fundamental construction has far wider application.
Theorems 3.2 and 3.3 appear in Grothendieck A), B) in the non-singular
quasi-projective case. Grothendieck A) showed that Theorem 3.3 (a) is true
when E is an affine bundle, with transition functions in the group of afiine
automorphisms. I. Vainsencher has also proved Theorem 3.3 in the singular
quasi-projective case. Example 3.2.22 was suggested by J. Harris.
For the study of Chern classes in topology the reader is referred to Milnor-
Stasheff A) or Bott-Tu A). The compatibility of the algebraic and topological
constructions will be verified in Chapter 19.

Chapter 4. Cones and Segre Classes
Summary
If X is a proper subvariety of a variety Y, the Segre class s(X, Y) of X in Y is
the class in A^X defined as follows: let C = Cx Y be the normal cone to X in y,
P(C) the projectivized normal cone, p the projection from P(Q to X. Then
s(X,Y) =
/so
When X is regularly imbedded in y, C = N is a vector bundle, and
These Segre classes have a fundamental birational invariance: if/: Y' -*Y
is a birational proper morphism, and X'=f'[(X), then s(X',Y') pushes
forward to s (X,Y).
The coefficient of [X] in s (X, Y) is the multiplicity of Y along X.
Segre classes will be used in one of our later constructions of intersection
products, and in several intersection formulas.
This chapter contains the construction of Segre classes for general cones,
and for general closed subschemes of a scheme. The birational invariance is a
special case of a general proposition describing the behavior of Segre classes
under proper push-forward and flat pull-back.
Segre classes arise naturally in many areas of algebraic geometry. Some of
these occurrences are discussed in the examples and in the last two sections,
which are not required for later chapters.
4.1 Segre Class of a Cone
Let C be a cone over a scheme X, i.e., C= Spec (S'), where S' is a sheaf of
graded ^-algebras; we assume #x~* S° is surjective, S' is coherent, and S' is
generated by S'. Let
P(C 01) = Proj E* [z])
be the projective completion of C, with projection q:P(C ffi 1) -> X, and let
<^A) be the canonical line bundle on P(C®\). For more about cones, see
Appendix B.5.1-5.4.
4.1 Segre Class of a Cone
71
The Segre class of C, denoted s{C), is the class in A*X defined by the
formula
Proposition 4.1. (a) If E is a vector bundle on X, then
where c(E) = 1 + cl(E) +... + cr(E) is the total Chern class of E, r = rank (?).
(b) Let C\,..., C, be the irreducible components of C, m, the geometric multi-
multiplicity of C, in C. Then
Proof, (a) Since [P(E ® 1)] = q*[X], the above definition of s(E) agrees
with the definition of c(E ® I) n [X] in § 3.2. And c(E ® 1) = c(E) by the
Whitney sum formula.
(b) Since each C, is open and dense in P (C, ffi 1) (cf. Appendix B.5.3),
from which the assertion follows. Note that each s(Cj) is a cycle class on the
support of C,, and the equality in (b) is understood in the sense of Convention
1.4. ?
Example 4.1.1. For any cone C,s(C® 1) = s(Q.
Example 4.1.2. Assume C is a purely n-dimensional cone, and P(C,) is not
empty for each irreducible component C, of C Then
where p:P(C) -*X is the projection. (Use Lemma 1.7.2 with Proposition
4.1 (b); by Appendix B.5.2, c, (^A)) n [P(C ® 1)] = [P(Q].)
Example4.1.3. Although one looks for components of a normal cone CXY
over "special" subvarieties of X (cf. Example 4.2.2), such components need not
be present. For example, if Y=A2", and X is the union of two n-planes
meeting transversally at the origin, then CxY has two (reduced) irreducible
components, and the fibre of P(CXY) over the origin is isomorphic to
Example 4.1.4. Let Y= A", the space of linear maps from A to A", with
coordinates (x,y), I S / S «, \SjSm, with m ^ n. Let Xa Y be the subspace
of maps of rank < n (cf. Lemma A.7.2), and let C = Cx Y. For each
@ = (<i,..., in), iSii<...<i,sm, let ?(,) be a variable, and let Sm be the
corresponding minor of (x,y). Sending ?w to <5(/) imbeds P(C) in Ix IP",
TV = C?) -1. The graded ring of C is reduced. By the Plucker relations, P (C) is
contained in Xx G, where G is the Grassmannian of n-planes in A. In fact,
= {(<p,L) | Image « <= L}.

72 Chapter 4. Cones and Segre Classes
Example 4.1.5. If C and D are cones on X defined by sheaves of algebras S'
and T\ the cone CxxD is defined by S'®,XT'. If C is a cone and ? is a vector
bundle, we write C ffi ? in place of C xxE. In this case
(Reduce to the case where C is irreducible and P(C) * 0. Let p be the projec-
projection from P{C®E) to X. There is a regular section of p*E®0{\) on
P(C®?), whose zero-scheme may be identified with P(C). By Example
3.2.16,ifr=rank(?),
[P(Q] = cr(p
n
Therefore
Example 4.1.6. ?xac? sequences of cones. Consider cones C=Spec(S'),
C'= SpecEv) on a scheme X, with S°= SP> = 0X- A morphism q>: C -* C is
given by a homomorphism ^: 5"' -> 5' of sheaves of graded ^-algebras. Let ?
be a vector bundle on X, E = Spec (Sym <?), <? locally free. Given morphisms
<p: C -* C, y/: E -+ C, we call the sequence
if p: Sv -> 5" is injective, ip: S'-> Sym <? is surjective, and if, locally on
X, there is a locally free subsheaf S of S1 that maps onto S by i/?, such that the
induced map
is an isomorphism.
(a) Basic properties of this concept are: (i) The last condition in the defini-
definition is independent of the choice dig. (ii) Exactness pulls back under flat base
extensions X'-+X. (iii) If <p and y/ are given, and (*) becomes exact under a
faithfully flat base extension, then (*) is exact. In particular, it suffices to
verify exactness after base extension to Spec @x,x) f°r all xeX.
(b) If "^ is a cone over X x A1 which is flat over A1 and C, is the restriction
ofitolx {t}, t = 0, 1, then
(c) For an exact sequence (*) of cones,
= c(E)ns(C)eA*(X).
(The proof of (a) is straightforward, (b) follows from the fact that i*, s{iS) =
s(C,) for ? = 0, I, /, the inclusion of X in XxA] at t; for a more general
statement see Example lO.l.lO. For (c), it suffices by (b) and Example 4.1.5 to
show that there is a cone 4 on XxA1, flat over A', with C\= C and
C0=C"ffi?. Let p: XxA *-> X be the projection. Define 4 to be the closed
4.2 Segre Class of a Subscheme
73
subscheme of p*(C xxC) locally defined by the homogeneous ideal in
p* (S'®*XS") generated by
{<p(s')®\-T(\®s')\s'eS1'} .
Using the local splitting of (*) one verifies that 4 is flat over A1.)
Example 4.1.7 (cf. Fulton-Johnson A)). A coherent sheafs on a scheme X
determines a cone
C(Sr) = Spec(Sym(Sr))-
One may define the Segre class of .T, s(y), to be the Segre class of its cone
C(T). If
0->3r'->S~->g->Q
is an exact sequence of sheaves, with ^locally free, then
where ?= Spec (Sym (?)) is the vector bundle whose sheaf of sections is #v.
(The corresponding sequence of cones is exact in the sense of Example 4.1.6.)
Example 4.1.8. Let C be a purely ^-dimensional closed subcone of a vector
bundle ? of rank r on X. Let sE be the zero section of ?, st the corresponding
Gysin homomorphism (§ 3.3). Then
s*E([C]) =
n
n sk.r+i(Q
in Ak-r(X). (Use Proposition 3.3, with /?= [P(C © 1)].) See Proposition 6.1 for
generalizations.
When X is non-singular and E = TX is the cotangent bundle, such cones,
with k = r, appear as characteristic varieties of holonomic ^-modules, and the
intersection with the zero section .^([C]) is an important invariant (cf.
Brylinski-Kashiwara A)).
4.2 Segre Class of a Subscheme
Let X be a closed subscheme of a scheme Y. Let C = Cx Y be the normal cone
C = Spec
where J^ is the ideal sheaf defining X in K The Segre class of X in Y, denoted
s(X, Y), is defined to be the Segre class of the normal cone C:
) = s(CxY)eA*X.

74
Chapter 4. Cones and Segre Classes
In case X is regularly imbedded in Y, so the normal cone is a vector bundle,
it follows from Proposition 4.1 that .s (X, Y) is the cap product of the total
inverse Chern class of the normal bundle with [X].
Lemma4.2. Let Y be a pure-dimensional scheme, Y\, ...,Yr the irreducible
components of Y, mt the geometric multiplicity of Y, in Y. If X is a closed sub-
scheme of Y, and X, = X{\ Y,, then
s{X,Y) = Y.^s{XhY)
in A* (X).
Proof. Let MXY denote the blow-up of YxA] along Xx{0}. Since X is
nowhere dense in Yx A', the varieties MX{ Y, are the irreducible components of
Mx Y, with multiplicities m{.
The exceptional divisor in MXY restricts to the exceptional divisor in MXlY:,
so, by Lemma 1.7.2,
[P{CxY@\)}^mi[P{CXlYi®\)\.
Capping by
ity. ?
' and pushing forward to X gives the asserted equal-
equalProposition 4.2. Let f:Y'—>Ybe a morphism of pure-dimensional schemes,
Xa Y a closed subscheme, X' = f~l (X) the inverse image scheme, g : X' -* X the
induced morphism.
(a) /// is proper, Y irreducible, and f maps each irreducible component of Y'
onto Y, then
g* (s (X\ Y')) = deg (Y'IY) s (X, Y).
(b) /// is flat, then
g*(s(X,Y)) =
Proof In (a), deg (Y'/Y) is defined to be ?[=1 m, deg(Y,'/Y), where Y/,..., Y/
are the irreducible components of Y', mi the geometric multiplicity of Y,'. By
Lemma 4.2, we may assume Y' is also irreducible.
Let M be the blow-up of YxA1 along the subscheme Xx{0}. The
exceptional divisor is P(Cffil). Similarly let M' be the blow-up of Y'xA1
along A" x {0}. There is an induced morphism F from M' to M, with
F*P(C® 1) = P(C® 1) as Cartier divisors. Let G be the induced morphism
from P(C®\) to P(C®\), and let q (resp. q') be the projection from
P(C® 1) (resp. P{C® 1)) to X (resp. X'). If ^A) is the canonical line bundle
on P(C® 1), G*0{\) is the canonical line bundle on P(C'@ 1).
In the situation of (a), F*[M'] = d[M],d= deg (Y'/Y), so
4.2 Segre Class of a Subscheme
by Proposition 2.3 (c). Therefore
g*s(X', T) = gmqi
75
' n
9 1)]
making use of the projection formula (Proposition 2.5 (c)).
Similarly for (b),
= s(X', Y') . D
Corollary 4.2.1. With the assumptions of Proposition 4.2(a), if X' is regularly
imbedded in Y', with normal bundle N', then
gt(c(N')-> n [X']) = deg (Y'/Y) s(X, Y).
Y is also regularly imbedded, with normal bundle N, then
gt(c(N')-i n [X']) = deg (Y'/Y) (c (TV) n [X]) .
r. Apply Proposition 4.1 (a). ?
Corollary 4.2.2. Let X be a proper closed subscheme of a variety Y. Let Y be
the blow-up of Y along X,X=P(C) the exceptional divisor, r\ : X -> X the
projection. Then
s{X,Y)= S (-II r,*(Xk)
/so
Proof. The self-intersection class Xk is meant in the sense of Defini-
Definition 2.4.2. The first formula follows from the fact that the normal bundle to X
in Y is the restriction of #?(X) to X, and the fact that multiplication by the
first Chern class of the normal bundle is the same as intersecting with the
divisor X (Proposition 2.6 (c)). The second follows from the fact that, when X
is identified with P(C), the normal bundle is dual to ^A); or one may apply
Example 4.1.2. ?

76
Chapter 4. Cones and Segre Classes
Remarks. When / is birational, i.e. deg {Y'/Y)=\, Proposition 4.2(a)
asserts the birational invariance of Segre classes: the Segre class of X' in Y'
pushes forward to the Segre class of X in Y. In case X a Y and X' cz Y' are
regular imbeddings — for example, if all four schemes are non-singular -
Corollary 4.2.1 gives a remarkable relation among the Chern classes of the
normal bundles.
Consider the situation of Corollary 4.2.1 when X is a non-singular point on
Y. A formula for the degree of/results:
deg(Y'/y)= I dNT'nlX'].
X'
This can be used in a situation where the general fibre of d is difficult to
describe, but a particular degenerate fibre is known very well. A procedure
like this was carried out by R. Donagi and R. Smith A) to calculate the degree
of a Prym map.
The first formula of Corollary 4.2.1 is useful also when X = P is a singular
point on Y. We will see in the next section that s(P, Y) is the multiplicity eP Y
of Kat P. For example, if/is birational, one has
This is essentially the procedure used by G. Kempf B) to calculate the
multiplicity of varieties of special divisors (cf. Example 4.3.2).
For XclY, Y non-singular, the Segre class s(X, Y), after correction by
c(TY), is independent of the imbedding (Example 4.2.6).
Example 4.2.1. Corollary 4.2.2 is valid for any purely n-dimensional scheme
Y and any closed subscheme X of dimension less than n. This may also be
deduced from the fact that P(Q is a Cartier divisor on P(C® 1),
and
q*[P(C®l)] =
Example 4.2.2. Let A, B and D be effective Cartier divisors on a surface Y.
Let A' = A + D, B' - B + D, and let X be the scheme-theoretic intersection of
A' and B' on Y. Assume that A and B meet only at one point P, which is non-
singular on Y, and that A and B meet transversally at P. Then
s(X, Y) = [D]+([P]-D[D]).
(To see this, let/: Y -* Y be the blow-up of Y at P, E the exceptional divisor.
Then X=f~' (X) =f*D + E . Therefore
s(X, Y) = g*[X] - g*(X- [X]) = [D]- g*(f*D ¦ [f*D] + 2f*D •[?] + ?• [E])
= [D]-D[D] + [P].)
One may also calculate the normal cone to X in Y; it contains a component
which lies over P.
Similarly if A and B have the same multiplicity m at P, and no common
tangents at P, then
s(X, Y) = [?]+ (m2[P]-D[D]).
4.2 Segre Class of a Subscheme
77
For arbitrary intersections of A and B, the answer depends also on how A and
B meet D (cf. Example 6.1.4).
Example 4.2.3. Under the hypotheses of Proposition 4.2 (a), if A'is regular-
regularly imbedded in Y with normal bundle TV, then
g*(c(g*N) n s(X', Y')) = deg (Y7Y) ¦ [X].
Example 4.2.4. Lemma 4.2 may fail if Y is not pure-dimensional (cf.
Example 1.7.1).
Example 4.2.5. Let X-, be a closed subscheme of a scheme y, for / = 1, ..., r.
Then
s(X{x...xXr, y,x...x Yr) = s(XuYi)x...xs(Xr,Yr).
(Reduce to the case where X, is a Cartier divisor in y, and use Example 3.2.8.)
Example 4.2.6. Canonical classes of singular varieties.
(a) Let X be a scheme which can be imbedded as a closed subscheme of a
non-singular variety M. Then the class
in A*(X) is independent of the choice of imbedding. If X is a local complete
intersection, then
c*(X) = c{TM\x)
n [X] = c(Tx) n [X].
Here Tx = TM \ x — Nx M is the virtual tangent bundle to X, a well-defined
element of the Grothendieck group of vector bundles on X (cf. § 15.1 and
Appendix B.7.6). (Two imbeddings are dominated by the diagonal, so it
suffices to compare i:X-* M withy : X -> P, M, P non-singular, when there is
a smooth <x.P->M with Qj=i. By Example 4.1.6, it suffices to show that
there is an exact sequence of cones
(*)
C.--0
with C,, Cj the normal cones to Xa M, X a p, and Tg the relative tangent
bundle to q. Let Y= XxMP, and consider the diagram
Y + P
X-r+M
with i',g' induced by /, q, and k = (\,j). Then k is a regular imbedding with
Ck = k* Tg' =/* Te. Since q is flat, Ce = g'* C,-, so k* Q = C,. The factoring of/
into /' ° k determines morphisms
To see that the resulting sequence (*) is exact, use Example 4.1.6 (a) to reduce
to the case where M= Spec (^4), A a complete local ring and P = Spec E), B a

78
Chapter 4. Cones and Segre Classes
formal power series ring over A; in this case the local splitting can be verified
directly.)
(b) If X is a plane curve of degree d, then
deg(co(*)) = $ c,(X) = 3d-d2 = 2x(X,0x).
X
We know no simple general formula for deg co(X) for a general curve. The
examples X= V(x2,xy,y2) or X = V(x2,xy, z2) in P3 show that deg(co(X)) is
not always 2x(X, 0X).
(c) (cf. Fulton-Johnson A)) If X a M as in (a), and jTxM = J'IJ~2 is the
conormal sheaf to X in M, the class
c(TM\x)ns(yrxM)
in A*Xis also independent of the imbedding, where s(jVxM) is the Segre class
of the sheaf jVxM. (This follows similarly from Example 4.1.7.) This agrees
with the canonical class in (a) for local complete intersections, but not in
general.
Example 4.2.7. Let X be a closed subscheme of Y, E a vector bundle on
Y, with Y imbedded in E by the zero section. Then
s(X,Y) = c(E\x)ns(X,E).
(Reduce to the case where Y is a variety and X is a Cartier divisor on Y.)
Example 4.2.8. (a) If X <-> Y^Z, and j is a regular imbedding with
normal bundle N,s(X,Y) is not always equal to c (i*N) n s (X, Z). For
example, let Z = P2, Y a. curve, and X a singular point on Y.
(b) If X^ Y <-» Z, and / imbeds X as a Cartier divisor on Y, with normal
bundle N, i*s(Y, Z) is not always equal to c(N) n s(X, Z). For example, let
Z be the cone x2+^2 = z2 in A3, Y the line x = z-y = 0, and X the point
@, 0, 0).
Example 4.2.9. (a) Local Euler obstruction. Let X be an ^-dimensional
variety. Let v: f -+ J be a proper birational morphism such that there is a
surjection of sheaves
with Q locally free of rank n on X. For example, X could be the Nash blow-up
of X. Let f=Qv. For any point P in A', define the local Euler obstruction
EupX by the formula
Eu,* = J c(T\v-HP))r>s(v-HP),X)-
v-'(P)
This integer is independent of the choice of X. (Since two such X are
dominated by a third, this follows from the birational invariance of Segre
classes.) This definition was made by Gonzalez-Sprinberg A) and Verdier, who
proved that it agrees with the original transcendental definition of MacPherson
A).
4.3 Multiplicity Along a Subvariety 79
(b) Mather Chern class. With v: X -> X as in (a), the Mather Chern class
in A*X is independent of the choice of X. (Use the projection formula.)
4.3 Multiplicity Along a Subvariety
For an irreducible subvariety X of a variety Y, the coefficient of [X] in the
class s(X, Y) is called the multiplicity of Y along X, or the algebraic multiplicity
ofXon Y, and is denoted ex Y. If codim(X, Y) = n > 0, then
exY[X] = q)f{
Here C = CxY,p and q are the projections from P (C) and P(C© 1) to X, Y is
the blow-up of Y along X, with exceptional divisor X=P{C). This defi-
definition is equivalent to the definition of the multiplicity of the local ring 0x.t
given by Samuel A) (cf. Example 4.3.1).
If X = P is a point, C = CPY is the tangent cone to P in 7, and
ePY= J c,
In this case ep Y is called the multiplicity of Y at P.
Example 4.3.1. Let A be the local ring of Y along X, J/ the maximal ideal
oSA,AlJ/ = k. Then
0, whose leading term is
dim
is a polynomial of degree n = codim(A", Y) in / for t
et"/n\, where e = ex Y (cf. Example 2.5.2).
Example 4.3.2. In this example C is a projective non-singular curve of
genus g; C(J) is the d1 h symmetric product of C, parametrizing effective divisors
of degree d on C; ?o is a fixed point of C; J is the Jacobian of C; m^ is the
morphism from C{<t) to7 which takes a divisor D to the divisor class of D — rfP0-
The following facts are assumed: (i) the (scheme-theoretic) fibres of ud are
the linear systems \D\=V'; (ii) if d>2g — 2, ud makes C^ a projective
bundle over J; (iii) if 1 S dS g,ud maps C(d) birationally onto its image Wd.
(a) If deg D = d, dim | D \ = r, then
(l +/!)9-rf+r n
where /; is the first Chern class of the canonical line bundle on | D \ = Fr. (This
follows from (ii) if d is large. For smaller d consider CM <= C(rf+-r) by

80
Chapter 4. Cones and Segre Classes
E -* E + sP0; the normal bundle to this imbedding restricts to a bundle on | D |
whose Chern class is A + hf.)
(b) From (a) and Proposition 4.2 (a) follows the Riemann-Kempf formula
(cf. Kempf B): the multiplicity of W^at ud(D) is
where r = dim \D\.
R. Smith has shown how a similar procedure can be used to prove an
assertion of Mumford (cf. Beauville C)) that a theta divisor in the inter-
intermediate Jacobian of a non-singular cubic three-fold in P4 has a singular point
of multiplicity 3. If F is the Fano variety of lines in the cubic three-fold, there
is a morphism of degree 6 from Fx fonto the theta divisor, so that the inverse
image of the singular point is the diagonal in FxF; calculating (Clemens-
Griffiths A), p. 326) that
J s2(TF) = JClG»2 - c2G» = 45 - 27 = 18,
F F
it follows that the multiplicity is 18/6 = 3.
Example 4.3.3. (a) (Schwarzenberger A)) With the notation of the previous
example, for d> 2g — 2 there is a vector bundle Ed on J with P(Ed) = C(d),
and whose canonical line bundle ^A) is the line bundle of the divisor Cw"]).
This gives a geometric realization of the Segre classes of Ed-
Let S be the involution of J which takes the class of a divisor D to the class of
K — Bg-2)P0-D, with K a canonical divisor on C. There is an exact
sequence
0-> ?„->• M-> <5*?*->0
for « large, with Ma successive extension of trivial line bundles. Therefore
j,(?.) = c,-(S*?:f) =(-!)'<:,(*•?„),
so
<:,(?„) n [7] =(- 1 )'$,(*,(?.) n [J]) = (- 1
If w, represents [Wg-j, and w, represents [<5(PF9_,)], this explains Mattuck's
formula
(Sw,)(E(-D'",) = i-
(b) (cf. Mattuck C)) The Chern class of the dth symmetric product C(d) of a
non-singular projective curve C of genus g is given by the formula:
c,(Tc»<)=(\ +xt)d-g+[/u*dw(t/(l +xt)).
Here x is the class of Od-]\ imbedded in C(d) by D -+ D + Po, and w(t)
= 2>,V. (If d>2g-2, C(d) = P(Ed), with c,(?rf) = w(?)"', and Example
3.2.11 applies. If the formula is known for C(<0, imbed C(rf"" in C(<" as above,
with 1 + x t the Chern polynomial of the normal bundle, to deduce it for
4.3 Multiplicity Along a Subvariety
81
Example 4.3.4. For any closed subscheme X of a pure-dimensional scheme
Y, and any irreducible component V of X, the multiplicity of Y along X at V,
denoted (ex Y)y, may be defined as the coefficient of [V\ in the class s(X, Y).
If V= X, we write simply ex Y. This multiplicity is the same as Samuel's
multiplicity e{q) of the primary ideal q determined by X in the local ring
A = fiv, y, i.e., if n = dim (/I) = codim (V, Y), then
l(A/q') = e(q) ¦ t"/n\ + lower terms
for t 5> 0. In general, if Yu..., Y, are the irreducible components of Y which
contain V, with geometric multiplicities m , m,, then, by Lemma 4.2,
Example 4.3.5. Continue the notations of the preceding example.
(a) If the residue field R{V) of A is infinite, Samuel A) has shown that
there are ax,..., an e q, generating an ideal q' = (fli,..., an) with dim {A/q') = 0
and
e(q') = e(q).
If the ground field K is infinite, such a, may be found among AMinear
combinations of a given set of generators for q. (See Zariski-Samuel A),
Vol. II, p. 294 for a proof.)
(b) If a,,..., an e A generate an ideal q' with dim {A/q') = 0, then
e(q') = eA(ai,...,an)
where the right side is the alternating sum of the lengths of the Koszul
complex defined by a,,..., an (Appendix A.5). For the proof in this geometric
context, see Example 7.1.2. For an algebraic proof, see Serre D), p. IV-12.
(c) If A is a Cohen-Macaulay local ring, then
If K is infinite, equality holds if and only if q can be generated by a regular
sequence of elements. (For q' = (at,..., an) cz q as in (a), at,..., an is a regular
sequence by Lemma A. 7.1, so
with equality when q' = q. If K is not infinite, make a base extension A B>KK'
with K' infinite, e.g. K' = K(T).)
(d) In this geometric context, e(q) =(ex Y)v= 1 if and only if A is regular
and q is the maximal ideal of A. Assuming (a), this will follow from
Proposition 7.2; for algebraic proofs see Samuel A) or Nagata B). This
criterion is not valid for arbitrary Noetherian local rings A, even if A is a
domain and q is the maximal ideal (Nagata B) Appendix A.I).
Example 4.3.6. Let f:Y'-> Y be a proper surjective homomorphism of
irreducible varieties, X a closed subscheme of Y, X' =f'[(X). Let V be an
irreducible component of X, and assume that each irreducible component V

82 Chapter 4. Cones and Segre Classes
of/ (V) has the same dimension as V. Then, by Proposition 4.2(a),
deg G7 7) (e* J> = Z deg (K7 F) (er F) r
v
where the sum is over the irreducible components V off'1 (V).
Example 4.3.7. Let /: Y' -> Y be a proper surjective homomorphism of
irreducible varieties. Let C be a subvariety of Y', V=f{V). If K' is an
irreducible component of/~'(K), define the ramification index of / at K',
<?,,.(/), to be the multiplicity of Y' along f~l(V) at K' (cf. Example 4.3.4). If all
irreducible components V of f~l(V) have the same dimension as V, then
For example, if Y is smooth over an algebraically closed field, and Q is a point
of ysuch that/ @ is finite, then
In particular, the sum of the ramification indices is independent of Q e Y.
For a notion of ramification index related to the separable degree of/ see
Gaffney-Lazarsfeld A). For other geometric interpretations of multiplicity, see
Example 12.4.5 and Mumford E).
Example 4.3.8. Assume X, and X2 are two subschemes of Y, each contain-
containing V as an irreducible component. Suppose there is a proper birational
morphism/: Y' -> Yso that/-' (X,) =/"' (X2). Then (<?*, y) ,, = (<?*, Y)y.
Example 4.3.9. Let Xbe a variety of dimension at least 2, Pa simple point
of X, %: X -> X the blow-up of X at P, E the exceptional divisor, ?7: ? -> P the
projection. Let Z) be an effective Cartier divisor on X, and let D be the blow-
blowup of D at P (the strict transform of D in X). If m is the greatest power of the
maximal ideal of 6)P x which contains a local equation for D, then
Therefore
0 = D ¦
in/lo^ = 2. This verifies the formula m — ePD.
Example4.3.10. If Xt is a subvariety of Y, for i= l,...,r, then the multi-
multiplicity of Xt x ... x Xr in Y[ x ... x Y, is the product of the multiplicities of X,
in y,. (Use Example4.2.5.)
4.4 Linear Systems
If a subscheme is the base locus of a linear system, its Segre class is related to
important invariants of the system.
4.4 Linear Systems
83
Let L be a line bundle on an n-dimensional variety X, and let K<= H"(X, L)
be an (r + l)-dimensional space of sections of L. For s e V, s =? 0, let Ds be the
zero-scheme of s, a Cartier divisor on X. Let
5= n d,
se y-m
be the base of the linear system. Let n: X -> X be the blow-up of X along 5.
There is a morphism
extending the morphism X — B -* P(KV) which takes x to the hyperplane of
divisors containing x. Indeed, if J? is the ideal sheaf of B in X, there is a
canonical surjection from the trivial bundle Vx to ,f®L. Therefore
Sym(Vx®L~l) surjects onto ®J", which gives an imbedding of X in
Proj(Sym(FA-® L"')) = Proj(Sym(K*)) = X
The morphism/is the projection to P(VV). From this description it follows
that
(
where ? is the exceptional divisor on X.
Define de.gfX to be the degree of f*[X] as an ^-dimensional cycle on
(
), i.e.,
= deg(Z//(Z))- J Cl
Proposition 4.4. PF/?/; ?/ie above notation,
Proof. By the above description of/*
, (L)—¦
E (-
i-o
using the projection formula and Corollary 4.2.2. ?
Example4.4.1. Suppose that n divisors D],..., Dn in the linear system cut
out B scheme-theoretically, together with a finite set S disjoint from B. In this
case deg/X may be interpreted as the weighted number of points in S, and
ixC^Lf is the total intersection number (Dl-... ¦ Dn). The remaining term
]Bc(Ly n s(B, X) therefore represents the contribution to this total intersection
number which is carried by B. This is a special case of a general formula to be
proved in Chap. 9.

84 Chapter 4. Cones and Segre Classes
Example4.4.2. Let 5<=P" be the rational normal curve. Let Kc//°(P"/B))
be the linear system of quadrics passing through B. Then deg/P" =
2"-(n2- n + 2). If" n = 4,/(P4) <= P5 may be identified with the Grassmannian
of lines in P3.
Example4.4.3. Let X be an irreducible hypersurface in P"+1 = P(K)
defined by an equation F(X0, ...,Xn+]) of degree d. The singular, or Jacobian
subscheme of X is the scheme J of zeros of the partial derivatives
f0, • • •, dF/dXn+1 . If X is the blow-up of X along J, we have morphisms
X
X
The image f{X) is called the dual variety of X and denoted Xv. In
characteristic zero, biduality holds, i.e., (Xv)v = X (cf. Kleiman A1)). In any
case, we call degft[X] the degree of the dual of X. Then
- 1)"-
where j> is the /-dimensional component of s(J, X). For example, if X is non-
singular, its dual has degree d(d— 1)" (cf. Example 3.2.21).
Example 4.4.4. Let X a P2 be an irreducible plane curve of degree d. Then
the degree of the dual is
where {ejX)P is the multiplicity of X along J at P (cf. Example 4.3.4).
Equivalently, if <j: X' -* X is the normalization of X in its function field, and
J' = q~' (J), then the degree of the dual is
(Use Proposition 4.2 (a).) For example, a node counts for 2, an ordinary
m-fold point for m(m— 1), an ordinary cusp for 3; a higher cusp, of the form
y = xf, with p < q, and /;, <? relatively prime (and not divisible by the charac-
characteristic) counts for (p— 1) q (cf. Walker A) IV.6). For an analogous result for
surfaces, see Example 9.3.8.
Example 4.4.5. Polar classes (cf. Piene C)). Let X" <= P"+I be an irreducible
hypersurface of degree d, over an algebraically closed ground field. For any
0 ^ k ? n, let Wk c P"+l be a linear subspace of dimension k— 1. The k'h polar
locus of X, with respect to Wk, denoted Mk, is defined to be the closure of
Wk).
Here TxX denotes the imbedded tangent space to X at x. For general Wk, Mk
has pure codimension k in X, and the class [Mk] e An-k{X) is independent of
Notes and References
Wk. Indeed, with h = c,
85
1)), Sj= s,(J, X) as in Example 4.4.3,
=(rf- 1)* A* n m - ? tf)(rf- \)>h> n JA_t+,.
The A: c/oy.? of X, denoted gk, is defined to be the degree of [Mk]. Therefore
k-\
Qk = d(d- \)k-Y, (W- l)'deg(j,,_t+I-).
i-O
(Let X, 7i,/be as in Example4.4.3, L =#P(y)(d— 1). Set
' (Wv)] = ^(c, (/* ^A))* n [Xj). The proof concludes as in
Then [Mk] = n,[
Proposition 4.4.)
Note that Mo = X, For A: = 1, if F is an equation for X, and W^ — y =
(y0:...: yn + J, let Fy = ??=o 7,5.F/5X,-. The hypersurface defined by Fy = 0 is the
classical polar hypersurface of X with respect to y. A non-singular point jc e X
lies in M, if and only if Fy(x) = 0. For the ?th polar locus, Xtcs may be
intersected with k polar hypersurfaces, choosing the points in general position.
For calculations of s0 and j,, and hence the polar classes, when X is a
surface in P3 with ordinary singularities, see Example 9.3.7. Generalizations to
higher codimension are discussed in Example 14.4.15.
Notes and References
The normal cone to a subvariety or subscheme, in the form of an associated
graded ring, was used by Samuel A) to define the multiplicity of a subvariety.
Samuel used these multiplicities to define intersection numbers (cf. Chap. 7).
Normal cones were used explicitly by Verdier E) for his construction of
Gysin maps. They were implicit in MacPherson's graph construction (cf.
Baum-Fulton-MacPherson A)).
Segre classes of cones and general subvarieties and subschemes were
defined in Fulton-MacPherson A). At least in the non-singular case, these
classes had been studied by B. Segre D) (cf. Example 3.3.4 and Corollary 4.2.2
above), and more recently by J. King C) and Lascu-Scott A).
A sheaf-theoretic version of the multiplicity formula of § 4.3, together with
results like Example 4.3.9, were given by Ramanujam A).
Example 4.3.2, which gives a result of G. Kempf B), was suggested by
J.Harris, and Examples 4.1.3, 4.1.4, 4.2.6 and 4.4.2 by R. Lazarsfeld.
Examples 4.1.6 and 4.2.6 were worked out jointly with A. Collino and R.
MacPherson.

Chapter 5. Deformation to the Normal Cone
Summary
If X is a closed subscheme of Y, there is a family of imbeddings X<^> Y,,
parametrized by t e P1, such that for / = 0 (in fact for t 4= oo) the imbedding is
the given imbedding of X in Y, and for t = oo one has the zero section
imbedding of X in the normal cone Cx Y. The existence of such a deformation,
together with the "principle of continuity" that intersection products should
vary nicely in families, explains the prominent role to be played by the normal
cone in constructing intersection products.
5.1 The Deformation
Let X be a closed subscheme of a scheme Y, and let C = Cx Y be the normal
cone to X in Y. We will construct a scheme M= MXY, together with a closed
imbedding of Xx P1 in M, and a flat morphism q : M -* P1 so that
xF
P1
commutes, and such that:
A) Over P'-{oo} = A1, e~'
imbedding:
') =
and the imbedding is the trivial
B) Over oo, the divisor Mx= q~' (oo) is the sum of two effective Cartier
divisors:
MP(C® \) + Y
where Y is the blow-up of Y along X. The imbedding of X=*Xx{oo} in Mx is
the zero-section imbedding of X in C, followed by the_ canonical open
imbedding of C in P(C® 1). The divisors P(C© 1) and Y intersect in the
scheme P(C), which is imbedded as the hyperplane at infinity in P(C® 1),
and as the exceptional divisor in Y.
5.1 The Deformation 87
In particular, the image of X in Mx is disjoint from Y. Letting M° = Mx Y
be the complement of fin M, one has a family of imbeddings of X.
XxF'^M0
p1
which deforms the given imbedding of X in Y to the zero-section imbedding of
X in CXY.
To construct this deformation, let M be the blow-up of 7xP' along the
subscheme Xx{co). Since the normal cone to Xxjoo} in YxF' is Cffi 1, the
exceptional divisor in this blow-up is P (C ® 1).
From the sequence of imbeddings
the blow-up of XxP[ along Xx{<x>} is imbedded as a closed subscheme of M
(Appendix B.6.9); since Xx{oo} is a Cartier divisor on XxF\ the blow-up of
XxF' along Xx{co] may be identified with XxF1, so we have a closed
imbedding
A'xP'^M.
Similarly from
the blow-up Y of Y along X is imbedded as a closed subscheme of M.
Since the projection from Yx P1 to P1 is flat, the composite
of the blow-down map from M to Yx P1 followed by the projection to P1 is
flat (Appendix B.6.7).
Since M-+yxP' is an isomorphism away from yx{oo}, assertion A) is
clear. The description B) of Mx = q~' (oo) as the sum of Cartier divisors will
follow from the explicit algebraic description given below; since we have
P(C © 1) and ?globally imbedded in M, it suffices to examine their structure
locally on Y.
Assume Y= Spec (/I), and X is defined by the ideal / in A. To study M near
oo, identify P'-{0} with A'= SpecA:[r], where K is the ground field. The
blow-up of yx A1 along Xx {0} is Proj E'), with
S" = (/, T)" =I" + I"~l T + ...+ AT" + ATn+' + ....
Proj (S') is covered by affine open sets Spec {S'(a)), where S\a) is the ring of
fractions
S-(a) = {s/a" j s e S"},
and a runs through a set of generators for the ideal (/, T) in A[T]. For a e I,
the exceptional divisor P{C®\) is defined in SpecE('a)) by the equation
a/1, a eS°, while Y is defined by T/a; since T= (a/1) • G7a), the description
of Mx as the sum of P (C® 1) and Y follows.
The complement of Y in the blow-up of yxA1 along Xx{0} is Spec(S'iT)),
where
S(y, ? ... ® I"T~"® ...®IT-X®A®AT® ...®AT"@ ....

Chapter 5. Deformation to the Normal Cone
This is the ring studied by Rees B), and Gerstenhaber A). The canonical
homomorphism from A[T] to S'{T) becomes an isomorphism after localization
at T, while °°
C" /T1 C **T\ In I fn+\
OGy 1 >->(T) — <±) 1 II
Remark 5.1.1. MacPherson's description of this deformation, as a special
case of his graph construction, is particularly vivid. Assume ? is a vector
bundle on Y, and 5 is a section of E whose zero-scheme is X. (If Y is quasi-
projective such E always exists, although its rank r may be larger than the
codimension of X on Y.) For each scalar X, the graph of Xs is a line in E @ 1.
This gives an imbedding
YxAlc-> P(E@l)xlPl,
(y, X) -> (graph of Xs (y), A: X)). The deformation space MXY is in fact the
closure of YxA' in this imbedding. We won't need this description; the con-
construction given above is simpler because it relies only on standard properties of
blowing up. The graph construction is more powerful, however, in that it
generalizes to general vector bundle maps, or to complexes of vector bundles (cf.
§18.1).
PIE) -
/
PIC)
Remark 5.1.2. From the point of view of deformation theory of varieties,
this construction would be called a deformation from the normal cone. This
terminology seems inappropriate here, since we always start with the im-
imbedding X^Y. The alternative, "specialization" to the normal cone, is
reserved for the associated homomorphisms of cycles or cycle classes (§ 5.2).
Example 5.1.1. Assume Y is purely ^-dimensional. Since M is flat over
P1, [e-'@)] -[(?"' (oo)], i.e.,
in An(M). Here /: K->Mis the imbedding of Y at / = 0, andy (resp. k) is the
canonical imbedding of P(C® 1) (resp. f) in Mover/= oo.
Example5.1.2. Assume X is regularly imbedded in Y so CXY=N is a
vector bundle. The imbedding of X in P(N ® 1) has several advantages over
the given imbedding of X in Y. For example:
(i) There is a retraction from P(N @\) io X.
(ii) There is a vector bundle ?, on P{N®\) of rank equal to the
codimension of X, with a regular section whose zero-scheme is X. Thus, on
P(N®l),Xis represented by the top Chern class of the bundle ?.
There is usually no such retraction or bundle for the given imbedding, even
if Y is replaced by an open subscheme containing X. The deformation to the
normal bundle may be thought of as an analogue in algebraic geometry of the
tubular neighborhood construction in topology.
5.2 Specialization to the Normal Cone
5.2 Specialization to the Normal Cone
89
Let Ibea closed subscheme of a scheme Y, and let C = Cx Y be the normal
cone to X in Y. Define the specialization homomorphisms
a:ZkY->ZkC
by the formula
for any fc-dimensional subvariety V of Y, and extending linearly to all fe-cycles;
note that CV^XV is a purely fc-dimensional scheme (Appendix B.6.6.), so it has a
fundamental cycle [Ci^F] by § 1.5.
Proposition 5.2. If a cycle a e Zk Y is rationally equivalent to zero on Y,
then a (a) is rationally equivalent to zero on C.
Therefore a passes to rational equivalence, defining specialization homo-
homomorphisms
a:AkY->AkC.
Proof. Let M° = A/J Y be the deformation space constructed in § 5.1, / the
inclusion of C in M°,j the inclusion of Yx A' in M. Consider the diagram
Ak+l C ± Ak+lM° 4 Ak+l (Yx A1) - 0
i I"
AtC* Ak(Y)
The row is the exact sequence of § 1.8. The map /* is the Gysin map for
divisors; the composite /*/* is zero since the normal bundle to C in M° is
trivial (Proposition 2.6 (c)); in fact, C is a principal divisor on M° — q~] @).
There is therefore an induced morphism from Coker(/») =Ak+i (YxA1) to
Ak C, and therefore a morphism from Ak Y to Ak C obtained by composing with
the flat pull-back from AkY to Ak+] (YxA1). To prove the proposition, it
suffices to verify that this composite takes [V] to [Cy(]XV]. First, pr*[F] =
[V x A1]. The variety M^nxV is a closed subvariety of MXY which restricts to
Vx A1, so
J*[M°vf]XV]=pr*[V].
The Cartier divisor C = M°C)q~'(oo) intersects M?nA-F in Cvf]XV, so
which concludes the proof. ?
The following example may provide a useful preview for the next few
chapters.
Example 5.2.1. Let i:X-+ Y be a regular imbedding of codimension d, with
normal bundle N = Nx Y. Define the Gysin homomorphism
i*:AkY->Ak.dX

90
Chapter 5. Deformation to the Normal Cone
to be the composite /* = s* ° a:
AkY 1+AkN A Ak-dX,
where s* is the Gysin homomorphism of Definition 3.3.
(a) If d= 1 (resp. / is the zero section of a vector bundle) this Gysin homo-
homomorphism agrees with that defined in § 2.6 (resp. § 3.3).
(b) If Y is purely /i-dimensional, /* [ Y] = [X].
(c) For all aeA*X, i* in (a) = cd{N) n a. (See § 6.3.)
(d) For any A:-dimensional subvariety V of Y
i*[V] = {c(N)ns(Vf)X, K)}*_,.
(e) If X is an ^-dimensional variety which is smooth over the ground field,
the diagonal imbedding d of X in XxX is a regular imbedding of codimension
n. This defines an intersection product on A*X:
ApX®AqX-
(See Chap. 8.)
Notes and References
Deformation to the normal bundle, or cone, has an interesting history. It has
appeared in at least three places:
A) For a non-singular subvariety I of a non-singular quasi-projective
variety Y, Mumford (unpublished, 1959), Jouanolou B), Lascu and Scott A),
B), and Lascu, Mumford and Scott A) used the blow-up of Yx P1 along
Xxfoo} to prove important formulas in intersection theory: the self-intersec-
self-intersection and key formulas, Riemann-Roch without denominators, and the formula
for blowing up Chern classes.
B) For an ideal / in a ring A, Gerstenhaber A) deformed A to the
associated graded ring ©/"//"+1. The algebra for this deformation had
previously appeared in Rees B).
C) For a regularly imbedded subscheme X of a variety Y, and a section s
of a vector bundle on Y whose zero-scheme is X, MacPherson (cf. Baum-
Fulton-MacPherson A), and Example 18.1.7) deformed the graph of Is to
X = oo, and identified the normal bundle to A' in Y at oo.
In A) the deformation was not at first explicit. In fact, considerable
simplification occurred when the rational equivalence between [Y] at 0 and
[P(N© 1)] + [?] at oo was used (cf. Lascu and Scott B)). The deformation in
B) had been used in algebraic geometry by Kleiman and Landolfi A),
Mumford, and others, but not in intersection theory. The graph construction
C) was used to solve problems of intersection theory. The identity of the three
approaches was established in the 1974/75 seminars of Kleiman and Douady-
Verdier(l).
Notes and References
91
When one realizes the role of normal cones in Samuel's construction of
intersection multiplicities in the case of proper intersections, and the role of
deformation to the normal bundle in proving excess intersection formulas, it
becomes reasonable to expect general intersection products to be constructed
using normal cones and bundles; this reasonable expectation, however, was ap-
apparently not formulated before these products were constructed.
Verdier E) used deformation to the normal cone, together with the Gysin
homomorphism for principal divisors constructed in Fulton B), to construct
specialization homomorphisms to the normal cone. He used these specializa-
specialization homomorphisms to construct Gysin homomorphisms for regular im-
beddings of arbitrary codimension. The present chapter follows Verdier's
exposition closely. Except for (d) and (e) of Example 5.2.1, all the results
appear in Verdier E), at least in the quasi-projective case.
Gau and Lipman A) have recently used deformation to normal cones to prove
the invariance of multiplicity under diffeomorphism.

Chapter 6. Intersection Products
Summary
Given a regular imbedding i:X-*Y of codimension d, a A:-dimensional
variety V, and a morphism /: V-* Y, an intersection product X ¦ V is con-
constructed in Ak-d(W), W=f'\X). Although the case of primary interest is
when / is a closed imbedding, so W = X D V, there is significant benefit in
allowing general morphisms /. Let g:W^>X be the induced morphism. The
normal cone CWV to W in V is a closed subcone of 3* A^ Y, of pure dimension k.
We define X- V to be the result of intersecting the A:-cycle [CjyK] by the zero-
section of g* NXY:
X-V=s*[CwV]
where 5 : IV-* g* Nx Y is the zero-section, and s* is the Gysin map constructed
in Chapter 3. Alternatively X • Kis the (k — rf)-dimensional component of the
l
c(g*NxY)ns(W, V)
where s{W,V)\% the Segre class of W in V.
If the &-cycle [Cw V\ is written out as a sum X "«/[CJ, with C, irreducible,
one has a corresponding decomposition X-V = '?mia.i, with af a well-defined
cycle-class on the support of C,.
If the imbedding of W in K is regular of codimension of', then
E= g* NXY/NWV is the quotient bundle, there is an excess intersection
formula
XV (E)[W]
(*)
Given i: X ~* Y as above, and a morphism /: Y' -
X' -4 y
X -> Y
Y, form the fibre square
There are refined Gysin homomorphisms
i':AkY' ^Ak-dX'
determined by the formula i'[V\ = X ¦ V for sub varieties Kof Y'.
In this chapter the fundamental properties of these intersection operations
are proved. After proving that r is well-defined on rational equivalence classes,
the most important of these properties are:
6.1 The Basic Construction
93
(i) Compatibility with proper push-forward (§ 6.2)
(ii) Compatibility with flat pull-back (§ 6.2)
(iii) Commutativity (§ 6.4)
(iv) Functoriality (§ 6.5).
For example, to calculate XV, by (i) it suffices to calculate X ¦ V for any V
mapping properly and birationally to V; one may blow up V along VO W to
reduce to a case where the excess intersection formula applies. A particular
case of (ii) is the assertion that the intersection products restrict to open
subschemes: one may often compute intersection products locally. An impor-
important case of commutativity asserts that intersections may be carried out before
or after specialization in a family; this will include a strong version of the
"principle of continuity" in Chapter 10.
When Y'= Y, v determines the (ordinary) Gysin homomorphisms
Functoriality (iv) refines the statement that (Ji)* = i*j* for i:X—*Y,
j: Y-> Z regular embeddings.
More generally, if f:X-> Y is a local complete intersection morphism,
there are Gysin homomorphisms /*, and refined homomorphisms f. These
Gysin homomorphisms are used to describe the group A* Y, when Y is the
blow-up of a scheme Y along a regularly imbedded subscheme. A new blow-
blowup formula describes the Gysin map from /4* Y to A* Y explicitly.
The rest of this book is based on this intersection product and the
fundamental properties proved in §6.1-§6.5. As in Chap. 2, the formal
properties can be motivated from topology. As we shall see in Chap. 19, a
regular imbedding X <-* Y of codimension d determines an orientation, or
generalized Thorn class, in H2d(Y, Y—X). The Gysin maps are the algebraic
geometry versions of cap product by this orientation class, or with its pull-back
to Y', if Y' maps to Y.
6.1 The Basic Construction
Let /: X -* Y be a (closed) regular imbedding of codimension d, and denote
the normal bundle by Nx Y. Let V be a purely A:-dimensional scheme, and let
/: V -> Y be a morphism. Denote the inverse image scheme f~l(X) by W, and
form the fibre square
W± V
n if .
X -> Y
Let N = g* NxY,a bundle of rank d on W, and let n : N -> W be the projecti on.
Since the ideal sheaf j*" of X in Y generates the ideal sheaf/" of Win V, there is

94
a surjection
Chapter 6. Intersection Products
This determines a closed imbedding of the normal cone C = Cw V as a
subcone of the vector bundle N:
W .
Since C is purely ^-dimensional (Appendix B.6.6), it determines a A:-cycle [C]
on N (§ 1.5). Let s be the zero section of the bundle N. Define the inter-
intersection product of V by X on Y, denoted X ¦ V (or X ¦ rV, or i' [V], see Example
6.2.1), to be the class on W obtained by "intersecting [C] by the zero section of N".
That is, set
A)
X-V = s*[C]
in Ak_d(W), where s*: Ak(N) -> Ak_d W is the Gysin homomorphism of
Definition 3.3; equivalently, XV is the unique class in Ak_d(W) such that
ti*(X-V) = [C] in Ak(N).
Proposition 6.1. (a) With the above notation,
X-V={c(N)ns(W,V)}k_d.
(b) If ? is the universal quotient bundle of rank d on P (N © 1), and q is the
projection from P(N ®\) to W, then
(c) If d-\, i.e. X is a Cartier divisor on Y, Vis a variety, and f is a closed
imbedding, then X ¦ V is the intersection class constructed in § 2.3.
Proof. Since P(C © 1) restricts to C on N, (b) follows from Proposition 3.3.
To prove (a), consider the universal exact sequence
Q-*fi(-\)-*q*N©\ ->? ->0
on P(N © 1). By the Whitney formula, c (?) c {fi(- 1)) = c(q*N). Therefore
q*{cM) n [P{C® 1)]) = {q*(c(O n [J»(C© 1)])}*-,
1)) n
which proves (a).
For (c), if K<= X, then C = W= V, so 5*[C] = cx{N)n[W] by Proposition
2.6(c) (cf. Example 3.3.1). If V<tX, then PK is the pull-back Cartier divisor
f*X, C= N, and s*[C] = \W\ = [f*X]. These prescriptions agree with those of
§2.3. D
Definition 6.1.2. Let Cx,..., C, be the irreducible components of the cone C,
and let m, be the geometric multiplicity of C, in C. Thus [C] =?'=! m,[C,] is
6.1 The Basic Construction
the cycle of C. Let Z,<= PKbe the support of C,, i.e.
95
a closed subvariety of W. The varieties Z,,...,Z, (which need not all be
distinct) are called the distinguished varieties of the intersection of V by X. Let
iV,- be the restriction of A^ to Z,. We have a commutative diagram
C,
Z,
Let 5, be the zero section of Nh and set
in Ak~d{Zj)- From the definition of i* and it, it is clear that a,- maps to ^*[C,]
in Ak_d W. Therefore r
XV = S^a,
i=i
in Ak_d W. We call the equation X- V = ? m; a, the canonical decomposition of
the intersection product.
If Z is a distinguished subvariety of W, the sum of those terms w,a, with
Z, = Z is called the equivalence of Z for the intersection of V by X, or the
contribution of Z to X ¦ V. It should be emphasized that a distinguished variety
Z may have any dimension from k — dio k, but that the contributions are
always cycle classes of dimension k — d; only if d\mZ = k-d is the equiva-
equivalence of Z a multiple of [Z] (cf. Chap. 7).
For any closed subset S of V, the part of X-V supported on S, denoted
(X-V)s, is the class in At.dS obtained by adding the equivalence of all dis-
distinguished varieties contained in S:
(X-Vf= ? in,*,.
z,^s
In the rest of this chapter we will study the classes X ¦ V, and show that
they satisfy the formal properties one expects for intersection products. The
individual terms in the canonical decompositions are more subtle, however.
Examples and classical applications will appear later, particularly in Chapters
9 and 16. A canonical decomposition occurs naturally in the proof of a formula
of Severi in Example 16.2.4. A dynamic interpretation of these equivalences
will be given in Chap. 11.
Example 6.1.1. The formulas of Proposition 6.1 are also valid for the
equivalences of the distinguished varieties:
cc,=
1)])
where ?, is the universal bundle on P (N, © 1), and q, the projection to Z,.
Example6.1.2. Define divisors Du D2 on P2 by Z), = 2A+ B, D2=A-?2B,
where A and B are lines meeting in a point P. Let X= DtxD2, F=P2xP2,
V= P2, / the diagonal imbedding of V in Y. Then A, B, and P are the

96
Chapter 6. Intersection Products
distinguished varieties for the intersection of V by X, and the corresponding
canonical decomposition is
where a (resp. /?) is any zero cycle of degree 3 on A (resp. B).
Example 6.1.3. Let Y= P2, X, the curve xy = 0, X2 the curve x = 0, P the
point x = y = 0. For the intersection product of X2 by Xx, only X2 is
distinguished, but for the intersection product of X{ by X2 both Z2 and P are
distinguished. (To see this, replace Y by the affine plane. If / is the ideal in
K[x,y]/(xy) generated by x, then
K[x,y,T]/(x,yT)-> ®I"/I" + 1,
sending T to x, is an isomorphism; thus the components of C are: the line
x = y = 0, which maps to P, and the line x = T= 0, which maps to X2.)
Similarly for the intersection product of the diagonal Avt by X{xX2 in
P2xP2, only X2 is distinguished, but for the intersection of XxxX2 by APi,
X2 and P are distinguished.
In each case, the canonical decompositions of the intersection classes may
be calculated either directly, or by the dynamic interpretation of Chap. 11
(cf. Example 11.3.2).
Example 6.1.4 (cf. Examples 4.2.2 and 11.3.2). Let A, B and D be effective
Cartier divisors on a non-singular surface X, with A and B assumed relatively
prime. Let A'=A + D, B'=B + D. Consider the intersection of the diagonal
A = X by A'x B' arising from the fibre square
A'Off-* X
1 I*
A' x B' -» X x X .
(a) The distinguished varieties for this intersection product are the irre-
irreducible components of D, and the points of A f) B. (Identify the blow-up of X
along A fl B with the blow-up of X along A' D B'.)
(b) If P is a point where A and B meet transversally, the equivalence of P for
the intersection class is
A + ord, (/>))[/>].
(c) Let E be an irreducible component of multiplicity m of D, and assume
A and B meet transversally at any common point P on E. Then the equivalence
of ? for the intersection class is
m(AE + BE + DE)~ ? ordP(D)[P].
PeAftB
(Use the blow-up of (a) to compute Segre classes.)
(d) When A and B are not transversal, the contributions may still be
computed by successive blow-ups, but the results are more complicated. For
example, if X= P2, and A, B, D are defined by polynomials y2— xz, y2+ xz,
and y — Xx (for some X e K) respectively, and Pq = @:0:1), the equivalence
of Po is 3 [Po] if/I * 0, but 4[P0] if X = 0.
6.2 Refined Gysin Homomorphisms 97
Example 6.1.5. With homogeneous coordinates (x, y, z, w, t) on P4, let
Ki = V(z3 — x y (y — 2 x), w)
V2 = V(w3-yx(x-2y),z).
The distinguished varieties for the intersection product of the diagonal P4 by
Vl x V2 in P4 x P4 are the lines x = z = w = 0, y = z = w = 0, and the point
x=y = z = H' = 0. Each contributes a zero-cycle of degree 3 to the intersection
product.
Example 6.1.6. In the situation of Proposition 6.1 (a), c(N) n s(W,V) =
X ¦ V + higher terms, i.e., {c(N) n s(W, V)}{ = 0 for i < k - d.
Example6.1.7. If the imbedding of Win Vis a regular imbedding of codi-
codimension d', with normal bundle N', then
X-V=cd_d,{NIN')r\[W].
(See § 6.3 for a generalization.)
Example6.1.8. Let Q be the universal quotient bundle, of rank d— 1, on
P(N). Assume that dim Wsk-l. Then
where p is the projection from P (N) to W. (Compare the proof of Proposition
6.1 (a) with Example 4.1.2.)
Example 6.1.9. Uniqueness of intersection products. The intersection product
X-yKin Ak_d{X), defined for any regular imbedding X c» Y of codimension
d, and any purely A:-dimensional subscheme V of Y, is characterized by the
following properties:
(i) ("normalization"). If Y is a vector bundle on X, X &-»• Y the zero section
imbedding, and V=n~l{W), where n:Y-+X is the projection, then
X-yV=[W].
(ii) ("continuity"). If Xx P1 -> .-y is a family of regular imbeddings, f a
subvariety of <&, with 'S' and iT flat over P1, then all the classes X ¦ YIV, are
equal. Here Y, and V, are the fibres of <$l and y over rational points f 6 P1.
(Apply (ii) to the deformation to the normal cones, i.e., <& = MJK,
6.2 Refined Gysin Homomorphisms
Let i:X-* Y be a regular imbedding of codimension d, and let/: J" -> Y be a
morphism. Form the fibre square
(*)
Y'
if
Y

Chapter 6. Intersection Products
i] :ZkY'
98
Define homomorphisms
by the formula
where X ¦ Vt is the intersection product constructed in the previous section. (Of
course, XVt is constructed as a cycle class on X'DK; following our usual
convention (§ 1.4) the same notation is used for its image in the larger
scheme X'.)
To see that i passes to rational equivalence, we give a variant of this
definition. The normal cone C'= Cx- Y' is a closed subcone of N=g*NxY.
Then /' is the composite
Zk Y' 1 Zk C - AkN ^ Ak.dX'
where a is the specialization homomorphism of § 5.2, the second map is
induced by the inclusion of C" in N, and s* is the Gysin map for the zero-
section i of A" in N (§ 3.3). By Proposition 5.2, a passes to rational equivalence,
so v does also.
The induced homomorphisms
are called refined Gysin homomorphisms. We also may write X ¦ Ya in place of
i'(a). If Y'= Y, f=idY, these are called simply Gysin homomorphisms, and
denoted i* instead of V,
i*:AkY-*Ak.dX.
A more precise notation for the refined Gysin homomorphism would be /*¦ or
/*; we prefer to use the single notation r for all these homomorphisms, taking
care to specify where they act.
Theorem 6.2. Consider a fibre diagram
X" -*
'I
X' ±
•1
X -7*
Y"
V
Y'
V
Y
with i a regular imbedding of codimension d.
(a) (Push-forward) Ifp is proper, and a e Ak Y", then
«'/>* (°0 = ?*(«'«)
in Ak~d
(b) (Pull-back) Ifp is flat of relative dimension n, and a. e Ak Y', then
in Ak+n-dX".
6.2 Refined Gysin Homomorphisms
99
(c) (Compatibility) If i' is also a regular imbedding of codimension d, and
a. e Ak Y", then
i'a = i"tx
in Ak-dX".
Proof, (a) and (b) follow from the corresponding properties of Segre classes
(Proposition4.2). For (a), one may assume a = [V']; let V=p(V). Let
N=g*NxY. Then
V pJK'] = deg(F'/K) {c(N) n s(X'f)V, V)}k.d
= {c(N)nq^s(X"f)V',V'))}k_,
= q*{c(q*N)ns(X"r)V',V')}k_d
The proof of (b) is similar and left to the reader. For (c), it suffices to observe
that, when /' is a regular imbedding of the same codimension as i, g*NxY is
the normal bundle to X' in Y'; indeed if./" and J"' are the ideal sheaves, the
canonical epimorphism of <jr*(X/X2) onto J''/Jr'2 must be an isomorphism,
since both sheaves are locally free of the same rank. D
Remark 6.2.1. If
•I I'
X r* Y
i
is a fibre square, with / and /' regular imbeddings of codimensions d, an
important case of (c) is the formula
/» = /'•(«)
for all a. e A* Y'. If /' is not a regular imbedding, or if i' is a regular imbedding
of codimension < d, then r(a) depends on /, not just on /' (cf. Theorem 6.3).
Remark 6.2.2. By the push-forward property (a), to calculate the intersec-
intersection product XV, it suffices to calculate X-V for any V which maps
properly and birationally onto V; for example, we may blow-up V along
X'OV to reduce to the case where X'OV is a divisor in K(or X'f]V=V).
A simple formula for the intersection product in this case will be given in the
next section. Note that even when one starts with subvarieties of Y, such
reductions are possible only if one has intersection products for varieties
mapping to Y.
An important special case of the pull-back property (b) is the case when
Y" is an open subscheme of Y'. For example, the part of an intersection
product i'a carried by a connected component of X' can be calculated by
replacing Y' by any open neighborhood of the component.
Example 6.2.1. The intersection product of § 6.1 also determines the class
v [ V\ for an arbitrary pure-dimensional scheme V, i.e.
i'\V] = X-V

100
Chapter 6. Intersection Products
in Ak_d(X'). (To see this, if C' = CXY', and W= VOX', it suffices to show
that
where a: Ak Y' -> Ak C is the specialization homomorphism. From the con-
construction of a given in the proof of Proposition 5.2,
If [V] = '?mi[Vi] is the cycle of V, then, as in Lemma4.2, [M^V] =
XmjM^^Fj]; an application of Proposition 2.6(d) completes the proof.) In
particular, if Y is pure-dimensional, then
i*[Y] = [X].
Example 6.2.2. If V is not pure dimensional, the above must be modified.
If M is the blow-up of Vx P1 along Wx oo, E=P{C®\) the exceptional
divisor, q;E^> Wthe projection, then
In this case, however, E ¦ [M] is not necessarily the same as [E] (cf.
Example 2.6.4).
However, if dim Kg«, and [V\n denotes the ^-dimensional component of
[V\, one always has the formula
<¦ ([n.) = ** (lO.) = <?*MO n [?]„)
where s is the zero-section of g* NXY. (Apply Example 1.7.3 to the inclusion
P(C®\) c+Mrf]vV.)
Example 6.2.3. If ? is a vector bundle of rank d on X, the zero-section sE is
a regular imbedding of codimension d, with normal bundle E, and the Gysin
homomorphism sE: AkE -> Ak-dX of this section agrees with the Gysin
homomorphism constructed in Definition 3.3. (It suffices to check that
s%[n~s V\ = [V], with n : E -> X the projection, and this follows from Theorem
6.2 (c).)
Example 6.2.4. Let {%) be a diagram as in Theorem 6.2 (a). Let S be a
closed subset of Y', S' = p~l (S), r the induced morphism from S' to S. Let V
be a subvariety of Y", V= p(V'). Then
The individual terms of the decomposition do not enjoy such a "birational
invariance", however.
Example 6.2.5. Consider the diagram (*) of the beginning of this section.
For any open subscheme Y^ of Y', and any ^-dimensional subvariety V of Y',
the intersection class XV in Ak_dX' restricts to X-Vo in Ak_d(X'o), where
Vo = V0 Y<;, X'o = XT)
S fl Y^, the restriction
In addition, for any closed subset S of Y', if So =
6.2 Refined Gysin Homomorphisms
101
takes (X-Vf to (X-VQf°. For example, if dim(S - So) < k -d, this restric-
restriction is an isomorphism, so the part of X ¦ V supported on S is determined by
the part of X-Vo on So.
Example 6.2.6. Let d be the diagonal imbedding of P" in P" x ... x IP" (r
factors). Let [k] denote the generator of Ak P" given by a A:-plane in P" (cf.
Example 2.5.1). Then the Gysin homomorphism d* is determined by the
formula d*([k{\ x ... x [kr]) = [I] where /= k, + ... + k,-{r- 1) n. (Take the
linear spaces in general position and apply Theorem 6.2(c)). For V,,..., Vr
closed subschemes of P", with V, of pure dimension kh it follows that
d'lVi x ... x Vr] is a cycle-class in Ai(f] Vfr whose degree is the product of the
degrees of the Vt:
Vtx...xVr]= II deg[V,].
i
(See Chapters 8 and 12 for more on Bezout's theorem.)
Example 6.2.7. Let ? be a vector bundle of rank d on a scheme Y, s a.
regular section of E, X=Z(s) the zero-scheme of s, i the (regular) imbedding
of X in Y. Let a be a fe-cycle on Y, Y' = Supp(a), X' = XD Y'. Then
in Ak-d(X'), where sE is the zero section of E. (Apply Theorem 6.2 (c) to the
diagram
(S)
X' -> Y'
I I
X ± Y
and to the analogous diagram with s and sE interchanged.)
Example 6.2.8. The results of this section extend with little change to the
case when i\X-* Y is a regular imbedding which is not assumed to be a
closed imbedding. Let / be factored into a regular closed imbedding io: X -* U
followed by an open imbedding U <= Y. Given a fibre square (*), define
r :AkY'-> Ak.dX'
to be the composite Ak Y'-> Ak U'-> Ak-dX', where ?/'=/"'(?/), the first map
is the restriction homomorphism, and the second is i'o. This homomorphism /'
is independent of the choice of U. (This is a special case of the Gysin map
constructed in § 6.6.)
Example 6.2.9. The operations of intersection theory are compatible with
field extension. For an algebraic scheme X over a field K, let XL denote the
scheme X®KL over L. For a A:-cycle a^J^riylV] on X, let aL be the A:-cycle
J^nvlVi] on XL. This determines a homomorphism a -> aL from AkX to
Aic(Xl), which is compatible with proper push-forward, flat pull-back, Chern
classes, and refined Gysin homomorphisms. (When L is a finite extension of

102
Chapter 6. Intersection Products
K, a -* aL is the flat pull-back for the projection XL -> X, in which case the
assertions have been proved in the text; the proofs for the general case are
similar.)
6.3 Excess Intersection Formula
Consider a fibre diagram
(S)
A"'
•1
X
Y"
1'
r
V
Y
with / (resp. /') a regular imbedding of codimension d (resp. d') and normal
bundle N (resp. N'). There is a canonical imbedding of N' in g*N (see § 6.1).
The quotient bundle
E=g*N/N'
is a vector bundle of rank e = d — d' on Jf'. We call E the excess normal bundle
of the lower fibre square.
Theorem 6.3 (Excess Intersection Formula). For any ol e Ak Y",
i'(a) = c€(q*E)ni'!(ci)
inAk-d{X"\
Proof. Let Q' = P{q*N' ®\),Q = P(q*g*N © 1), and let ?' and ? be uni-
universal quotient bundles on Q' and Q. There is a canonical imbedding of Q' in
0, with the canonical line bundle on Q restricting to the canonical line bundle
on Q. There results an exact sequence of bundles
o
r*(q*E) ->0
on Q; here r is the projection from Q' to X". We may assume a = [V], V a
subvariety of Y". Set P = P(CV()X»V®\). Using the Whitney formula and the
projection formula,
(ot). ?
Corollary 6.3. Z,e/
X-
6.3 Excess Intersection Formula
103
6e a fibre square, with i a regular imbedding of codimension d and normal bundle
N. Assume that i' is an isomorphism. Then
r'(oc) = cd{g*N) n a
for all a e A*Y''. D
This includes the self-intersection formula
for a 6/4* (.Y).
Remark 6.3. Given a diagram (*) as in § 6.2, and a class a. e Ak Y', if some
connected component X'o of X' is regularly imbedded in Y', then the excess
intersection formula may be used for the part of /'(a) supported on A^. As in
Remark 6.2.2, this follows by restricting to open subschemes of Y'.
The fact that intersection products commute with Chern classes is a formal
consequence of the properties proved so far.
Proposition 6.3. Let i: X -> Y be a regular imbedding of codimension d,
(*)
X'-^ Y'
I I
a fibre square, and let F be a vector bundle on Y'. Then for all a e Ak(Y'), and
)
in Ak-d-m (X').
Proof. Reduction Step. It suffices to find a proper morphism h : Y'
and a. 6 Ak(Y') with A* (a) = a, so that if we form the fibre square
Y',
X'ir*Y'
and set /= h*F, then
/' (cm (F) n 3.) = cm (I'* (F)) n /' (of)
inAk-d-m(X'). This follows from the commutativity of Gysin homomorphisms
and Chern class operations with push-forward (Theorem 6.2 (a) and Theorem
3.2(c)):
/'(cm (F) n A*dO = i1 {h*{cm(h*p n a)) = K i'(cm (F) n a)
= K (cm(?* (/)) n /'a) = cm (i'*F) n hi i>(d)
To prove the proposition, we first assume F is a line bundle, and m = 1. We
may assume a = [V\, Fa A:-dimensional subvariety of Y'. By the reduction

104
Chapter 6. Intersection Products
step, we may replace Y' by the blow-up of V along X' n V. Thus we may
assume Y' is a variety, a = [Yr], and X' is either a Cartier divisor on Y' or
A" = Y'. In case A" is a Cartier divisor, let E be the quotient of the normal
bundles on X', as constructed before Theorem 6.3. Then
, (F)noi) = cd.x (E) n /'!(c, (/) n a)
= ?„_,(?) n(c, (/'*/) ni" (a))
= c, (/'*/) n(c,_,(?)n/'! (a))
= c, (/'*/) n r (a)
(Theorem 6.3)
(Proposition 2.6 (e))
(Theorem 3.2 (b))
(Theorem 6.3).
If X' = Y' one uses similarly Corollary 6.3 with Theorem 3.2 (b).
For a general vector bundle F, and any m, let F' — i'*F, and form the fibre
square of projective bundles
"I I'
Let L be the canonical line bundle on P(F), L' its restriction to P(F'). Since
the Gysin homomorphisms commute with push-forward and pull-back (Theo-
(Theorem 6.2) and first Chern classes, we have
for all a,j. Since the Chern classes are defined as polynomials in the Segre
classes, the proposition follows. ?
Example 6.3.1. Given a fibre square as in the beginning of this section,
assume Y' is pure A:-dimensional. Then
i'[Y'] = ce{E)n[X'\
mAk-d(X'). (See Example 6.2.1.)
Example 6.3.2. If in a fibre square
X'-UY'
•I I'
X-r>Y
f, g, i and; are all regular imbeddings, then
g*Nx Y/Nx. Y' =zj*Nr Y/NrX.
In other words, the excess normal bundle is independent of the orientation of
the fibre square.
Example 6.3.3. Let Ibea scheme. For any point / e P", rational over the
ground field, let /, be the imbedding of X in X x P" at /, i.e., i,(x) =(x, t). Let a
6.3 Excess Intersection Formula
105
be a fe-cycle on Xx IP". Then the classes i*(a) in Ak_n{X) are independent of t.
(By Theorem 3.3, write a = ?"=o a, x [HJ], OLj 6 Ak-jX, Hj a ;-plane in P". Then
/•() )
Example6.3.4. (a) If E is a vector bundle of rank don a scheme Y, and s is
a regular section of E, then the inclusion / of the zero-scheme X = Z (s) in Y is
a regular imbedding of codimension d, and NXY is the restriction of ? to X. If
/: Y' -> Fis a morphism, form the fibre square
(*)
Then
for all a in A* Y'. (Form the diagram (J) as in Example 6.2.7; by Theorem
6.2(c), /!(a) = ^(a). Taking a = [F], V= Y', one has a fibre diagram
X'^U Y'
Y'—>f*E
Therefore .%(a) =
, and one concludes by Example 6.2.3.)
Example 6.3.5. Let 0 -> E -* F-* Gbean exact sequence of vector bundles
on a scheme X. Let n : F -* X be the projection, and let g = rank G. Then
[?] = c,(**G)n[/l
in A*F.
Example 6.3.6. The refined Gysin homomorphism is uniquely determined by
the excess intersection formula and the push-forward property. (If a = [V],
Kc, y, blow up V along VOX'.)
Example 6.3.7. Consider a fibre square
XY
•I I'
with / a regular imbedding of codimension d. Assume that cd(NxY) = 0; for
example, Nx Y might have a nowhere vanishing section. Then there is a unique
"specialization" map
o:Ak(Y'-X')->Ak-d(X')

106
which makes the diagram
Chapter 6. Intersection Products
AkX'JUAk Y'J^ Ak(Y'- X') - 0
I'1 „'-
Ak-dX'
commute, where j' is the inclusion of Y' — X' in Y'. (The row is exact by
Proposition 1.8, and /!/'i(a) = cd{g*Nx Y) n a by Theorem 6.2(a) and Corol-
Corollary 6.3.)
In case d = 1, and a = J] ni [ ^] is a cycle on Y' — X', then
where Pj is the closure of F; in J".
6.4 Commutativity
In this section we prove that the refined Gysin homomorphisms defined in
§ 6.2 commute with each other. This will be done by blowing up to reduce to
the case of divisors, which was proved in § 2.4.
Theorem 6.4. Let i: X -> Y be a regular imbedding of codimension d,
j: S -* T a regular imbedding of codimension e. Let Y' be a scheme, f: Y' -> Y,
g : Y' -* T two morphisms. Form the fibre diagram
X" -> Y"
«*)
X'
i
X
Y'
Y
i.e., each of the three squares is a fibre square. Then for all a. 6 Ak Y',
in Ak-d.eX".
Proof. Reduction Step. Let a. s Ak Y'. Suppose h : Y' -> Y' is a proper
morphism, a. e Ak(Y'), with A*(a) = a. Form the fibre diagram
X"-*
I
X' ->
I
X -+
Y"-*
4
Y' ^*
9
if
Y
s
T
where f=f°h, g = g ° h. If we prove that f i'{ti) = i'j'{ti) in ^.^^X"),
it follows from Theorem 6.2(a) thaty!/!(a) = i'j'(a) in Ak^d_e(X"). Indeed, if
6.4 Commutativity 107
p. X' ->¦ X',q : Y" -* Y", r : X" -* X" are the morphisms induced by h, then
By linearity, we may assume a = [V], V a subvariety of Y'. Applying the
reduction step to the inclusion of V in Y' we may assume V= Y'. Let
h: Y' -> Y' be the blow-up of J" along X'. By the reduction step again, we
may assume either X' is a Cartier divisor on Y', or that Jf' = Y'. Similarly
blowing Y' up along Y", we may assume Y" is a Cartier divisor on Y', or
K" = Y'.
In case A" = Y', let iV' (resp. TV") be the pull-back to X' (resp. X") of the
normal bundle to X in Y. By Corollary 6.3 the homomorphisms /' are the
Chern class operations of capping with cd(N') or cd{N"). Therefore
/ /'(a) =f(cd(N') n a) = crf(JV") n/(a) = /'/(ot)
by Proposition 6.3.
Thus we may assume X' and (by symmetry) Y" are Cartier divisors on Y'.
Let ? (resp. F) be the excess normal bundle on X' (resp. Y") constructed in
§ 6.3 for the square
X'
1
Then by Theorem 6.3, and using the notation of Remark 3.2.2,
f /'•(«) = c,_, (/) n/!(o-, (?) n j"(ot))
= ce_| (F) n (q_i (?) ny'! /" a) (Proposition 6.3).
For the Cartier divisors X', Y" on Y' we have the fundamental equation
inAk-2(X") (Theorem 2.4). Now
and
Therefore i"j''{a) =j'' /''(a), so
c,_, (/) n (q_, (?) ny"«" a)
n (c'_, (F) n i"j" a),
using the commutativity of Chern classes (Theorem 3.2 (b)). Reversing the
previous argument, the right side is /'/' (a), which concludes the proof. ?

108
Chapter 6. Intersection Products
6.5 Functoriality
We show that the refined Gysin homomorphisms for a composite of regular
inbedding is the composite of the refined Gysin homomorphisms of the factors.
Theorem 6.5. Consider a fibre diagram
(**)
A"
4*
X
I9
Y
Z'
Z.
If i (resp. j) is a regular imbedding of codimension d (resp. e), then j i is a regular
imbedding of codimension d + e, and for all a 6 Ak Z',
(/01 («) = ''(/'(«))
Proof. (For another proof, see Example 17.6.3.) The regularity of the
imbedding/ i follows from Lemma A.5.2.
We first consider the case where Z=E is a vector bundle over Y, Z' = E' is
the pull-back bundle g*E,f is the natural map from g*E to E, and/ (resp./')
is the zero-section imbedding of Y in E (resp. Y' in E'). Let n (resp. rf) be the
bundle projection from E to Y (resp. ?' to Y').
There is a canonical isomorphism of cones on X':
To see this, let J be the ideal sheaf defining X' in Y, and let 3F be the sheaf
of sections of (?')v, so that ?' is Spec(S"), S" = Sym(^). Then the ideal / of AT'
in ?' is generated by J <=. S° and by S1. Then
S" ® Sn+l
so
and
©...©
which proves A).
In particular, the normal bundle Nx E is a direct sum of Nx Y and i*E.
Pulling back to X', this gives
h*NxE=h*NxY®i'*E'.
B)
Let q be the projection from h*NxE to A*iV^ Y, r the projection from h*Nx Y
\.oX';sorq is the projection from h*NxE to A".
Assume also that Y' is an irreducible variety. From A) and B) it follows
that
q*[Cx,Y'] = [Cx,E']
6.5 Functoriality 109
in A^ (¦?')• From the construction of the refined Gysin homomorphism we have
and
'] = {rq)*{JiY\E'\.
Since (rq)* = q*r* is one-to-one (Theorem 3.3 (a)), the preceding three
formulas give
C) U0'[E'] = il[Y']
in A* (X1). It now follows for any Y', and any a 6 ^4t?', that
For by Theorem 3.3 (a), we may assume a = n'*P, and by linearity that
/?= [V], V an irreducible subvariety of Y'. By Theorem 6.2(a) we may replace
Y' by V. Then a = [?'] and/!a = [r] by construction, and D) therefore
follows from C).
To prove formula D) in the general case, we may assume a = [Z'], Z' an
irreducible variety (Theorem 6.2(a)). Let M = MfZ and M' = M^Z' be the
deformation varieties constructed in § 5.1. For any / 6 P1, rational over the
ground field, let <p, be the imbedding of {;} in P1, and form the fibre diagram
(*)
Here k is the composite of the inclusion ; x id of Xx P1 in Y x P1 and the im-
imbedding of Yx P1 in M constructed in § 5.1. Since the fibres M', of M' over / in
P1 are Cartier divisors on A/',
A"
A"
A"
X
4
X
4
X
w
P1
P1
->a/;->
4
¦*+ A/' •**
4
-> A/ .
W
i«
P1
E) <?[[*/'] =
Consider the fibre diagrams
¦'] if t 4= co
[CrZ'] if f=oo.
A" x {;} -> A/;
1 1
X x{t}% M,
1 1
with x, =/; if ; 4= go, and xx =Ji where /is the zero section imbedding of Y
in NYZ. By Theorem 6.2 (c),
But by the special case considered above,
G) (Ji)'[CrZ'] = i'f[CrZ'] = i'/[

110
Chapter 6. Intersection Products
the last equation from the construction of/. The conclusion then follows from
F) and G), provided we know that x'(p\[M'] is independent of t. But by the
fundamental commutativity result of § 6.4, applied to diagram (*),
(8) x'<p][M'] = <p)x][M'].
Let p=x[M'] ? At(X' x P1). The required claim follows from the fact that for
any fieA+(X'x P'), the elements
v\(fl)eA,{X'x{t})=A,(Xt)
are independent of t (cf. Example 2.6.6 or 6.3.3). ?
The following result is formally similar, but more straightforward to prove.
Proposition 6.5. Consider a fibre diagram
(**)
X'
4*
X ¦
Y'
I9
Z'
(a) Assume that i is a regular imbedding of codimension d, and that p and pi
are flat of relative dimensions n and n — d. Then i' is a regular imbedding of
codimension d, p' andp' i' are flat, and for a 6 AkZ'
{p'i')* (a) = i'* {p'*a) = i'p'*a
in Ak+n-dX'.
(b) Assume that i is a regular imbedding of codimension d, p is smooth of
relative dimension n, and pi is a regular imbedding of codimension d— n. Then for
all asAk Z',
(/>/)'(«) = «V*a)
in Ak+n^dX'.
Proof, (a) The assertion that i' is a regular imbedding of codimension d
follows from Lemma A.5.3. To prove the formula, we may, by our standard use
of the projection formula, assume that a = [K], V= Z', V a variety. Then
p'*[V\ = [ Y'\ (p' i')* [ V\ = [X'] by the definition of flat pull-back, and
by Theorem 6.2 (c) and Example 6.2.1.
(b) Form the fibre square
»1
'*Y
[p
Z
Construct a corresponding square over this, induced by base extension from Z'
to Z; denote corresponding schemes and morphisms in this square by cor-
corresponding letters with primes. The morphism i determines a section s: A"-> W
with qs = idx. Since q is smooth, s is a regular embedding (Appendix B. 7.3).
6.5 Functoriality
Therefore
i'p'* a = sfp'*a
= s[q'*(pi)'a
111
(Theorem 6.5)
(Theorem 6.2 (b))
(Proposition 6.5 (a))
since q's' is the identity on X'. D
Corollary 6.5. Let E be a vector bundle of rank d on X, n: E -* X the projec-
projection. Any section s of E is a regular imbedding, and
s*:AkE-*Ak-dX
is the inverse isomorphism to n*. In particular, s* is independent of the choice of
section s. If Z(s) is the zero-scheme of s, and aeAk(E), the class s'(<x) in
Ak-d{Z(s)) maps to (V) (a) in Ak^d(X).
Proof. By the proposition, s*n* = {ns)* = id; since n* is surjective, both
are isomorphisms. The last statement follows from Theorem 6.2 (a). D
Example 6.5.1. (a) Let;: V-> X, j: W-> X be regular imbeddings, and let a
be a cycle on X. Then
in At{VC) WC\ |a |). Here (;' xj)'(a) is constructed by intersecting a, diagonally
imbedded in X x X, by the subscheme VxW. In case X is pure dimensional,
and a = [X], this reads:
V- W=W-V={Vx W)-Ax.
(Theorem 6.4 gives the first equality. For the second, form the diagram
V —^-> X
4 Is
Vx W UJ >VxX -^ XxX
1 4
W '—> X
and apply Theorem 6.5 and Theorem 6.2 (c).)
(b) The analogous formula is valid for more than two factors. For example,
let D\,...,D, be effective Cartier divisors on a scheme X. Let i be the product
imbedding of Dx x ... x ?>, in X x ... x X. For any &-cycle aonl
in Ak_r(D1[)...0Dr[)\<x\). Here Dl-...Dra. is constructed inductively by
the process of § 2.2.
Example 6.5.2. Let ij'.Xj-* Yj be regular imbeddings of codimensions dh
j=\,..., r. Let fj = Yj -> Yj be morphisms, at eAkl(Yj). Then ;'ix ... x ;, is a
regular imbedding of A^x ... xXr in Y\X ...xYr, of codimension ^dit and
(;,x... x;r)!(a,x ...xa,) = /j(ai)x...x /-(a,)

112
Chapter 6. Intersection Products
in AYM(k,-dl){X'ix....x.X'r), X) = XjXYl Y\. (One may assume r=2. Factoring
i]Xi2 into A^-, x B) ° ('ixly,), one may assume /2 = 1 r2- Then the assertion
follows from Theorem 6.2 (c).)
Example 6.5.3. Commutativity was used in our proof of functoriality;
conversely, using Cartesian products, functoriality implies commutativity.
Given the situation of Theorem 6.4, form the fibre square
X'
1
XxS
- r
1 if.s)
->Yx T .
= (/xy)! (a) =
Then
(Factori xj into (/ x 1T) {Ixxj) and into A Yxj) (i x ls).)
Example 6.5.4. In the situation of Proposition 6.5 (b), one may prove more:
if Vis a subvariety of Z', and W = p'~'(V), then the intersection classes XZV
and XYW have the same canonical decompositions. (There is a surjection
from h*NxY onto h*NXZ (Appendix B. 7.5), such that Cx^wW 's the inverse
image of GVnrK)
6.6 Local Complete Intersection Morphisms
In this section, for simplicity1, all schemes will be assumed to admit closed
imbeddings into schemes which are smooth over the ground field. For ex-
example, all quasi-projective schemes are allowed. Any morphism f:X-*Y then
admits a factorization into a closed imbedding i:X-*P followed by a smooth
morphism p; P-> Y. For example, if y" is a closed imbedding of A" in a smooth
M, one may take P = YxM,i= (f,j), and p the projection from Yx M to Y.
A morphism f'.X-* Y is called a local complete intersection (l.c.i.)
morphism of codimension d if / factors into a (closed) regular imbedding
i.X^-P of some (constant) codimension e, followed by a smooth morphism
p: P -* Y of (constant) relative dimension e — d. This notion is independent of
the factorization of/ into a closed imbedding followed by a smooth morphism
(Appendix B.7.6).
For any l.c.i. morphism f:X^>Y of codimension d, and any morphism
h:Y'^Y, form the fibre square
(*)
7
1 What is actually needed for this treatment is that morphisms under consideration have
factorizations into closed imbeddings followed by smooth morphisms, and that these may
be chosen compatibly whenever a composite of morphisms is considered, as in the proof
of Proposition 6.6 (c).
6.6 Local Complete Intersection Morphisms
Define a refined Gysin homomorphism
113
as follows; Factor/intop ° i as above, and form the fibre diagram
X'±p>?+ r
(**) n 4 4*
Then/?' is smooth, and we define
i = i'(/>'•«)
for azAk Y'. When Y' = Y, we write/* for/1.
Proposition 6.6. (a) The definition of f is independent of the factorization
off
(b) Iff is both a l.c.i. morphism and flat, then f'=f*.
(c) The assertions stated in Theorems 6.2, 6.4, and 6.5 for regular imbeddings
are valid for arbitrary l.c.i. morphisms. The excess intersection formula of
Theorem 6.3 holds if the excess normal bundle E of a diagram (*), with f and f
l.c.i. morphisms, is defined to be
E=h'*NxP/Nx.P'
with P, P' as in (**); this definition of E is independent of choice of factorization.
Proof, (a) If X-^ P\ A Y is another factorization of/ compare them both
with the diagonal:
X
(i. '0
¦ P, v/l
Y.
The assertion then follows from Proposition 6.5 (b).
(b) follows from Proposition 6.5 (a).
(c) The extension of Theorems 6.2 and 6.4 are obvious, since the
analogous statements are true for smooth (flat) morphisms as well as regular
imbeddings. For the functoriality, if f:X-> Y and g : Y-+ Z are l.c.i. morph-
morphisms, one chooses factorizations fitting into a commutative diagram
1
R
4
Q
with the vertical morphisms smooth, the horizontal morphisms regular im-
imbeddings, and the square a fibre square; for example; if P = Yx M as in the
first paragraph of this section, one may take R= Qx M. Then the functoriality
(gf)'-fg' follows from Theorem 6.5, using Theorem 6.2(b) to go around the
square.

114
Chapter 6. Intersection Products
To see that the excess bundle E is well-defined, by the diagonal construc-
construction used in (a) it suffices to compare factorizations f=pi and f=pqj where
p, q are smooth, i,j are regular imbeddings, and qj = i. Consider the diagram
X' ¦** Q -> P'
n i i
X -* Q ~> P .
There are exact sequences (Appendix B.7.5)
0
/
Since
pulls back to T^/p*, there results a canonical isomorphism
h'* (Nx Q)/Nx- Q' = h'* NxP/Nr P' ¦
The excess formula then follows from Theorem 6.3 and Proposition
6.5 (b). ?
For more on Lei. morphisms see § 17.4.
6.7 Monoidal Transforms
Let A" be a regularly imbedded subscheme of a scheme Y, of codimension d,
with normal bundle N. Let Y be the blow-up of Y along X, and let X = P(N)
be the exceptional divisor. We have a fibre square
(*)
X-1* Y
H 1/
X-* Y .
Since NjtY=tfN{—\), the excess normal bundle E is the universal quotient
bundle on P{N):
E = g* N/Nx Y= g* N/<CN (- 1).
We assume that Ycan be imbedded in a non-singular scheme2. The same is
then true for Y (Appendix B.8.2). Then / is a l.c.i. morphism of relative
dimension zero, in the sense of the preceding section. Indeed this is a local
assertion, and the local case is proved in Lemma A.6.1.
Proposition 6.7. (a) (Key Formula). For all x e AkX,
f*i*(x)=U(cd.](E)ng*x)
2 The precise assumption needed is that / factor into a closed imbedding followed by a
smooth morphism.
6.7 Monoidal Transforms
in Ak Y.
(b) Forally&AkY,Uf*y = y inAkY.
(c) If xsAkX, and g*x=j*j*x = 0, then x = 0.
(d) Iff 6 Ak Y, and f*y=j*y = 0, then y = 0.
(e) There are split exact sequences
0^>AkX-i>AkX®AkYJ>AkY
115
0
with a(x) = {cd-\ (E) n g* x , -;*x), and §{x,y) =j*x + f*y. A left inverse for
a is given by (x, y)^> g* (x).
Proof, (a) By Theorem 6.2(a) and Theorem 6.3 (cf. Proposition 6.6),
/* i* x =j*fx =/* (q_, (?) n g* x).
(b) One may assume y=[V\, for Fa subvariety of Y. If VczX, then
y = ;* x, x = [ V\, and by (a),
f*f*y =f*j* (crf-,(?)n0*x)
='*9* (c<i-i (E) r'9*x) = '*x
(cf. Example 3.3.3). If V $ X, let V c Y be the blow-up of V along Vf]X. By
Theorem 6.2 (b),
f*[V] = [V]+jt(x)
where x is a class supported on VOX. Therefore
/* /* [V] = /„ [V] + fj.(x) = [V) + i, g.(x).
But gt(x) is a class supported on VOX, and dim(Vf)X) < k, so gt x — 0, as
required.
(c) By Theorem 3.3(b), x = ??=o ^(^A))' n g*xt for some x^A^X.
Then by Propositions 3.1 (a) and 2.6 (c)
and
d-2
E
/=0
By the uniqueness assertion of Theorem 3.3 (b), all x, = 0, so x = 0.
We next verify that any y e Ak Y can be written in the form
for some xeAkX. Indeed, y — f*f*(y) restricts to zero on Y—X (Theorem
6.2 (b)), so it is the image of an element of AkX (Proposition 1.8).
(d) If f*y=j*y= 0, then by the preceding formula, y=j*(x). Therefore
= f*y = 0. Set
By (a), /, x' =j*x- f* i* (g, x) = y. But
g* (x') = g*x-g* (q-, (?) n ^* (^* x)) = 0
(Example 3.3.3). Therefore, by (c), x'= 0, so/= 0.

116
Chapter 6. Intersection Products
(e) The surjectivity of 0 was verified before the proof of (d). That 0 a = 0 is
precisely (a). That the given map is a left inverse to a is Example 3.3.3.
Finally, suppose
y*x + /V = 0.
By (b), y = - f*j*x = - i* g* x. Define x' as in the proof of (d). Then g* x' — 0,
and
j*x' =/,x-f* ;„ (g*x)=jtx + f*y = 0.
By (c), x'= 0, i.e. x = cd.x (E) r\ g* (g*x), so (x,y) = a (g* x), as required. D
Theorem 6.7 (Blow-up Formula). Let V be a k-dimensional subvariety of Y,
and let V a Y be the proper transform of V, i.e. the blow-up of V along Vf] X.
Then
f*[V] = [V] +jJc(E) n g* s(VDX, V)}k
in AkY.
Proof. If Vc X, then V= 0, s(Vf]X, V) = [V], and the formula reduces to
the key formula. So we assume W= VC\X 4= V. By Proposition 6.7(d), it
suffices to show that the two sides of the formula agree after applying /*,
and/*.
When /* is applied to the left side, one obtains [K], by Proposition 6.7(b).
Since /*[K] = [F], and /*./*='*?¦, to see that the two sides agree after
applying /* it suffices to note that
By the projection formula, since s (W, V) is a class on W, the left side of this
equation is the image of a class on W. But dim W< k, so the ^-dimensional
component of this class must be zero.
Now we apply y* to each of the three terms in the blow-up formula:
j*f*[V\ = g* i*[V] = g
= g*
n s{W,V)}k.d)
and W is the exceptional divisor in V. For the second
where (= C\ (<^v(
term,
Since c (E) = c (g* N) ¦ c @N(- 1))"' by the Whitney formula,
/*U({c(E)ng*s(W,V)}k=-{(-c(E)r,g*(s(lV,V))}k-i
jsl
Setting /? = [ W\ m — k-\, and regarding the last three equations, it suffices to
prove that
k-l
sO
for all pEAmX. This is a formal identity, valid for any projective bundle. To
verify it, by Theorem 3.3 we may assume /?= (q n g*a, q § d—\, a eA*X.
Notes and References
By Proposition 3.1,
g*(L Cn 0) = <?,(! C'n g*a\ = s(N) n a
The left side of the identity is therefore
g*N)s(g*N)ng*a\ =
117
l;B0
as required. D
Corollary 6.7.1. If X = P is a point in Y, then
where L is a k-dimensional linear subspace of E = Wjc~\ K the residue field of
#pj, andep Vis the multiplicity of P on V(% 4.3). ?
Corollary6.7.2. //dim VOX^k-d, then
f*[V] = [V\.
Proof. In this case g*s(W,V) is supported on g~'(W), which has
dimension S k - 1, so the other term is zero. ?
Example 6.7.1. The key formula, and the more general blow-up formula,
are also valid in refined versions: if K'-> Y is any morphism, and V is a
^-dimensional subvariety of Y', then
f[V] = [V] +jt{c(E)ng'*s(VC]X; V)}k
in Ak(X'), where X' = XxrY', X'=XxYY' and the induced morphisms are
denoted by primes. Likewise, f*f'y' = y' for any y' in A* Y'. (The proofs are
the same as those given in the absolute case. For a formalism which includes
these generalities, see § 17.5.)
Notes and References
The construction and formulas for the general intersection product of § 6.1
were first given in Fulton-MacPherson (IK. The properties of the refined
Gysin homomorphisms were sketched in Fulton-MacPherson C), where
the formalism of bivariant theories provided a useful guide (cf. Chap. 17).
Gysin homomorphisms for regular imbeddings (the case Y'=Y) had been
constructed by Verdier E).
The construction of refined intersections via specialization to normal cones
that we use in § 6.2 follows Verdier closely. Our proof of functoriality (§ 6. 5) is
also modelled on Verdier's proof for unrefined Gysin maps.
3 To be precise, this paper relied on the intersection theory which had been developed
previously for non-singular quasi-projective varieties.

118
Chapter 6. Intersection Products
The excess intersection formula (§ 6.3) has many precedents. In the non-
singular case, the self-intersection formula had been proved by Mumford in
1959, the key formula by Jouanolou A) § 4.1; see also Lascu-Mumford-Scott
A). A topological version of the general formula was given by Quillen A);
J. King B), D) gave an analytic analogue. Illusie asked us if such a formula was
known for rational equivalence. Such an excess intersection formula was given
in Fulton-MacPherson A) and by H. Gillet (unpublished).
The formula for intersection classes in terms of Segre classes of cones first
appeared in Fulton-MacPherson A). As mentioned in the notes to Chap. 4,
such classes were constructed in many cases by B. Segre, who stressed the im-
importance of blowing up to simplify problems in intersection theory.
The extension from regular imbeddings to l.c.i. morphisms follows the
formalism of [SGA6]. Kleiman A2) has also developed and applied this
extension.
In the case of smooth quasi-projective varieties, most of Proposition 6.7
was proved by Jouanoulou A) § 9, by essentially the same calculations; the
case of codimension 2 had been done by Samuel. See also Beauville A) Prop. 0.1.3.
The blow-up formula of Theorem 6.7 is apparently new, even in the non-
singular case.
Example 6.1.4 comes from R. Lazarsfeld.
Chapter 7. Intersection Multiplicities
Summary
As in Chap. 6, consider a fibre square
fF-> V
4 4/
with ; a regular imbedding of codimension d, V a /c-dimensional variety.
If Z is an irreducible component of W of dimension k — d, the intersection
multiplicity i(Z,X-V; Y) is defined to be the coefficient of Z in the intersec-
intersection class X-Ve Ak_d(W). The intersection multiplicity is a positive integer,
satisfying
i (Z, X -V;Y)^ length (<9Z,W).
Examples show that this inequality may be strict; equality holds, however, if
#z, v is a Cohen-Macaulay ring.
On the other hand, the criterion of multiplicity one asserts that
i(Z, XV; Y) is one precisely when <9ZV is a regular local ring with maximal
ideal generated by the ideal of A" in Y.
The standard properties of intersection multiplicities, worked out in the
examples, follow from the basic properties of the general intersection product
which were proved in Chapter 6.
7.1 Proper Intersections
Consider, as in § 6.1, a fibre square
W+* V
si 4/
X -> Y
with i a regular imbedding of codimension d, V a purely A:-dimensional
scheme. Let
C=CWV, [q=2>,[C,],

120 Chapter 7. Intersection Multiplicities
C, the irreducible components of C, and let Z, be the support of C,; Z\, ...,Zr
are the distinguished varieties of the intersection.
Lemma 7.1. (a) Every irreducible component of W is distinguished.
(b) For any distinguished variety Z,
k-d^ dimZ S k.
Proof, (a) follows from the fact that the support of Cw V is W, for any
closed subscheme W of a scheme V. Since C,- is an irreducible subvariety of
g*NxY which projects onto Z,, if iV,- is the restriction of g*NxYto Z,, then
Therefore
dimZ, s dim C, ^ dim (AT,) = dim Z,+ d.
Since C, is ^-dimensional (Appendix B.6.6), (b) follows, n
If dimZi=k-d, the inclusion C/ciNj of irreducible ^-dimensional varie-
varieties must be an isomorphism. In particular, the class a, obtained by intersecting
[C,] with the zero-section of Ni is just [Z,], and the equivalence of 2T, for the
intersection is w,[Z,].
Definition 7.1. An irreducible component Z of W=f~'(X) is a proper
component of intersection of V by X if dim (Z) = k — d. The intersection multi-
multiplicity of Z in X • V, denoted
i(Z,X-V; Y)
or simply i(Z,X-V), or ;'(Z), is the coefficient of Z in the class X-F in
Ak_d(W). Equivalently, the equivalence of Z for the intersection class is
i(Z,XV; Y)[Z]
If Nz is the pull-back of NXY to Z, then i(Z, X-V;Y) is the coefficient of Nz in
the cycle [C] of the cone C=CWV.
Let ,4 =<^z> ^ be the local ring of V along Z, and let J <z ,4 be the ideal of
W; ^4/./ has finite length when Z is an irreducible component of W.
Proposition 7.1. Assume Z is a proper component of W. Then
(a) l^i(Z,X- V;Y) S l(A/J), where l(A/J) is the length ofA/J.
(b) IfJ is generated by a regular sequence of length d, then
i{Z,X-V;Y) = l{A/J).
If A is Cohen-Macaulay (e.g. regular) the local equations for X in Y give a
regular sequence generating J, and the equality in (b) holds.
Proof. Let N=g*NxY. The restriction Nz of N to Z is an irreducible com-
component of N. Since N is a vector bundle over W, the coefficient of Nz in the
cycle [A7] is the same as the coefficient of Z in the cycle [W], which is 1{A/J).
Since C is a closed subscheme of N, the coefficient of any irreducible
component of N is no larger in [C] than it is in [N] (Lemma A.I.I). Since the
coefficient of Nz in [C] is i(Z,X- V; Y), (a) follows.
7.1 Proper Intersections
121
If J is generated by a regular sequence of d elements, replacing V by an
open subscheme which meets Z (which doesn't effect the intersection multi-
multiplicity, by Theorem 6.2 (b)), we may assume the imbedding of W in V is
regular of codimension d. Then C is a sub-bundle of N of rank d, so C= N, and
the coefficients of Nz in [C] and in [A7] coincide.
The last assertion of the proposition follows from Lemma A7.1. ?
The inequalities in (a) may be strict, as shown by Macaulay (Example
7.1.5).
Example 7.1.1. Let (e^. V)z be the multiplicity of V along W at Z, as
defined in Example 4.3.4. Then
i(Z,X-V;Y) = (ewV)z
i.e., the intersection multiplicity defined here agrees with Samuel's.
Example 7.1.2. Let al,..., ad be the images in A of a regular sequence of
elements defining X in Y (locally, in an open set which meets / (Z)). Then
where ^(a) = Z?=o ^(#i(K*00)). with K,(a) the Koszul complex defined by
a,,...,ad (Appendix A.5). Serre D) IV.A3 showed generally that XaC) gives
Samuel's multiplicity. We sketch an alternative proof, by induction on d. If
d = 1 it says
where the sum is over the minimal primes p of A; this is a special case of
Lemma A.2.7. For the inductive step, localize so that one has a fibre diagram
Wcz
I
X a
We
I
X'c
- v
{
- Y
with X'cY and XcX' regular imbeddings of codimension d—\ and 1, and
local equations pulling back to a-i,..., ad in A and a\ in Alifli,..., aj) respec-
respectively; we may also assume the localization is sufficient so that Z is the only
irreducible component of W, and all the irreducible components W't of W
contain Z, and therefore have dimension k — d+ 1. Letp, be the prime ideal of
A corresponding to W',. By induction
.., ad)).
By functoriality (§6.5), X-YV= Xr(X'-Y V). Let Hk= Hk(K*(a2,.
Then
i(Z,XV; Y) = Z eApi(a2,...,ad) lA{A/Pi+axA)
= Z (- l)*/^((tf*)*) • /^//>,+ M)

122
Chapter 7. Intersection Multiplicities
by Lemma A.2.7 and Example A.5.1. (Note that each Hk has support in
V(a2,..., ad), so Hk is an J-module for A=A/(a%,..., 0%), some m > 0 - in
fact m= 1 will do; Lemma A.2.7 may be applied over the one-dimensional
ring A.)
Example 7.1.3. With the notation of the preceding example, the following
are equivalent:
(i) i(Z,X-V;Y) =
(ii) J is generated by a regular sequence of length d.
(iii) at,..., ad is a regular sequence in A.
(iv) Hk(Ki, (a)) = 0 for all k > 0.
In particular,; = / if and only if A is Cohen-Macaulay. Algebraic proofs are
given by Serre D)IV. To prove directly that (i) implies (iii), the main point is
to show that the equality of cycles [C] = [N] implies that C = N, at least after
replacing V by an open subset which meets Z. (Indeed, if A is an Artin local
ring, and Q is a homogeneous ideal in A[Tt, ...,Td] whose localization at the
minimal prime is zero, then 2 = 0.)
Example 7.1.4. Let
K=A4, X=
\ — x3,x2 — x4), V= V(x1x3, xtx4, x2x3, x2x4) .
Then V is purely 2-dimensional, [V] = [V{xl,x2)] + [J/(x3,x4)]. The intersec-
intersection number of the origin in XV is 2, while l(A/J) = 3 (cf. Hartshorne E)
p. 428.)
Example 7.1.5. Let Y= A4, X= K(x,, x4), and let Kc A4 be the image of
the finite morphism <p from A2 to A4 given by
(p{s, t) = {s\ s3t, st\ z4).
The origin P is a proper component of the intersection of Kby X.
(i) I(V) = (x,x4-x2X3,x2X3-xi, x2x2-x^, x2x4-x2x,).
(ii) [XC\V] = 5[P],l(A/J) = 5.
(iii) i(P,XV; Y) = 4.
(For (iii), note that p*[A2] = 4[K]. Apply Theorem 6.2(a) to the situation
V{s\ z4) -> A2
xnv -* v
i i
X -> Y .
giving that ^(r[A2]) = i <p*[A2] = AX¦ V. Since A2 is regular, Proposition
7.1 (b) gives
i(Q,X- A2; Y) = l(K[s, t]/(s\ t4)) = 16 .
with Q the origin in A2. Therefore 16 = 4i(P, X- V; Y), as required.) Note that
the kernel of multiplication by x4 on /4/Xj/l has length 1, which accounts for
the difference between (ii) and (iii).
7.1 Proper Intersections
123
Example 7.1.6. Without regularity assumptions, irreducible components Z
of X D V may have dimension smaller than dim V— codim(X, Y). The standard
example is Y = V{x1 x4 — x2 x3) c A4, X = J/(xj,x2), F= K(x3,x4).
Example 7.1.7 (Commutativity). If V-> Y is also a regular imbedding, then
by Theorem 6.4,
i{Z,XV;Y) = i{Z,V-X;Y).
Example 7.1.8 (Associativity). Let i:X-*Y factor into a composite
i':X->X',j:X'-> Y of regular imbeddings. Let W\,...,W'r be the irreducible
components of /"' (X1) which contain Z. If Z is a proper component of IV,
then Z is a proper component of the intersection of each W'h by X on X', each
W'k is a proper component of the intersection of V by X' on Y, and by
Theorem 6.5,
i(Z,X-V; Y)=
V; Y).
Example 7.1.9 (Projection formula). Let g : V -* V be a proper surjective
morphism of ^-dimensional varieties, and let Zx,...,Zr be the irreducible
components of g~l{Z). If each Z, and Z have dimension k-d, then by
Proposition 6.2 (a),
deg(K'/K)- i(Z,XV; Y) = ? deg(Zj/Z)-i(ZJtX-V; Y).
Example7.1.10. (a) Let D\,...,Dd be effective Cartier divisors in a A>
dimensional variety V. An irreducible component Z of P)A °f dimension
k — d is called a proper component, and the intersection multiplicity
i(Z,Dr...-Dd;V)
is defined to be the intersection multiplicity of Z in the intersection of V— A v
by D\ x...xDd:
DD, -> K
1 Is
Di x...xDd-> Vx... x K .
Equivalently (cf. Example 6.5.1), i(Z,Dx ¦... -Dd; V) is the coefficient of Z in
the intersection cycle ?>, ¦... ¦ Dd in /^(f) ?>,) (Definition 2.4.2). If A is the
local ring of Kalong Z, and a, is a local equation for ?>, in ,4, then
i(Z,Dr...Dd;V)= eA(au...,ad).
\{A is Cohen-Macaulay, then
For example, if d = k, and Z is a simple point on V, the intersection multi-
multiplicity is given by the length.
(b) Let Di,...,Dd be hypersurfaces in AdK defined by polynomials
fi,..., fd, and assume P = @,..., 0) is an isolated (i.e. proper) point of inter-

124
Chapter 7. Intersection Multiplicities
), ...,/„))
section of DD,. Then
i(P,Dt ¦... Dd;
If .?=€, one may replace formal power series by convergent power series in
the last formula. (Note that modules of finite length are not altered by comple-
completion, cf. Zariski-Samuel A) VIII.2.)
Example 7.1.11. Let Kbe an ^-dimensional variety, P a simple point on V,
n: V-* V the blow-up of V at P, E the exceptional divisor. For an effective
Cartier divisor D on V, let D be the blow-up of D along P, i.e. the proper
transform of D in V. If Du ... ,Dn are divisors such that DiD^E is finite,
then
i(P,Dr...-D.;V)=I[e,(Dd+ Z
QeE
• Dn; V) .
If n = 2, and ?>, and D2 meet properly at P, it follows that the intersection
multiplicity i(P,D\ ¦ D2;X) is the sum of the products of multiplicities of D\
and ?>2 at all infinitely near points, a result of M. Noether. (By Example 4.3.9,
n*Di = Di+ep(Di)E. Write out the product of the A, and push forward to V.)
Generalizations will be given in Example 12.4.8.
Example 7.1.12. The equivalence of a distinguished variety Z of the
minimal dimension k — d is always a positive multiple of [Z]. If dim Z > k — d,
the equivalence of Z may be represented by negative cycles. For example, if Y
is the blow-up of a surface at a simple point, and X=V= E is the exceptional
divisor, then Z=E is the only distinguished variety, and its equivalence is
-[P], Pa.point onE.
Example 7.1.13. Let ? be a vector bundle of rank r on a purely «-dimen-
sional scheme X, s a section of E, Z(s) the zero-scheme of s:
(*)
Z(s)
i
X
1°
where sE is the zero section. If Z is a proper component of Z(s), i.e.
dimZ=«-/-, then the intersection construction from (*) determines an inter-
intersection multiplicity ;(Z). If A is the local ring of A" at Z, the stalk of E at Z is a
free ,4-module with an induced section sA, which determines a Koszul complex
A*{sA) (Definition A.5). Then
Example 7.1.14. Let/:A"-> C be a morphism from a smooth ^-dimensional
variety to a smooth curve C. The tangent map df: Tx ->/* 7*c corresponds to
a section j of Tx®f*Q]c. If x 6 A" is an isolated zero of this section, the
intersection multiplicity of x in the intersection of s (X) by the zero section (as
in the preceding example) is called the multiplicity of x as a critical point of /,
and denoted fix(f). If / is given in local coordinates by a function
7.1 Proper Intersections
125
/(zi,...,zn), then
Hx(I) = /(fix,x/{df/dzu ..., df/dz.)) .
For a discussion of this multiplicity from an analytic and topological point of
view, see Milnor C) and Orlik A), cf. Example 14.1.5.
Example 7.1.15. Let /: J"-> Y be a proper surjective morphism of varie-
varieties. Let X' be a subvariety of F, X=f(X'); assume that dimA"=dimA", and
X' is an irreducible component of/ (X), and that X is regularly imbedded in
Y. Then the ramification index of/ at X' (Example 4.3.7) is given by an inter-
intersection multiplicity
This applies in particular if / is finite and X is a simple point of Y.
Example 7.1.16. Fractional intersection numbers on normal surfaces, (cf.
Mumford (l)II(b), Reeve B)). Let n:X-+ V be a resolution of a singular
point P on a surface V, n~l(P) = ?1LJ.-.U?P connected, as in Example 2.4.4.
For an irreducible curve A on V, there are unique rational numbers Au ..., 1,
so that if A is the proper transform of A on X,
for all /. Set A'=A + Xl,?,-6 Z,A"Q= Z,1"®ZQ. This extends to a homo-
morphism a -> a' from Z, Kto ZxX^, satisfying
(i) [?>]' = [n* D] for any Cartier divisor D on V.
(ii) If ^4 is positive and contains P, then all the 1, are positive.
(For (i), (n*D ¦ E,)x= (D ¦ nifE^v= 0. For (ii), with ?>,, Z as in Example 2.4.4,
let A'=A + 'YJ jijDj, with /i, minimal among the /ij, fii=Q. Then
0 = A'-D,* Zjfij(Dj¦ D,)X ^ MZjiDj¦ D,)x = ~MZ-D,)x ^ 0; the connected-
ness of E then implies that all fij are zero.)
For any two one-cycles A, B on V which meet only at P, set
(This is defined since \A'\C)\B'\ a E, which is complete.) This intersection
number is symmetric and bilinear; it is non-negative if either A or B is
positive, and positive if A and B are positive and pass through P. If A = [D] is
the Weil divisor of a Cartier divisor D, then
j(P,AB) = (DB)vel,.
This definition of multiplicity is independent of the resolution. (If q:X^X
blows up a point on X, q*A' is perpendicular to all exceptional components,
and (q*A'¦ q*B')x= (A'-B')x.)
If A" is a quadric cone with vertex P, and A and B are generating lines of
the cone, then j {P,AB) = \ 12.
Example 7.1.17. Let C be an irreducible curve on a scheme X, D an effec-
effective Cartier divisor on X, with C not contained in the support of D. Let

126
Chapter 7. Intersection Multiplicities
/: C—> X be a finite morphism which maps an irreducible curve C" birational-
ly onto C. Then
i(P,DC;X) = ? ord
(Use Theorem 6.2(a).) For example if C'= P1, f*D is given by a polynomial,
and the intersection multiplicities are given by multiplicities of roots of this
polynomial.
7.2 Criterion for Multiplicity One
Let Z be a proper component of the intersection of V by X on Y. Let
A = 0z.v, J the ideal in A generated by the ideal of X in Y, and let m be the
maximal ideal of A.
Proposition 7.2. Assume that Vis a variety. The following are equivalent:
(i) i(Z,XV;Y) = l.
(ii) A is a regular local ring, and J—m.
Recall that a d-dimensional local ring is regular if its maximal ideal has d
generators, which necessarily form a regular sequence (Lemma A.6.2). Since J
always has d generators, the regularity of A follows from the assertion that
J = m.
Proof. The implication (ii) => (i) is a special case of Proposition 7.1, since
l{A/m) = \. We prove (i) => (ii) by induction on d. The assertions are un-
unchanged if Kand Y are replaced by open subschemes which meet Z and /(Z)
respectively. Therefore, we may assume that Z is the only irreducible com-
component of W, that Y is affine, and X is defined in Y by a regular sequence in
the coordinate ring of Y.
If rf= 1, then X- V= [W]. The coefficient of Zin[fV] is lA(A/J), which can
be 1 only if 7= m.
Let d>\, and assume (i) => (ii) for smaller d. Assume first that A is a
normal domain. Let X' be the divisor on Y defined by the first of the equa-
equations defining X. Form the fibre diagram
Cartier divisor on V and
In particular W can have only one irreducible component, which contains Z,
and this component appears in the cycle [W] with coefficient 1. In other
4
X -7?
w->
4
X' ->
V
4
Y .
Since dim W= k-d, W'=?V, so W is a
X'-YV=[ W'\ By functoriality (Theorem 6.5),
Notes and References
127
words, a\A has only one minimal prime ideal p containing it, and
lAt(Ap/aiAp) = \.
Since A is a normal domain, p is the only prime ideal associated to atA
(Lemma A.8.1), so atA is ^-primary. Since a{Ap = pAp, it follows that a^A=p.
Therefore W is a variety, and A/a^A the local ring of Z on W. By induction,
the images of a2,..., ad in A/axA form a regular sequence generating mla\A,
so a\,..., ad form a regular sequence generating m.
Returning to the general case, it remains to show that A must be normal.
Let g: V-* V be the normalization of V in its function field. Let h be the
induced morphism from g~l(fV) to W. By Theorem 6.2(a), since g is proper
If;' [V] = ?J= j m,. [ZJ, this gives
Therefore r = m\= deg(Z\IZ) = 1. The local ring A' of Z, in K is the integral
closure of A in its field of fractions. The case of (i) => (ii) proved above applies
to V, so J generates the maximal ideal rri of A'; in particular, mA' = m'. Since
deg(Z,/Z) = l, the canonical map from Aim to A'/m' is an isomorphism.
Therefore A'=A + mA'; since A' is finite over A, Nakayama's lemma implies
that A=A', so A is normal, as required. D
Example 7.2.1. It is not enough to assume that V is a pure-dimensional
scheme in Proposition 7.2. For example, let Y= A2, X= v(y), V= V{xy, x2).
Then X and V meet properly at the origin, and the intersection multiplicity is
1, but the local ring of Kat the origin is not regular. However, if all associated
primes p in A have dimA/p = d, then the intersection multiplicity is one only if
A is regular and J is the maximal ideal. (Since X- V=X- [V] (Example 6.2.1),
A can have only one minimal prime p, and l(Ap) = 1; i.e. pp= 0; since elements
outside p are assumed to be non-zero divisors, p = 0, and Proposition 7.2
applies.)
Nagata has extended this to general local rings A whose completion is
unmixed (cf. Nagata BL0.6 and Huneke A)). Nagata has given an example
of a local Noetherian domain whose multiplicity is one without being regular
(cf. Nagata B) Appendix Al).
Notes and References
The problem of assigning a multiplicity to an isolated solution of n polynomial
equations in n variables can be traced back near the beginnings of algebraic
geometry, although clear statements did not appear until relatively recently.

128
Chapter 7. Intersection Multiplicities
Two points of view, which remain vital, can be found in the work of Newton
and his contemporaries:
A) The dynamic approach, where the multiplicity of a solution is the
number of solutions near the given solution when the equations are varied. For
example, a point of tangency of a line with a curve is a limit of intersections of
nearby secant lines.
B) The static approach, where the multiplicity is obtained without varying
the given equations. For n = 2, Newton and Leibnitz showed how to eliminate
one of the variables, obtaining a polynomial equation whose roots give the
abscissas where the equations have common solutions, the multiplicity
question is likewise reduced to the multiplicity of a root of a polynomial in
one variable.
In 1822, Poncelet A) made the dynamic point of view quite explicit with
his "principle of continuity". Rules of this type were given for calculating
intersection multiplicities, e.g. by Cayley A), Halphen A), Schubert A), and
Zeuthen C). A useful summary of this era is given by Zeuthen and Pieri A).
We will discuss these principles in Chap. 11.
Elimination theory and the calculation of resultants also received consider-
considerable attention; the names of Euler, Bezout, Cayley, Sylvester, Kronecker, and
Hilbert should at least be mentioned. This is discussed by Salmon B) and
B. Segre (8); cf. Example 8.4.13.
In 1915 Macaulay A) gave a static definition in terms of the length of a
ring modulo an ideal, and proved Bezout's theorem for n hypersurfaces in
P".
The intersection of more general varieties than hypersurfaces in rc-space
was taken up, from the dynamic point of view, by Severi, Van der Waerden,
and Weil in the 1930's. In 1928 Van der Waerden B), borrowing an example
from Macaulay (Example 7.1.5 above), showed that the naive definition using
length would not always work. Van der Waerden C) also pointed out in 1930
that the Poincare-Lefschetz intersection theory in topology includes a notion of
intersection multiplicity for complex varieties, since they can be triangulated.
Severi's treatments (cf. Severi G) for a summary) were almost entirely
geometric. Van der Waerden A), Weil B), and Barsotti A) developed algebraic
notions of specialization to make such geometric ideas rigorous, not relying on
geometric intuition, and valid over general ground fields.
Chevalley A), in 1945, gave an important new definition of intersection
multiplicity in terms of completions of the local rings; his theory was therefore
equally valid in the analytic or formal case. He also gave a criterion for multi-
multiplicity one, which includes that given in § 7.2. Samuel A) gave the first
definition valid for a general Noetherian local ring A. As in Examples 4.3.1 and
4.3.4, he defined a multiplicity eA(J) for an ideal J primary to the maximal
ideal. Samuel proved many basic properties for this multiplicity, including its
agreement with Chevalley's.
We can only mention a few of the very many subsequent treatises on multi-
multiplicities in general local rings. The books of Nagata B), Northcott B), and
Kunz A) may be consulted for this literature. Nagata proved that when J is
Notes and References
129
generated by d elements, d=A\mA, then eA(J) = l(A/J) precisely when A is
Cohen-Macaulay (cf. Example 7.1.3). Nagata A) also generalized Chevalley's
criterion of multiplicity one: if A is unmixed, then eA(J) = 1 if and only if A is
regular and J is maximal. Nagata also gave an example to show that the
criterion may fail for general local rings.
Lech A) proved a remarkable asymptotic formula:
eA(au...,ad) =
lim /{A/{a\\ ..., a'j))/!, ¦... ¦ td
m(/i) -»ao
which he used to prove the associativity formula for multiplicities (cf. Exam-
Example 7.1.8).
In 1957 Serre D) showed that eA(ax,..., ad) is the alternating sum of the
lengths of a Koszul complex (cf. Example 7.1.2); or an alternating sum of
lengths of Tor modules, this definition, unlike previous algebraic definitions
extends to intersections where neither factor is defined by a regular sequence.
Several authors, beginning with Kleiman E), have constructed other ideals J'
!in A so that eA(J) = l(A/J'); one such is worked out and applied to Bezout's
theorem by Vogel A). Teissier A), B) has given some interesting new multi-
multiplicity formulas.
The definition of the intersection multiplicity i(Z,XV) given in this
chapter is also a length — the length of the local ring of the normal cone
Cyf]XV at the component lying over Z. Since normal cones are constructed
from associated graded rings @Jm/Jm+\ it is not hard to see that this defini-
definition agrees with Samuel's (Example 7.1.1). This calculation of intersection
multiplicities occurs implicitly in Verdier E), and in Fulton-MacPherson A),
B). Basic properties of intersection multiplicities, in this geometric context,
follow from trie properties proved for more general intersections in Chap. 6;
other than the algebra in Appendix A, none of the previous multiplicity theory
is required. The proof of the criterion of multiplicity one in § 7.2 is new, to our
knowledge.
It should be emphasized that all of the above constructions of intersection
multiplicities, with the notable exception of Serre's Tor definition, are valid
only when one of the varieties being intersected is regularly imbedded in the
ambient space. Intersection multiplicities for arbitrary varieties on a non-
singular variety are defined by reduction to the diagonal, as discussed in the
next chapter.

Chapter 8. Intersections on Non-singular Varieties
Summary
If Y is a non-singular variety, the diagonal imbedding <5 of Y in YxY is a
regular imbedding. For x, y e A* Y, the product x ¦ y e A* Y is defined by the
formula
x ¦ y = 5*{x xy) .
Setting Af Y = An-P Y,n = dim Y, this product makes A* Y into a commutative,
graded, ring, with unit [ Y].
If/: X-*¦ Y is a morphism, with Y non-singular, the graph morphism jy
from A' to A' x Y is a regular imbedding. For x e A*X,y e A*Y, define
xyy=y*f(xxy) e A*X.
This product makes A*X into a graded module over ^4*7. If X is also non-
singular, setting
defines a homomorphism/* : A* Y -> ,4*Ar of graded rings.
Using the refined operation y) in place of y*, x yy has a canonical
refinement in AJf(\x\[) f~i{\y\)). In particular, if V and W are sub varieties of
a non-singular variety 7, the intersection class V- W is defined in ^4ra(FnW),
m = dim F + dim W — dim K Any m-dimensional irreducible component Z of
VOW has a coefficient in V-W, called the intersection multiplicity, and
denoted i(Z, V- W; Y). The expected properties of these intersection products
and multiplicities follow readily from the general properties proved in
Chaps. 6 and 7.
Bezout's theorem, in its simplest form, states that A*(P")^1[h]/(h"+'),
where h is the class of a hyperplane. A deeper analysis of intersections on
projective space will be given in Chap. 12.
8.1 Refined Intersections
A variety Y will be called non-singular if it is smooth over the given ground
field. For our purposes, the important point (Appendix B.7.3) is that the
8.1 Refined Intersections
diagonal imbedding
131
5:Y-> Yx Y
is a regular imbedding of codimension n, n = dim Y.
The (global) intersection product is the composite
where <5* is the Gysin homomorphism (§ 6.2). We write x -y = 5*(x xy) for
x,yeA*(Y).
More generally, if A' is a scheme, Y a non-singular variety, /: X -* Y a
morphism, then the graph morphism
yf:X^Xx Y,
y/(P)=(P,f(P)), is a regular imbedding of codimension n, n = dim(Y).
Define a cap product
denoted
y ® x '->/*(>') n x ,
by the formula/*(j/) n x = y*(x xy). When/ is the identity on Y, this is the
previous product. This product is also denoted by x -fy. If X is also non-
singular, we write/*>> for/*y n [X].
By using the refined Gysin homomorphisms <5! and y} in place of <5* and y*,
these products can also be refined. If x and y are cycles on a non-singular
\ \ j
variety Y, with supports
square
and \y |, then x e A* jx |, y e A* \y |. Form the fibre
i
YxY
i
Y
We have 6'{x x y)e Aif{\x\(]\y\). This product, also denoted x-y, maps to
the corresponding global product in A*(Y). If x = [V\, y= [tV], for V, W pure
dimensional subschemes of Y of dimensions k, I, we write the refined product
V-W=[V]-[W]eAk+l_n{VOW).
Recalling the procedure of Chap. 6.1, this product V- W is constructed as
follows. The normal bundle to the diagonal imbedding of Y in Yx Y is the
tangent bundle Tr to Y. Let T be the restriction of Tr to VOW, s the zero
section of T. The normal cone C = CV^W(Vx W) is a (k + ()-dimensional
subscheme of T, and
V- W=s*[C\,
i.e., VWis the intersection of the cycle of C with the zero section of T.
Similarly, if/: X -> Y, Y non-singular, x a cycle on X, y a cycle on Y, the
cap product f*y n x has a canonical refinement in ^4^A jc | D/-1(ly|))> which
we denote by x-fy:
x-fy = y<f(xxy)eAlf(\x\C)f->{\y\)).
These products have the following common generalization.

132
Chapter 8. Intersections on Non-singular Varieties
Definition 8.1.1. Let f.X-* Y be a morphism, with Y non-singular of
dimension n. Let px '¦ X' -> X, pr: Y' -* Y be morphisms of schemes X', Y' to
Xaxid Yrespectively, and let x e AkX',y e A/ Y'. Form the fibre square
Define
4
= y}(x xy)
Xx Y .
" *rYr),
where x xy eAk+t{X'x Y') is the exterior product (§1.10) and y} is the
refined Gysin homomorphism (§ 6.2). When X' = X, Y' = Y, these are the
global products; when A'' = | jc j, Y' = \y\, the preceding refinements.
The following proposition proves the expected formal properties of these
refined products. In this proposition, it is assumed that each named variety
X, Y, Z, Yh comes equipped with a morphism px: X' -* X, p Y: Y' -> Y, etc.
and a class x e A* X', y e A* Y', etc.
Proposition 8.1.1. (a) (Associativity). Let X
singular. Then
Z,
Z rcorc-
inA*(X'xYY'xzZ').
(b) (Commutativity).
: X ~* Yt, Y-, non-singular, i = 1, 2.
(c) (Projection formula). Let X -> Y ^* Z, with Z non-singular. Let
f':X'-* Y' be a proper morphism such that prf = / px; let f" =/' x z\z. be the
base extension. Then
f*{x-gfz)=fi{x)-gz
inA*(Y'xzZ').
(d) (Compatibility). Let f: X ~* Y, with Y non-singular and let g : V -* Y'
be a regular imbedding. Then
g'(xyy) = xyg'y
in A* (X' xyV).
Proof. For (a) consider the fibre square
Xx Y
lx*y.
lyi*U
Xx YxZ.
The canonical map from X' x Y' x Z' to X x Y x Z induces a fibre cube lying
over this square. Then
yg)'(x xyxz)
\z)'(\xx yg)'(xxyxz)
8.1 Refined Intersections
133
by two applications of Theorem 6.2 (c). Now by Theorem 6.4,
G/X lz)!(l;rx yg)'(x xyx z) =(l^x yg)''{yf x \z)'(xxyxz)
= 7gA(xyy)xz)=(xyy)-gfz,
using Theorem 6.2 (c) again.
Similarly (b) follows by applying the commutativity theorem (§ 6.4) to the
fibre square
X ^-* XxY2
M J.WlXll-2
Y,xX ,-^j Y.xXx Y2
and the class yt x x x y2 in A* (Y\ x X' x Y'2) .
For (c), apply Theorem 6.2 (a) to the diagram
Y'xzZ'-» Y'xZ'
4 4
Y -j YxZ.
This gives the formula/; (x) -g z =fi' (y'g(x x z)). From the fibre square
X —>X x Z
a i/xi,
Y~irYxZ '
Theorem 6.2 (c) gives
yg(x x z) = ygf(x x z) = x -^z ,
which concludes the proof of (c).
For (d), apply Theorem 6.4 to the diagram
X'xYV -+X'x V -* V
4 4
xYY'-+X'x Y'
4 4
X y* XxY
Y'
U
The following corollary follows from (d).
Corollary 8.1.1. If Y is non-singular, and j: V-* Y is a regular imbedding,
and x is a cycle on Y, then
x-[V}=f(x)
inA,(\x\0V). D
Corollary 8.1.2. ///: X -> Y, with X and Y non-singular, and /)<= A'x Y is
the graph off, then for cycles x on X,y on Y,
x-fy = {xxy)-[rf]

134
Chapter 8. Intersections on Non-singular Varieties
in A^(\x\)(]f '(lyl)). In particular, for cycles x, y on a non-singular Y, xy
is the intersection product of xy. y with the diagonal Ar on Y y Y.
Proof. Apply Corollary 8.1.1 with j = yf the imbedding of X= Ff in
XxY. ?
Corollary 8.1.3. Letf:X—> Y, Y non-singular, x a cycle on X. Then
Xy[Y] = X.
Proof By (c), one may assume x = [X], X a variety. Then x y[ Y] =
y* [Xx Y] = [X] since y/ is a regular imbedding. ?
Definition 8.1.2. Letf-.X-* Ybe a morphism from a purely m-dimensional
scheme I to a non-singular n-dimensional variety Y. For any morphism
g : Y" -* Y, define a refined Gysin homomorphism
f :A
with X' = X xYY', by the formula
Ak+m.n X'
Proposition 8.1.2. (a) If f is also flat, then f{y)—f'*{y), where f is the
induced morphism from X' to Y'.
(b) If f is also a l.c.i. morphism, then '/' agrees with the homomorphism
constructed in § 6.6.
Proof, (a) follows from Proposition 6.5 (a). For (b), if i": X -> P, p : P -> Y
factors/, with p smooth, i a regular imbedding, then we have a factorization
> a x Y -» /
of/, and the conclusion follows from Proposition 6.5 (b). ?
The functoriality of these refined Gysin homomorphisms follows from
Proposition 8.1.1 (a), the projection formula from Proposition 8.1.1 (c), and the
commutativity of these Gysin homomorphisms with l.c.i. Gysin homo-
homomorphisms from Proposition 8.1.1 (d). Similarly all other formal properties
of refined Gysin homomorphisms given for l.c.i. morphisms in Chap. 6 are
valid for morphisms to non-singular varieties. These results will all be
subsumed in Chap. 17, so we do not write them out here.
Example 8.1.1. Let V be a closed subscheme of a non-singular variety Y
such that the imbedding i of Fin Y is a regular imbedding. Then
[V]-y=i*i*(y) eA*Y
for all y e A* (Y). (Use Corollary 8.1.1.)
Example 8.1.2. Both classes in part (a) of Proposition 8.1.1 (a) are equal to
¦x '(f.gf) (>>xz)' as we" as t0 (x'g/z)'/y- (The morphism (fgf) from X to
XxYxZ is the composite of morphisms given in the proof of (a); use
Theorem 6.5.)
Both classes in (b) are equal to x -y^ {yx x y2).
8.1 Refined Intersections
135
Example 8.1.3. If in Proposition 8.1.1 (d) one assumes that g is flat instead
of a regular imbedding, then
where g': X' x Y V -* X' x r Y' is induced by g.
On the other hand, if g is assumed to be proper, and v e A* V, then
g*(xyv) = xyg*(v) .
Formula (d) is also valid for g : V ~* ran arbitrary l.c.i. morphism as in
§6.6.
Example8.1.4. If/: Xt -* y, are morphisms, with Y, non-singular, / = 1, ...,r,
then
(x, x ... x xr) •(/¦,x...x/,)(>'i x...xyr)=(xl -Ayt) x...x(xr -fryr)
in A* ((X\ xY, Y\) x...x(X'rxYr Y'r)). (Use Example 6.5.2.)
Example8.1.5. Let Y be non-singular, Vu..., V, regularly imbedded
subschemes. Then the intersection product of Vx x... x V, by the diagonal
A = Y in Yx...xY is the same as the intersection product of A by
Vx x ... x V,. (Use Theorem 6.4.)
Example 8.1.6. Let/: X -> Y, Y non-singular, and let ? be a vector bundle
on Y. Then, for all /,
xy(ci(E)ny)=(cl(f*E)nx)-fy
inA*(X' xrY'). (Use Proposition 6.3.)
Example 8.1.7. Projection formula. Let /: X -* Y be a proper morphism of
non-singular varieties, and let x (resp. y) be a cycle on X (resp. Y). Then
/* (* yy) =/* M ' y
in A^{f(\x\H\y\), where /' is the induced map from | jc| f\f~1(\y I) to
/(l*l)n \y\. In particular
/•(/'W^) = r/,W
in A* (Y). (Use Proposition 8.1.1 (c).)
Example 8.1.8. If/: X -> Y is a morphism, with Y non-singular, there is a
fibre square
X^- XxY
with 5 and jy regular imbeddings of the same codimension. With x and y cycles
as in Definition 8.1, it follows (Theorem 6.2 (c)) that
x yy = y}(x x y) = 5' (x x y) .

136
Chapter 8. Intersections on Non-singular Varieties
In this sense all the intersections of this section are intersections with a
diagonal.
Example 8.1.9. Let Y be non-singular, and let <5, be the r-fold diagonal
imbedding of Yin Yx ... x Y. for cyclesyx,...,yr on Y, define
y^ ... ¦yr=5\(yxx...xyr)
y\- ¦¦¦¦y, = y\-{yr ¦¦¦¦y,)-
(See the proof of Proposition 8.1.1 (a).)
Example 8.1.10. Let I be a non-singular closed subvariety of a non-
singular variety, / the inclusion of X in Y. Then for any cycles y,, y2 on Y,
in A^XO \yt\ 0 |y2l)- (The intersection products in the first two formulas are
taken on Y, the third on X.) In particular, if X is a hypersurface on Y, and Vx,
V2 are subvarieties of Y not contained in X, then
in A^(X0 Vx 0 V2), the first intersection taken on Yy the second on X.
Example 8.1.11. Let V, W be subvarieties of a non-singular, n-dimensional
variety Y. If the diagonal imbedding of the intersection scheme VO W into
V x W is a regular imbedding of codimension n, then
V-W = [V][W] = [V0W]
in A^VOW). This equation holds if the imbedding of F fW in F x W is regular
of codimension n on an open set of each component of V 0 W (cf. Remark 6.2.2).
This is valid in particular if V and W are non-singular varieties meeting transver-
sally at generic points of VOW.
Example 8.1.12. If X is an ^-dimensional non-singular variety, and
A c X x X is the diagonal then (Corollary 8.1.1 and Corollary 6.3)
A ¦A=cn(Tx)n[X].
In particular, the degree of A-A is the topological Euler characteristic f cn(Tx).
Un=l,\A-A = 2-2g(d. Example 3.2.13). x
Example 8.1.13. Consider a fibre square
X7Y
with Y and Y' non-singular, and g a closed imbedding of codimension d and
normal bundle N. Then for x e A* X', y e A,, Y',
h* (x) yg*(y) = hm(cd(J'*N) n (x -ry))
in A* X. (Use Theorem 6.3.) This formula corrects Lemma 2.2D) of Fulton B).
8.2 Intersection Multiplicities
8.2 Intersection Multiplicities
137
Let Y be an ^-dimensional non-singular variety. Let V and W be closed
subschemes of Y of pure dimensions k and /. From the fibre square
VOW -> Vx W
i i
AY=Y-t YxY
and Lemma 7.1, it follows that every irreducible component Z of VOW has
dimension at least k + I— n. One says that Z is a proper component of the
intersection of Fand W, or that Fand W meet properly at Z, if
dim Z = k + I— n .
If Z is proper, the coefficient of Z in the intersection class V ¦ We Ak+l_n(V f] W)
is called the intersection multiplicity of Z in V- W, and denoted i(Z, V ¦ W\Y).
In other words,
i(Z, V-W;Y) = i(Z, A r ¦ (Fx W); Yx Y),
W),
where the right side is defined in § 7.1. Setting T = Tr\v^w, and C = Cvr] W(V
i(Z, VW; Y) is the coefficient of T\z in the cycle [C].
If Y° is an open subscheme of Y which meets Z, and V°=V0Y°,
W° = W0Y°,Z° = Z0Y°, then by Theorem 6.2 (c)
i(Z, V-W;Y) = i{Z°, V°-W°;Y°).
If Y is a singular variety, but Z is not contained in the singular locus of Y, the
preceding formula, with Y° the non-singular part of Y, defines i(Z, V- W; Y).
If every component of VOW is proper, the intersection class V-We
Ak+l_n(V0 W) is a well-defined cycle
V- W=YJi(Z, V- W\ Y)[Z].
z
Proposition 8.2. Assume Z is a proper component of VOW. Then
(a) lgi(Z, V-(]
(b) If the local ring of V x W along Z is Cohen-Macaulay, then
i{Z,V-W;Y)=Wz<V[]W).
(c) If V and W are varieties, then i(Z, V-W; Y)= 1 if and only if the maximal
ideal of (9Z r is the sum of the prime ideals of V and W; in this case (9Z v and (9ZW
are regular.
Proof, (a) and (b) follow from Proposition 7.1. By Proposition 7.2 (and
Example 7.2.1 if Fx W is not a variety), the intersection number is 1 precisely
when there is an open subset U of Yx Y, meeting Z, such that the diagonal A Y
intersects U0(Vx W) scheme-theoretically in the (reduced) variety Z. Since
Ar0{VxW) is scheme-theoretically isomorphic to VOW, this implies the
equivalence stated in (c). To see that (9ZV and Ozw are regular, let A = Ozr,

138
Chapter 8. Intersections on Non-singular Varieties
and let p, q and m be the prime ideals of V, W and Z in A, and let K = Aim.
From the exact sequence
0 -> (p + m2)/m2 -> m/m2 -> m/(p + m2) -* 0
and the identification of m/(p + m2) with MyiZ/(My:ZJ, it follows that
dim*((p + m2)/m2) ^ dim (Y) - dim (V) ,
with equality if and only if^V.z is regular (Lemma A.6.2); similarly for W.
Since p + q = m,
m2)/m2 =mlm1.
Since (dim(r)-dim(K))+(dim(y)-dim(W)) = dim(r)-dim(Z), the
spaces on the left must have the maximum dimensions, and the conclusion
follows. ?
Remark 8.2. If the ground field is algebraically closed, and Z is a (closed)
point, the last displayed equation decomposes the cotangent space of Y at Z
into a direct sum of the cotangent spaces of V and W at Z. The intersection
number is one precisely when Fand W are non-singular and meet transversally
at Z When dim (Z) > 0, the condition says that F and W are generically non-
singular along Z, and generically meet transversally along Z.
Example 8.2.1. If Vx,..., Fr are pure-dimensional closed subschemes of a
non-singular variety Y, an irreducible component Z of D Vt is a proper
component if dim(Z) = ? dim(F,)-(r- l)dim(y). The intersection multi-
multiplicity
is the coefficient of Z in the class F, •... ¦ Fr = 5\{ F, x ... x Fr), where <5 is the
/¦-fold diagonal imbedding of y in Yx...xY. Proposition 8.2 is valid for r
factors.
Example8.2.2. Let Vx,..., Vn be hypersurfaces in an ^-dimensional variety
Y. Assume: P is a (closed) point of Y, rational over the ground field; P is a
non-singular point of Y and of each VL; P is a component of D Ff; all of the F;
are tangent to each other at P. Then
i{P,Vx
F,; Y) i= 2"
(Use Example 8.1.10.)
Example 8.2.3 (cf. Samuel B)). Let K be a field of characteristic p * 0,
with an element a in K with no pth root in K. Let 7 be the affine plane over K
with coordinates x, y, and let F= F(>>), W= F(/ - xp + a), Z = F(>>, x' - a).
Then
(a) Z is a proper component of the intersection of F with W, and
i(Z,V-W\ Y)=\,
(b) If the ground field is extended to a field containing a />th root of a, the
intersection still has one irreducible component, but the intersection number
becomes p. (Note that 0Z w is regular, but W is not smooth over Spec (AT)
at Z)
8.2 Intersection Multiplicities
139
Example 8.2.4. If Y, V, W, Z are as at the beginning of this section, and the
imbedding of Fin Y is a regular imbedding, then the intersection multiplicity
i(Z,VfV;Y) defined in this section agrees with that defined in §7.1.
(Use Corollary 8.1.1, with x = [W].) Similarly if F,,...,Fr are regularly
imbedded subschemes of a non-singular Y, and Z is a proper component of
DFj, then the intersection number i(Z,Vl-...-Vr; Y) obtained by intersect-
intersecting F( x... x F by the diagonal, is the same as that obtained by intersecting
the diagonal by F, x ... x Vr (cf. Example 8.1.5). In particular, if F,,..., V, are
hypersurfaces in Y, the intersection number defined here agrees with that
defined in Example 7.1.10.
Example 8.2.5. Let f:Y'-*Ybe a finite surjective morphism of non-
singular varieties. Let V, W be irreducible subvarieties of Y', V=f(V),
W = f(W). Let Z be an irreducible component of VOW, and assume
there is only one irreducible component Z' of f'l(Z) which is contained in
either V or W. Assume that Z is a proper component of VOW, and that
/ is etale at the generic point of Z'. Then V and W meet properly at Z', and
deg(F'/F) •
, V-W; Y) = deg(Z'/Z) • i(Z', V'-W; Y').
(The assumptions imply that (/ x /)"' (Ar)f](V x W) is equal to ArC\{V x W)
in a neighborhood of Z'. Therefore i(Z', V ¦ W'\ Y') is the coefficient of Z' in
the class <5!(F'x W) where <5 is the diagonal imbedding of AY in YxY (cf.
Theorem 6.2(c)). Then apply Theorem 6.2(a) to the diagram
V Xy W -*
1
VOW -
1
F'x
1
Fx
i
W
W
Ay -f YXY)
Example 8.2.6. Let Y be a subvariety of PN, V, W subvarieties of Y, Z a
proper component of VO W which is not in the singular locus of Y. Then for
a generic projection /: Y -* P", with n = dim Y,
i(Z, V- W; Y) = i(f(Z),f(V) -f(W); P").
(The hypotheses of Example 8.2.5 are satisfied by a generic projection.) This
was one of Seven's methods for reducing intersection multiplicities on general
varieties to intersections on projective space (Severi (9) p. 203). A similar
construction was used by Chevalley A) in his algebraic definition of intersec-
intersection multiplicities.
Example 8.2.7. The intersection number is given by the length as in (b)
whenever Fand W are Cohen-Macaulay schemes, i.e., all their local rings are
Cohen-Macaulay. Indeed, the Cartesian product Fx W is then Cohen-Macaulay
([EGA]IV.6.7.3).

140
Chapter 8. Intersections on Non-singular Varieties
8.3 Intersection Ring
For an n-dimensional, non-singular variety Y, set
With this indexing by codimension, the product x® y -* x ¦ y of §8.1 reads
i.e., the degrees add. Similarly if/: X -* Y, the cap product^ ® x ->f*(y)nx
reads
2
If A'is also non-singular the pull-back y -*f*{y) preserves degrees:
f*:A'Y-> A?X.
Let I eAaY denote the class corresponding to [ Y] in An Y. Set A* Y = © A' Y.
We sometimes write A(Y) in place of A* Y or A*Y when the grading is
unimportant.
Proposition 8.3. (a) If Y is non-singular, the intersection product makes
A* Y into a commutative, associative ring with unit 1. The assignment
Y~*A*Y
is a contravariant functor from non-singular varieties to rings.
(b) Iff: X -»¦ Y is a morphism from a scheme X to a non-singular variety Y,
the cap product
">Aa-nX
makes A*X into an A* Y-module.
(c) Iff: X -* Y is a proper morphism of non-singular varieties, then
for all classes x on X, y on Y.
Proof. The associativity and commutativity of A* Y follow from Proposi-
Proposition 8.1.1 (a) and (b), with /=/=# the identity map of Y; that 1 is a unit
follows from Corollary 8.1.3. The functoriality of the pull-back follows from
Proposition 8.1.1 (a), with x = [X], >' = [!']. The projection formula (c) is a
special case of Proposition 8.1.1 (c), with Y=Z. Proposition 8.1.1 (a), with g
the identity on Y= Z, gives the formula
f*(yz) nx =f*z n (f*y n x)
for x e A* X, y, z e A* Y. This formula shows that A* X is an A* y-module;
setting x = [X], it shows that/* preserves products. ?
Remark8.3. If Yis non-singular, andyt,...,yr are cycles on Y, then
y,- ¦¦¦ ¦yr=S*(yix...xyr)
8.3 Intersection Ring
141
where <5r is the diagonal imbedding of 7 in Yx...x Y (r factors). This
follows by writing <5r = (<5r_i x 1 y) ° <52 and applying Theorem 6.5 (or see
Example 8.1.9).
The ring A*Y is often called the Chow ring of Y; the notation CH*{ Y) is
also common.
Example 8.3.1. Let/: X -* Y be a morphism, with Y non-singular, X pure
dimensional. If/is flat, or a regular imbedding, or a l.c.i. morphism, then the
cap product/*y n [X] agrees with the Gysin pull-back/*>> constructed in § 1.7,
§ 6.2, or § 6.6, respectively. (Use Proposition 8.1.2.)
Example 8.3.2. If X •? Y ^ Z, with Y, Z non-singular, then (Proposition
8.1.1(a))
(gf)*(z) n x =f*{g*z) n x
(ora.l\xeA*X,z?A*(Z).
Example8.3.3. Let P = P(ch(?,),..., cir(Er)) be a polynomial in Chern
classes of vector bundles on a non-singular variety Y. Suppose that
P n [Y] = 0 in A*(Y). Then for all morphisms f:X~* Y, and all x eA*X,
/*(/») n x = 0 in AMX. (By Example 8.1.6,/*(/») n x = x ¦/(/»n [Y]).) There is
therefore no loss of generality in identifying P with its image in AMY. This
aspect of Poincare duality will be formalized in Chap. 17. If P is homo-
homogeneous of isobaric degree m, we regard P e Am Y.
Example 8.3.4. Let ? be a vector bundle of rank r on a non-singular variety
Y, let X= P(E), with projection^ : X -* Y. Then
¦... + cr(E))
^r(l)). (This follows from
A*X ^
as graded rings. Here ? corresponds to c,
Theorem 3.3 (b).)
Y is an isomorphism of non-singular varieties,
Example 8.3.5. If f:X
with inverse/", then
In particular,/* preserves products. (By the projection formula,/*(/*)/) = y.)
Example8.3.6. Let Y be non-singular, yteA*Y, U,.a Y open subschemes
such that y-, restricts to 0 in A*U-,, for i=\,...,r. Then >v ... ¦ y, e A* Y
restricts to 0 in A* (I/, U... U Ur). In particular, if the Ut cover Y, then yx ¦... ¦ yr = 0.
(Let Y; = Y - U:. By Proposition 1.8, y-t is represented by a cycle on Y{'. The
refined intersection product of § 8.1 then gives a class on D Y{ which represents
Example 8.3.7. Let X and Y be non-singular varieties. Then the exterior
A*X®A*Y±A*(XxY)
is a homomorphism of rings, preserving the grading. (Use Example 8.1.4.) If
Y= IP", this homomorphism is an isomorphism.

142
Chapter 8. Intersections on Non-singular Varieties
Example 8.3.8. Let A"=P'xP', /: X -* P3 the Segre imbedding. The
induced map from /4'P3 = Z to A](X) = Z®Z takes 1 to A, 1). For any
surface H in P3 of degree d, i*[H] = (d, d). Therefore no irreducible curve C on
X, of bidegree (d, e) with d 4= e, can be — even set-theoretically — the inter-
intersection of A" with any surface in P3.
Example 8.3.9. Let
9i if
X-f Y
be a blow-up diagram as in § 6.7, with Y, X, and therefore Y, X non-singular.
The ring structure on A{ Y) is determined by the following rules:
(i)f*yf*y'=f*(yy')-
(ii) J*W •./*(*')=7*(
Example 8.3.10. Let Y be a non-singular surface, and let Y
blow-up of Y at r points. Then
At Y=At y©tz[?f]
Y be the
with E, the exceptional divisors, and AaY=AQY. The product is given by:
[E,] ¦ [E,] =-[/»,], P,eE,; [E,] • [?,] = 0 for i+j, f*[D] • [?,] = 0, f*[D] f*[D>]
=f*([D][D']).
Example 8.3.11. Let V be a complete surface, n: X -* Fa resolution of
singularities of V. Assume that n~* (P) is connected for all Pe V (V normal,
for example). For any irreducible curve A on V there is a unique' one-cycle A*
supported on the exceptional locus of X, with rational coefficients, such that if
A is the proper transform of A, then
for all irreducible components ?, of the exceptional locus (Example 7.1.16). Set
A' = A + A*. There is a unique homomorphism, denoted a -> a', from At Fto
At(X)q, which takes [A] to [A'], for irreducible curves A. (Use Example
7.1.16(i).). This determines a product
by a • fi = n*(a' ¦ ft), which is symmetric, bilinear, and independent of choice
of X. For any Cartier divisor D on X, [D] ¦ /? is the image of the integral class
D ¦ $ (defined in §2.3) in A0V.
Example 8.3.12. If X = Y/G is a quotient variety of a non-singular variety
Y by a finite group G of automorphisms of Y, then A*Xq may also be made
into a ring. Indeed, in this case one has an isomorphism (Example 1.7.6)
8.3 Intersection Ring
143
so A^ Xq is the ring of G-invariants of A(Y)q. In fact, if V, W are subvarieties
of X, one may construct a refined intersection class V-W in An{VC\ W
m = dim V + dim W — dim X; with the notation of Example 1.7.6,
where r\ is the projection from n~' (Vf] W)to V D W. In particular, for any m-di-
mensional component Z of VOW, one has a rational intersection number
i(Z,VW;X), the coefficient of Z in the class K-W (cf. Matsusaka B),
Briney A)). Note that the product on X is determined so that n*(a - b) =
7t*(a) • n*{b), and 7t*(Gt*a) • c) = a • 7t*(c), for cycles a, b on X, c on 7. For a
proof that these definitions are independent of the presentation of I as a
quotient variety, see Examples 17.4.10 and 16.1.13.
Mumford G) has constructed a ring structure on ^4*^ when X is the
moduli space of stable curves of genus g, in characteristic zero. He uses the
fact that such X is locally (in the etale topology) a quotient of a non-singular
variety by a finite group, and X is globally a quotient of a Cohen-Macaulay
variety by a finite group, which dominates the local charts.
Example 8.3.13 (cf. Fulton B) § 3). If Xis a quasi-projective scheme, one may
define a graded ring A*X by
A*X=\unA*Y
where the limit is over all pairs (Y,f) with Y a non-singular quasi-projective
scheme,/a morphism from A* to Y. This is a contravariant functor from quasi-
projective schemes to graded rings. There are (Example 8.3.2) cap products
APX<E> AqX A Aq_pX, with a projection formula /,(/'xoj') = xnf^x' for a
proper morphism f:X'-*X.
Any vector bundle E on X has Chern classes c,(?) in A'X, satisfying the
formal properties of § 3.2, such that for any/: X' -* X, x e A^X',f*ci(E) nx'
is the class c,(/*?) n x' defined in § 3.2. (The essential points for this are the
following facts (loc. cit. § 3.2): (i) There is a morphism /: X -* Y, Y non-
singular, and a vector bundle E on Y, with/*? s E; (ii) If/': X -> Y', E' is
another, there is a non-singular Z, g : X -* Z, h: Z -> y, /j' : Z -+ y with
hg=f, h' g=f, and h* E = h'* E'; (iii) Any exact sequence of vector bundles on
A' pulls back from an exact sequence on some non-singular Y.)
More generally, for any scheme X which admits a closed imbedding in a
non-singular scheme, one may define A*X to be lim A* Y, the limit over all
X -* Y, Y non-singular. One has the same properties as in the quasi-projective
case. For (i), one uses Lemma 18.2.
Another "cohomology theory" to pair with A* is discussed in Chap. 17.
Example 8.3.14. Ruled varieties (cf. B. Levi A)). Let Y be a non-singular
projective curve, let ? be a vector bundle of rank d on Y, and let X = P{E),
p:X->Y the projection. Let V be a vector space of dimension m + \,
P(V)^Wm, and let/: X -> P{V) be a morphism, with/V,,(l) =^?A) ® p*L
for some line bundle L on Y; the fibres A^ of p are imbedded as linear
subspaces of P(V). Set C = c,(^?(l)), A = c,(/V,/(l)). For any subvariety W

144 Chapter 8. Intersections on Non-singular Varieties
of A'of dimension wset
n{W) = \ V n [W] = deg(W/f(W)) ¦ deg(/(W)).
x
(In characteristic zero, deg (W/f( W)) is the number of rulings, i.e., fibres Xy of
p, which pass through a general point of/( W).) Set
k(w) = \c~l[x,\-\w\ = la* -p*[y}-[w].
X X
(k(W) is the degree of^HWasa subscheme of Xy = P^ \ for generic y e Y.)
Then
] = k{W) hd-w + p*(a) hd~w-[ ,
where a is a zero-cycle on Y of degree n (W) — k( W) • N, with N = n (X). (Use
Theorem 3.3 (b).) If Wu..., Wr are subvarieties of X with ? codim(W,) = d,
then
i-1 \y*i
where n,¦ = n(W,), /t, = /t @Q, # = n (JSQ.
Let Y be a non-singular curve of genus g in P (V) = Pm, and let X be the
ruled surface of tangent lines to Y, i.e. X = P(E), where E is the "principal
parts" bundle, constructed to make the following diagram have exact columns
and rows:
0 0
1 i
0->^-> E -> TY -0
II i 1
r(l) -> Tr-lr -> 0
1 4
In this case N = n(X) = 2g - 2 + 2 deg(Y). U m = 2, N is the degree of the
dual curve.
For more on ruled surfaces, see Beauville B) III.
8.4 Bezout's Theorem (Classical Version)
Intersections on projective space IP" are particularly .simple, as well as
important for applications.
We saw in § 3.3 that ^(P") = Z, with generator [Lk], Lk a ^-dimensional
linear subspace of P", k = 0, 1,...,«. For any &-cycle a on P", the degree of a,
deg (a), is defined to be the integer such that
a = deg(a)-[L*]
8.4 Bezout's Theorem (Classical Version)
145
in AkW. Equivalently, deg(oc) = |P.c1(E(l))* n a. This follows from the fact
that cl((P{l)f n [H] = 1 (Proposition 3.1 (a) (ii)).
Write A" F" = An_d P".
Proposition 8.4 (B6zout's Theorem). Let a.ieAdiW, i=\,...,r. If
d] + ... + dr ^ n, then
deg (a i ¦... -a,) = deg(ai) •... -deg(a,).
Proof. From the preceding considerations, the ring homomorphism from
Z[h]/{hn+]) to A*P" which takes h to c,(^(l)) n [P"] = [Ln~'] is an iso-
isomorphism, from which the proposition follows. (For a more direct proof, see
Example 6.2.6.) D
If Vt,..., Vr are pure-dimensional subschemes of P" which meet properly,
i.e. the irreducible components Z1,...,Zt of [)Vt all have codimension equal
to ?codim(F;, P"), then
t
7-1
where i(Zj, K, •... ¦ Vr\ P") is the intersection multiplicity (cf. Example8.2.1).
In this case Bezout's theorem says that
A)
I i(Zj, Vr...-K- P») ¦ deg(Z7) = n deg(F,).
7-1 i-1
For example, if H is a hypersurface, and V is a subvariety of P" not
contained in H, then
HV=[H[)V]=ZmJ[Z]]
™y-deg(Z7.) = deg(//)-deg(K).
with
B)
Another important case is the one considered originally by Bezout: Let
Hu..., Hn be hypersurfaces in P? with only a finite number of common
points. For each PeHit let 0P(OHi) = (Pp^/Qii,... ,hn) be the local ring of
P" at P modulo the ideal generated by local equations ht for Ht at P (i.e., the
local ring of the scheme 0 Ht at P). Then
n
C) 21 dimK(Ep(ni/i)) = Y\ deg(H,).
P 7-1
Indeed, in this case each Ht is Cohen-Macaulay, so by Proposition 8.2 (b)
For a (closed) point P, degP = [R(P): K], and
dimK (9P(f)Hi)= /(<P,(nH,)) ¦ [R(P):K]
by Lemma A 1.3. Thus C) follows from A).
Example 8.4.1. Let Y be an ^-dimensional variety, smooth over a field K. If
K),..., Fr are pure-dimensional subschemes of Y which meet properly in a

146
Chapter 8. Intersections on Non-singular Varieties
finite set of points, then
P Y
if Hi ,...,//„ are hypersurfaces meeting properly, then
J
P Y
Example 8.4.2. (a) If s (resp. /) is the class of a hyperplane on P" (resp. P),
then (Example 8.3.7)
A*(P"x P) = Z[s, t]/(sn+\ r+l).
(b) If Hu..., Hn+m are hypersurfaces in P" x P, and //, has bidegree
(a,-, A,-) (i.e., [//,] = a,:• s + bt ¦ t), then
J[ff,]-...-[^+B] = Sfl/.----fl<.V •¦¦¦*/..
where the sum is over all permutations (/,,..., in,j\, ...,jm) of (\,..., n+m)
with i, < i2 < ... < in andy, < ... <jm.
(c) if A is the diagonal in P" x P", then
in ^"(P"xP"). (Write [A] = X a;^''""'. and intersect both sides with [L x M],
where L and M are linear spaces of complementary dimensions meeting
transversally at a point. Alternatively, the composite of the canonical maps
) c* pr* {Tp- (- 1))
corresponds to a section of pr* GV») ®^A, — 1) whose zero-scheme is A; the
top Chern class of this bundle is ? ¦s' '""'•)
Example 8.4.3. (a) Let <p: P" x Pm -* PN be the Segre imbedding, N =
nm + n + m. If w is the hyperplane class on P", then <p*u = s + t. The degree of
the image of q> is (" + m\
(b) If t/rP"-*P'" is the m-fold Veronese imbedding, N = (n + m] -1,
and s, u are hyperplane classes on P" and P", then y/*u = m ¦ s. If V is a
^-dimensional subvariety of P" of degree d, then y/( V) has degree d ¦ mk.
Example 8.4.4. Let Fbe a subvariety of P" x P of dimension k, so
m= E a./s'-'f-J;
the ay are the bidegrees of K Define a variety F' c PntItl as follows.
Geometrically,
V'={(XxQ:Xxi:...:Xxn:fxy0:...:fxym)eF"+'"+l\(X)x(y) e V,{.k:p) eP1}.
Algebraically, if / is the bihomogeneous ideal in K[X0,..., Xn, Ya,..., Ym]
defining V, then the ideal of V is generated by those elements in / which are
8.4 Bezout's Theorem (Classical Version)
147
homogeneous in all variables. Then V is a variety of dimension k + 1, and
(cf. Van der Waerden G)).
deg V = S au
Example 8.4.5. Let V, W be irreducible subvarieties of P" of dimensions
k, I respectively. Let J(V,W)c: P2n+1 be the ruled join of V and W, i.e. let
P" (resp. PJ) be the linear subspace of P2"+l where the last (resp. the first)
n+ 1 coordinates vanish; regard F<= Pf, Wa P2, and define J(V, W) be the
union of all lines in P2n+I from points of Fto points of W. In the notation of
the previous example, J(V, W)=(Vx W)'. If p, q are the ideals of Fand W,
then the ideal of J(V, W) is the ideal in K[X, Y] generated by all f(X),fe p
and g(Y), g e q.
Let L be the linear subspace of P2"+' defined by
L={(xQ:...:xn:y():...:yn)\xi = yi for O^iSn).
The imbedding i:P"-*P2"+1, which takes (x0: ...:xn) to (x0:... :xn: xQ:... :xn),
maps P" isomorphically onto L, and determines an isomorphism of schemes:
V[)W^L0J(V, W).
Moreover, if P?"+1 = P2n+1 - (P^ UP"), the canonical projection p from P20n+i
to P" x P" maps L isomorphically onto the diagonal A, and maps Lf]J{V, W)
to A f](V x W). This projection p is smooth - in fact an (A1 — {0})-bundle. The
claim is that
V-W=L-J(V,W)
in Ak+l_n(VC\ W). To prove this consider the diagram
,Pjj"+1 +-MV, W)
P"
P"xP"
Vx W
where J0(V, W) = J(V, W)C\F20"+i. By the definition of VW, and Proposi-
Proposition 6.5 (b),
V-W = d'[V x W] = iop'*[V x W] = i'0[J0(V, W)).
And to[Jo{V,W)] = t[J(V,W)] = L-J(V,W) by Theorem 6.2(b) and Corol-
Corollary 8.1.1.
In fact, by Example 6.5.4, the canonical decomposition of the intersection
class A -{Vx W) is the same as the canonical decomposition of the intersection
class L J(V, W). Note in particular that deg7(F, W) = deg(F) • deg(W), as
stated in Example 8.4.4.
A similar discussion is valid for the intersection of r varieties in P"; the
ruled join is a subvariety of Pr<"+1)"'. In summary, all intersection products on
projective space can be realized by an intersection of a subvariety by a linear
space (cf. Gaeta A)).

148
Chapter 8. Intersections on Non-singular Varieties
Example 8.4.6. Let Vl,..., Vr be subvarieties of P". Let Z, ,..., Z, be the
irreducible components of Vx f]... f] Vr. Then
In particular, the number of irreducible components of Vj is at most the
product of the degrees. (By the preceding example, one may assume r = 2 and
K, is a linear subspace. Write K, as an intersection of hyperplanes; by
induction one may assume K, is a hyperplane. Then either Vx => V2, or Vx ¦ V2
= X mi [Z,], from which the inequality is clear.) A refinement of this inequality
will be discussed in § 12.3. For an application, see Bass-Connell-Wright A)
p. 293.
A typical classical application of Bezout's theorem is a proof that an
irreducible subvariety X a P" of degree d, not contained in a hyperplane,
satisfies
dxmX + d^ n+l .
Indeed, taking generic linear sections, one is reduced to the case where A" is a
curve. There is a hyperplane through any n points, which contradicts Bezout's
theorem if n > d and the n points are on the curve.
Example8.4.7. Let A" be a projective scheme. Let V{,..., Vr be sub-
varieties of X, and let dt be the degree of V, with respect to some imbedding of
XmJPN.UVlf]...r\ Vr is finite, then
(Use Example 8.4.6.) Note that this upper bound depends only on the
(numerical) equivalence classes of the [ V].
Example 8.4.8. Let L be a linear subspace of P", V a subvariety of P", Z a
proper component of L f] V. If dim Z = k, then for a generic linear subspace
Afcpof codimension k, and a point PonZflM, and L = Lf] M, then P is a
proper component of L f] V and
i(Z,L- K;P") = i(P,L'- V; P").
(Choose M transversal to Z at P and apply associativity of intersection
products.) Together with Examples 8.4.5 and 8.2.6, this shows how intersection
multiplicities on arbitrary non-singular varieties are determined by intersec-
intersection multiplicities of varieties of complementary dimension in P", with one
factor a linear subspace.
Example 8.4.9. Let K,,..., V, be subvarieties of P" with
(a) The intersection f]rl=l V{ cannot be empty.
8.4 Bezout's Theorem (Classical Version)
149
(b) Assume that each V) has odd degree. Then Vlf]...f]Vr must contain
an m-dimensional variety of odd degree. (The intersection class Vl ¦... ¦ Vr is
represented by a cycle of odd degree on 0 V{.) In particular, for at least one
irreducible component Z of 0 V), the restriction of (9(\) to Z is not the square
of any line bundle.
Example 8.4.10 (cf. Fulton-MacPherson B)). Given V, W <= P", with
dim V + dim W = n, there is no way to assign integers to the irreducible
components of Vf]W so that the sum is the Bezout number, at least if one
requires the assignment to be preserved by automorphisms of P". For
example, let n = 4,
V= V(x]- X| x2(x2- 2x,), x4)
W= V(x\ - x2 x, (xi - 2x2), x3).
Then Vf] W is the union of two lines; the involution interchanging xl and
x2, and x3 and xA, takes V to W and interchanges the two lines; thus each
would have to be assigned the same number, but the Bezout number 9 is odd.
(In fact, the point of intersection of the two lines is a distinguished variety for
the intersection of Vx Wby A.)
Example 8.4.11. Let V be, & subvariety of P? of dimension d, with K in-
infinite. Then there is a linear space LcFof codimension d which meets V
properly, and teg(V) = Y.i(P,L-V;VK)-[R(P):K\.
p
(Choose, inductively, d hyperplanes H1,...,Hd so that Ht meets all compo-
components of K fl H, 0... 0 H; _: properly.)
Example 8.4.12. A Bertini theorem. Let X be an ^-dimensional subvariety
of Pm, over an algebraically closed field. Let 0 < k < m, and let G be the
Grassmannian of ^-planes in Pm (cf. § 14.7).
(a) There is a non-empty open set U c G such that for all L in U,
-m^0, or
dim(X(M) = n + k-m
= 0 it n + k-m <0. (Set
I={(x,L)eXxG\x € L).
The projection from / to X is smooth of relative dimension k(m — k), so
dim G) — dim (G) = n + k — m. The generic fibre of I -* G therefore has the
indicated dimension.)
(b) Let A'o be the smooth locus of X. For x e Xo, let Tx c Pm be the
projective n-plane tangent to X at x. If n + k ? m, there is a non-empty open
set U a G such that every L in U meets A'o transversally, i.e.
dim(Txf]L) = n + k-m
for all x e Xo f]L. (Set
J, = {(x, L) € Xo x G | x € L, dim (Tx 0 L) = n + k - m + /},
/= 1, 2,.... The projection from J; to Xo is smooth, and a dimension count
shows that dim Jt < dim G.)

150
Chapter 8. Intersections on Non-singular Varieties
(c) Set k = m — n. There is an open set U c G such that every L e U meets
X transversally in deg(X) points. (Apply (b) to XQ, (a) to the components of
X—Xo. More generally for X, Y subvarieties of Pm of complementary
dimension, X meets a(Y) transversally, in deg(X) deg(y) points, for generic
a € Aut (Pm) (Appendix B.9).
Example 8.4.13. Resultants. Fix positive integers dl,...,dn. The hyper-
surfaces of degree d, in P" are parametrized by P"\ mi = ('jj"nj—1. Let
xo,...,xn be homogeneous coordinates on P", t'^ homogeneous coordinates on
P', so that T-, = Yj t'(j) *w defines the universal hypersurface in P" x P'. Set
T=Pmix...xPm", and definedc P"x Tby
y=((x,(l,...,(")|J'j((U) = 0 for i=l,..., n}.
By projecting to P", one verifies that V is a smooth irreducible subvariety of
P" x T of codimension n. The fibre V, of T over t in T is the intersection of the
corresponding hypersurfaces. Set
L ={(x)€P"|xo = x1=O},
so T° is a non-empty open subvariety of T, ^° is open in ir, and closed in
(P" - L) x T°. Let p be the projection from P" - L to P1, p(x) = (xo:*i)- Let
Since "V° is proper over T°, the induced morphism / from V° to 5?° is proper, so
5?° is a closed subvariety of P1 x T°. In fact, / maps "V° birationally onto 01°, i.e.
Indeed, if T°° consists of those t e T° for which the corresponding hypersurfaces
meet in d^ ¦ ...¦ dn points with distinct projections to P\ / restricts to an isomor-
isomorphism over T°°. In particular 5?° is a hypersurface in P1 x T°. Let
R = R(xo,xutl,...,t")
be the equation, unique up to scalars, for the closure of 0P in P1 x T; R is
called the resultant. For any particular hypersurfaces F,, ..., Fn of degrees
d\,...,dn, the resultant R(FX,..., Fn) is obtained by substituting the cor-
corresponding coefficients /',...,/" into R. The resultant is characterized, up to
multiplication by polynomials in /',...,/", by being homogeneous of degree
df ... ¦ dn in xa,x\ (cf. below) and vanishing at points ((xo:xi), (/',..., t")), in
any algebraic closure of the ground field, where the F, (;', x) have common
zeros not on L. (For some of the many classical methods for calculating
resultants, see Salmon B) pp. 66-98, and Van der Waerden D)§XI. For a
modern discussion see Jouanolou D) and Lazard A).)
Theorem. Let V{ a P" be the hypersurface defined by equation F{, i = 1,..., n.
Assume f] V, 0 L = 0, and 0 Vx is finite. Then, for any Q e P1,
(•) ordQ(R(Fu...,Fn))= E i{P,V,-...Vn;W").
PrL
Notes and References
151
To prove this, let t: Spec(K) -¦ T be the point corresponding to F,,..., Fn.
Let Y° be the subscheme of (P" - L) x T° defined by F,(x, t'). Form the fibre
diagram:
C\Vt -> V° x...x V° -> P1
I i I
f° ->ffx...xf; -+FxT°-> T°
I
P"
¦ Spec(X)
1 x ••• x
i
P" X...X
By Theorem 6.4, and Theorem 6.2 (b), (c),
t'[r°\ = t' d'[-r° x... x r°] = &'¦ t'[r° x... x r°\
in A0(f] Vt). Let /' be the morphism from 0 V, to P1 induced by p. By Theorem
6.2 (a),
/¦'([Pi] • ••• • IK]) = /* t'[r°\ = t'U[r°) = t\0t°\.
Since R(Flt..., FJ 4= 0, t'\0t°\ is the cycle determined by the divisor of
R(Ft,..., Fn). The equality of these cycles in A0(f (D VJ) gives (*).
If one does not assume 0 Vt finite, but only that (~) Vi is contained in a
finite number of fibres of p, the same proof shows that ordg(i?(F1,..., FJ) is
the total contribution of the intersection class V[ ¦... ¦ Vn which is supported
on OViOp-'iQ).
Adding formula (*) over QeP1, for any V^,..., Vn which meet properly
(off/.) shows that R has degree dx ¦... ¦ dn in x0, xy.
Notes and References
The procedure of reduction to the diagonal, i.e., of intersecting two cycles on a
non-singular variety by intersecting their exterior product with the diagonal,
has played an important role in intersection theory. One may detect its
presence in the nineteenth century theory of correspondences (cf. Pieri A)).
Apparently Weil B) in 1946 was the first to use this principle in modern
geometry, although Lefschetz C) had made extensive use of cycles on product
manifolds.
Following Lefschetz's model in topology, and ideas of Severi in algebraic
geometry, the construction of an intersection ring A*Y for a non-singular
projective variety has usually proceeded in two separate steps: A) a theory of
intersection multiplicities was developed (see the notes to Chap. 7), so that
properly intersecting cycles had a well-defined intersection cycle; B) one
showed that two cycle classes have representatives which meet properly, and
that the resulting product is well defined up to rational equivalence. This is the
approach followed by Samuel C), Chow A), and Chevalley B). Earlier

152
Chapter 8. Intersections on Non-singular Varieties
descriptions of A* Y along these lines were made by Severi (9), Todd B), F),
Segre D), and Hodge and Pedoe A), among others.
In the present version, sketched in Fulton-MacPherson A), the intersection
product of two cycles on a non-singular variety is constructed directly, in one
step, as a well-defined rational equivalence class on the intersection of the
supports of the two cycles. In addition to its simplicity, a primary advantage of
this approach is that it leads to useful formulas for intersection classes in cases
where the intersection is not proper. There is also the benefit that the
construction works equally well on non-projective varieties; the construction of
the rational equivalence ring for regular algebraic schemes has also been
carried out using higher A^-theory (see Chap. 20).
In the projective case, for varieties V, W of complementary dimension,
Murre A) showed how to assign an intersection number to each connected
component of Vf] W, thus answering a question of Weil. His method, based
on the moving lemma, would apparently also lead to the class V- W in
At(V(~] W) that we have constructed in § 8.1, when the ambient variety is
projective.
The original theorem of Bezout A) concerned the number of solutions of n
polynomials in n variables; precedents can be found in the work of Jacques
Bernoulli, Euler, Braikenridge, and Maclaurin (cf. Berzolari B) and Zeuthen-
Pieri A)). Now its name is attached to a number of theorems concerning inter-
intersections of arbitrary of cycles on projective space — and often to more general
situations whenever an intersection ring of a variety is explicitly computed.
Poncelet's approach to Bezout's theorem was to deform the varieties to be
intersected until they are unions of linear spaces, using his principle of
conservation of number to know this didn't change the number of solutions.
This approach, suitably justified, is the one followed in this chapter; indeed,
once the intersection product is known to be well-defined on rational
equivalence classes, Bezout's theorem is evident.
The first modern algebraic — and perhaps the first fully complete —
treatment of Bezout's original theorem was given by Macaulay A). For the
relation to resultants, see Example 8.4.13. A discussion of Bezout's theorem
emphasizing its algebraic aspects is given by Vogel A); his methods also yield
the conclusion of Example 8.4.6.
For intersections on P", a further reduction beyond reduction to the
diagonal is possible; the intersection of V and W in P" is equivalent to the
intersection of the ruled join of V and W by a linear space in P2n+1
(Example 8.4.5). This construction was made by Gaeta A) to compare the
Chow point of a product cycle with Chow points of the factors. An algebraic
analogue has been used by Boda and Vogel A). The geometric construction
given in Example 8.4.5 we learned from Deligne; this construction has been
useful in the study of the topology of projective varieties (cf. Fulton-Lazarsfeld
A)). The result of Example 8.4.6 was discovered and proved with MacPherson
and Lazarsfeld, in answer to a question of Kleiman.
The intersection products x yy of § 8.1 generalize those of Serre D) V. C 7,
who defined such products and stated similar formal properties in the case of
proper intersections.
Chapter 9. Excess and Residual Intersections
Summary
If X c> Y is a regular imbedding, Kc 7 a subvariety, we have constructed
(§6.1) an intersection product XV in Am(Xf]V), where m = dim V —
codim (X, Y). If a closed subscheme Z of X H V is given, the basic problem of
residual intersections is to write X ¦ V as the sum of a class on Z and a class on
a "residual set" R. There is a canonical choice for the class on Z, namely
{c(N)ns(Z,V)}m
where TV is the restriction to Z of NXY, and s (Z, V) is the Segre class. Our
problem is therefore to compute this class on Z, and to construct and compute
a residual intersection class R in Am(R), for an appropriate closed set R such
that Z{JR = Xf]V, with
X- V= [c{N)r\s{Z, F)}m+R.
If m = 0, and R is a finite set, knowing XV and {c(N) n s(Z, V)}0 gives a
formula for the weighted number of points of R. This is the basis for
applications of the excess intersection formula to enumerative geometry.
In case Z is a (scheme-theoretic) connected component of X f] V, R is the
union of the other connected components; since At(XC\V) = At(Z)® At(R),
the above decomposition is part of the construction of Chap. 6. Computa-
Computations, applications, and a few of the many classical examples are considered in
§9.1.
The general case is considered in § 9.2. In the main theorem Z is assumed
to be a Cartier divisor on V; in this case there is a natural scheme structure on
the residual set, which can be used to construct R. If Z is arbitrary, one blows
up V along Z to reduce to the divisor case.
An important and typical application of the residual intersection theorem
is to the formula for the double point cycle class of a morphism, which is given
in § 9.3.
9.1 Equivalence of a Connected Component
Let Y be a scheme, A", <-> Y regularly imbedded subschemes, 1 S / ^ r, and V a
k-dimensional subvariety of Y. The intersection product
AV...-AV V

154
Chapter 9. Excess and Residual Intersections
is a class in Am( f] Xt f] V), m = dim V — Y!i= 1 codim (Xt, Y). It is constructed by
the procedure of §6.1, applied to the diagram
A'ix.T.xA', <-> Yx.r.x Y.
If Z is a connected component of f) ^, H K we write
(Xr...Xr- V)zeAm(Z)
for the part of X\ ¦... ¦ X, ¦ V supported on Z, and call it the equivalence of Z
for the intersection Xx ¦... ¦ Xr ¦ V.
Proposition 9.1.1 Let N, be the restriction ofNXl Y to Z. Then
(Xv...-Xr- K)z=|ric(JV,)ns(Z, V)\ .
IfZ is regularly imbedded in V, with normal bundle Nz V, then
(*,-... ¦*•,• V)z=\tlc(Ni)c(NzV)-^[Z]\ .
1,-1 \m
If V and Z are non-singular, then
(Xr ... ¦ Xr- V)z=\flc(N,) c(Tv\zy' c(Tz) n [
Proof. The first assertion follows from Proposition 6.1 (a) and the Whitney
sum formula (Theorem 3.2 (e)). The second follows from Proposition 4.1 (a).
The last uses the identification of Nz V with the quotient of tangent bundles
Tv\ z/Tz (Appendix B.7.2) and the Whitney sum formula. ?
The global intersection class Xx ¦...¦ XrV is always the sum of the
equivalences for the connected components of f) Xt f] V. The following case
suffices for many enumerative applications.
Proposition 9.1.2. Suppose m = 0, and f] Xt [) V consists of a connected
component Z together with a finite set S. Then the degree of Xx ¦ ...¦ XrV is
deg((Xl-...X,V)z)+ ZKP,Xl-...XrV;Y)[R(P):K],
PeS
where [R(P):K] denotes the degree of the residue field of P over the ground
field. ?
In enumerative applications, deg (Xt ¦... ¦ X, ¦ V) is known for global
reasons, e.g. Bezout's theorem. Knowing the equivalence of Z then predicts the
(weighted) number of residual points. Following the classical terminology,
when m = 0, we also call the degree of (Xx¦... ¦ Xr ¦ V)z the equivalence of Z in
the intersection product. If V= Y, we write (A^ • ... • X,)z in place of
(Xv...-Xr-V)z.
9.1 Equivalence of a Connected Component
155
Notation 9.1. For an arbitrary imbedding Z <= Y, and any k, s(Z, Y)k
denotes the ^-dimensional component of the total Segre class s(Z, Y). We
regard s(Z, Y)k as an element of AkZ, or of AkZ' for any closed subscheme Z'
of Y which contains Z.
Example 9.1.1. Let Xl,...,Xr be Cartier divisors in an r-dimensional
variety Y. Suppose a connected component Z of f] Xt is a curve. Then
T
t v . . v \Z X1 v . r. (*y v\ _i_c/ v\
If Z is a scheme-theoretic complete intersection of Cartier divisors Dx,..., Dr-\,
then r-1
s(Z, Y) = [Z]-ZDj-[Z].
Therefore '"'
/ y . • Y \Z — (Y -L. 4- X D D } ' D ' ' D
For example, if Y= P', and deg A', = nit deg D, = dh then the equivalence of Z
is
On the other hand, if Z and Y are non-singular, then
5(Z, Y) = [Z] + c, (Tz) n [Z] - c, G>|z) n [Z].
Therefore
(X,-...-
X,-Cl (Trjj -
, [Z].
For example, if Y = Pr, and deg A", = nh the equivalence of Z is
(ii)
where g is the genus of Z.
Note that the equivalence of Z need not be a multiple of the degree of Z.
Example 9.1.2. A non-singular curve of genus 1 and degree 5 in P3 can
never be the scheme-theoretic intersection of three surfaces in P3 (such a curve
cannot lie on a quadric, say by Example 15.2.2. But there are no solutions to
the equation
nx n2 n3 = 5 (n, + rc2 + n3 - 4)
in integers n, ^ 3.)
Since a linear system of degree 5 and dimension 3 on a curve of genus 1 is
not complete, such curves are not linearly normal. The existence of such a
curve which was not the intersection of three surfaces was posed by Peskine
and Szpiro A) Problem 7.3. Different examples have been given by Rao A).
Many other curves which are not the complete intersection of three
surfaces can be constructed by the same method. For any degree and genus,
equation (ii) of the preceding example severely restricts the possibilities for
the degrees of the three surfaces which cut it out.

156
Chapter 9. Excess and Residual Intersections
Example 9.1.3. The twisted cubic curve in P3 may be written as the inter-
intersection of three quadrics. If three surfaces have a twisted cubic curve as their
scheme-theoretic intersection, the surfaces must all be quadrics. (If the degrees
are n]t n2, n3, then
«, n2n3 = 3(n, + n2 + n3-4) + 2 .
The only solution to this equation in integers 1= 2 is n, = n2 =  = 2.)
Similarly, if n > 3, the rational normal curve in P" cannot be written as the
scheme-theoretic intersection of any n hypersurfaces. This should be con-
contrasted with the following facts, valid over an algebraically closed ground field:
(i) Any algebraic set in P" is the set-theoretic intersection of n hyper-
hypersurfaces (Eisenbud-Evans A)).
(ii) Any subscheme of P" which is locally a complete intersection is the
scheme-theoretic intersection of n + 1 hypersurfaces. (Choose d so large that
J~%0(d) is generated by its sections, where S is the ideal of the subscheme;
apply the construction of § 4.4 to this linear system, obtaining a morphism /
from P" to IP. If Hi,..., Hn+l are hyperplanes in IP so that f]Htr\ f(P") = 0,
then the corresponding hypersurfaces in the linear system cut out the
subscheme.)
Another general fact explains why the equivalence of a connected com-
component is studied mainly for zero-cycles:
(iii) If X\,..., X,, V are subvarieties of P", and m = dim V~Y, codim (X,)
> 0, then n^iH V is always connected (Fulton-Hansen A)).
Example9.1.4. (a) If three surfaces in P3, of degrees n,, n2, n3, contain a
line as a connected component, the equivalence of this line is nx + n2 + n3 — 2.
The three quadrics x0 x2 = x], X|X3 = x2, Xo*i = x2x3 meet in the line
x, = x2 = 0 and in 4 points outside this line.
(b) The equivalence of a non-singular rational quartic curve in the
intersection of four quadrics in P4 is 14, leaving two points of intersection
outside the curve.
Example 9.1.5. Let Xlt...,Xr be hypersurfaces in IP, nt = deg(Zf), and
assume a connected component of C\X, 's a non-singular surface Z. Then the
equivalence of Z in Xt ¦¦¦¦ Xr is
deg(c2) +(* - ,- 1) • deg(c) +(jV - "(^ ') +
where n = ?'= l n,, N = Y.i<j »,¦ "j, and c, = ct(Tz).
"
• deg(Z) ¦
Example 9.1.6 (cf. Severi B), James A), Semple and Roth A) p. 228). Let
X\,...,X, be divisors on a non-singular /--dimensional variety Y. Suppose a
non-singular curve Z is a set-theoretic connected component of f]Xt, but that
the multiplicity of Z on X-, is m,. Let n: Y -* Ybe the blow-up of Y along Z, E
the exceptional divisor, and write
9.1 Equivalence of a Connected Component 157
Assume that X[ f]... f] X'r f] E = 0. Set m = mx ¦... ¦ mr, m; = mjm{. Then
r
(•) (Xl-...-X,)z=Y.miXrZ+m(cl(Tz)-cl(TY\z)).
If Y = P', and «, = deg Xit g = genus (Z), then the equivalence of Z is
m,«,-m(r+ l)j ¦ deg(Z) + m-B-2g).
(Let r\ be the projection from E to Z. One may shrink Y so that f) X[ = 0. Then
0 = ^((Tt*^, - m, ?)•...• (n*Xr - mr?))
= (Xr...-Xr)z-fJmlXrs(Z, Y)i-ms(Z, Y)o.)
It is necessary to make the above assumption that the X{ do not have
infinitely near intersections over Z, i.e., that n^i'H? = 0. In case all m-, = 1,
this corresponds to our previous assumption that the X: cut out Z scheme-
theoretically. For example, if Y = P3, and XUX2, X3 are defined by equations
x2=0; x}=xox2, xxx3 = xQx2,
and Z is the line xt = x2 = 0, then Z is the set-theoretic intersection of the Xit
and m, = 1 for all i, but
although degft^! ¦ X2 ¦ X3)z) = n, n2 n3 = 4.
If X[Cl... X'r f]E + 0, one must add the image of (X[ ¦... ¦ X'rf to the right
side of (*). In the above example, the X[ meet transversally at one point.
Example 9.1.7 (cf. B. Segre A) for dim Y=3, Todd C) for dim Y= 4). If
Y, X\,..., X,, and Z are non-singular, and Z is a curve, and m = 0, then
(Here Kv= — cx(T^ is the canonical divisor class, and for Z c K, (Ky-Z)y
= -c,GV|z)n[Z].)
If 7 is a 4-fold, and A",, A are surfaces, the equivalence of Z is
(tf * • Z)Xl + (Kx, ¦ Z)x, -(KyZ)Y-Kz.
For example, if Y = P4, Xx is the projection of the Veronese surface, X2 is a
plane, and Z a conic, then the equivalence of Z is 3; there is therefore one
residual point outside Z.
If there are a finite number of points which are singular on Z, Xx, and X2,
such that the proper transforms in the blow-up Y of Y at these points become
non-singular, one may use the formula in Y to deduce the equivalence on Y.
For an example where Xt, X2 are surfaces with improper nodes, see Todd C).
Example 9.1.8. The plane conies are parametrized by P5. The conies
tangent to a given line / form a hypersurface H, in P5 of degree 2. If 5 lines

158
Chapter 9. Excess and Residual Intersections
/,,...,/5 are given, no three passing through any point, then the Veronese
surface Z of double lines (Z = IP2) is a scheme-theoretic component of f] H^.
Therefore the equivalence of Z for if,, •... • H,5 is
= 160- 180 + 51 =31
where h is the class of a line in Z = P2(cf. Example 3.2.15 (b)). Therefore there
is one residual point, which represents the unique conic tangent to the given
five lines.
Example 9.1.9 (Fulton-MacPherson B)). The conies tangent to a given
conic D form a hypersurface HD of degree 6 in P5. Let n: P5 —* P5 be the
blow-up of P5 along the Veronese Z, with exceptional divisor E. Then
n*HD = 2E + GD for some divisor Go- If Dx,..., Ds are given conies, such that
(i) no three pass through any point, and (ii) there is no line with two points on
it such that each D,- is either tangent to the line or passes through one of the
points, then f] GD C\E = 0. The equivalence of Z for the intersection HDl ¦... ¦ HDs
is therefore
j(l + 6B/j)MB3 - 24 • 9h + 25 • 5lh2) = 4512 .
z
Assume in addition: (iii) no two of the five conies are tangent, and (iv) the
pairs of lines, each tangent to two of the conies, do not intersect on the fifth
conic. Then all the points of f]HD[ outside Z are isolated points which
represent non-singular conies C, and the intersection number of the hyper-
surfaces at the point corresponding to C is i(Q, where
i(C)=riD-#(CnDi)).
Since the total intersection number is 65 (by Bezout),
Z'(Q = 65- 4512 = 3264.
c
In particular, if the D, are chosen in general enough position so that all
i(C) = 1, then there are 3264 non-singular conies tangent to the 5 given conies.
(These calculations hold in all characteristics but 2. In characteristic 2 the Ho
are cubics, and a simpler calculation shows that the number 3264 should be
replaced by 51. See Example 10.4.3 or Vainsencher C) for a different approach
in characteristic 2.) For generalizations, see § 10.4.
Example 9.1.10. The class (Xx ¦... -Xr)z in A+Z depends only on the
rational equivalence classes of Xu ...,Xr in AtY: if X\ is rationally equivalent
to Xt, and Z is also a connected component of C\Xl, then
However, contrary to an assertion of Segre D) p. 20, the class may change —
even numerically — if Z is replaced by a rationally equivalent variety. For
example, if Y = P3, Z is a twisted cubic, and Z' is a plane elliptic curve, and
surfaces X^X'j are taken of large degree d,, then
, • X2
- (X[ -X'2-X'3)zl=2(g (Z') - g (Z)) = 2 .
9.1 Equivalence of a Connected Component
159
Example9.1.11. Let Y be an r-dimensional variety, Xl ,...,Xr^l divisors
on Y, such that XlC]..-C]Xr_l is one-dimensional. Let Z be a curve which
is contained in Xl[]...f}Xr_l. The number i of intersections of Z with the
rest of Xt fl... f) Xr_ l is given by the formula
i = deg{c(JV)nj(Z,y)}0
where N = ®NXlY\z. Note that {c(N) r\s(Z, Y)}x represents the contribution
of Z to the intersection product; it is interesting to see a geometric interpreta-
interpretation for a lower dimensional term. To justify this interpretation, let n:Y -*Y
be the blow-up of Y along Z, E the exceptional divisor, rf.E-*Z the
projection. Write
Then deg(A"i •... -X'r-X'E) is a reasonable geometric definition for the
number i, and
1*(
¦E) = Zxrs(Z,
, Y)o,
which gives the formula stated.
If y=P', degA",= rti, and Z is non-singular of degree e and genus g, then
..r
Example9.1.12. Suppose r — 1 hypersurfaces in Pr, of degrees nu ..., nr~u
intersect in a union of two non-singular curves Z and Z' which intersect
transversally (i.e., with distinct tangents) at i points. If e,g (resp. e',g') denote
the degree and genus of Z (resp. Z'), then
@
e + e' = nx- ...¦ nr_.
(ii)
(iii)
Thus from nu...,nr-i, e, and g one may calculate e',g' and ('. (Bezout's
theorem gives (i), the preceding example gives (iii). Writing (iii) with the
roles of Z and Z' reversed gives an equation equivalent to (ii).)
For example, if Z is a twisted cubic contained in two quadrics, Z' will be a
line meeting Z in two points. If Z is a curve of genus 1 and degree 5 contained
in two cubic surfaces, this predicts that Z' will be a rational curve of degree 4
meeting Z in 10 points.
Example9.1.13 (Le Barz E)). Let Hilb^P* be the Hilbert scheme of k-
tuples of points in PN, AlkVN the subvariety consisting of ^-tuples which are
aligned. For any subvariety X of P*, the intersection

160
Chapter 9. Excess and Residual Intersections
corresponds to the k-fo\d multisecant lines to X. However, any line L
contained in X gives an excess component of this intersection. For example,
suppose A" is a surface in P4 containing a finite number of lines, and L is a line
in X not meeting any singular point of X. Then the equivalence of L for the
above intersection is ( 5 j where m is the self-intersection number of L
on X. We refer to the papers of Le Barz for details and other examples.
Example 9.1.14. Herbert's multiple point formula. Let f:X-* Y be a proper
immersion of non-singular varieties. Assume that / is completely regular, i.e.,
for any distinct points x, € X with the same image y e Y, the images of the
tangent spaces TXl X are in general position in Ty Y. Let
and Xk =f~l(Yk), with their reduced structures. Then
C) [*t] = /*[yt-,]-c,(v/)nMt_1]
inAm_ik(X), with m = dim Y,m- d= dimA", and Vf=f*TY/Tx. Therefore
These formulas are due to Herbert A). Ronga C) deduces (*) from the excess
intersection formula. Let
Let Xk = Xk/Sk, X'k = Xk/Sk_lt the symmetric group Sk-] acting on the last
k — 1 factors. There are canonical proper immersions fk:X'k-*X with
ifk)*[X'k] = [Xk] and Qk'-
Ywith (gk)*[Xk] = [Yk]. There is a fibre square
X'k Jl XL-! > Xk
/, 11/,-, I Is.
X — Y .
From the excess intersection formula (Proposition 6.5),
f*(9k)*[Xk] = (fk)*[X'k] + C/i-i
where E is the excess normal bundle. In fact, E is the pull-back of v; to X'k_x,
and (*) results.
For a general discussion of multiple point formulas, including some cases
where ramification may be present, see Kleiman (8), A2), and Ran B).
9.2 Residual Intersection Theorem
Definition 9.2.1. Let DcW^V be closed imbeddings of schemes. Assume
that D is a Cartier divisor on V. There is a closed subscheme R of W, called
9.2 Residual Intersection Theorem
161
the residual scheme to D in W (with respect to V), such that
W = D(JR
and, moreover, the ideal sheaves on V are related by
Indeed, the inclusion D<=W means that <f(W) czJ'iD), so that every local
equation for W is uniquely divisible by a local equation for D; the quotients
give local equations for R.
Proposition 9.2. Let Da Wa Vbe closed imbeddings, with Va k-dimensional
variety, and D a Cartier divisor on V. Let R be the residual scheme to D in W.
Then, for all m,
s(W, V)m = s(D, V)m
j=o
(-DY ¦
inAm(W).
Proof. Assume first that W^ V. Let n: P-> V be the blow-up of V along R.
Let W=iC{(W), R = n-](R), D = n~[(D) = n*(D), so that W=D+R as
Cartier divisors on Y. Let r\ be the induced morphism from W to W. Set
d=k — m. By the definition of Segre classes (cf. Corollary 4.2.2) and the
projection formula (Proposition 2.3 (c))
= s(D, V)m+
• s(R, V)a+J,
which is the required equation since s(R, V)m+d= 0. If W=V, the required
equation reads
{[V]}m = s(D,V)m+(-D)"-m-[V],
which amounts to the definition of s (D, V). ?
Theorem 9.2 (Residual Intersection Theorem). Consider a diagram
R
D-Z-tW-^ V
I9 I'
X—.Y
with the square a fibre square, ij, a, b closed imbeddings, and V a k-dimensional
variety. Assume:
(i) / is a regular imbedding of codimension d;
(ii) j a imbeds D as a Cartier divisor on V;
(iii) R is the residual scheme to D in W.

162
Chapter 9. Excess and Residual Intersections
Let N = g*NXY, and(t(—D) =j*#y(—D). Define the residual intersection class
R in Ak-d(R) by the formula
(*) R = {c(N®#(-D))ns(R,V)}k_d.
Then
X-V={c(N)ns(D,V)}k-d+R
in Ak-d W,
Proof. Set r = k - d. By Proposition 6.1 (a),
d
A) X- V= {c(N) ns(W, V)}r= ? c,(N)ns(W, V)r+i.
{c(N) ns(D, V)}k.d= E c,(N) ns(D, V)r+i,
i=Q
By definition,
B)
and, by Example 3.2.2,
d
C) {c(N®J(~D))ns(R, V)}r= E cp{N®0(-D)) n s(R, V)r+p
= Z ilidZ^CiWc^i- D))"'1 ns(R, K)r+p
(d - A
= I c,(N) E [a T ' (-DV n s(R, V)r+i+J .
i = 0 \j = 0 V J ' J
From Proposition 9.2, for m = r+ i, it follows that the right side of equation
A) is the sum of the right sides of B) and C), as required. ?
Notation 9.2. For a power series P in Chern classes of vector bundles,
denote by (P)m the term of P of isobaric weight m.
Corollary 9.2.1. In addition to the assumptions of the theorem, assume that R
is regularly imbedded in V of codimension d'. Then
XV= c(i/ja) n [D] + c(i/jb; D) n [R],
where
In particular, ifd'= d, then
X-V=c(i/ja)n[D] + [R).
Proof. Since s {R, V) = c (NR V)~l n [R],
{c(N®J(-D))ns(R, V)}k.d= {c(N®#(-D)) -c(NRvyl n [R]}k-d,
from which the corollary follows. ?
The residual scheme R is always locally defined in V by d equations - the
quotients of local equations for A" in Y by an equation for D in V. In particular,
if V is Cohen-Macaulay, and codim (R, V) S d, then R is regularly imbedded
9.2 Residual Intersection Theorem
163
in V of codimension d (Lemma A7.1), and the last formula in the corollary
applies. If Kis not Cohen-Macaulay, the theorem gives the following result.
Corollary 9.2.2. In the situation of the theorem, assume that codim (R, V) = d.
Let RU...,R, be the irreducible components of R, and let <?, = (<?/; V)Rt be the
algebraic multiplicity of Valong R at Rt (§ 4.3). Then
Definition 9.2.2. Let ZcWcV be closed imbeddings, with V a variety,
and Z^V. The residual set to Z in W is defined as follows. Let n: V-+ V be
the blow-up of V along Z. Let D = n'x (Z) be the exceptional divisor,
W=n~\W). Then Da Wa V, so the residual scheme R to D in tT'( W) is
defined. Define the residual set R to be the image of R in V: R = n(R). (R
may be given the image scheme structure, but this will play no role here.)
Consider a diagram R
W
I9
X-
V
lf
Y
satisfying the conditions of Theorem 9.2, but without assuming Z is a Cartier
divisor on V. Define the residual intersection class R in Ak-dR by the formula
R=
, V)}k.d
where r\ is the induced morphism from R to R. In other words, R is the image
of the residual intersection class on R defined in Theorem 9.2.
Corollary 9.2.3 (Residual Intersection Formula). With the above assump-
assumptions,
X-V={c(N) n s(Z, V)}k-d+ R.
Proof. By Theorem 6.2 (a), X- V is the push-forward of A"- V. The corollary
follows from the theorem, applied to V, and the identity
c{N)r\s(Z, V) = 77* (c{rj*N) n s(D, V)),
which follows from the projection formula and Proposition 4.2 (a). ?
A more general residual intersection theorem together with a geometric
explanation for the terms involved, will be given in § 17.6.
Example 9.2.1. If Z is a connected component of W, the residual scheme R
is the union of the other components, and
is the contribution of R to the intersection product X- V.
Example 9.2.2. In the situation of Theorem 9.2, let E be the restriction of
tf(—D) to R. The normal cone CR V is canonically imbedded as a sub-cone

164
Chapter 9. Excess and Residual Intersections
of E. lfsE is the zero-section of E, then
R= {c(N®#(-D)) ns(R,
Thus the residual class R has a canonical decomposition: R = X "»,-v|
where C, are the irreducible components of CR V, nt/ is the geometric multi-
multiplicity of C, in CRV. (For the first statement, the equation S(X)-0v =
\f{K) determines surjections
where h = g b is the morphism from R to X. Therefore
which corresponds to an imbedding of CR V in h*N ®0y(— D) \R, as asserted.
The second statement follows from Proposition 6.1 (a).)
Example9.2.3. In Corollary 9.2.1, the class c(i/jb;D) is the top Chern class
of the bundle E/NRV, where E is the restriction of N®0(-D) to R (cf.
Example 9.2.2).
Example 9.2.4. Let A, B, D be effective divisors on a non-singular surface
X, with A and B relatively prime. Let A' = A + D, B' = B + D. Then the
residual scheme to D in A'OB' (with respect to X) is the scheme Af]B. For
the intersection of the diagonal Ax in XxX by A'xB', the residual
intersection theorem says that
(A' x B') ¦ Ax =
')) n{D - D ¦
= (A' D + B' D-DD) + A- B.
Note that this decomposition of the intersection class differs from the
canonical decomposition using distinguished components (cf. Example 6.1.4).
Example 9.2.5. Let V= P2, y=P4, X=V2 the plane in P4 defined by
the vanishing of the first two coordinates. Let / be the morphism from V to
Y given by
f(xo:x]:x2) =(x$
Then W=f~' (X) is the scheme x4, = xjj x i = 0. If D c V is the triple line
defined by xl = 0, then the reduced point R = @:0:1) is the residual scheme
to D in W. Therefore
X- V=
= 15
[*]= 16 [/>].
where P is a point of D. Note that the image / (V) is a subvariety of P4 of degree 4
which meets X geometrically in a triple line; since ft(X- V) = X ¦ /„ [V] =
4 X ¦ [f (V)], the above equation is compatible with Bezout's theorem.
Example 9.2.6. Let X = A1, Y = A2, /: X -> Y by f(t) = (a(t), b(t)). Then
(/ x /)"' (Ay) contains Ax, and the residual scheme R is the scheme defined by
9.3 Double Point Formula
165
fli(/i, t2) and bi(t\, t2), where
a(h) - a(t{) =(t2- t{) a^tu t2),
and similarly for bs. The scheme-theoretic intersection of R with Ax is defined
by the derivatives a'(t) and b'(t). Off <d.r, # is the double point set, while
Rf]Ax measures ramification. (See Example 9.3.12 for a generalization.)
Example 9.2.7. Let Y be a non-singular variety, and let Z c y be a union
of t curves Zl ,..., Zt, such that no point lies on three of these curves, and if
P € ZjHZj, (=t=j, then i* is simple on Zi and Zj with distinct tangent lines.
Then
(Blow up yat ni<j(Z,nZj), and use Proposition 9.2.)
If Xl ,... ,Xr are divisors on Y, dim y = r, and Z is a connected component
of 0 Xt, and each Z( is non-singular, then
(Xx
¦ X,)z=
i ~ c. (TV)) • Z + ?
/
+ I [Z,
If Y= Pr, n,= deg A",, ^,= genus(Zj), and N is the total number of points in
more than one Z,, then the equivalence of Z is
n,-(r+ 1)
• deg(Z) + ? B - 2g,) + N .
(cf. Semple and Roth A)IV.8.3).
Example 9.2.8. For any Z c We V, one may define a residual scheme R to
Z in Why defining the ideal sheaf J~(R) on affine open sets of V to be the
ideal of functions which multiply all functions in J~(Z) into elements ofJr(W),
i.e. (cf. Peskine and Szpiro A)),
S(R)/S(Z) = Hom^z^w) ¦
This agrees with the description in the text if Z is a Cartier divisor on V. The
residual class R constructed in Definition 9.2.2 is a class in Ak_dR. It would be
valuable to have conditions which would imply that R is the cycle [R]
associated with this scheme structure, or to compute R using this scheme
structure.
9.3 Double Point Formula
Let /: X" -* Ym be a morphism of non-singular varieties of the indicated
dimensions, with A" complete. Then (/x/)"' (A Y) contains Ax, and the residual
set is the set of double point pairs, which we denote by D'(f). Let Xx X be the
blow up of Xx X along Ax, n the projection from X~X to Xx X, E — P(TX)

166
Chapter 9. Excess and Residual Intersections
the exceptional divisor. Let F = (fxf)°n be the induced morphism from
XxX to YxY. The double point scheme D(f) is defined to be the residual
scheme to E in FH {A Y). We have a diagram as in § 9.2:
D(f)
P(TX) =
¦XxX
>Y xY.
One may verify (cf. Fulton-Laksov A) Prop. 4) that the points of D(f) are
those point pairs (xt, x2) with x, 4= x2 and f(X[) =f(x2), together with those
tangent directions in P(TX) on which the induced tangent map vanishes. By
definition, D'(f) = n(D(f)). We define the double point set D(f) <= X to be
the image of D' (/) by the first (or second) projection from X x X to X. Let r\
be the induced morphism from D (/) to D(f).
Define ?>(/) e A2n-m{D(f)) to be the residual intersection class, defined
by formula (*) of Theorem 9.2. If D(f) has the expected codimension m in
X~X, then 6 (/) = [D (/)]. Define the double point class D (/) in A2n-m (D (/))
Theorem 9.3 (Double Point Formula). With the above notation,
V(f)=f*f*[X]-(c(f*Tr)c(Txy%_nn[X]
Proof. By Corollary 9.2.3, since NAr(Yx Y) = Ty,
Ay(XxX) = {c(f*TY)ns(Ax,XxX)}2n-m+n*[t>(f)]
in A2n-m(Xx X). Let/) be the first projection from Xx X to X, and apply />„ to
this equation. The first term on the right projects to
the second to D (/). It suffices to show that
p,(Ay(XxX))=f*f,[X}.
To see this, recall that f*f*[X] = y}([X] xf*[X]), where yf is the graph
imbedding of X in X x Y. Consider the fibre diagram
XxjX > XxX
'I
X -
'1 " ['••
Y > YxY.
s
I"'
X x Y
By Theorem 6.2(a) and (c),
yj ([X] x/,[X]) = /,;{y}[Xx X}) = pi,(<5![Xx X]) = p*{A Y ¦ (X x X)) . ?
9.3 Double Point Formula
167
Example 9.3.1. If/ is a closed imbedding, then D(/) = 0, and the formula
reduces to the self-intersection formula. From the self-intersection formula one
knew that the right side of the double point formula restricts to zero on
X—D(f), so it comes from some class on D(f). The residual construction
produces such a class D (/) explicitly.
Example 9.3.2. If n= 1, and m = 2, and / maps X birationally onto its
image X in Y, then
where 4 is the subscheme of X whose ideal sheaf is the conductor of X over X.
This may be proved inductively by comparing both sides for/:Ar-> Y and
f':X-> Y', where Y' is the blow-up of Y at a singular point of Y (see Fulton
F) § 4 for details). The analogous formula, for /: X" -> Y"+[ finite and
birational onto its image, is still open, although it follows from Theorem 9.3 or
Kleiman (8) that the two cycles are rationally equivalent on X. An example of
Artin and Nagata A) (cf. Fulton F) § 2.4) shows that the conductor and D(/)
may disagree if m > n+ 1.
Example 9.3.3. Let C c Y = P2 be a nodal curve, and let X be the
normalization of C. The double point formula for the induced morphism
f: X -* Y gives the classical formula
(a) 2d=ni-?>n + 2-2g
relating the number d of nodes, the degree n of C, and the genus g oiX.
(b) The mapping df: Tx^>f*Tj« is injective, with quotient bundle Nf. In
fact,
Nf S f*#p, (n)®tfx (- D)
where D = D (/) is the double point divisor on X.
(c) The nodes on C impose independent conditions on curves of degree
k s n — 3. (It is enough to prove this for k = n — 3. The vector space L curves of
degree n — 3 passing through the nodes is a subspace of
H°(X, Nf®f*# (- 3)) = #°(X, Kx).
Clearly dim L^(n- \)(n- 2)/2 - d, and this integer is g = dim H°(X, Kx)
by (a).)
(d) The linear system cut out on X by curves of degree k ^ n — 3 passing
through the nodes of C is complete. (The dimension of the space of such curves
is
(k+\)(k + 2)/2 -d-(k-n+\)(k-n + 2)/2,
which is k n — 2d+ [ — g, the dimension predicted by Riemann-Roch (Exam-
(Example 15.2.1). By (b)J*tfP,(k) ®^(- D) is non-special.)
Example 9.3.4. Let /: X" -> Y"+l be a finite morphism, with X and Y non-
singular, and assume that the sets
A = {x e X | dfx : TxX -* Tf(x) Y is not injective},
B = {x e X | 3 x' * x", distinct from x, with /(*') =/(x") =/(*)}

168
Chapter 9. Excess and Residual Intersections
have dimensions at most n - 2. Let D (/) be the double point set, endowed
with its reduced scheme structure. Then
ID (/) = [/>(/)].
(Let Xo = X-(f~' {f (A)) u B). It suffices to show D(/) and [?)(/)] agree on
Xo. Over Xo, D(f) is smooth and maps isomorphically to D(f).)
Example 9.3.5. Let X" be a non-singular hypersurface in a non-singular
variety Y"+[, n>\. Let X be the blow-up of X along some non-singular
variety, and let D be the exceptional divisor. Then D is the double point set
D (/) for the induced morphism / from X to Y, and D has the expected
codimension. However D(f) does not have the correct codimension, and in
fact
is a negative cycle.
Example 9.3.6 (cf. Johnson A) and Fulton-Laksov A)). With/: X" -> Ym as
in this section, but with X singular, the same procedure produces a double
point class D (/) in A2n-m (D (/)), and then
D(f)=f*MX]-{c(f*TY)ns(Ax,XxX)}2n-m
in A2n-mX.
Example 9.3.7. Let /: X2 -> Y3 be a morphism of non-singular complete
varieties over an algebraically closed field K of characteristic * 2. Assume that
/maps X finitely and birationally onto its image X <= IP3, and assumethat the
singularities are all ordinary, i.e., the singular locus of X is a curve D, with a
finite number / of triple points Q\,...,Qlt and a finite number v of pinch points
Pi,..., P.. The completion of the local ring of X is isomorphic to:
Klxux2,x,1/{x,x2)
f n~v- \- v- Ti/^v- \- \- \
TV \\_-\\ , -^2> -*3_1K V I ¦^¦2-^3/
KlXl,x2,x3J/(x22-x2x3)
at a general point of D
at a triple point Q,
at a pinch point P,.
The curve D is non-singular except for the triple points. The double curve
D <= X is non-singular except at the 3/ triple points Q\,-..,Qi, lying over the
triple points of D; each Q, is a node on Z). There is one point P, on D mapping
to each P,. The induced map/: Z) -> D is generically two to one, with simple
ramification points at the P,. (A local analytic equation for/at a pinch point is
given by_/(/,, h) = (/,, f, /2, $.)
Let 7 be the singular subscheme of A': if g(xu x2, x3) is a local equation for
X in 7, then 7 is defined by the vanishing of the three partial derivatives of g;
more intrinsically^ the ideal of 7 is the second Fitting ideal of Q\ (cf. Piene
D)). Let J =/"'(/). Then J contains D, and the residual scheme to D in J is
the reduced scheme of pinch points {P,,..., Pv}. From Proposition 9.2, one has
(a)
s(J,X) =
(-D- D+
9.3 Double Point Formula
From the double point formula (cf. Example 9.3.2),
In addition,
(c)
/*[5] =/*[X] ¦ D - D ¦ D + ? [Q,] -
(To prove this, first consider the case when there are no triple points. Then
D, D are non-singular,/ (D) = D, and by the excess intersection formula (and
Corollary 8.1.1),
= cl(f*NBY/NDX)n[D]
= (c\(f*Tr) ~ c, (f*TB) - c, (Tx) + c,
n
the last step from the ramification formula (Example 3.2.20 or Example
9.3.12). Combine this with (b) to get (c). For the general case, let n: X' -* X
(resp. q: Y' -> Y) be the blow-up of X at the 2, (resp. the Q,), and apply the
previous case to the induced map/' from X' to Y', noting that
(ory = [5] or [X]. Use Proposition 6.7 to compute g*y.) It follows that
(d) s (J, X) = [D] + If* [D] -f*[X]D-Z [Q,] + 2 ? [Pj]).
Therefore, by Proposition 4.2 (a),
(e) s(J, X) = 2[D] + (-X-D-3t [&] + 2
Example 9.3.8. Let/: X -> X <= Y be as in the preceding example, but with
Y = P3. By Example 4.4.3, the degree of the dual of X is
d(d- IJ - deg s{J, XH- 2{d- 1) deg 5G Jp),
where d is the degree of X. By Example 9.3.7, one derives the classical formula
for the degree of the dual, where e is the degree of the double curve D, and /, v
are the number of triple points and pinch points on X. (For another proof,
generalizations, history, and examples, see Piene B).)
Example 9.3.9. In the preceding example, if the double curve of X is a line,
there are 2d- 2 pinch points on the line, and the degree of the dual is
(cf. Salmon A) § 20).
d(d- \J-Gd-\2)

170
Chapter 9. Excess and Residual Intersections
Example 9.3.10. Let X be the surface in P3 with equation x2 jc3 = x\ x0. The
double curve is a line with two pinch points, so the degree of the dual is
3-22-5-4 = 3.
In fact, the dual of X is isomorphic to X.
Example 9.3.11 (cf. Salmon A)). Assume the ground field is algebraically
closed of characteristic zero. Let X <= P3 be an irreducible surface with
equation F(x0, xu x2, x3) = 0, of degree d. For a point P=(!o'-J-i'-h'h)?lP:l,
the polar surface of X with respect to P is the surface VP of degree d - 1 with
equation 3
X A, • dF/dXi = 0 .
i-O
A point Q is in X f) VP if and only if the (projective) tangent space to X at Q
contains P. If Pl, P2 are two general points of P3, then
Xf)VP,C)VP2=:S{JT
where S is the singular locus of V, and T is a finite set of non-singular points Q
such that the tangent plane to V at Q contains the line through Px and Pi.
Thus the cardinality of T is the degree of the dual surface. If X has ordinary
singularities, as in the preceding example, the equivalence of the singular
curve for the intersection X ¦ VPl- VPl is
Cd~4)e-3t + 2v
where e is the degree of the curve, / the number of triple points, v the number
of pinch points.
Example 9.3.12 Ramification Formula. Let f:X"-* 7™ be as in this section,
and let R(f) = D(f)DE. Define R(/)e A2,.m.l (R{f)) by the formula
Let /?(/) be the image of R(f) in X, and let R(/) e A2n-m-t (R(f)) be the
push-forward of R(/); R(/) is the ramification class. Then
inA2n-m-\ (X). (By Theorem 9.2 one has
where F is the induced morphism from E to Y. Intersect both sides by the
divisor E and push down to X. If n > 1, then n* {A Y ¦ E) = A Y ¦ n* E = 0, and
the formula
0 = -{c(f*TY)ns(Ax,XxX)hn-n,-i
results. If n = m= 1, E= Ax, and A Y ¦ E = c, (/* TY), E- E = c](Tx), and the
formula reads
(cf. Example 3.2.30).) As in Example 9.3.6, this extends to the case when X
may be singular.
Notes and References
171
Example 9.3.13 (Johnson A)). Let X" be a subvariety of WN, and let
/m: A™-> Pm be a general linear projection. Let D(/m) and R(/m) be the
double point and ramification classes of/, and let //= c, (^(l)) be a
hyperplane section. Then
(*) H ¦ lD(/m) - D(/m+i) = R(/m)
in A2n-m-](X). (This is a formal calculation, using Theorem 9.3 and Example
9.3.12.) If TVs 2«, and a projection to some Pm is unramified, it follows that
the projection is an imbedding. This remarkable discovery sparked the
development of a general theory, which includes a proof that an arbitrary
morphism X" -> Pm with X" projective and irreducible, and m <2n, cannot be
unramified unless it is an imbedding (Fulton-Hansen A)). For example, the
number v of pinch points in Example 9.3.8 must be positive unless the surface
is non-singular.
J. Hansen A) has recently given a geometric explanation for the identity
(*)•
Example 9.3.14. With /: X" -> Ym as in this section let_? (/) =f(D (/J) be
the image in Y of the double point set. There is a class E>(/) e A2n-m (D (/))
so that the push-forward of D(/) by the canonical map from D(f) to D{f) is
This may be seen by an alternative construction of 1D(/), following Ran
B). For a non-singular V, let V™ denote the blow-up of Vx V along Ay. Let
Z = XxY. There are imbeddings of X[1] and X^xY in Zpl induced by the
graph imbedding of Jin Ix y and by the diagonal imbedding of Y. Then
?>(/) is the intersection class of X[2] and X[2] xY on 7S\ (If/ is the projection
from Z121 to YxY, then X^xY is the residual scheme to the exceptional
divisor of Z!2J in/"'(Ay); this follows from Appendix B. 6.10, applied to the
inclusions XxY^>XxXxY~+Zx.Z induced by the diagonal imbeddings of
X and Y. The fact that ID (/) = r [X®], where i is the imbedding of X[2] x Y in
ZP-\ follows from the construction of D(/); Theorem 17.6 gives a general
formula which implies this.)
For V non-singular, let K*2) be the quotient of F12' by the involution which
reverses the factors, i.e., K*2) is the Hilbert scheme of length 2 subschemes of
V. As above, one has imbeddings of Xi2) and Xi2) xY in ZB), and one may
construct their intersection class, denoted D(/). Since the quotient mapping
from Zl21 to ZB) has degree 2, ?>(/) pushes forward to 2t>(f) (by
Theorem 6.2). If one defines D(/) to be the push-forward of_?)(/) by the
projection from X'2' x Y to Y, then D (/) is the required class on D (/).
For a similar approach to higher multiple point formulas, see Ran B).
Notes and References
We point out only a few of the landmarks in the extensive literature on
residual intersections and multiple point formulas. The interested reader may
find many additional references, and hundreds of examples, in the cited works.

172
Chapter 9. Excess and Residual Intersections
To our knowledge, the first excess intersection problem was formulated in
1847 by G. Salmon A), although precedents can be found in Jacobi A). To
find the degree of the dual of a surface with a multiple curve, arguing as in
Example 9.3.11, he was led to the general problem of finding the "equivalence"
of a curve for three surfaces which contain it, so that the equivalence, plus the
number of intersections of the surfaces outside the curve, totals the product of
the degrees of the surfaces. These ideas are developed further in Salmon-
Fiedler A).
More general formulas of this kind were given by Cayley B), C), who also
showed that many enumerative problems - such as the problem of how many
curves in a-family are tangent to a collection of given curves — can be
formulated as problems of excess intersection (cf. Example 9.1.9, and § 10.4).
Cayley also pointed out that these excess intersection problems can be very
difficult. Noether C) extended the work of Salmon and Cayley.
The impressive accomplishments of the great enumerative geometers such
as Chasles, Schubert, and Zeuthen, did not follow the ideas of Cayley. In
modern terms, we can see that what they did often amounts to doing
intersection theory on suitable blow-ups of the original spaces; in such blow-
blowups the proper transforms of the hypersurfaces corresponding to given
geometric conditions will meet properly. In fact, however, these parameter
spaces were not mentioned explicitly, and the intersection theory was done in a
purely formal, symbolic manner. It is, nevertheless, difficult to reconcile the
clarity and precision of Cayley's formulation with the debate still raging half a
century later over such basic foundational problems as the principle of
continuity. It should be noted that there was a concurrent dichotomy between
the synthetic and the analytic approaches to geometry, cf. Chasles D), p. 1168.
Residual intersection problems were considered by several others in the
nineteenth century, notably Noether, Pieri, Caporali, and Bertini. Quite
general intersections in projective space were considered by Severi B). B.
Segre A) and Todd C) generalized to the case of general ambient varieties of
dimension 3 and 4, and extended Severi's results from numerical results to
equalities modulo rational equivalence. Todd F) gave a version of a residual
intersection theorem in higher dimensions, and Segre D) gave general
formulas for equivalences. Many applications may be found in the books of
Salmon C), Enriques-Chisini A), Semple and Roth A), and Baker A), B).
The classical approach to the problem may be illustrated by the original
case considered by Salmon: to calculate the equivalence of a curve which is a
component of the intersection of three surfaces in P3. Two of the surfaces
contain the original curve Z together with another curve Z'. First the degree
and genus of Z', and the number i of intersections of Z with Z', were
determined from the invariants of Z and the surfaces (cf. Example 9.1.12). The
number i was then subtracted from the total (Bezout) number of intersections
of the third surface with Z', to obtain the number of intersections of the three
surfaces outside Z.
Such inductive proofs make a number of implicit assumptions. For
example, it is apparently assumed that the residual curve Z' is non-singular, or
at least reduced, and meets Z transversally. If not, one must count the points of
Notes and References
173
Zf]Z' with multiplicities; but then the equality of different calculations of
the number i, on which the proof is based, is not at all obvious. Similar
problems are met in many of the applications of intersection theory by
classical geometers.
The problem encountered in this discussion, to compare properties of Z'
with those of Z, is interesting in its own right. It has received considerable
attention under the name of linkage, or liaison. For an approach to this
problem using homological algebra, see Peskine and Szpiro A), and Rao A).
Other properties of residual schemes are proved by Artin and Nagata A), cf. Huneke
C). Hironaka A) has used residual constructions for smoothing cycles.
The use of the modern excess intersection formula avoids the problems of
general position arising in classical inductive proofs. The approach followed
here is related to the method of Segre D); it confirms his conjecture that the
general equivalence (Xt ¦... ¦ Xr)z can be calculated purely in terms of the
"covariants" (Segre classes) of the varieties involved (but see Example 9.1.10).
Double point formulas also have a long history, going back to Clebsch's
formula for the genus of a plane curve (cf. Example 9.3.3). Severi B) gave
general numerical formulas for projections to projective spaces, which were
rediscovered by Holme A) and Peters and Simonis A). Todd F) gave a
formula for the rational equivalence class of double point cycles, based on his
residual intersection formula. The extension to morphisms to general non-
singular varieties, and the simple approach followed here, was made possible
by the residual construction of the double point scheme made by Laksov C).
Extensions to singular varieties have been made by Holme, Johnson, and
Fulton'and Laksov (see Example 9.3.6). Kleiman A2) has a version of the
residual intersection theorem which he uses to prove formulas for triple points
and higher multiple loci (cf. Example 17.6.2). We refer to the article of
Kleiman (8) for more on the history of multiple point formulas. Ran B) has
recently given a new application of the residual intersection formula to prove
secant and multiple point formulas.
Most of the applications to classical problems are newly worked out in the
examples of §9.1, although the case of conies tangent to five conies
(Example 9.1.9) was included in Fulton-MacPherson B), and the application to
multi-secants (Example 9.1.13) is from Le Barz. Example 9.3.3 was suggested
by R. Lazarsfeld, and Example 9.3.14 by Z. Ran. The application to curves in
P" which cannot be the scheme-theoretic intersection of n hypersurfaces
(Examples 9.1.2, 9.1.3) is, apparently, new.
The residual intersection theorem in § 9.2 is based on the construction of
Laksov C). The original theorem made rather stringent requirements on the
residual scheme, which have gradually been removed as technique in inter-
intersection theory has improved (cf. Fulton F), Fulton-Laksov A), Kleiman A2),
Fulton-MacPherson C)). The present Theorem 9.2 is the first to make no
assumptions on the residual scheme. Corollary 9.2.1 is also new when the
residual scheme has larger dimension than predicted; Corollaries 9.2.2 and
9.2.3 have not appeared before.

174
Chapter 9. Excess and Residual Intersections
Similarly, Theorem 9.3 is the first to allow the double point locus to have
arbitrary dimension. The proof follows Fulton-Laksov A), and is based on
Laksov's fundamental construction.
The calculation of Segre classes of singular subschemes in Example 9.3.7 is
apparently new; numerical formulas, when K=P3, had been given by Piene
B).
The construction and simple proof of the ramification formula in Example
9.3.12 is also new. It generalizes the formula of Johnson A) to arbitrary
morphisms, without assumption on the ramification set.
It should be remarked that there have occasionally been alternate proposals
for assigning numbers to components of intersections which have larger than
the expected dimension. Severi G) has discussed this when the expected
dimension is negative, Samuel A) has given a definition using his algebraic
multiplicity. These definitions, whatever their virtues, all violate the principle
of continuity. They are therefore quite a different sort of notion than that
considered in classical intersection theory, and in this text.
Chapter 10. Families of Algebraic Cycles
Summary
If T is a non-singular curve, and p: s -> T is a morphism, any (k+ l)-cycle
a = ? n/P^i] on ¦& determines an algebraic family of ^-cycles a, on the fibres
«,= I
Rationally equivalent (k + l)-cycles on 3^ determine rationally equivalent
^-cycles in each fibre. The basic operations of intersection theory preserve
algebraic families. For example, if ^ is smooth over T, and {a,} and {/?,} are
algebraic families of cycles, then the intersection products a, ¦ /?, also vary in an
algebraic family. These facts are consequences of the general theorems of
Chap. 6, and the recognition of a, as the image of a by the refined Gysin
homomorphism constructed from the diagram
Y, ><&
I i'
In this formulation, T may be replaced by any variety of arbitrary dimension,
with / a regular point of T. This provides a simple method for studying
algebraic equivalence.
The principle of continuity, or conservation of number, has two parts. First,
in an algebraic family of zero-cycles, on a scheme which is proper over the
parameter space, all the cycles have the same degree. Second, as mentioned
above, the operations of intersection theory preserve algebraic families.
Refined intersection theory yields an improvement over classical formula-
formulations of this principle. For example, the ambient variety need not be complete;
all that is necessary is that the locus of intersections is proper over the
parameter space. This is useful for applications to enumerative geometry,
when the ambient space is a space of non-degenerate geometric figures. In the
last section an example of this kind is worked out: the formula for the number
of curves in an r-dimensional family of plane curves which are tangent to r
given plane curves in general position, in terms of the characteristics of the
family, and the degrees and classes of the given curves.

176
Chapter 10. Families of Algebraic Cycles
10.1 Families of Cycle Classes
In this chapter T will denote an irreducible variety of dimension m > 0. The
notation "/ e T" will be used to denote a regular, closed point of T
(Appendix B.I). By abuse of notation we write {t} in place of Spec(x(/)), where
x(t) is the residue field of the local ring of Tat the point, and we denote by
t:[t)-*T
the canonical inclusion of Spec(x(/)) in T. The assumption that the point is
regular means that / is a regular imbedding of codimension m.
Script letters will be used to denote schemes over T, with corresponding
Latin letters, subscripted by /, denoting the fibre over / e T. If p: & -> T is
given, then
Y, = p-](t);
Y, is regarded as an algebraic scheme over the ground field x(t).
Any (k + m)-cyc\e a on '&, or more generally any rational equivalence class
a 6 Ak+m'J/ determines a. family of k-cycle classes a, e Ak(Y,), for all t e T, by
the formula
a,= t'(cc)
where t':Ak+m& -+Ak Y, is the refined Gysin homomorphism defined from
the fibre square y ,. qy
by the construction of § 6.2. We may say that a, is the specialization of a at /.
More precisely, if a = p^] where y is a subscheme of '& of pure dimension
k + m, then
[] {(
where V, = r{\Y,, and s(V,,r) is the Segre class of V, in Y. (This follows
from Proposition 6.1 (a) and the fact that the normal bundle to {/} in T is
trivial.) In particular, if V, is ^-dimensional, then p^], is a well-defined positive
&-cycle supported on V, (cf. Example 10.1.1). On the other hand, if T c Y,,
then p^, = 0.
If T is a curve, and a is a (k + l)-cycle on &, then each a, is well-defined as
a &-cycle on Y,. For if a = X Wip^/], withX,- a variety, then we define
Thus all components of a which do not map dominantly to T are simply
discarded. In case dimT> 1, a general (k + m)-cycle <x=?«,|>',] on* will
not determine a &-cycle on 7,, unless one assumes that dim (F,), = k for all i.
If f:3f-* -V is a morphism, within —> Tas above, we denote by
the induced morphism on fibres over t e T.
10.1 Families of Cycle Classes
177
The next proposition states that this specialization in algebraic families is
compatible with the main operations of intersection theory.
Proposition 10.1 (a) Iff: S~ -> & is proper, and a is a (k + m)-cycle on 2",
then
inAk(Y,).
(b) Iff: 3T -*'& is flat of relative dimension n, and a is a (k + m)-cycle on ??,
then
mAk+n(X,).
(c) If i:S~ -* y is a regular imbedding of codimension d, such that
i,:X, -> Y, is also a regular imbedding of codimension d, f:V-*'& is a morphism,
and a is a(k + m)-cycle on T, then
(d) If E is a vector bundle on '2/, with restriction E, on Y,, and a is a (k -+ m)-
cycle on 1/, then
Proof. Since specialization is a refined Gysin homomorphism, the proposi-
proposition follows from corresponding compatibilities of refined Gysin homo-
morphisms, namely Theorem 6.2 (a), Theorem 6.2 (b), Theorem 6.4, and Pro-
Proposition 6.3. ?
Corollary 10.1. Assume T is non-singular, t e T is rational over the ground
field, and "J/ is smooth over T of relative dimension n. If a e Ak+m (&),
inAk+i-n(Y,)-
Proof. Let 5 (resp. 5,) be the diagonal imbedding of ^ \n <&*.<& (resp. Y, in
Y,xY,), and i, the imbedding of Y, in*. Then the two imbeddings (i,x.i,) ° 5,
and 5 ° i, of Y, in &x'y are equal, so by functoriality (Theorem 6.5, Example
6.5.2)
5* (i* a x HP) = 5* (i, x i,)* (a x p) = i* 5*{axp).
Since (!/ is flat over T, i*< (y) = y, for any cycle y on * (cf. Example 10.1.2), so
the displayed equation is equivalent to the required a, ¦ fi, — (a ¦ /?),. D
Remark 10.1. These results allow one to deduce identities involving rational
equivalence classes for special values of / from the corresponding identities for
general values, even if T is not a rational variety. For example, given Y
nonsingular, a, b, c e A (Y), to prove that c = a ¦ b, it suffices to find * -> T as in
Corollary 10.1, with Y, = Y, and a.,p,yon'& with a,= a, /?, = b, y, = c, such that
y = a ¦ p. If this is achieved with a meeting /? properly, the identity y = a ¦ Ji can
be verified generically on T. For a simple application see Example 10.1.9.
Corollary 10.1 is also valid for the product of more than two cyles, as the
proof shows.

178
Chapter 10. Families of Algebraic Cycles
Example 10.1.1. If y is a purely (m + ^-dimensional closed subscheme of
¦y, and each irreducible component W< of V, has dimension k, then
where e,=
^, is the multiplicity ofT'along V, at (•F, (Example 4.3.4).
Example 10.1.2. Assume that the imbedding i, of Y, in ^ is a regular
imbedding of codimension m, m = dim (T). For any (& + m)-cycle a onS^,
a, = if (a)
in 4t(T,), where if is the Gysin homomorphism of §6.2 (Use Theorem
6.2 (c)).) This holds, for example, whenever^ is flat over T (cf. Example A.5.5).
Example 10.1.3. (a) If a is a family of cycles on YxT, the cycles a,, as t
varies in T, need not be rationally equivalent. If Y= T is a projective non-
singular curve of positive genus, and a = [A], where A is the diagonal, then a, is
never rationally equivalent to ar if t + /'. The cycles a, are rationally equivalent
if 7" is unirational, however (Example 10.1.7).
(b) If a and /? are cycles on Yx T which are rationally equivalent on Yx T,
then a, is rationally equivalent to /?, on Y for all / e T. (The refined Gysin map
t' preserves rational equivalence.) Doubts about the validity of this fact, or at
least the ability to prove it, gave rise to some of the criticism of Seven's
methods (cf. Van der Waerden F)).
Example 10.1.4. Assume f :%'-*'& is a l.c.i. morphism of relative codimen-
codimension d, while f,:X,-+ Y, is a l.c.i. morphism of relative codimension d'. Let
e = d— d',E the excess normal bundle of the diagram
i I
Then for all (k+ m)-cycles a on
in Ak-d(X,). In particular, if d = d', then /;((*,) = (/'(«))». (See Proposition
6.6.)
Example 10.1.5. Assume T is non-singular, t is rational over the ground
field, />:,y-> T is smooth of relative dimension n, and f-.S"^1^ is a morphism.
Let a eAk+m3~, fi eAi+m». Then
in /tt+/-n(^Q- The product on the left is that of Chap. 8, since f,:X,-*Y,, and
Y, is non-singular; on the right yf:S'^' S~x<y is the graph of /, a regular
imbedding since & is non-singular. (Consider the fibre square
4 I'
10.1 Families of Cycle Classes
179
Factor the lower homomorphism into Y, -> Y, xY, -»'yx "& and into
y,->*->.-j'x.f, and apply Theorem 6.5. In fact, one sees from this that the
two classes agree in Ak+I_n((\ix\ n/~'(|/J|)),).)
Example 10.1.6 (cf. Samuel C)). Let T be a unirational variety over an
algebraically closed field.
(a) Any two points of T can be joined by a chain of rational curves. (By
standard reductions, it suffices to prove the following case. If n: 7"-> Am is the
blow-up of Am along a subscheme Z, with exceptional divisor E, and t e E,
then there is a morphism /i: A1 -+T with h @) = / but h (A1) * ?. To see this,
let C be a non-singular curve, # : C -> r a morphism with # (P) — t, g (C) ct ?.
Let X\,..., xm be the coordinate on A, and let z,,..., zr be the generators for
the ideal of Z. Choose an integer N larger than ordPGt^)*(zy)) for 1 ^jsr.
Choose h:A'-+Am so that the power series expansion of each h*(xf) at 0
agrees with that of {ng)*{xi) through order N. Then h = nh, with h as
required.)
(b) If T is complete, then Ao T= 7L, generated by the class of any point of T.
(Use Theorem 1.4.)
(c) If T is not complete, then Ao T= 0. (If T is open in f, and P eT-T,
then [P] generates A0T, but the restriction from A0T to A0T is surjective and
maps [P] to 0.)
It had been conjectured by Severi that a complete non-singular surface T
with AqT= Z must be rational, but a counter-example is given by Bloch-Kas-
Lieberman A).
Example 10.1.7. Let 7" be a non-singular variety over an algebraically closed
field, such that any two points of T can be joined by a chain of rational curves,
(a) Let Y be a scheme, a a (&+m)-cycle on Yx T. Then all the cycle classes
a, eAii(Yx{t})=AicY are equal. (Using Theorem 6.5 one may reduce to the
case where 7" is an affine open subset of A'. In this case the pull-back
q*:AkY-*Ak+l(YxT)
is surjective, where q : YxT'-» Y is the projection (Propositions 1.9 and 1.8). If
i, is the imbedding of Y in YxT at /, and a = q*fi, then if (a) = if q*fi =
(b) Let X, Y be schemes, and let
be a morphism, i.e. a family of morphisms f,:X-*Y parameterized by T.
Assume that the morphism
F:XxT-+YxT
defined by F(x, t) = (f,(x), t), is proper. Then each /, is proper, and the
induced morphisms
(f)AXAY
are independent of / e T. (This follows from (a), since, by Proposition 10-1 (a),
(//)¦(«) = (^(«x [71)),.)

180
Chapter 10. Families of Algebraic Cycles
From (a) is follows in particular that the criterion for rational equivalence
in § 1.6 can be enlarged to include families of cycles parametrized by arbitrary
unirational varieties (cf. Samuel E)).
Example 10.1.8. If y-+ T is given, and a is a positive (k + m)-cyc\e on '»,
then each class a, can be represented by a non-negative cycle. (Choose, in a
neighborhood of t, m principal divisors D\,...,Dm whose intersection is {/}.
By construction, intersecting with a divisor whose normal bundle is trivial
preserves non-negative cycles.) See Chap. 12 for generalizations.
Example 10.1.9. Let C be a non-singular projective curve, C("' its «th
symmetric product, whose points are effective divisors of degree n on C. For
an effective divisor A on C, set
XA = {DeCW\D^A}.
If deg/f = a, XA is the image of the imbedding of C("-a) in C(n) which takes E
to A + E. For any effective divisors A, B on C,
(*) [XA] ¦ [XB] = [XA+B]
in An_a_b(XA(\XB), a = deg(A), b = deg(B). (If A and B are disjoint, the
intersection is transversal. For the general case let T= C(<1), with / e T cor-
corresponding to A. Let S" (resp. SrB) be the image of the inclusion of C-"~a)y.T
(resp. C'-'-^r) in C^xT which takes (E,D) to (E + D,D) (resp. to
(E + D + B,D)). Then
[3"
as cycles on C{n) x T, since the intersection is transversal over the open set of T
consisting of divisors which are disjoint from B. Since S~ specializes to XA, and
%"b to Xa+b, aX t eT, (*) follows from the preceding equation and Corollary
10.1.) The corresponding equality in Aa+b(Cin)) was proved by Mattuck A),
using a linear equivalence between A and A'—A", for divisors A' and A" which
are disjoint from B.
Example 10.1.10. Given S"
7'then
T, and a cone 4 over S", such that € is flat over
in A* (X,) for / e T. (This follows the definition of Segre classes in § 4.1 and
Proposition 10.1 (b), (d), and (a).)
10.2 Conservation of Number
The following proposition is the basic fact on which the principle of continuity
depends.
Proposition 10.2. Let p:&-*Tbe a proper morphism, T an m-dimensional
variety as in § 10.1. Let a be an m-cycle on '». Then the cycle classes a, eA0(Yt)
all have the same degree.
10.2 Conservation of Number
Note that deg(a,) is calculated by regarding Y, as a scheme over x(
is the induced morphism from Y, to {/} = Spec (x (/)), then
181
): if P,
Proof. Let;?* (a) = N ¦ [T], N e Z. Then by Proposition 10.1 (a),
/•«¦(«.) = (/>¦(«)), = N-[T}, = N- [{/}].
Therefore deg (a,) = N, for any t eT. ?
We have seen in Proposition 10.1 that our basic intersection operations
preserve families of cycle classes. Combined with Proposition 10.2, if a
sequence of these operations are applied to families of cycles, resulting in a
family of 0-cycles whose support is proper over T, then the degree of these
zero cycles will not vary in the family. The following theorem suffices for most
applications.
Theorem 10.2. Let i: X -> *& be a regular imbedding of codimension d,
p : °y -*T aflat morphism, such that pi is also flat. Letf: "V —> ®)be a morphism,
and form the fibre square
Assume thatWis proper over T. Then for any (d+m)-cycle a onT, the degree of
the 0-cycle classes
in A0(W,) is independent oft. More generally, if P is a polynomial of weight e in
the Chem classes of a collection of vector bundles on W, and P, denotes the same
polynomial irt the Chem classes of the restrictions of these vector bundles to W,,
and a is any (d+ e + m)-cycle class on T, then
is independent oft eT.
Proof. By Proposition 10.1 (c) and (d),
The assertion therefore follows from Proposition 10.2. ?
Corollary 10.2.1. Let Y be a scheme, and let 3^i <= Yx T be effective Cartier
divisors which are flat over T, i = \, ...,d. Let a be a d-cycle on Y. Assume that
is proper over T Then
is independent oft.
deg (G/,),-.. ¦¦(#*),-a)

182
Chapter 10. Families of Algebraic Cycles
Proof. Apply the theorem to the situation
i
, x ...
> r
> (YxT)x...x(YxT)
= Supp(a) xT,5is the diagonal imbedding, and a = a x [T] eAd+my. O
Corollary 10.2.2. Let Y be a non-singular n-dimensional variety. Let a, be
r
{ki+m)-cycles on YxT, \Si^r, with ?&,= (/•-!)«> and assume that
H Supp(ot;) is proper over T. Then
is independent oft.
Proof. Apply the theorem to the situation
0 Supp (a,) (-Suppf^) x ... x Supp(ar)
i I
Gx7) > {YxT)x...x{YxT)
with <5 the diagonal, a = a! x ... x ar. ?
Example 10.2.1. Let / :#¦->'#• be a regular imbedding of codimension d,
^<= v a subscheme with S",'?/, y flat over T. Assume that there is a non-empty
open subset T° a T such that X, meets F, properly for all t eT°.
(a) For any t eT, X,-y, V, is represented by a non-negative cycle. (Since S"
meets T properly, 3T-ry is represented by a non-negative cycle. Then use
Proposition 10.1 (c) and Example 10.1.8.)
(b) Assume that 3C{\Y is proper over T, and dim(f) = d + m. Let
N = deg(X,-Vt) for teT". Then for any teT, X,f]V, is either positive
dimensional, or a finite set of cardinality 5g N. CC ¦ Y is an effective m-cycle
whose support is ?Tlf. If X,f]Vt is finite, (X-Y), is an effective cycle of
degree N whose support is X,f] Vt.)
In fact, (b) follows from a more general assertion. Assume that the ground
field is algebraically closed, that dim(f) = d + m, and 9C{\Y is separated
over T. Suppose M is an integer such that X, f] V, has at most M points for all
t e T°. Then, if X, f] V, is finite, it has at most M points. (The essential point
for this is that each irreducible component H^ of SC 0 "V has dimension at least
m. One may reduce to the case where T is a non-singular curve, and each yrt is
a curve mapping dominantly to T. In this case it is elementary to prove that
the number of points in C^), is no more than the separable degree of the
extension R (?r,)/R (T). See [EGA]IV. 15.5 for generalizations.)
As in Corollary 10.2.2, analogous results hold for families of intersections
of pure dimensional schemes on a non-singular variety.
Example 10.2.2. Let lt,..., /4 be four general lines in P2, and let P be a point
of P2 not on any /;. If P is not on one of the three "diagonal" lines joining pairwise
intersections of the four lines, then there are precisely two non-singular conies
10.2 Conservation of Number
183
tangent to all the /; and passing through P. If P is on one diagonal there is one
such conic, while if P is on two there are none.
Let Hi <= P5 be the hypersurface of conies tangent to /;, and let HP be the
hyperplane of conies passing through P. For general P, Hl[\...P[H^.C\Hr
consists of the Veronese of double lines (whose equivalence for the intersection
is 14, cf. Example 9.1.8), and two points where the hypersurfaces meet trans-
versally. As P approaches one (or two) diagonals, one (or two) of these points
approach the Veronese.
In such circumstances, classical geometers maintained the conservation of
number by counting limiting solutions as well; here one (or both) doubled
diagonal would be counted as a conic. Since it is not always obvious from the
statement of an enumerative problem which degenerate solutions should be
counted as "limiting", this contributed to the controversy over the principle.
If the ambient space Y is taken to be the open subspace of P5 correspond-
corresponding to non-singular conies, then the intersection of the five hypersurfaces, as P
varies, is not proper over P2 — U*.i ',• For the validity of Theorem 10.2 it is
necessary that the "intersection scheme" iV be proper over the parameter
space T.
Example 10.2.3 (cf. Study A)). Over an algebraically closed field, identify
the set of 4-tuples of points on P1 with P4 as usual: the divisor ??=i [(s;: J,)]
corresponds to (x0:... x4) if 2*=0 xj & T*~' = n?= i ih S - st T). Let T be the
open set in P4 consisting of distinct 4-tuples. The set G of automorphisms of P1
is identified with an open subset of P3, the automorphism with matrix (yu)
corresponding to (yu : yl2: y2l : y22). Let
-T= {(o,D) e GxT\ a(D) = D}.
Then y is a closed algebraic subset of GxT, and for an open set T° of T, Tt
has 4 distinct points. However, for / e T corresponding to 4-tuples with cross
ratio -1 (resp. a non-trivial cube root of 1), V, has 8 (resp. 12) points (cf.
Semple and Roth A)XI.6).
This does not contradict the principle of continuity, since V cannot arise as
a proper intersection; y is locally defined by 4 equations, and has irreducible
components of dimension 3 as well as 4.
Example 10.2.4. Let Y2 <= P3 be the cone x] = x2 + xj, with vertex
P= A : 0:0:0). Let Fbe the line x, = 0, jc3 = jc2. Let T= A1, with coordinate u,
and let
Sf^YxT
be the family of Cartier divisors x2= uxo- The general X, is a non-singular
curve meeting V transversally in one point, but A'o is the union of two lines,
each meeting Fat the point P. By Corollary 10.2.2, such a phenomenon cannot
occur on a non-singular variety Y. Note that in this example the intersection
product Xo ¦ Fof Fby the Cartier divisor Xo is 1 • [P].

184 Chapter 10. Families of Algebraic Cycles
Example 10.2.5 (cf. Zobel C)). Let Y3<=P* be the cone xlx4 = x2x3. Let
ra YxA1 be the family of lines defined by the equations
X] = 0, x3= 0, x2= ux0,
with u the coordinate on A1. Let V <= Y be the plane xl = x2, x3 = x4. Then
for general t e A1, F, and F' are disjoint, while Fo meets F' in the vertex of the
cone. Again, such jumps can occur only at singular points.
Example 10.2.6. Let T = A\ with coordinate », and let 7=P2. Let
% a Y x P1 be the family of lines
3r={(x:y:z)x(u)\y=ux}.
Let Va Y be the line >» = 0. The classes X, ¦ V e Ao (V) are all equal as / varies
in T. As happens when a proper intersection degenerates to an improper inter-
intersection, the normal cones to X,f] V in F, and the distinguished varieties for X,- V,
do not vary continuously. This phenomenon will be studied in Chap. 11.
Example 10.2.7. In Theorem 10.2, let 3T,,... ,yrr be the connected com-
components of^ Let a be a (d-t-m)-cycle on^, and let
with Xi(t)eA0((lVi),). For any i such that
independent of/. (Writer- a = Y, h, k eAm
Example 10.2.8 ("Compact supports"). For any scheme X, define
is proper over T, deg(l,(f)) is
\tnen -*/(') = (-*<)/•)
where the limit is over all closed subschemes Z of X which are complete. If X
is complete, A%X=AifX. For any morphism f:X-+Y, there are induced
homomorphisms
U:A%X^A%Y
which are functorial. In particular, mapping X to a point, elements of A%X
have a well-defined degree.
In addition, there are functorial Gysin maps
for morphisms f-.X^Y which are flat and proper, or l.c.i. and projective.
Chern classes of vector bundles on X operate on A*X. If Y is non-singular, and
f:X-*Y is a morphism, there are cap products A%X® A*Y'-»A^X'. When X is
the complement of the singular locus of a projective variety, such groups have
been studied by Collino B).
Example 10.2.9. Let Y be a scheme, S"t c^YxT regularly imbedded of codi-
mension d,, with 3f< flat over T, 1 ^ i S r. Let /:X-> Yx T be a morphism, and
let tT" = H/"' (#;). Let a be a (/c + ™)-cycle on n Let e = k - ?1= t d,. Then
there is a (e + /n)-cycle [i on if such that
(Z,),-...-(JIG),-(X, = ?,
10.3 Algebraic Equivalence
185
in ^4P ((^,) for all t eT. If P is a polynomial of degree e in Chern classes of
vector bundles on^, with restriction P, to Wt, and ^is proper over T, then
deg (P,n
•«,))
is independent of / e T.
10.3 Algebraic Equivalence
In this section the parameter spaces T will be non-singular varieties, and t eT
will denote a point which is rational over the ground field. If A1 is a scheme,
and a is a (A: + m)-cycle onXxT, m = dimT, then Xx{t} = X for /e T, and the
class a, constructed in § 10.1 is a class in
Definition 10.3. Let A" be a scheme. A ?>cycle a on X, or a class a eA^X, is
algebraically equivalent to zero, written a~aig0, if there is a non-singular
variety T, a cycle a eAk+m(XxT),m = dim 7", and points /], t2e Tsuch that
a=a.,-a.,,
in AkX. Two ^-cycles are algebraically equivalent if their difference is
algebraically equivalent to zero (cf. Example 10.3.2). From the following pro-
proposition, the cycles algebraically equivalent to zero form a subgroup of 2kX.
The group of algebraic equivalence classes will be denoted BkX:
BkX= ZfcAV~aig.
Proposition 10.3. The cycles algebraically equivalent to zero form a subgroup
of the group of all cycles on a scheme. This subgroup is preserved by the basic
operations:
(a) Proper push-forward (§ 1.4)
(b) Flat pull-back (§ 1.7)
(c) Refined Gysin homomorphisms (§ 6.2)
(d) Chern class operations (§ 3.2).
The groups 5*X therefore satisfy the same formal properties as A*X.
Proof. For the first statement, suppose a and b are two ^-cycles on X which
are algebraically equivalent to zero. Suppose a = ah— a,2 as in the definition,
and b = /?„,— /?„, for a (k + «)-cycle /? on Xx U,n = dim U. Set
y=ax[U]-fix[T],
a (k+m + «)-cycle on Xx Tx U. It suffices to show that
in /ifcX This follows immediately from the identity

186
Chapter 10. Families of Algebraic Cycles
for any t e T, u e U. This identity in turn follows from the functoriality of
refined Gysin maps (Theorem 6.5): if t (resp. u) denotes the morphism from
the point Spec (K) to T (resp. U), then (/, u) is the composite AT x u) ° /, so
(« x [?/])<,.„) = /!(A r x «)'(« x [?/]) = t'{a) = a,.
That the four operations (a)-(d) preserve algebraic equivalence follows from
the corresponding parts (a) - (d) of Proposition 10.1. ?
Example 10.3.1. If Y is non-singular, the classes in A*{Y) which are alge-
algebraically equivalent to zero form an ideal, so B* (Y) is a commutative, graded,
ring with unit. If /:X-* Y is a morphism, 5*X is a 5* K-module; if Z is also
non-singular,/* : 5*7-> B* X is a homomorphism of graded rings.
Example 10.3.2. Assume the ground field is algebraically closed. Two k-
cycles a, a' on a scheme Z are algebraically equivalent in the sense of Defini-
Definition 10.3 if and only if there is a non-singular variety T, of dimension m, with
(& + m)-dimensional subvarieties t; of XxT, flat over T, |s/sr, and points
tl,t2eT, such that
r
In addition, one may achieve this with Ta. projective, non-singular, curve. (If a
and a' are algebraically equivalent, by connecting points in parameter spaces
by chains of curves, one first constructs r non-singular affine curves Th sub-
varieties^ cZx 7", projecting dominantly to Tt, and tn,tn e Tsuch that
Here one uses the fact that if /: T-* 7" is a morphism of non-singular
varieties, / e T, and a is a cycle on XxT', then, by Theorem 6.5, a/@ = {fa),.
Taking closures of the ^ in Xxfh where f, is a projective non-singular
completion of Tit one may have the same equation with all T< projective. Set
T = 7", x ... x 7V
/,• = /y X,..X/,j, 7=1,2
to achieve the required equation, with dimT= r (cf. Baldassarri A)VI.7). To
obtain r = 1, by induction, it suffices to show that any two points on the
product of two curves can be joined by an irreducible curve; for Rosenlicht's
simple proof of this last fact, see Weil EI: Lemma 5.)
Example 10.3.3. Two positive cycles a, a' on X are algebraically equivalent
if and only if there is a positive cycle b on X such that a + b and a' + b are
members of an irreducible family (Chow variety) of positive cycles. (Use
Example 10.3.2 and proceed as in Example 1.6.2.)
Example 10.3.4. If y<= X, U = X-Yas in Proposition 1.8, then the sequence
is exact. With this and Proposition 10.3 other general calculations for A* (e.g.
§ 3.3, § 6.7) extend without change to 5*.
10.4 An Enumerative Problem
10.4 An Enumerative Problem
187
A typical problem in enumerative geometry (cf. Schubert A)§ 1) is to find the
number of geometric figures in a given family which satisfy certain conditions.
Such a condition would usually require the geometric figure to have a certain
relation with a given configuration, which is assumed to be in general position.
If a parameter space for the given family is a non-singular, r-dimensional
variety S, those figures satisfying a given condition form a closed subset of S,
which will be a hypersurface if the condition is "simple". The problem
becomes one of finding the number of points in the intersection of r hyper-
surfaces H\,..., Hr in S. This problem of intersection theory is seldom
straightforward, however, because one wants to count only non-degenerate
solutions to the problem. The parameter space S for non-degenerate solutions
will usually not be compact (complete). If the family is compactified to S, the
hypersurfaces H\,...,H, may intersect in the locus S-S of degenerate solu-
solutions, even with an excess component.
Consider for example the problem of how many (non-singular) plane
conies are tangent to five given lines. If the conies are parametrized by P5,
those tangent to a given line form a quadric hypersurface. The five hyper-
hypersurfaces all contain the Veronese of double lines, whose equivalence for this
intersection problem is 31. Of the 25=32 solutions predicted by Bezout's
theorem, only one corresponds to a non-singular conic (cf. Example 9.1.8). Of
course, the fact that there is one such conic can also be seen easily by
considering the dual problem.
One may proceed similarly to count the number of conies tangent to five
given conies. The conies tangent to a given conic form a hypersurface of degree
6, and the equivalence of the Veronese is 4512, leaving
65-4512 =3264
as the required number (cf. Example 9.1.9).
In general it is difficult to calculate the equivalence of an excess degenerate
component directly. For example, if the plane curves of degree d are param-
parametrized by a projective space of dimension d(d+ 3)/2, the locus of curves with
multiple components, for large d, is a complicated singular subset.
A classical procedure for such problems is to degenerate the given figures
to simpler ones, where the number satisfying the simpler conditions can be
computed, and then to appeal to the principle of conservation of number. Of
course, for this to be valid, there must be some compactness assumption. One
method is to construct a better (non-singular) compactification of S, where the
intersections are proper and correspond only to non-degenerate solutions. In
the example of conies, the space of "complete conies" — the blow-up of P5
along the Veronese, is such a compactification.
Using refined intersections and the version of continuity given in § 10.2, it
may not be necessary to construct such a compactification. If the degeneration
of the given figures is parametrized by a variety T, what is essential is that the

188
Chapter 10. Families of Algebraic Cycles
intersection of the corresponding varying hypersurfaces form a schemed which
is proper over T. Or, if a compactification 5 is used, the components of yr
contained in SxT should not meet any components of yr contained in
(S-S)xT. In these cases, by Corollary 10.2.1, the degree of the intersection
cycle of the varying hypersurfaces in 5 will be independent of / e T. Of course,
if non-degenerate solutions (in S) approach degenerate solutions (in 5 — S) as
/ varies, this will no longer be true (cf. Example 10.2.2).
We will illustrate these ideas by sketching a solution to the following
problem:
Given an r-dimensional family of plane curves, and r curves in general position
in the plane, how many curves in the family are tangent to the r given curves?
The solution is as follows. Let Du...,Drbe the given curves, «,- the degree
of ?>,, and m, the class of Z), (i.e., the number of lines passing through a given
general point and tangent to ?>, at some simple point). If the ?>, are in general
position, the number of curves in the family tangent to Z),,..., Dr is
r
II (m, n + rii v).
i-\
This formula is to be interpreted as follows. Expand the polynomial formally:
r
jif
so
/•={1 r}
card (P) = k
and substitute for /xkvr~k the corresponding characteristic of the family, i.e.,
Hkvr~k= the number of curves in the family passing through k general points
and tangent to r— k general lines.
For example if the given family is the family of all conies, then (in
characteristic not 2, cf. Example 10.4.3)
Thus the number of conies tangent to D i,..., ZM is
No + 2 Nt + 4tf2 + 4Nj + 2N* + N5
with Nk as above. In case the D, are all non-singular conies (in characteristic
not 2), n, = m, = 2, Nk — (i,) ¦ 25, and the number is
Vv
25A+2-5 + 4-10 + 4-10 + 2-5+ 1) = 3264.
In this problem it shall be assumed that the curves D[,...,Dr have no
multiple components, and that the ground field is algebraically closed of
characteristic zero. The proof proceeds in several steps.
10.4 An Enumerative Problem
189
Step 1. The set of lines in P2, denoted P2, is identified with a projective
plane with homogeneous coordinates a, b, c; the point (a : b : c) corresponds to
the line
(*) ax + by + cz = 0.
The incidence correspondence I <= P2 x P2,
I={(P,L)\PeL),
is the non-singular 3-fold defined by the global equation (*). If one defines the
bundle E on P2 by the exact sequence
0
,93 (*>¦*)
Ip2 >
0,
then I=P(E). If X (resp. Q is the pull-back of c,
then
(resp.
to I,
and A* (I) is a free Z-module with basis
This follows from the isomorphism (Example 8.3.4)
A*(P(E)) = A*(P2) [(]/(C2 + Cl(?) ( + c2(?)),
and the formula c (E) = c (^pjA)) ' = 1 — X + X2.
For a line Min P2, set
M ={(P,L)e I\L = M),
M" = {(P,L)e I\P e M}.
For a point Q in P2, set
Q' ={(P,L)eI\P=Q},
Q"={(P,L)eI\QeL).
Then X = [M"], C = [Q"], X2 = [Q'}, C2 = [M']. From this one sees also that
Step 2. Let D be a curve in P2 with no multiple components. Define the
(reduced) curve D' <= I to be the closure of
{(P, L) ? 11 P is a simple point of ?>, and L is tangent to D aiP}.
If h and m are the degree and class of D, then
(**) [D'\ = n [A/'] + m [?>'] = « ?2 + m X2
in /42/, where Mis a line, Q a point, as above. This can be seen by intersecting
[?>'] with the dual basis X, ( of A{{I). For a general line M, D' meets M"
transversally in n points (P,, Lf), where P, ? MOD, and L, is the tangent to D
at Pj\ this shows that the coefficient of ?2 is n. The dual argument gives the
coefficient of X2. (The fact that the intersections are transversal follows from
the general considerations in Step 3 below.)

190
Chapter 10. Families of Algebraic Cycles
The rational equivalence (**) may be constructed explicitly. Choose a
general point Po and a general line M. Let Q\,..., Qm be the points where the
tangent lines to D through Po meet M.
By gradually deforming D onto M, projecting from Po, we construct a
rational equivalence from [?>'] to the cycle
(**')
W'\ + E [0].
We may assume Po = @ : 0 : 1) and M is the line z = 0; let F(X, Y,Z) = 0 be
the equation for D, and let F{, F2, F3 be the partial derivatives of F. For
t e A', / 4= 0, let a, be the automorphism of P2 given by
a,(x:y:z)=(x: y.tz).
Then a,(D) is defined by F{tX, tY,Z) = 0. As / varies in A1 - {0}, there is a
family of curves at{D)' in /. There is a unique extension of this family to a
surface &' in / x A1, flat over A1. The claim is that the fibre 7N of &' over / = 0
is a curve whose cycle is n [M'] + X [Q'i\- To verify this, it suffices to show that
D'q is set-theoretically equal to the union of M' and the Q'it for then the
previous calculation (**) determines the coefficients. If / 4= 0, a point in a,(D)'
will have the form (x : y : tz) x(a : b : c) with ax + by + ctz = 0, F(x, y, z) = 0,
and
(a:b:c)=(tFl(tx, ty, tz): tF2(tx, ty, tz): F3(tx, ty, tz)),
with not all Fj(x,y, z) = 0. The displayed equation is equivalent to the three
equations
aF1(x,y,z) = bFl(x,y,z),
aF} (x, y,z) = tcFi (x, y, z),
bF3(x,y,z) = tcF2(x,y,z).
One may set / = 0 in these equations to find D'o. For the point (x:y:0) x(a:b:c)
to be in D'o either a = b = 0, i.e., the point is in M'\ or there must be a z so that
F(x, y, z) = 0 and F3 (x, y, z) = 0, with (x : y : z) simple on D, i.e., (x : y: 0) is
one of the Qh so the point is in Q'j. (Singular points of D also give rise to
points satisfying the above equations, but since only a finite number of lines
through singular points are limits of tangents to nearby simple points, they
cannot produce curves in D'o.)
Step 3. Let & <= P2 x 5 be a family of plane curves, flat over S, with 5 a
non-singular, r-dimensional variety. Assume the general curve Xs of the family
has no multiple components. Let 5° be any non-empty open subset of 5 such
that Xs is reduced for every s e S°. For example, if the general curve of the
family is non-singular, S° may parametrize the non-singular members of the
family.
Define the variety %~{r) to be the locally closed subvariety of / x ... x 7 x 5°
(r copies of 7):
ST(r) = {(Pu L{) x ... xGJr, 7_,) x s | each P, is a simple point of Xs, and L, is
tangent to Xs at 7>,-}.
10.4 An Enumerative Problem 191
Then J"(r) is smooth, of dimension 2r. Let
(p:&(r) -> 7x ... x 7 (r factors)
be the projection
Given reduced curves D|,..., 7), in P2 form the fibre square
W >D\ x...x D'r
(***) 1 1
Let G be the product of r copies of the automorphism group GL(P2) of P2.
Since G acts transitively on 7 x ... x 7, for an open set of a = (u,,..., <r,) e G, if
D, is replaced by cr,G),), the above diagram is differentiably transversal
(Appendix B.9.2), and W consists of N (reduced) points. Moreover, if
# c P2 x ~S^ is any compactification of the family, and S"(r) is the closure of
?"(r) in 7x ...x 7x~Sr5~, and Z is any closed subset of S"(r) of dimension less
than 2r (containing all of 3T(r) -3T(r)), then
for an open set of a e G. In particular, discarding any proper closed subset
from 5 does not affect the number N of solutions.
Step 4. Now construct a degeneration of each 7), to a multiple line as in
Step 2, using a different general point and line for each curve. The product of
these degenerations is parametrized by Ar. We have a diagram
¦w —^x.-.x^;—>Ar
with the square a fibre square. Since %~(r) is complete, ^is proper over A'.
The fibres of ^ over /,=A,..., 1) and /0 = @,...,0) are finite, and disjoint
from any given proper closed subset Z of J"(r), for generic position of the D,
and the points and lines. Therefore if the above degenerations are restricted to
an open neighborhood T of {/0, /|} in Ar, then yf will be proper over T and
disjoint from Z. Now by Corollary 10.2,
deg(iT(r) ;D\ x ... x D'r) = deg(iT(r) ;(E\ x ... x E',))
where D', (resp. E'i) is the fibre of &\ over 1 (resp. 0). The left side is TV, the
number of tangencies. Writing out the E\ according to (**'), the right side
expands to give the required formula.
One additional point must be verified. This intersection actually gives the
number
card {{Px, Lx) x ... x (Pr, Lr)xse 3T(r) | L, is tangent to 7), and Xs at 7*,}.
To obtain the original solution, one should check that, with the general
position assumption, each 5 6 5 will have at most one point in the above set;
any bitangents can be put in a proper subvariety Z as above. Similarly, one
sees that the TV tangents that occur in the solution are all simple tangencies of
3T(s) and each D,.

192 Chapter 10. Families of Algebraic Cycles
For other approaches to this problem, we refer to Grayson C), and Fulton-
Kleiman-MacPherson A), cf. Example 14.7.18.
Example 10.4.1. The class of a non-singular plane curve of degree n (in
characteristic zero) is n(n- 1) (cf. Example 4.4.5). The number of conies
tangent to five non-singular curves of degree n, in general position, is
N=ni({n-\M+\0(n- IL + 40(« - lK + 40(n- 1J+ 10(n- 1)+ 1).
If n = 2, this gives 3264. If n = 3, N= 168,399, while if n= 11, N = 39,312,710,151.
Example 10.4.2. A circle in the plane is a conic passing through the two
"ideal" points A : ± l/^T: 0). The circles form a P3, with a(x2 + y2) + bxz
+ cyz + dz2 = 0 corresponding to the point (a : b : c : d). The circles tangent to a
given line, and the circles tangent to a given circle, form quadric surfaces in P3.
There are 8 circles tangent to three circles (in general position), or to two
circles and a line, or to a circle and two lines, but there are only 4 circles
tangent to three lines in general position. (In the last case, the double line
z2 = 0 corresponds to a point in the intersection of the three quadrics, and the
intersection multiplicity of the three quadrics at this point is 4.)
The number of circles tangent to three curves of degrees nl,n2,n3 and
classes w,, m2, w3, in general position, is
where ^3= \,fi2v = 2, ^ v2=4, and v> = 4.
When the three given curves are real, this analysis does not say how many
real circles satisfy the conditions. For lines and circles in various positions this
number is easy to determine, and the numbers given above can all be achieved
by appropriate real configurations. In general counting the number of real
solutions to enumerative problems appears to be very difficult.
Example 10.4.3. In characteristic two, the class of a nonsingular conic is 1.
The characteristics of the family of all plane conies in characteristic two are
fis = n* v = fi2 v2 = 1, jx2 v3 = n v4 = v5 = 0 .
The number of conies tangent to five given conies in characteristic two is
C" + 2vM = 1 + 52+ 10-4=51
(cf. Example 9.1.9, and Vainsencher C)).
Example 10.4.4 (cf. Example 14.7.18 and Fulton-Kleiman-MacPherson A)).
The analysis of this section extends readily to finding the number of varieties
in an r-dimensional family of varieties in VN which are tangent to r given
varieties in general positon in P". For example, given an r-dimensional family
of curves in P3, the number tangent to r given surfaces in general position is
r
II (r»i v + «,- q)
Notes and References
193
where «, is the degree of the ;th surface, m, its first class (i.e. the number of
points in a general plane section at which the tangent plane passes through a
fixed general point), and the characteristic v' q'~' is the number of curves in the
family tangent to / general lines and r - i general planes. For the family of all
(plane) conies in P3, these characteristics were found by Chasles, cf. Schubert
A)§ 20:
v" = 92, v7e=116, vV=128, v5 e3 = 104, vV=64, vV = 32,
ttf=\6, ve7=8, e8 = 4.
Thus the number of conies tangent to 8 general quadric surfaces is
B v + 2fj)8 = 4,407,296.
Similarly, there are 666,841,088 quadric surfaces in P3 tangent to 9 given
quadrics in general position.
A method for calculating the characteristics for the family of all quadrics of
dimension m in P" was given by Schubert D), based on the beautiful geometry
of complete quadrics. This was reconsidered by Semple A) and Tyrell A), and
recently by Demazure, Vainsencher, De Concini and Procesi.
Notes and References
The principle of conservation of number, known also as the principle of special
position, or the principle of continuity, has a long and stormy history. From the
time of Poncelet, it has been the basic tool of enumerative geometry. Early
justifications were based on the fact that the number of solutions of a
polynomial equation, when properly counted, remain constant when the
coefficients of the polynomial vary continuously.
Controversy has arisen when the principle has been asserted as a general
law regarding variations of geometric conditions. For each such formulation,
counterexamples were produced (cf. Examples 10.2.3 and 10.2.5), followed by
the addition of new hypotheses to rule out such examples, and so forth. When
the principle has been founded on intersection theory on appropriate param-
parameter spaces, these difficulties have disappeared. Severi D) made this point in
1912 to settle the dispute then raging. As we have mentioned, Cayley B) had
advocated an intersection-theoretic approach to enumerative problems in 1868,
before most of the controversy arose. Perhaps the desire to keep geometry
"pure", unsullied by coordinates or parameter spaces, contributed to the
resistance to this idea.
The articles of Zeuthen and Pieri A), Berzolari B), Dieudonne A) and
Kleiman G) are recommended for discussions of the role of the princi pie of
continuity in enumerative geometry, and for additional references.
It should be pointed out that there is not a single principle of continuity for
all situations. As intersection theory develops, those constructions which can be

194
Chapter 10. Families of Algebraic Cycles
proved to vary in families lead to stronger formulations of the principle. For
example, Theorem 10.2 gives a version which is valid on singular and non-
complete parameter spaces. Severi A1) attempted definitions of intersections
on non-complete varieties, or on varieties "modulo" a closed subset. Seven's
ideas have been examined by Zobel B), D); the ideas developed in § 10.4
resemble those of Zobel.
The theory of algebraic equivalence was developed along with rational
equivalence. Modern constructions were given by Weil E) and Samuel C) for
non-singular projective varieties, using the definition of Example 10.3.2. The
use of refined Gysin maps in the present version gives a simpler construction,
as well as more general results. The basic reason for this is that, with excess
intersections allowed, we can work directly with cycles on the ambient space of
a family even if they do not meet all fibres properly.
Results similar to Examples 10.1.6 and 10.1.7 are stated in the seminar of
Samuel E).
Chasles A0) gave the formula for the number of conies in a one-parameter
family tangent to a given curve. Halphen, Schubert, and nearly every other
enumerative geometer, gave formulas for the number of varieties in a family
tangent to given varieties. The general formula of § 10.4 for plane curves
appears in Zeuthen C)§ 165. The proof in § 10.4 represents joint work with
MacPherson. For the generalization to higher dimensions see Zobel D) and
Fulton-Kleiman-MacPherson A).
This tangency problem is only one of an unlimited number of enumerative
problems that can be solved using intersection theory. In addition to the
classical literature already referred to, the reader will find hundreds of
examples in Lemoyne A). Recently enumerative geometry has played an
important role in moduli problems on curves, especially by Arbarello,
Cornalba, Griffiths and Harris A), Harris and Mumford A), Eisenbud and
Harris A), Harris A), and Mumford G).
In 1866 Chasles E) posed the problem of finding the characteristics ^ v> for
the family of all plane curves of degree n . For n = 3 and n = 4 answers were
given by Maillard and Zeuthen; verifying their answers is the subject of some
current research. For n > 4, the problem remains open.
For references to the transcendental study of the variation of algebraic
cycles, see § 19.3.6.
Chapter 11. Dynamic Intersections
Summary
Let X c, Y be a regular imbedding of codimension d, with normal bundle
NXY; let V be a fc-dimensional subvariety of Y, W=Xf]V, N the restriction
of Nx Y to W, and C <= N the normal cone to W in V. In Chap. 6 the intersec-
intersection class X- V in Ak_d(W) has been constructed to be s%[C], where
sN: W -> N is the zero-section.
If X ?-* Y is imbedded in a family S" <-* Y x T of regular imbeddings, with T
a non-singular curve, 0 e T, Xo = X, andTcyxTisa deformation of V, then
there is a closed set lim (X. f] Vt), contained in W, and a class we denote
\\m(X,-V) in Ak.A\\mX,V\V\ which refines XV, i.e., maps to X-V in
The Kodaira-Spencer homomorphism for the deformation Sf determi nes a
section s:, of N, and hence a class s:r[C] in s7f] (C), which also refines X¦ V. In
fact,
and, by these inclusions,
lim (A",- K,)
/»o
V.
If X, meets V, properly for generic t, then lim (AT, f]V,) has dimension k — d,
so lim (X, ¦ V,) is a well-defined cycle representing X ¦ V. If dim sJr (C) = k — d,
/-»o
this limit cycle must be Sy-[C], in which case the limit cycle is determined by
infinitesimal data.
This allows a dynamic interpretation for the distinguished varieties and
their equivalences, which can be useful for calculations. For any closed subset
Z of X, let (X- V)z be the part of X- V supported on Z (§ 6.1). If NXY is
generated by its sections, there is an open set T(Z) of sections such that for
each j e F(Z), sl [C] is a (k — rf)-cycle, and the part of sl [C] which is supported
on Z is precisely (X ¦ V)z. Thus (X ¦ V)z is represented by the part of the limit
cycle lim (X, ¦ V,) supported on Z, for generic deformations, i.e., deformations
'-•0
whose characteristic section is in F(Z). Knowing (X ¦ V)z for all Z is the same
as knowing the equivalences of the distinguished varieties.

196
Chapter 11. Dynamic Intersections
For arbitrary cycles on an arbitrary non-singular quasi-projective variety,
there is a moving lemma which asserts that the given cycles are rationally
equivalent to cycles which meet properly. Intersection cycles constructed by
this procedure represent the intersection classes on the ambient variety, but,
unlike the cycles arising from geometric deformations, they do not give
refinements of the intersection products. The moving lemma has historical
importance as the foundation for previous constructions of intersection
products, and it can be used to simplify verifications of some intersection
formulas.
11.1 Limits of Intersection Classes
In this chapter, the ground field will be assumed to be algebraically closed. We
consider families parametrized by a non-singular curve T equipped with a
given point 0 ? T. Let T* = T- {0}.
For a scheme trover T, the fibre of trover/ e T is denoted W,. Set
Let yf be the closure of yr* in yr. The limit set lim W, is defined to be the
fibre of^ over 0, i.e., "°
When W has irreducible components contained entirely in Wo, lim W, will be
smaller than Wa. If W, is purely A>dimensional for generic t e T*, then
lim W, is a purely A>dimensional (or empty) closed subscheme of Wo. Indeed,
;-»0
by discarding a finite number of fibres, one may assume all components of 7?*,
and hence also W", map dominantly to T and have dimension k + 1; the fibres
over 0 are Cartier divisors, so of dimension k.
Let cn = Y,ni[yi] be a (fc + l)-cycle on it*, with X, an irreducible
subvariety of TUT*. We define the limit cycle lim a,, a A>cycle on lim W,, as
r ,, /->0 /->0
follows:
where f'L is the closure of Tt in
-Ti over 0.
1, and
is the scheme-theoretic fibre of
Proposition 11.1. (a) If a and a' are rationally equivalent (k + \)-cycles on
yr*, then lima, and lim a', are rationally equivalent k-cycles on lim W,.
1-0 t — 0 1-0
(b) Ifd is any cycle on yf which restricts to a on W*, then lim a, is rationally
equivalent to a0 on Wo. Here a0 denotes the cycle constructed in § 10.1.
11.1 Limits of Intersection Classes
Proof. Let; be the imbedding of 0 in T. There is a diagram
Ak+\(W0)-^Ak+l(?r') ^ Ak+i(yr*) -> o
197
where /' and / are the inclusion of W'o and T* in yr'. The row is the exact
sequence of § 1.8. The composite i!/"i is zero, since (Theorem 6.2(a) and
Corollary 6.3) r /; is multiplication by the top Chern class of the pull-back to
W(, of the normal bundle to 0 in T, and this normal bundle is trivial. Therefore
there is an induced homomorphism a as indicated with cry'* = i' (see
Example 6.3.7 for generalizations). With this notation, lima, is a (a). This
1-0
proves (a), since a is well-defined on rational equivalence classes. The same
argument, using Wva. place of yr', proves (b). D
For any class a € Ak+l (yr*)t we will write lim a, € Ak (lim w\ for the class
o \o /
constructed using the preceding procedure for a cycle which represents a. Note
that this limit class is constructed from the class a on the punctured total space
>^*, not from the classes a, on the fibres.
Consider a fibre square
ar -»¦»
of schemes over T. Assume that / is a family of regular imbeddings of
codimension d, i.e., ; and all fibres /,: X, -> Y< are regular imbeddings of
codimension d. Assume that V is flat over T of relative dimension k, and Jis a
closed imbedding. Then for all t e 7" we have the intersection class
In particular, Xo- Ko ? Ak-d(W0). By Proposition 10.1 (c), X, ¦ V, = (gr -Y),,
where ?r--y e Ak+l.d(TT). We define the limit intersection class lim (X, ¦ V,) in
Ak-dl\im W, by setting
lim (X,- V,) = lim
10 0
k_d (lim XtC\K).
\i->0 /
Corollary 11.1. The inclusion of lim(X,f)V,) in X0C\V0 maps lim(X,-K\
toX0-V0. '-0
Proof. Apply (b) of the proposition to a = J" • T. ?
If X, meets V, properly for generic teT*, i.e., dim(X,C\ V,) = k - d for
generic teT*, then \\m(X,C\V) has pure dimension k-d. The class
\im(X,-V^ is therefore a well-defined (fc — d)-cyc\t on limA^DK, which we
(-¦0 ,-»0
call the limit intersection cycle. Since J"* • T* is a positive cycle whose support
is yr*y the limit intersection cycle is a non-negative cycle whose support is
lim (X, f) V,), and which represents Xo ¦ Vo.

198
Chapter 11. Dynamic Intersections
Example 11.1.1 (Fulton-MacPherson B)). Care must be taken when passing
to limits when cycles have negative coefficients, due to possible cancellation in
the limit. Let T= A1, and define families of plane curves .< 88 by
s/= V(x (y- tx) + /z2) c= P2 x T
89 = V(y-tx)<=F2xT.
Let L = V(x), M= V(y). Set a = [s/\ - [8$]. Then
For all t 4=0,
= [A:0: !)] + [(- 1 : 0: 1)]-[@:0: 1)].
[M] are both well-defined zero-cycles, but they
Thus lim (a, ¦ [A/1) and (lim a,)
/-»o \/-»o /
are not equal as cycles. If one modifies a, by adding the constant cycle
[ V(x — z)] + [ V(x + z)], one obtains an example of the same phenomenon where
both limit cycles have the same support. Of course, the two cycles are
rationally equivalent on M, since a, is rationally equivalent to a0.
Example 11.1.2. Let 5 be a section of a vector bundle ? on a scheme X, and
let Kbe a subvariety of E. Form the fibre square
I
XxA1
V
where i(x, X) ={X ¦ s(x), X), and / is the product imbedding. Then W is the
union of i"'(F)x A1 and si'(K)x0, so Wo = si1 (V), while the limit set W'o is
s~](V). The limit intersection class is s'[V] e At(s'] (V)). By Corollary 11.1,
s' [ K] maps to s'e [ K] in A* (si1 (V)). This gives another proof of Corollary 6.5.
11.2 Infinitesimal Intersection Classes
Consider a family of regular imbeddings
of codimension d, deforming a given imbedding io-.X^ Y, X=Xa. Let
A = No T be the tangent space to T at 0, and let Ax be the trivial line bundle on
A" with fibre A; equivalently, Ax = Nx^. From the fibre diagram
X-
i
• Y
I
'YxT-
0
I
T
one sees that Nx(YxT) = NxY® NX%'=NX Y ® Ax (Appendix B.7.4). From
the inclusions A"c=j"e YxT one has an inclusion of normal bundles
11.2 Infinitesimal Intersection Classes
NX3T^ Nx{YxT). This takes the form
199
Ax -^ Nx Y ®AX
where q:Ax~* Nx Y is the characteristic, or Kodaira-Spencer homomorphism.
Fixing a basis d/dt for A, g(d/dt) is a section of NXY, which we call the
characteristic section of the deformation.
Let ycz Yx T be a closed subscheme, flat over T of relative dimension k. Let
V= Ko, W= ATI K, and let JV be the restriction of NXY to W. Let % be the
section of N induced by the characteristic section of NXY. Let C= Cw Kbe the
normal cone to Win K, a closed subcone of N.
We call the class
s:'f[C]eAk-d(s7,>(C))
the infinitesimal intersection class.
Theorem 11.2. With the above notation
iim(x,nv;)c=.s-i(C)<= xnv.
These inclusions take the limit intersection class to the infinitesimal intersection
class, and the infinitesimal intersection class to the intersection class:
by the induced homomorphisms
Corollary 11.2. If dim s~r (C) = k- d, the limit intersection class is a well-
defined non-negative (k — d)-cycle, which depends only on the characteristic
homomorphism:
lim (X, ¦ V,) = s.HQ
in Ak^{sl (Q). ? '-°
Proof of the Theorem. Let 5 = g(d/dt) be the characteristic section of NXY.
We deform the given imbedding of A" in Y to the imbedding of X in Nx Y
given by s, much as in § 5.1. Let 9> (resp. S) be the blow-up of Yx T along
A"x0 (resp. of^along W), and let3s°, (resp. 3°) be the complement of BIXY
(resp. Blw V) in 9 (resp. I). Since Xc: 2C<=YxT,2C0 Y=X, and X is a Cartier
divisor on 3C, there is an inclusion i of 3C = B\X9C in 3P°, which over t = 0, is
the inclusion of X in NXY given by the section s:
X-
l
r-
+ Nx
I
* 9>a
y-
_>
0
!'•
T .
Here /0 denotes the inclusion of 0 in T. In fact, the inclusion of 3" in 9* induces
the inclusion oiX=P(Ax) in P(Nx(Yx T)) = P(NXY 8 Ax) determined by
(q, 1), where q is the characteristic homomorphism. Choosing a basis d/dt for
A identifies the complement of P(NXY) in P(NXY®AX) with NXY. The

200
Chapter 11. Dynamic Intersections
imbedding of y in YxT induces an imbedding of S in &, and hence of 9" in
&". One obtains a fibre diagram
Since %V\2- maps properly to &T\y, and
lim B" D .2), maps
r-0
isomorphically over T*,
(% f] j2H , and
lim (SC f] 2).
l->0
onto lim{X,C\VX Since
1-0
(9C f) .2)o projects (isomorphically) onto 5^' (C),
lim (X,nK)<= %'(?)•
(-•0
For any section sx: W -> iV, %' (C) c W. Likewise, the fact that s^ [C] maps
to i«[C] = (tc^)[C] in Ak.d(X(]V) holds for any section of a bundle (Corol-
(Corollary 6.5, or Example 11.1.2); since XV=s'li[C], s!x[C] = X-Vin Ak_d(XC\V).
To prove that lim (X, • V,) = s;AC] in Ak-d(sp{C)), we apply the com-
/ -»o
mutativity theorem (§ 6.4), giving
by Theorem 6.2(c). Since r[S>°] restricts toiT* -r* over 71*,
in Ak-d(sp(C)) by Proposition 11.1 (b), which concludes the proof of the
theorem. ?
11.3 Limits and Distinguished Varieties
Let X cy Y be a regular imbedding of codimension d, V <= Y a pure A>
dimensional subscheme, with k - d ^ 0. Let W= Xf] V, C = Cw V<z N, N the
restriction of NXY to W, 7t: A^ -> ^F the projection, sN: W-> N the zero section.
Let C\,...,C, be the irreducible components of C, /n, the geometric multi-
multiplicity of C, in C, so [C] = Y, w,[C,], and
11.3 Limits and Distinguished Varieties
201
in Ak_d(Xf] V). The varieties Z, = 7t(Cr) are the distinguished varieties for the
intersection (§ 6.1). For any closed subset Z of X, set
(JT-F)Z= E m^[C,] ?^_rf(Z).
For any cycle a = E "/[^1 on ^ with ^ irreducible, and Zcl closed, set
«z= E
Proposition 11.3. Assume that F is a finite dimensional vector space of sec-
sections of NxY which generates NxY. Let Z be a closed subset ofX. Then there is a
non-empty open subset f(Z) off such that
(i) For all s e F(Z), dim^C) = k-d, so s'[C] is a well-defined (k-d)-
cycle, and
(
(ii) For any deformation Sf^>Yy.T of
whose characteristic section
belongs to F(Z), lim X, ¦ V is a well-defined (k - d)-cycle, and
(MmX,-V\z={X-V)z
in Ak-d(Z).
Proof. Apply Serre's Lemma (Appendix B.9.1) to the subvarieties C, ,...,Cr
of the bundle N and the closed subsets W and Z of the base. This gives an
open set F (Z) of F such that
(a) dim s-'(O = k-d for all/,
(b) dim ($-l (QriZ)^fc - d -1, if C,¦ * tT1 (Z).
Therefore, if 5?T(Z), dimi"'(O = A:-d, so s'[C] is a well-defined (k-d)-
cycle. From (b),
E
since i"'[C,] can have no components in Z if C, <t 7c (Z). Since ^![C,]
represents sN[C] (Corollary 6.5), (i'[C])z is a representative cycle for (X- V)z,
which proves (i). Assertion (ii) follows from (i) and Corollary 11.2. D
Remark 11.3. Assume NXY is generated by a subspace F of sections, and
each section in F is characteristic for some deformation. Then the proposition
gives a dynamic interpretation for the classes (XV)Z:
is the part of the limit intersection cycle supported on Z, for a generic
deformation STczYxT, i.e., a deformation whose characteristic section is in
F(Z).
Knowing the classes (X- V)z for all closed subsets Z of X is equivalent to
knowing the equivalences of the distinguished varieties Z, of the intersection

202
Chapter 11. Dynamic Intersections
of V by X, i.e., the canonical decomposition of X- V. This may be described
explicitly as follows. For each point P e X, set
Thus / (P) is zero unless P is a distinguished variety, in which case, / (P) is the
contribution of P to X- V (cf. Definition 6.1.2). Dynamically, i(P) is the com-
component of the limit cycle lim X, ¦ V supported on P, for a generic deformation 3"
1-0
of X^-> Y. For each irreducible curve C <= X, set
j(Q = (X-V)c,
fee
Thus ;(Q = 0 unless C is a distinguished variety, in which case ;(C) is the
contribution of C to X- V; j(Q is the component of the limit cycle limAV V
supported on C, for generic 3", while / (Q is the part of lim X, ¦ V supported on
C, but not on any distinguished points of C. Inductively, for any irreducible
sub variety Z of X, set
i(Z)=j(Z)- Z HZ1),
the sum over all proper subvarieties Z' of Z. If i(Z)#=0, then Z is a distin-
distinguished variety, and ;(Z) is the contribution of Z to X- V. Conversely, if all
the contributions are positive - see § 12.2 for sufficient conditions - the
distinguished varieties Z are determined by the nonvanishing of i(Z).
Dynamically, i(Z) is the part of the generic limit cycle which is supported on
Z, but not on any proper distinguished subvariety of Z.
Suppose in addition that k = d, and that a generic section in F is character-
characteristic for a deformation 3" such that X, meets Ktransversally for generic /. Then
for each distinguished variety Z, the degree of its contribution ;(Z) is the
number of points of X,f)V which approach Z, but not any proper distin-
distinguished subvariety of Z, for a generic deformation 3". Equivalently, for any
closed subset Z of X,
deg(X- V)z is the number of points of Xtf]V which approach Z,
for a generic deformation 9C.
Example 11.3.1 (Severi A5), Lazarsfeld A)). Let Hu...,Hd be hyper-
surfaces in P", defined by forms Ft,..., Fd of degrees nu...,nd, and let V be
a pure ^-dimensional subvariety of P". Consider the intersection
d
i = 1 i i
H, V V H .-> P" V V P"
11 I A. . . . A. AI ? Jl A. . . . A Jl
Let X= H, x ... x Hd, Y= P" x ... x P". Then (Appendix B.7.4)
11.3 Limits and Distinguished Varieties
203
The space
F= {(G|,..., Gd) | G, is a homogeneous polynomial in Xo, ...,Xn of degree «,}
gives a space of sections of NxY satisfying the conditions of Proposition 11.3.
The deformation J"c= Yx A1 defined by
3-={(Pl,...,Pd,t)eYxA*\Fi(Pi) + tGi(Pi) = 0 for i=\,...,d)
is a deformation whose characteristic section is (G,,..., Gd). Using such defor-
deformations, one has the dynamic interpretation for the distinguished varieties and
their intersections given in Remark 11.3. In particular this proves a theorem
stated in Lazarsfeld A)§ 2, that the classes defined dynamically by Severi and
Lazarsfeld agree with the refined intersection classes of Fulton-MacPherson
(D,B).
Example 11.3.2 (B. Segre C), Lazarsfeld A)). Let Hu H2 be plane curves
defined by forms
F FA F FA
where A\ and A2 have no common factors. Let C be the normal cone to
Hl flH2 in P2. Define F as in the preceding example. If .s = @^ G2) is a sec-
section in F such that Ax G2 —A2 G, and F have no common factors, then s'[C\ is a
well-defined cycle:
(*)
s'[Q
-A2Gl)-V(F)
- V{A2) ,
where the cycles on the right are intersection cycles of properly intersecting
curves. In particular, for any deformations Ft+ tG, + /2G; + ... with G,, G2 as
above, the limit cycle is well-defined and given by (*). Contributions of the
distinguished varieties may be deduced from this description (cf. Example
6.1.4). (Consider the deformations (//,¦), = F,+ tG,, teA1. lfTcP2xA' is
the intersection scheme, the closured of the restriction oiWXo A1 —{0} has
equations
FA + tG0 AGAG0
FAi + tGi=0, A\G2A2G{=0.
If/, a,-, g, are local equations for F, A-,, G, in a local ring 0 for P2
exact sequence
at a point, the
0
» 0
remains exact after tensoring withtf/(alg2 — a2g\). This implies that 3^' is flat
over A1, and that [W'o] is given by formula (*). See Lazarsfeld A)§ 3 for
details.)
Segre's formula (*) is also valid for effective curves on an arbitrary non-
singular surface; the equations for the curves should be replaced by sections of
appropriate line bundles.
Example 11.3.3 (Lazarsfeld A)). Consider the plane curves Hu H2 defined
by the equations x2y = 0 and xy2 = 0.
(a) The distinguished varieties (for the intersection of the diagonal by
H\*.H2) are the lines x — 0, y = 0, and the point @:0:1). Each contributes a
zero-cycle of degree 3 to the intersection. (This may be seen from formula (*)

204
Chapter 11. Dynamic Intersections
of the preceding example, for F = xy, A{ = x, A2 = y, G, and G2 generic cubic
forms.)
(b) For the deformations
),= V((x-t2y)(y2-t2x2))
none of the nine points of (H1),f)(H1), approach the distinguished point
@ : 0:1) as / -> 0. The characteristic section does not satisfy the condition of the
preceding example. Contrary to an assertion of Severi A5), the contributions of
distinguished varieties may not be described as the minimum over all deforma-
deformations of hypersurfaces. In higher dimensions there are similar examples with
irreducible hypersurfaces, see Lazarsfeld A) p. 283.
(c) If the curves Hi and Hi are deformed by the action of the projective
linear group acting on P2, i.e., (//,¦),= (a,-), (//,), for (<r,), generic curves in
GL(P2) converging to the identity, 5 of the points in (H]),r\(H1), will
approach @:0:1) as t -* 0, while two points approach each of the lines
x = 0, y = 0. The characteristic sections of such deformations do not generate
the normal bundle to H\ xH2 in P2 x P2.
(d) If the intersection class HXH2 is constructed instead from the diagram
HlDH1^HixH1
i i
zV -P2xP2
the contribution of @:0:1) is 5, while each line contributes 2. (In this case
deformations by automorphisms of P2 give enough sections to span the normal
bundle Tr'.)
(e) If the intersection class Ht ¦ H2 is constructed from the diagram
I
H,
I
P2,
by deforming H, to V(x2y + t G) for a generic cubic G, one sees that only the
lines x = 0, y = 0 are distinguished, with x = 0 contributing a zero-cycle of
degree 3, >> = 0 a zero-cycle of degree 6.
Example 11.3.4. Let H1 = V(FAL), H2 = V(FA2) be as in Example 11.3.2.
Let f be a point in V(F)f]V(Al), but P ^V(A1). Consider deformations
(Hf),, (H2), such that (#,), passes through P for / infinitesimal, i.e. P e V{GX),
where (G,, G2) is the characteristic section. For generic (G,, G2) satisfying this
condition, the limit cycle contains P with multiplicity equal to the multiplicity
of Fat P. (Use Segre's formula (*).)
For example, of the four points of intersection of the two conies
xy + t(ax + by)z + t2(...) =0,
x(mx + ny+pz) + t(cx + dy + ez) z + t2(...)= 0 ,
only one approaches @ : 0 :1) as t -> 0, if the constants a, b, c, d, e, m, n, p are
generic (p + 0, e + pb will do). Severi A2) works out this example, and
discussed the subtlety of such limit problems.
11.4 Moving Lemmas
205
Example 11.3.5. The results of § 11.2 and 11.3 extend without essential
change to the case where the scheme Kmaps to Y by an arbitrary morphism /,
not necessarily a closed imbedding, with W = f~l (X).
11.4 Moving Lemmas
Let a, b be cycles on a non-singular variety X over an algebraically closed field.
If there are families of cycles {a,}, {/?,} parametrized by a non-singular curve T,
with a0 = a, /?0 = b, then
(i) a ¦ b = lim a, ¦ /?,
t- o
inA(X). Indeed, if a, /? are the cycles on XxT giving the families, then a,/3( =
(a • P), (Corollary 10.1), and the conclusion follows by Proposition 11.1 (b). If
a, meets /?, properly for general t, the right side of (i) gives a cycle which
represents a • b. If T is a rational curve, then
(ii)
a ¦ b = a, ¦
for all / e T. For in this case a, and /?, are rationally equivalent to a and b
respectively, and we have seen that the intersection product preserves rational
equivalence.
Such dynamic interpretations of the intersection product are particularly
valuable when the deformations arise naturally. Typical situations are: A)
divisors may move in their linear systems; B) subvarieties may move in
algebraic families of subvarieties (cf. Example 10.1.9, or § 11.3); C) if X
parametrizes a family of varieties, cycles which parametrize varieties in a
certain relation to given geometric figures will move when the geometric
figures are deformed (cf. § 10.4); D) if an algebraic group acts on X, it will
move all the cycles on X (cf. Example 11.4.5). In general, however, a positive
cycle may not move in any family of positive cycles; for example, a curve on a
projective surface with negative self-intersection number cannot be algebrai-
algebraically equivalent to an effective divisor not containing the curve. In its basic
form, the moving lemma asserts that, if non-positive cycles are permitted,
cycles can be moved to meet other cycles in the expected dimension.
Two cycles, a, /? on X are said to meet properly, if for each variety V (resp.
W) which appears with non-zero coefficient in a (resp. /?), V meets W
properly, i.e. dim (Vf) W) = dim V+ dim W— dim X.
Moving lemma. If X is non-singular and quasi-projective, and a, ft are cycles
on X, then there is a cycle a.' rationally equivalent to a such that a' meets ft
properly.
In particular the intersection product on A (X), for X non-singular and
quasi-projective, is uniquely determined by the knowledge of products of
properly intersecting cycles. Historically, moving lemmas were used for con-

206
Chapter 11. Dynamic Intersections
structing the intersection product on A(X); for this, a more delicate statement
is required, to know that the rational equivalence class of a' ¦ /? is independent
of the choice of a'. Since we do not use the moving lemma either for founda-
foundations or applications of intersection theory, we refer to the examples and literature
for the proof.
Example 11.4.1. Let X" be a non-singular quasi-projective variety over an
algebraically closed field, f:X-*U a closed imbedding into an open sub-
scheme U of a projective space Pm. Let V, W be irreducible subyarieties of X.
For a linear subspace L of Pm of dimension m — n — l, let CL(V) be the cone
over the closure of f(V), with vertex L, in Pm, and let CL be the intersection
ofCi(P)withf/.
(a) For generic L, CL meets X properly, and this intersection is generically
transversal along V, i.e.,
f*[CL] = [V) + yL,
where yL is a cycle on X which does not contain V.
(b) If dim(VTl W) = dim V+ dim W— n + e, with e>0, then, for generic
&m{Vif\W)<&.m(Vf\W).
The moving lemma follows by induction on the number e, and the fact that
cycles on Pm can be moved (Appendix B.9.2). (a) and (c) are proved by
counting constants, much as in Example 8.4.12. For details, as well as the
extension to arbitrary ground field, see Roberts A).
Example 11.4.2. Let A" be a variety over an algebraically closed field. We
say that two cycles a, /? on X meet transversally if for each variety V (resp. W)
appearing in a (resp. /?) with non-zero coefficient, each irreducible component
P of Vf) W is simple on X, and V and W meet transversally on an open set of
P, i.e., i(P, V- W\X) = 1. If X is non-singular and quasi-projective, and a, /? are
cycles on X, then there is a cycle a' on X, rationally equivalent to a, such that
a' meets /? transversally. (In the situation of Example 11.4.1, if V and Wmeet
properly, the cycle yL meets W transversally, at least after replacing the
imbedding of X in P" by a suitable Veronese imbedding. See Hoyt A) for
details.)
Example 11.4.3. With the notation of Example 11.4.1, let P be a proper
component of the intersection of V and W on X. Then, for generic, L, P is a
proper component of the intersection of CL and W on U, and
i(P,VW;X) = i{P,CL-W;U).
(By the refined projection formula of Example 8.1.7, f*[CL]-x[fV] maps to
[CLYv[W] in A*(CLf)W).). If F, CL, Ware the closures of P, CL, W in F",
then _ _ _
i(P, V- W;X) = i(P, CL- W\ Pm).)
This method was used by Severi (9) to determine arbitrary intersection multi-
multiplicities from the case where the ambient variety is projective space (cf.
Example 8.2.6).
11.4 Moving Lemmas
207
Example 11.4.4 Uniqueness of intersection numbers. We have defined an
intersection number i(P, VW\ X) for a component P of the intersection of
subvarieties V, W of a non-singular variety X, when P is proper, and simple
on X. If every component of Vf)W is proper the intersection cycle V-W =
Yj'(P> V' W\X)[P] is defined. The following properties determine the inter-
intersection numbers:
(i) If X° is an open subscheme of X meeting P, and V, W°, P° are the
intersections of A with V, W, P, then
i(P°, V° ¦ W°\X°) = i(P, V- W;X).
(ii) (Projection formula). If /:X-* Y is a closed imbedding of non-singular
varieties, C is a subvariety of Y meeting X transversally in a variety V, W is a
subvariety of X, and P is a proper component of Vf) W on X, then
i(P, V- W;X) = i(P, C- W;Y).
(iii) (Continuity). If T is a non-singular curve, y<z Xx T a subvariety, flat
over T, W a. subvariety of A" such that each V, meets W properly on X,
meets Wx T properly on Xx T, and
for all t e T; here, for a cycle a on Xx T, a, denotes the cycle on X defined in
§10.1.
(iv) (Multiplicity one) If Kand Wmeet transversally in P, then
i(P,V- W;X) = \ .
(To see that (i) — (iv) determine the intersection number, use (i) and (ii) and
the argument of Example 11.4.3 to reduce to the case where X is an open
subscheme of projective space, and P is the only component of Vf) W. Put V
in a family^so that the generic V, meets Wtransversally. Theny meets Wx T
transversally, and V- W= (T- (Wx T))o is determined by (iii) and (iv).)
Note that by the moving lemma the intersection product characterized by
(i) — (iv) uniquely determines the intersection product on A*X for any non-
singular quasi-projective variety X.
Example 11.4.5. Suppose a rational algebraic group G acts on a variety X;
e.g., G could be a product of general linear groups GL(«). For g e C let
(pg:X^Xbt multiplication by g. Then, for all g e G,
is the identity. (Use Example 10.1.7 (b)). If G acts transitively on X, then X is
non-singular, and if V, W are subvarieties on X, then (pg{V) meets W
properly for generic g e G (Appendix B.9.2). Therefore the cycles <pg{V) -(W)
represent the class V- W'mAif(X).
Example 11.4.6. Let X <=. A3 be the singular surface defined by x2 = yzz. Let
V be the >>-axis. There is no 1-cycle a on A" which is rationally equivalent to
[V], which meets Vproperly. (There is a finite morphism/: A2->A" which is

208
Chapter 11. Dynamic Intersections
an isomorphism off V. If a meets V properly, a =/* (a') for a 1-cycle a' on A2.
Since At(A2) = 0, a~/»@) = 0, but A,(X) = Z/22, generated by [K], cf.
Example 1.9.5.)
Example 11.4.7. The moving lemma fails on general singular varieties, even
if rational coefficients are allowed. Let Ic P4 be a cone over a non-singular
quadric surface Q c P3, with vertex P. Let L c X be the plane spanned by a
line in Q and P. Then no non-zero multiple of [L] is rationally equivalent to a
cycle which does not pass through P. (The homology class of L in /f4 (A1; Q) is
not dual to any cohomology class in H2(X;<Q), i.e. cl(L) =? c r\ cl(X) for all
c e H2 (X; Q). Indeed, cycles not meeting P do determine cohomology classes
on X (cf. Goresky A)), and H2(X,1) is generated by the cohomolgy class of Q,
i.e., by a hyperplane section.) This example was considered by Zobel C).
If A1 is a quotient of a non-singular quasi-projective variety by a finite
group, one can prove a moving lemma for cycles with rational coefficients (cf.
Examples 1.7.6 and 8.3.12). Mumford G) has extended this to certain varieties
- including moduli spaces of stable curves — which are locally such quotients.
Example 11.4.8. Let f:X-*Y be a morphism, with Y non-singular, n-
dimensional, and quasi-projective.
(a) If a is a &-cycle on X, /? an /-cycle on Y, then there is cycle /?' on Y,
rationally equivalent to j3, such that a meets /?' properly, i.e. dimflalf)/ |/T|)
= k + l—n. In this case the (k + I - n)-cycle a y/?' is defined, and represents
the class a y/? in A*X. (Stratifying the restriction of / to the components of
a |, one reduces this to the usual moving lemma, cf. Fulton B) §2.3.). We
know from § 8.1 that the rational equivalence class of the cycle a y/?' does not
change if /?' or a is replaced by a rationally equivalent cycle on Y or X. As
Demazure pointed out, the proof of this last fact in Fulton B) §2.3 Prop. B) is
faulty. Indeed, most foundational treatments of intersection theory based on a
moving lemma have failed to take care that all auxilliary constructions
preserve properness of intersections.
(b) Suppose X is a variety, / is dominant, and Y° is open in Y,
X°=f-i(Y°), such that the restriction f :X°^Y° is flat. For any sub-
variety V of Y whose intersection V° with Y" is non-empty, and which meets
Y— Y° properly, the class /*[F] in A%X, defined in §8.1, is represented by
the cycle [(/T'(KO)]. (Use Propositions 8.1.1 (d) and 8.1.2(a).) By the moving
lemma of (a), this determines the Gysin pull-back /*: A^ Y ->/!„ X.
Example 11.4.9 (Severi). Any &-cycle a on P" can be written as the proper
intersection of n - k divisors D |,..., Dn_k:
These divisors need not be effective, even if a is positive, cf. Example 9.1.2.
The proof uses the cone construction of Example 11.4.1. See Samuel BI1.6.4
for details.
Notes and References
Notes and References
209
The dynamic nature of intersection multiplicities is evident in all classical dis-
discussions of the subject. Seven's definitions of these multiplicities were
dynamic, realizing non-transversal or non-proper intersections as limits of
transversal intersections. Severi G) points out that these constructions anti-
anticipated their use in topology. Severi used the cone construction of Example
11.4.1 for this dynamic interpretation. Samuel C) and Chow A) used this con-
construction to prove moving lemmas, on which they based their construction
of the intersection ring ("Chow ring") of a smooth projective variety. Similar
proofs of moving lemmas have been given in Chevalley B), Samuel E), and
Roberts A). Murre A) and de Boer A) used the same cone construction to
show how to assign intersection multiplicities to connected components of
Vf] W, when V and W have complementary dimension on a non-singular pro-
projective variety. Severi A5) compares static and dynamic approaches to inter-
intersection multiplicities.
The first three sections of this chapter are a presentation of published
(Lazarsfeld A)) and unpublished work of R. Lazarsfeld, who began by
analyzing and correcting an idea of Severi A5). Limits of intersections were
also studied by B. Segre C), who gave the formula of Example 11.3.2 for the
surprisingly non-trivial case of varying curves on a surface.
Weil B) Appendix II gave an axiomatic characterization of intersection
numbers. His axioms included an associative axiom and a stronger form of the
projection formula than the one needed in Example 11.4.4. W.-L. Chow
(unpublished) has reworked his intersection theory, based on a strong moving
lemma as in Example 11.4.2.

Chapter 12. Positivity
Summary
We have constructed intersection classes by intersecting a cone C in a normal
bundle N with the zero-section. If ? #i,-[C,-] is the cycle of C, the intersection
class has a corresponding decomposition into ^m,a,, a, e/4* (Z,),
Z, = Supp(C,). If the bundle N is suitably positive, one can deduce corre-
corresponding positivity of the intersection classes, even if the intersections are not
proper.
Assume for simplicity that the restriction N, of N to Z, is generated by its
sections. Then a, is represented by a non-negative cycle. If Ni is also ample, a, is
represented by a positive cycle. If N, ® V is generated by its sections, for an
ample line bundle L, then using L to compute degrees, the degree of a, is
bounded below by the degree of Z,.
For intersections on a non-singular variety X, the positivity of its tangent
bundle will imply corresponding positivity for all intersection classes on X. For
X=V", V\,..., V, subvarieties, a refined Bezout's theorem follows:
F, ¦
Vr = Z m, a,,, II deg (V,) = ? m, deg («,) ? 2>, deg (Z,),
where the Zf are the distinguished varieties; all irreducible components of
f]j Vj are included among the Z,.
There are also applications to intersection multiplicities. For example, if
V],..., Vr meet properly at a non-singular point P of an n-dimensional
variety X, and V,c X are the blow-ups at P, then
Here the intersection class V, ¦... ¦ V, is in A0(E), E= F" the exceptional
divisor. The degree of V,-... ¦ V, is always non-negative, and one has
lower bounds as in the refined Bezout's theorem, e.g.,
deg(K,
where Wl ,... ,WS are the irreducible components of the intersection
C]jP(CpVj) of the projective tangent cones. Such positivity is noteworthy since
12.1 Positive Vector Bundles
211
the Vi may have excess intersections, and general intersections on X can be
negative. There are similar inequalities for proper intersections of divisors on
a possibly singular variety.
Notation. A cycle Y. ni [^] on a scheme X is non-negative if each nt is non-nega-
non-negative, and positive if, in addition, at least one nt is positive. Let AfX (resp. A? X)
denote the set of classes in AkX which can be represented by non-negative (resp.
positive) cycles. Thus
Both sets are closed under addition.
Let L be a line bundle on a complete scheme X. For a &-cycle or cycle
class a on X, the L-degree of a, denoted degi (a), is defined by
degi(a) = Jc,(L)*na.
x
If Kis a subvariety of X, the L-degree of V, degA( K), is defined by
degi(K) = degi([K]) = Jc, (L)diral/n [F].
12.1 Positive Vector Bundles
A line bundle L on a scheme X is ample if there is a positive integer m and a
finite morphism/from X to a projective space P" such that L®m =/*^p»(l).
In fact one may then find such mj with /a closed imbedding, although we do
not use this fact.
Lemma 12.1. If L is an ample line bundle on X, and a is a non-negative (resp.
positive) k-cycle on X, then
degi(oc) is 0 {resp. degi (a) > 0).
Proof. With/: X -> P", Z,® =/*^(l) as above,
mk degi(a) = f c, (/*^A)) n a = J c, (^A))* n/» (a).
by Proposition 2.5(c). Since push-forward by finite morphisms preserves
positivity of cycles, we are reduced to the familiar fact that every subvariety V
of projective space WN has positive degree. This can be verified by induction on
the dimension k of V, as follows. If k > 0, choose a hyperplane H which meets
Kbut does not contain V. Then the intersection cycle
H-V='Zi(W;H-V;FN)-[W]
is a positive cycle which represents 0^(9A)) n [V], so
degF=deg(HF)>0
by induction. Q

212
Chapter 12. Positivity
In particular, if X carries an ample line bundle, then 0 $ At X. If X is not
projective, however, it may happen that 0 e AX.X (cf. Example 12.1.1).
A vector bundle E is ample if the line bundle 0E^ A) on P(EV) is an ample
line bundle. In case E has a finite dimensional vector space V of sections which
generate E, this is equivalent to saying that the induced morphism from P(EV)
to the projective space P(VV) is a finite morphism. Any quotient bundle of an
ample bundle is ample, and any direct sum of ample bundles is ample. If ? is a
vector bundle, and L an ample line bundle such that E ® V is generated by
its sections, then E is ample.
Theorem 12.1. Let E be a vector bundle of rank r on a scheme X, n.E-^X
the projection, sE: X -> E the zero-section. Let Vbe a k-dimensional subvariety of
E,k^r.
(a) IfE is generated by its sections, then
s*E[V]eAtr(X).
(b) If L is an ample line bundle on X such that E ® V is generated by its
sections, then q
where W\,..., Wq are the distinct irreducible components ofsEl(V).
(c) IfE is ample and generated by its sections, then
(d) IfE is ample, and L is any ample line bundle on X, then
degL(s*E[V])>0-
Proof, (a) By assumption there is a surjection q: F -> E from a trivial
bundle /'onto E. Let s = sE, t = sF denote the zero sections. The morphism q is
smooth of relative dimension n — r, rt = rank(F)> so g*[F] = [q~1 (V)]. Since
S = Q o t,
s*[V] = t*q*[V\ = t* Iq
by Proposition 6.5 (b). Thus we are reduced to proving the assertion for F; i.e.,
we may assume E is trivial.
The proof proceeds by induction on the rank r of E. Write E=G® 1,
where G is trivial of rank r — 1, let j be the inclusion of G in E :j(g) = (g, 0);
and let u be the zero section of G. Then s —j ° u, so (Theorem 6.5)
s*[V] = u*(j*[V]),
so it suffices to show thaty*[F] e Af.t G. The classy* [F] is constructed by
intersecting Fby the effective divisor G on E (Definition 2.3). If F<t G, then
j*[V] is represented by the non-negative intersection cycle G-V on Gf]V. If
Vcz G, then./* [F] = 0 since the normal bundle to G in E is trivial.
(b) Mapping a trivial bundle F onto E ® Lv determines a surjection from
F® L onto E. Arguing as in (a), one is reduced to the case where E = F® L,
F trivial of rank r. The proof proceeds by induction on r.
12.1 Positive Vector Bundles 213
Assume r=\. In this case sE imbeds las a Cartier divisor on E= L. If
V^sE(X), then^?'(F) = V, and
sE[V] = cl(L) n [F].
Therefore deg^(s*E [F]) = \ c^ (L)k~[ ¦ c,(L) n [F]= degi(F), which is positive
by Lemma 12.1. If Fmeets the zero section properly, then
where m, is the (positive) intersection multiplicity. Therefore
degz. s*E [ F] = X m, degi(W) is ? degi (W) .
To complete the case r= 1, we verify that Kmust meet the zero section. If not,
let Kbe the closure of Kin P(L © 1), using the canonical imbedding of L in
P(L © 1), and let/: F-> X be the projection. Since V does not meet the zero
section, Fis contained in the complement of sE(X) in P(L © 1), which may be
identified with Lv. Therefore/is an affine as well as projective morphism, so/
is finite (Appendix B.2.4); in particular, f*L is an ample line bundle. How-
However, the pull-back of V to Lv is trivial, so f*L is trivial. A trivial line bundle
on a positive dimensional variety cannot be ample, however, as follows from
Lemma 12.1.
For r> 1, we proceed by induction on r. Choose a splitting F—F' ©1,
with F' trivial of rank r — 1, so that if G = F' ® L, andy is the induced imbed-
imbedding of G in E, theny(G) meets V. (For the existence of such a splitting the am-
pleness of L and positive dimensionality of Fare irrelevant. One may assume F
is a point, and restrict L to an open subset of X on which L is trivial, in which
case the assertion is obvious.) Ify'(G) meets Fproperly, and V\,..., Vf are the
irreducible components ofj~'(V), with intersection multiplicities mu...,mp
and u is the zero section of G, then
so
Since sE1(V) = U'[=lu~l(Vj), each Wi appears among the irreducible compo-
components of some u~l(Vj). By the inductive assumption for G,
/-1 /-1
which concludes the proof in this case. Otherwise Fis contained in j(G), so
j*[V] = cl(r,*L) n[V]
where r\ is the projection from G to X, since r;*L is the normal bundle to G
in E. Therefore

214
Chapter 12. Positivity
by Proposition 6.3. Therefore degLs*E [K] = degi u* [K]. Since u'] (V) = s'Ex(V),
the inductive assumption applies to give the desired inequalities.
(c) By (a), it sufficies to show s* [V] 4= 0. It suffices to prove this after base
extension from the ground field to its algebraic closure, so we may assume the
ground field is algebraically closed (cf. Example 6.2.9). For any section s of E
such that s(X) meets C,
s*E[V] = s*[V] = sE[C]
where C is the normal cone to V along Vf]s(X) (Example6.2.3 and
Corollary 6.5). Thus we may assume C is an irreducible cone. The proof
proceeds by induction on the dimension of X, being trivial if dim X = 0. We
may assume C projects onto X. We assume also C =t= E, since the claim is
trivial when C = E.
Suppose there is a section s of E such that s(X) meets C, but s(X) <t C. Let
X' = s-l(C). Then
*[C] *[C]
where C is the normal cone to C along X'. Since C has smaller support, one is
done by induction.
Suppose there is no such section, i.e. any section either misses C or is
contained in C. Let F be the space of sections, g the canonical surjection from
the trivial bundle XxF onto E, H the kernel of g. By assumption, g~l (C) is a
"constant" cone, i.e. g (C) = XxD where D is a cone in F. For each x e X,
the fibre H(x) of H at x is contained in D, and satisfies H(x) + D <= D. But
for any cone D in a vector space F, L = {v e F \ v + D a D] is a linear sub-
space of F which is contained in D. Therefore H is contained in a proper triv-
trivial subbundle XxL of XxF. Therefore E surjects onto the trivial bundle
Xx(F/L). From the definition of ampleness, however, an ample vector
bundle on a positive dimensional variety cannot have a trivial quotient bundle.
(d) If V is a subcone of E, this is proved in Fulton-Lazarsfeld C)
Corollary 2.2. The proof, in addition to intersection theory, uses constructions
of Bloch-Gieseker A), and relies on the Hard Lefschetz theorem; it will not be
repeated here. If Kmeets the zero section of E, then
s*E[V] = sE[C],
where Cis the normal cone to s}1 (V) in V, so the previous case applies.
To show that V must meet the zero section, we deform to the zero section
and apply the previous case. Let T= A1, T* = A1 - {0}. Let y* c E x T* be
the image of Vx T* by the closed imbedding (v, A) -> (A v, A). Let y be the
closure ofr* in Ex T. Then V is the fibre K, of y over 1, and the fibre Ko
of y over 0 meets the zero section at least in the projection of V on X. Then
[K,] = [V0] inA*E (say by Example 10.1.7), sosE[V] = s*E[V0], and it suffices to
apply the preceding case to an irreducible component of Fo which meets the
zero section. ?
Example 12.1.1. (a) Let X= A1 x P1. Then [{0} x P1] ~ 0, so Af X = A\X.
(b) Hironaka has constructed a complete, non-singular variety X over C
with a positive 1-cycle rationally equivalent to 0, so AfX = A$X (cf.
Hartshorne E) p. 443). Such a variety cannot be projective, of course.
12.1 Positive Vector Bundles
215
Example 12.1.2. Let i imbed lasa Cartier divisor on Y, and let V be a
^-dimensional subvariety of Y, k S 1. There are three possibilities for the
intersection class X- V = i*[V] in A^^X):
(i) X meets V properly. Then i*[V] is a positive cycle supported on X(~) V.
(ii) X doesn't meet V. Then i*[V] = 0.
(iii) X contains V. Then i*[V] = ct(L) n [V] where L = i* <9Y(X) is the normal
bundle.
These cases can all arise when Y = L is a line bundle on X, and / is the zero
section.
Example 12.1.3. (a) Let ? be a vector bundle of rank r on a scheme X,
which is generated by its sections, and let n : E -> X be the projection. Then
(ii) If E is ample, then A^(E) = n*(At-rX). (These are equivalent to (a)
and (c) of the theorem.)
(b) If ? is an ample vector bundle of rank r on X, and Vcz E is a subvariety
of dimension S r, then s~' (V) =? 0 for any section ^ of E. (Apply (d) and
Corollary 6.5.) In particular, 0 i A\E if k S r.
Example 12.1.4. The conclusion (c) of Theorem 12.1 may not hold if E is
not generated by its sections. Let D be a divisor of positive degree on a
projective non-singular curve X, which is not linearly equivalent to an effective
divisor, let E = L=0x(D), and let V= X imbedded by the zero section. Then
E is ample, and E ® V is generated by its sections, but sE [ V\ = [D] is not in
AtX.
In the situation of Theorem 12.1 (c), some positive multiple of sE[V] is in
At-r(X). (Argue as in the proof of (b).) We do not know if this is true in the
situation of (d).
Example 12.1.5. Following Sommese A), a line bundle ion a complete
scheme X is called n-ample if there is a positive integer m and a morphism
/: X-> P" such that L®m=/*^p»(l), and all fibres of/have dimension m n. A
vector bundle E is n-ample if ^?v(l) on P(EV) is n-ample. Note that 0-ample
is the same as ample.
If E is n-ample and generated by its sections, and V is a subvariety of E,
with dim(F) S rank(?) + n, thensE[V\ is mA%(X). (Argue as in (c).)
If L is an n-ample line bundle, and E ® V is generated by its sections,
with V as above, some positive multiple of sE[V] is in A%(X) (cf.
Example 12.1.4).
Example 12.1.6. Existence of degeneracy loci (Fulton-Lazarsfeld C)). Let E,
F be vector bundles of ranks e,fon a variety X. Assume
Horn (E, F) = EV ®F
is an ample vector bundle. Let Z be a subvariety of Hom(?, F) of codimension d.
If dim X ^ d, then for any homomorphism a : E -> F of vector bundles,
{xeX\a{x) e r}

216
Chapter 12. Positivity
is a non-empty closed set, all of whose components have dimension at least
dim (X) — d. For example, if Xk is the variety of maps of rank ^ k, then Ik has
codimension (e — k) (/— k) (Lemma A.7.2), so
Dk(a) = {x e X\ rank o(x) S k}
is non-empty whenever dim X S (e-k) (/— k). (Apply Theorem 12.1 (d) to
the section of ?v ® F determined by a.) This last result first appeared in
Fulton-Lazarsfeld B).
Similarly suppose ? is a vector bundle of rank r, L a line bundle, such that
S2E ® L (resp. A2? ® L) is ample, and
a : Ev -> ? ® L
is a symmetric (resp. skew-symmetric) vector bundle homomorpism (cf. Exam-
pie 14.4.11). If dim X ^ V j ) (resp. ( 2 J , with k even), then
non-empty. See Example 14.4.13 for refinements.
Example 12.1.7. Polynomials in Chern classes of ample vector bundles
(Fulton-Lazarsfeld C)). Let ? be a vector bundle of rank e on X. Let X be a
partition of an integer n in integers ^ e:
A,-= «. Define the Schur polynomial Ax(E) by
an isobaric polynomial of weight n in the Chern classes of E. Examples include
cn(E), for X = («, 0,..., 0), and (- 1)" sn(?), for A = A,..., 1), cf. § 14.5.
Let a be a positive &-cycle on X, with k^n.
(a) if ?is generated by its sections, then Ax(E) n a. e Af-n(X).
(b) If ?is ample and generated by its sections, then Ax(E) n a e At-a(X).
(c) If ? is ample, and L is any ample line bundle on X, then
degi(zlA(?)na)>0.
(One may assume X is a variety, and a — [X]. If Kis the trivial bundle of rank
n + e on X, there is a subvariety Q of the bundle H = Hom(V, ?), with
codim(?2, H) = n, such that zl^(?) n [AT] = 5M[?2]. This is proved in Ex-
Example 14.3.2. Since H satisfies the same positivity assumptions as ?, the
conclusions follow from Theorem 12.1 (a), (c), and (d).)
In fact positive linear combinations of the polynomials Ax are the only
polynomials which have positive degree for all ample bundles on all varieties.
The first few Ax are:
2, c\ -
c2 - c3,c\ -
note that c\ — 2c2 is not such a positive combination.
Example 12.1.8. Let ? be a vector bundle which is generated by its sections
on an n-dimensional variety X over an algebraically closed field. Let p be a
12.1 Positive Vector Bundles
217
positive integer. Assume that 0 ^ A%-p (X), which holds for all p if X is
projective. Then the following are equivalent:
(i) cp(E)n[X] = 0.
(ii) there is a trivial subbundle F of ? such that rank (E/F) < p.
(iii) Cj(E) n a = 0 for all j S p, all a e A*X.
(Realize cp (?) n [X] as a dependency locus of generic sections of E, cf.
Example 14.4.3.)
The implication (i) => (iii) fails for affine varieties (cf. Kumar-Murthy
(I))-
Example 12.1.9. Let ? be an ample vector bundle of rank r on a scheme X
over an algebraically closed field K. If Kis a subvariety of ? with dim V> r,
then r(V,<Sv) = K. (Let Z=V[)sE(X). If /: V -> A1 is a non-constant
morphism, then /(Z) is a finite set. For general X e A1 —/(Z), /"'(A)
contradicts Example 12.1.3 (b).)
Example 12.1.10. The ampleness of ? on all of .Vis not needed for the truth
of (c). Let ? be a vector bundle, on a projective variety X over an algebraically
closed field, which is generated by its sections. Following Lazarsfeld, define
the disamplitude locus Damp(?) to be the set of points x e X such that for
some irreducible curve Z through x, E\z has a trivial quotient line bundle.
Equivalently, if Kis the space of sections of ?, Damp(?) is the projection in X
of the locus in P(EV) where the canonical map from P(EV) to i>(Kv) is not
finite, which shows that Damp (?) is a closed subset of X. Call ? generically
ample if Damp (?) =1= X.
If C is an irreducible subcone of ?, with dim (C)S: rank (?), and the
support of C is not contained in DampfJQ, then s*[C] e A%(X). (The proof is
similar to the proof of Theorem 12.1 (c).) It follows that Schur polynomials in
Chern classes of generically ample vector bundles are positive.
If X cz G| (P4) is the Fano surface of lines on a cubic 3-fold (cf. Example
14.7.13), one can show that the universal quotient bundle, and the dual of the
universal subbundle on Gi(P4) restrict to generically ample, but not ample,
vector bundles on X.
Example 12.1.11. Let ? be a vector bundle of rank r on a scheme X over an
infinite field K, which is generated by sections su...,sN. For t = (tt,..., tN)
? KN, let s, be the section X '/¦*< of ?. Let Kbe a purely m-dimensional sub-
scheme of ? Then for generic teKN, dims^\V) = m — r (or s^'(V) is
empty), and
(If q: Xx AN -> ? is the surjection determined by su ...,sN, then s,= q o /()
where i,(x) = (x,t). By Proposition 6.5 (b), s\[V] = i\[W], with W=q'](V).
Let Tu ..., TNbs the coordinate functions on AN. Given any closed subscheme
W of XxAN, one shows by induction on N that for generic t e KN, the
functions T\ — t]: ...,TN— tN form a regular sequence on 0W- When N = 1 and
X is affine, this says that T\ — tt should not belong to any of the associated

218
Chapter 12. Positivity
primes of the ideal of W\ since any proper ideal of A[T{[ contains T\ — t\ for at
most one t, e K, at most a finite number of /( need be discarded.)
This argument also shows that a generic section s, of ? is a regular section.
12.2 Positive Intersections
Consider our standard intersection set-up: a fibre square
W—U V
X—-+Y
with i a regular imbedding of codimension d, N the pull-back of NXY to W, V
a purely ^-dimensional scheme, C the normal cone to W in V, [C] = ? fHi[Ci\
its cycle on N, _
X- V=YJmiai
the canonical decomposition of the intersection cycle, with
Z,-= s~n (C,); the Z, are the distinguished varieties (not necessarily distinct).
Theorem 12.2. Fix a distinguished variety Z,, and let Ni denote the restriction
ofNtoZt.
(a) If Ni is generated by its sections, then a, is represented by a non-
negative cycle on Z,.
(b) // Ni ® V is generated by its sections for some ample line bundle L on
Z,, then
degi(a,) S degi(Z,) > 0.
(c) If Ni is ample and generated by its sections, then a, is represented by a
positive cycle on Z,.
(d) IfN, is ample, and L is any ample line bundle on Z,, then
degi(a,)>0.
Proof. By the construction of the a,, these are immediate consequences of
Theorem 12.1. ?
In particular, if N itself satisfies any of these three positivity assertions, all
the terms in the canonical decomposition have the corresponding positivity.
Let X be a non-singular, n-dimensional variety, and let K,,..., V, be pure-
dimensional subvarieties of X, with
12.2 Positive Intersections
Construct the intersection product Vt ¦... ¦ V, from the diagram
n v » v x x v
219
X
X
X
with canonical decomposition V\ ¦... ¦ Vr = ? /w,oc,, a, e Am(Z,), Z, distin-
distinguished. Let T-, denote the restriction of the tangent bundle Tx to Z,.
Corollary 12.2. Fix a distinguished variety Zt.
(a) IfTj is generated by its sections, then a{ e A% (Z,).
(b)If L is an ample bundle on Z, such that 7",® V is generated by its
sections, then
degi(a,) i= degi(Z,) > 0.
(c) IfTi is ample and generated by its sections, then a, eA% (Z,).
(d) IfTi is ample, and L is any ample line bundle on Z,, then
degi(a,)>0.
Proof. This follows from the theorem, and the fact that the normal bundle
to 5 is the sum of (r — 1) copies of Tx (Appendix B.7.4). ?
It should be emphasized that, in (b) degL(af) denotes the degree of a, as a
cycle of dimension m, while degL(Zj) is the degree of Zf as a variety, which
may have dimension larger than m. Note also that each irreducible component
of fl Vj appears as a distinguished variety, regardless of its dimension.
If Tx (or its restriction to 0 Vj) is generated by its sections, the intersection
class can be represented by a non-negative cycle. If conditions (b), (c), or (d) hold
for Tx (or its restriction to f] Vj), corresponding positivity holds for each term in
the canonical decomposition.
Example 12.2.1. (a) The class of non-singular varieties X for which Tx is
generated by its sections includes all projective spaces, Grassmannians, flag
manifolds (complete or partial), and abelian varieties. If an algebraic group G
acts transitively on a variety X, in characteristic zero, then Tx is generated by
its sections; the same holds in positive characteristic provided one assumes
that, for x e X, the morphism fx:G->X, f*(g)=g ¦ x, is smooth. (The
derivative of fx at e maps TeG onto TXX. In characterstic zero, fx is always
smooth.)
(b) Let I be a smooth hypersurface of degree d in P"+l, over an
algebraically closed field of characteristic not dividing d. If n == 2 (resp. n = 1),
then Tx is generated by its sections if and only if d ^ 2 (resp. d S 3). (If F is a
form defining X, define the bundle E to be the kernel of the vector bundle
homomorphism
There results an exact sequence 0
Tx may be computed.)
E -* Tx -> 0, from which sections of

220
Chapter 12. Positivity
(c) The property of having tangent bundle generated by its sections is
inherited by arbitrary Cartesian products and arbitrary open subschemes of
varieties with this property.
(d) If X=1P", L = tf(\), then Tx is ample, generated by its sections, and
Tx ® V is generated by its sections (Appendix B.5.8). By a theorem of Mori
A), P" is the only variety with an ample tangent bundle.
For X + P", the restriction of Tx to certain subvarieties may be ample, in
which case positivity can be deduced for corresponding contributions to inter-
intersection products.
(e) If X" c Pm is non-singular, and Tx ® 0(- 1) is generated by its sections,
then X is a linear subspace, or a plane conic. (One may induct on n, taking
generic hyperplane sections, or appeal to Mori's theorem.) In particular, Tx is
not a direct summand of 7V>» \x unless X is linear.
Example 12.2.2. Let G be the Grassmannian of n-planes in PN. To give a
morphism /: X -> G is equivalent to giving a subbundle 5 of rank n + 1 of the
trivial bundle Xx AN+S. If Q denotes the quotient bundle, then (cf. Appendix
B.5.8) /* TG = Horn(S, Q) = Sv ® Q. The bundle /* Tc is generated by its
sections. It is ample if and only if for each flag
0<
with dimL=l, AimH = N, the set {x e X\ L <= S(x) <= H} is finite. (If
V=AN+l, P(f*T%) = P(Hom(Q, S)) has a canonical map to P(Hom(V, V));
the above set is the largest possible fibre of this map.)
Example 12.2.3. If the imbeddings X<^ Y, K-» 7 can be deformed to
%"-*»,y-* ?/ over a parameter space T as in Chap. 11, such that for generic
t, X, meets V, properly, then X- V=\im(X,- V,) can be represented by a non-
l-»0
negative cycle. For example, this gives another proof that the intersection
product preserves non-negativity on a homogeneous space.
Example 12.2.4 (cf. Fulton-Lazarsfeld D)). Let i-.X^Ybe a regular
imbedding of codimension rfsuch that the normal bundle NXYis ample.
(a) If a is a &-cycle on Y with k S d and i* (a) = 0, then a is not
algebraically equivalent to any positive &-cycle whose support meets X. (Use
Theorem 12.2(d) and Proposition 10.3(c).)
(b) If an algebraic group acts transitively on Y, then any subvariety of Y of
dimension at least of must meet X. Over C, this was proved by Liibke A).
Example 12.2.5. Assume an algebraic group acts transitively on the
variety Y. Let Xbe a variety,/: X -* Ya morphism such that/*TY is an ample
vector bundle. Then for any subvariety Vof Y with dim X + dim V ^ dim Y,
In particular, f(X) must meet V. (Apply the previous example to the graph
imbedding ys: X -> Xx Yand the subvariety Xx FofA'x Y.)
12.2 Positive Intersections 221
More generally, if/*7Yis n-ample (cf. Example 12.1.5) and
dim Ks dimF- dimX+ n,
then/*[K] 4=0.
Example 12.2.6. Consider a residual intersection situation, with notation as
in Theorem 9.2. If the restriction of N ® ff{- D) to R is generated by its
sections, then the residual intersection class R. is represented by a non-negative
cycle on R. Conclusions as in Theorem 12.2(b), (c), (d) may likewise be drawn if
N® <S(- D) is suitably ample. (Use Example 9.2.2.)
Example 12.2.7. Let Hl ,...,Hd be effective Cartier divisors on a scheme AT,
and let V be a purely ^-dimensional subscheme otX.L&tW = H1C\...V\HiC\V,
and form the intersection class
Hd- V in Ak_d Wusing the diagram
W > V
I'
H1 x ... x Hd > X x
X.
Let H\-...-Hj- V=Yjrn'a-i be the canonical decomposition, a,-e Ak-d{Z),
Z, distinguished.
(a) If the restriction of each line bundle #x(Hf) to Z, is generated by its
sections (resp. and ample) then a, is in Af-d(Zi) (resp. At-d(Zi)). If each such
restriction is ample, then degi(oc,) > 0 for any ample L on Z,.
(b) Suppose W contains a divisor D of the form D = H f] V, for H an effec-
effective divisor on X. Let R be the residual scheme to D in W with respect to
V, and let R e Ak-d(R) be the residual intersection class. If the restriction of
each ^x(Hj-H) to R is generated by its sections (resp. and ample), then
If X=W\ and degfysdegtf for ally, then R e Af.d(R). If degtf;>
deg H for ally, then R e ,4w(R), and
deg (R) is ?deg(R,),
where R,,..., Rq are the irreducible components of R.
Example 12.2.8. Given V,,..., Vr<=X as in Corollary 12.2, assume Tx is
generated by its sections. Let L be a line bundle on X. Then there are
irreducible m-dimensional subvarieties Wl,..., Ws in 0j=1 KJ; and non-negative
integers nx,...,ns,so that
(Let E"i[^] be a non-negative representative for K; •...• Vr.) For X = P",
L=CA), this implies that f] Vj must contain m-dimensional varieties of small
degree.

222
Chapter 12. Positivity
Example 12.2.9. In the situation at the beginning of § 12.2, let Z be an
irreducible component of W, and let
<x(Z)= 2>,<x,
z,-z
be the contribution of Z to X ¦ V. Assume that Nz ® Lv is generated by its sec-
sections, for an ample line bundle L on Z. Then
where (ewV)z is the multiplicity of V along W at Z (Example 4.3.4). If^2,K is
Cohen-Macaulay, then
(ii) degL ot(Z) is /(^z,».) • degi(Z).
(Let /?= Y*mi[P(Ci)]> tne sum over those components C, whose support is Z.
We may assume dim (Z) = k — r, 0 < r ^ d. If g is the universal quotient
bundle on i» (N2), and /> the projection to Z, then
By the formula for Chern classes of a tensor product (Example 3.2.2) and the
fact that Q ® p*V is generated by its sections,
cf. Fulton-Lazarsfeld D). Inequality (ii) follows from (i) and Example 4.3.5 (c).)
Example 12.2.10. Let X<-* Y be a regular imbedding of projective varieties,
such that NXY is ample. Let/: Y -> 5 be a surjective morphism onto a variety
S. lff(X) 4= 5, then dim (/(JO) = dim (JO. (Following / by a projection, one
may replace 5 by Pm. Consider linear L <= Pm with dim (L) + dim (/(JO)
= m-\. If dim(/(JQ) < dim(JQ, then dim (/"'(?)) is dimG) - dim(JT) for
all L. But some/~'(Z,) are disjoint from X, and others meet X, which leads to
a contradiction of Theorem 12.2(d).)
Example 12.2.11. There is a strengthening of (c) and (d) of Theorem 12.1
and 12.2:
(i) Let L be an ample line bundle on X, E a vector bundle of rank r on X
such that SmE ® Lv is generated by its sections for some m > 0. Let C be an
irreducible cone in E, with Z=Supp(Q. Let & = dim(C), /=dim(Z), and
assume that k^r. Then
m*"'-'• e(C) • degi(Z),
where e (C) is the multiplicity of C along its zero section.
12.3 Refined Bezout Theorem
223
(ii) In the situation of Theorem 12.2, if SmNt® V is generated by its
sections, with L ample on Z,, then
degi(a,)ism-dira<z')-degi(Z,).
One therefore has the analogous sharpening of Corollary 12.2. Note that
the condition that SmE ® V be generated by its sections for some m > 0 and
some ample L is equivalent to the ampleness of E, so this result implies (d).
For a sketch of the proof, see Fulton-Lazarsfeld D); when m = 1, the argument
of Example 12.2.9 suffices.
12.3 Refined Bezout Theorem
On P" there is an exact sequence (cf. Appendix B.5.8)
Thus Tf is ample, generated by its sections, and 7V>. ® #(— 1) is generated by
its sections, so all the results of the preceding section are valid for arbitrary
intersections on P".
Theorem 12.3. Let V\,..., Vr be pure dimensional subschemes ofW, with
r
m = Y, dim(K,)-(r- l)niO.
Let Z\,..., Z, be the distinguished varieties of the intersection,
;=i
the canonical decomposition, a,eAm(Z,). Then a, is represented by a positive
cycle on Z,, and
deg(a,)Sdeg(Z,).
In particular,
r I I
IIdeg{V,) = Y,m,deg(a,) is ?»>,¦ deg(Z,) > 0. D
j-1 ;=• i i-1
Example 12.3.1. Let Wl,..., Ws be the irreducible components of 0 Vj. Since
each W{ is distinguished,
fl deg(Fj) ^ ? m, deg(H^) S ? deg(^),
j=l i-l i=l
as we saw in Example 8.4.6. (To recover this inequality also in case m < 0,
imbed P" linearly in P", N= n— m, and intersect the cones over the Vj with
vertex LN'"'' disjoint from P".)

224
Chapter 12. Positivity
Example 12.3.2. (a) Suppose certain of the distinguished varieties
Z\,...,Zq of the intersection of Vt,..., Vr on P" are known, together with the
corresponding m, and deg (a,). If
./= i
then ZY,...,Zq are the only distinguished varieties. In particular, f]rj=i Vj =
= Z,U...UZ,.
(b) Suppose Hl,...,Hn are hypersurfaces in F1, df = deg(Hf), and a smooth
curve Z of degree d and genus g is a scheme-theoretic connected component of
C]"J=l Hj. Suppose that
7= i
y-1
Then Z = nj=1 Hr (Use Example 9.1.1 (ii).)
(c) The canonical imbedding of a non-hyperelliptic curve of genus 5 is a
curve of degree 8 in P4. If the curve is not trigonal, an argument using
Riemann-Roch shows there are three quadrics whose intersection contains the
curve as a component. It follows that the curve is the complete intersection of
the three quadrics (cf. Griffiths-Harris A) p. 535).
Example 123.3. Consider an intersection of r hypersurfaces H\,...,Hr in
P", dt = deg Hj, with a purely ^-dimensional subscheme Vof P". Let Z\,..., Z,
be the distinguished varieties for the intersection product Hx ¦ ...H,- V con-
constructed as in Example 12.2.7. Let Hy ¦...¦ Hr- V — ?|=1 mlai be the canonical
decomposition. If dimZf = k — r + et, and d = min(d,,..., dr), then
deg («,)?</'• deg (Z,).
(Apply Theorem 12.2. (b) with L=0(d), noting that degi(/i) = dm deg/i for an
m -cycle /?.)
Example 12.3.4. Suppose a non-singular curve Z is an irreducible com-
component of the intersection of three surfaces H\, H2, Hj, in P3. Let d,¦= deg Hi,
d = deg Z, g = genus (Z). Even if Z is a transversal intersection of two of the
surfaces at generic points of Z, the postulated contribution of Z to the inter-
intersection H) ¦ H2- Hi, i.e.,
(I d,-A) d+ 2 - 2g = JII A + »j) n J(Z, P3)
y-i
may be greater than the actual equivalence of Z to the intersection.
For example, let H, = F(x2x3), #2 = P(*i*3), ^3= P(*i *2>- The postu-
postulated contribution of each of the three lines in the intersection is 4, but the
total intersection number is only 8. In fact, the actual contribution of each line
is 2; the point of intersection of the lines is also distinguished, and contributes
12.3 Refined Bezout Theorem
225
2 to the total intersection. (By the previous example, the contribution of each
line must be at least 2. By symmetry they must all be the same. By positivity of
all contributions, they cannot be more than 2.)
Example 12.3.5. The bound deg (a,) ^ deg (Z,) is seldom sharp when Z, has
excess dimension, i.e., dim(Z,) > m (cf. Example 12.3.3). It is possible for all
inrreducible components W\,..., W,oiVj\a have dimension m + e, e > 0, and
y-i /-i
(Thus all Z, are irreducible components, m,= 1, and deg a,¦= deg (Z,).) This
happens if there is a linear space P"~e containing all the varieties K,, and the
Vi meet generically transversally as subvarieties of P"~c. However, this is
essentially the only way this can happen. If none of the varieties K;,..., Vr is
contained in a hyperplane and Wl,..., Ws are the irreducible components of
f]J=1 Vj, Lazarsfeld has shown that
C)
where e = max(dim Wt) — Y,]=i dim(^) + (r — 1)n. An extreme case, when
V= K, is a variety spanning P", and V2 is a point on V, is the well-known
inequality
deg(F)i?Codim(K, P") + 1
(cf. Example 8.4.6). For V\ = V a curve, V2 a line, (*) says that no secant line
can meet Kin more than deg (V) - n + 2 points.
Example 12.3.6. Assume X, Y, Z are non-singular varieties of P", with
Z = X f) Y (scheme-theoretically), and
dimZ= dim X+ dim Y- n + e.
Let E be the excess normal bundle for the intersection
Z > Y
I I
For each z e Z, let A, be the span of the (projective) tangent spaces of X and Y
at z, so A: is a (n — e)-plane in P". For 0 ^ i' ^ e, and a general (e - i)-plane L
in P" set
5,-= {z eZ\Az meets L}.
Then 5, has a natural scheme structure, and for general L, codimfS1,, Z) = /,
and
(i) c,(.

226 Chapter 12. Positivity
(Let P" = P (V); L corresponds to a subspace L of V. Consider the diagram
I
0
0.
Then 5, is the locus where the composite Z,®^A) -> E has rank ^ e — i;
tensoring by ff{— 1), this is the locus where e— i+ 1 sections of E(— 1) become
dependent (cf. Example 14.4.3).)
(ii)
(ce{E) = c,
(iii)
(- 1)) = E c, (^A))-' ?,(?(- 1)).)
deg(X- Y) = deg(Z) + ? deg([SJ).
In particular, deg(Z) = deg(AT- Y) if and only if Az is a constant (n — e)-plane.
By Example 12.3.5, this can happen only when X and Y are contained in an
(n — e)-plane.
In case X= Y= Z, the 51, are the polar varieties, and E — A^P"; (i) gives
the relation between the polar classes [S,] and the Chern classes of X (cf.
Example 14.4.15), while (iii) specializes to the equation
e
(deg(*)J=Ideg[S,]
i-0
if 2 dim(X)^n, e=n- dim(X).
Example 12.3.7. In the situation of Theorem 12.3, let Z be an irreducible
component of C\Vjt and let a(Z) = ^miai, the sum over those a, such that
Z, = Z. Then
@
dega(Z)ise(Z)-deg(Z),
where e(Z) denotes the multiplicity of Vy x ... x Vr along D Vj at Z (Example 4.3.4).
If F,,..., V, are Cohen-Macaulay schemes, then
(ii)
dega(Z)a/(Z)-deg(Z),
where /(Z) is the length of the local ring of f] V, along Z. In particular, if
Vx,..., Vr are Cohen-Macaulay, then
(iii)
the sum over the irreducible components Z of (~]Vj. (Apply Example 12.2.9.)
Note that (iii) may fail if one of the Vj is not Cohen-Macaulay, even when the
intersection is proper (Example 7.1.5).
12.4 Intersection Multiplicities
12.4 Intersection Multiplicities
227
For pure-dimensional subschemes Vx,..., V, of a non-singular variety X, with
X-=i dim (f<) = (r- \)n, the intersection class Vt -...- Vr in Ao((] Vt) is con-
constructed from the diagram
X > X x ... x X .
i
Assume P is an isolated point of C]ri=l Vt, which is rational over the ground
field. The intersection number i(P, Vx ¦... • Vr\ X) is the coefficient of [P] in the
class Vx- ...¦ Vr. Shrinking X if necessary, we assume the V, meet only at P.
Let n:X^>X be the blow-up of I at ?,? the exceptional divisor,
rj: E -> P the induced map;?' = P(TpX) is a projective space P"~' over the field
K=x(P) with 0(X) =tf(-E)\E. Degrees of subvarieties and cycles on E are
calculated as usual on P".
Let V,-c X be the blow-up of V, at P. The exceptional divisor is
= P{CPVi),
where CPJ^ is the tangent cone to Vt at P. Let ePVi denote the multiplicity of
Vt at P (§4.3). Note that H[=1 V^DUi ?tC)E = C)Ui P(CPVt).
Theorem 12.4. (a) With the preceding assumptions,
inA0(P) = Z.
(b) There are varieties Wl ,..., Ws, whose union is f]'=1 PiCpV^, and positive
0-cycles Pj on WS, and positive integers mjs satisfying:
A) Vr---K = I.j^mjpjinA0(C]Vi).
B) For any Wj which is an isolated point of r\P(CPVi), Pj = [Wj] and
V
C) For all j= I,..., s, deg # S deg (Wj) > 0.
Corollary 12.4. With notation as in the theorem,

228
Chapter 12. Positivity
The term Z"i; deg(PF;) is at least as large as the sum of the degrees of the
irreducible components offYi=i P(CrV{). In particular,
with equality if and only iff]ri = 1P (CP Vt) = 0. ?
Proof of the theorem. (Another proof is sketched in Example 12.4.6.) The
proof of (a) uses the deformation to the normal bundle (§ 5). Let M (resp. AQ
be the blow-up of Xx P1 (resp. F, x P1) along Pxco, denote the exceptional
divisor of this blow-up by X' (resp. F0, so X' = P(T,X®1), F/=P(CPF, 0 1).
We have a diagram of closed imbeddings:
X' 3 V{
'1 I
VczX > M 3 M,
X 3 Vi
with i the imbedding at 0,j, k the canonical imbeddings of A" and X in M; i,j,
and k are inclusions of non-singular Cartier divisors on M. We have equalities
of cycles
* WA = [ F], j* [M,] = [ Ffl, k* [M,] = [F].
For q = 1,..., r, let iq be the inclusion of VqC\...C\Vr in Mqf]... C\Mr induced
by i. Using the refined projection formula (Proposition 8.1.1 (c)),
'"¦•([ Vi]---[VrD = 'i.O* W\\-[v1\-...-[vr])
= [Mt] • /2,([F2] ¦... ¦ [Vr]) = ... = [M,] •... • [J|/_,] ¦ ff.[Kr]
in A0(f]Mi). Similarly, if j, (resp. kq) is the inclusion of Vq'C\...C\ Vr' (resp.
•... • [Kf]) = [Mi] •... • [M-i] ¦ *r.[
A basic identity from the deformation is (cf. Example 5.1.1)
in Aif(Mr). Substituting this in the preceding three equations yields
¦ ... ¦ [V,]) =./,.([Ki] ¦... • [F;]) + ^,.([F,] ¦ ... ¦ [Vr])
in AofflMj). The projections M -» X x P1 ->A' induce a proper morphism p
from flM, to fl F,. The desired formula (a) follows by applying p^ to the last
equation. Indeed, [F/] = [P(CPF, 0 1)] has degree e,(K;) on r = P(r,J91)
= P", by the definition of multiplicity. By Bezout's theorem, [F^] ¦... • [F/] is
represented by a 0-cycle of degree OI=i ep(^i) 0n -^ •
12.4 Intersection Multiplicities
229
We turn now to (b). Unfortunately, the normal bundle to the diagonal
imbedding S of X in Ix... x X is not ample (see Example 12.4.7). We find
another intersection which yields the same class, but which has an ample
normal bundle. For simplicity of notation, assume r = 2; if r > 2 the same
proof is valid, replacing each two-fold Cartesian product by the corresponding
/¦-fold product, e.g. Vx x V2 by V\ x ... x Vr. Consider the diagram
Y <- W
X
X
X x X .
Here l:Y -+ X x X is the blow-up of X x X along Ex E, and PF is the blow-up
of F, x F2 along P(C,K,) x P(CPV2). Since J (? x E) = E is locally principal,
5 factors into X ° n, as shown. Shrink X if necessary so X is affine and its
tangent bundle, which is the normal bundle Ns to 5, is trivial.
Since Vix__V2 meets ExE in />(CPF1) x P(CPF,), which is regularly
imbedded in Vtx F2, with the same codimension as the imbedding of E x E in
XxX, W= X'l{V\ x F2); by a simple case of the blow-up formula (cf.
Corollary 6.7.2)
mAit{W). Since /r A! = <5! (Theorem 6.5),
(ii) F1-F2 = ?![FlXF2] = /i![^].
We need also a formula for the normal bundle N^ to pi:
(iii) ^=7r*
which will be discussed after completing the proof of (b). Since N$ is trivial,
and n~x(W) cE=P"'[, and 0(-E) restricts to ^A) on E, Theorem 12.2(b)
applies to the intersection diagram
P^)
I I
producing a canonical decomposition n'[W] = X-W = Yfj=imjPj and distin-
distinguished varieties Wj satisfying A) and B) of (b). Although we cannot assert
that this is the canonical decomposition obtained by intersecting V\ x F2 by the
diagonal of X x X, (b) C) follows from the fact that the two classes are equal
mAa{C\V{).
It remains to verify (iii). Let Fbe the exceptional divisor in Y for the map X.
Let a : Y -» Xx X be the composite of X and nxn. We claim that fx imbeds X

Chapter 12. Positivity
'(<5(X)), i.e., their ideal sheaves are
230
as the residual scheme to F in a"
related by
(iv) X(or'(<5(X)) =
Equation (iii) follows from (iv) by the construction of normal bundles (cf.
Example 9.2.2). Identity (iv) may be verified by a straightforward calculation
in local coordinates. More conceptually, it may be seen as follows. Let Yx be
the blow-up of Xx X along P = Px P, with exceptional divisor /"j. There is a
commutative diagram of birational morphisms:
XxX a
XxX
with y an isomorphism in a neighborhood of n (X). Identity (iv) therefore pulls
back via y from the analogous identity on Y{:
This last formula simply expresses the fact that yn{X) is the proper transform
of * in 7, (Appendix B.6.10). ?
Remark 12.4. For the intersection of divisors, on a possibly singular variety,
a similar theorem is proved more simply in Example 12.4.8.
Example 12.4.1. A particular case of the corollary is the criterion for
multiplicity one (Proposition 8.2 (c)):
with equality if and only if all the Vt are non-singular and meet transversally
at P.
Example 12.4.2. If F,, V2 are curves on a surface X, the curves F,, V2 must
meet properly in X, and one may repeat the process. One recovers the formula
ofM. NoetherB):
the sum over all infinitely near points Q of X
Example 12.4.3. The varieties Vx,..., Vr may not intersect properly, even
though VX,...,V, do. IfX=A4, F, = F( y2 - x\ w), V2 = V(w2 - z\ y), then
Vl and F2 meet properly at the origin P, but V, f] V2 = PiCpVjHPiC^)
is a line / in P(TPX) = P3. In fact, er(V,) = 2, i(P, V- W; A4) = 9, and Vx ¦ F, is
represented by a 0-cycle of degree 5 on /.
Example 12.4.4. The equation (a) of the theorem (and its proof) can be
generalized as follows:
12.4 Intersection Multiplicities 231
Let V\,..., Vr be pure-dimensional subschemes of a non-singular X", with
m=Y, dim(F,)-(r- l
i-i
Let P be a non-singular subvariety of X, n : X
Let W= nf= i Ff, W' = n-1(W), and r\:W ^
X' = .P(iVpX®l), i; = X'fir, and g the projection from X'w to W. Let
P; c X be the blow-up of F, along K-flP, and let F/ c X' be />(Cv,npFi © 1).
X the blow-up of X along P.
the map induced by n. Let
in Am{W). If Wt, is an irreducible component of W of dimension m, equating
coefficients of [IV^] in this equation gives a formula for the intersection
number i(Wt; V{-... ¦ Vr; X); if W^ = P has dimension m, then
where ^ is the coefficient of [P] in r;* (F, ¦... • Fr).
Example 12.4.5 (cf. Samuel B) II § 6.2b). Let F be a subvariety of Y, with
Y affine or projective space over an algebraically closed field, and let W be a
subvariety of F Then
(a)
= rrimi(W,XV; Y),
the minimum taken over all subvarieties X of Y such that W is a proper
component of Xf]V. (Find X an intersection of hypersurfaces so that
ew(X) = \, and P(CwV)f]P(CwX) has no irreducible component mapping
onto W.) If W= P is a point, one need only consider linear spaces X
through P.
(b) eyv{V) = 1 if and only if f-Fis not contained in the singular locus of V.
Example 12.4.6. Let i: X c, Y be a regular imbedding of schemes of
codimension d, V c Y a subvariety of dimension k, and assume W = X f) V
has only one irreducible component P, of dimension k — d. Assume the
imbedding of P in X is a regular imbedding. Let X, ?, F denote the blow-ups
of X, y, F along P. Let D be the exceptional divisor in F, rj: D -> /> the
projection. Then
in Ak-d(P) = Z. (Apply Theorem 9.2 to the residual intersection diagram
D
R
i
I
x
I
y,

232
Chapter 12. Positivity
giving XYV= {c(N) ns(D, F)}w+R. The first term gives er(V)-[P]. By
Example 9.2.2, or § 17.6, R = X-f V.)
This general formula implies Theorem 12.4(a). (Take Y = Xx...xX,
F= V[X ... xVr. Use Example 4.3.10 to calculate the first term, and argue as in
the end of the proof of Theorem 12.4(b) to interpret the second.)
Example 12.4.7. The normal bundle to the diagonal imbedding of X in
XxX, i.e., the tangent bundle Tg, does not restrict to an ample bundle on E.
There is an exact sequence
0 -> TE -> Tg\E -» NE X -> 0
and NEX = (9(-l) on E = F"~1. If C is the normal cone to Vl xV2 along
f^riFj, then C is a closed subscheme of T%\E. If one knew that the
irreducible components C, of C met 7^ properly, one could deduce the
positivity of the corresponding contributions from the ampleness of 7^; this
would strengthen part (b) of the theorem. On the other hand, if C/c: TE, its
intersection with the zero section in 7>|? would be strictly negative.
Example 12.4.8. Intersection multiplicities for divisors. For effective divisors
Di,..., Dd on an ^-dimensional variety X, the intersection class D\ ¦... ¦ Dd in
I is the class constructed from the diagram
nD; > x
by the procedure of §6.1. Alternatively, this class may be constructed
inductively:
Dr.-.Dd=Dl{D2-...Dd),
using the simple description of § 2.3 (cf. Example 6.5.1).
Assume P is an isolated point of the intersection of Dx,..., Dd on X. The
intersection multiplicity i(P, D, ¦... ¦ Dd;X) is the coefficient of [P] in the
class D\ ¦... ¦ Dd. Shrinking X if necessary, we assume P is the only point in
D;. Let n:X^>X be the blow-up of X at P, E the exceptional divisor,
rj: E -> P the map induced by n. Regard E as an algebraic scheme over the
residue field x(P) of P. The bundle <^A) =<^(— E)\? is an ample line bundle
on E, which we use to compute degrees of subvarieties and cycles on E.
Theorem. Let 7i*D, = n,E + R, for some positive integers n,, and some
effective divisors R: on X. Then
(a)
zv...
Let Rl ¦... ¦ Rd = ?|=! njj <x; be the canonical decomposition for the intersection of
the R, on X, a, 6 A0{Zi), Zit...,Z, the distinguished varieties. Then, for
(b)
deg(a,-)^deg(Z,).
12.4 Intersection Multiplicities
Corollary. With notation as in the theorem,
t
i(P, Dr...Dd;X) = nr.--- nd- eP(X) + X m,deg(oc,)
233
In particular,
, deg (Z,).
with equality if and only if f] Rt = 0.
(For the proof of (a) write i?,= n*D,— n,E, expand R\ ¦... ¦ Rd, and use
the projection formula to calculate 7*(R]¦...¦ Rd), noting that rj^(Ed) =
(-\)d-xer(X)-[P] by §4.3; (b) follows from Theorem 12.2(b) and the
ampleness of the restriction of 0(Ri) to E. We refer to Fulton-Lazarsfeld D)
for details.)
The theorem also generalizes to the proper intersection of d divisors on a
variety of dimension > d, much as in Example 12.4.4.
The last inequality of Corollary 12.4 may be deduced from the preceding
Corollary, by writing the diagonal as an intersection of divisors, in a neighbor-
neighborhood of P.
Example 12.4.9. In the situation of the preceding example, let A be the local
ring of Zat P, m its maximal ideal, and let/ be a local equation for Dt at P. A
natural choice for the integers «, are the largest integers such that / 6 mn\ In
this case the inequality
(*)
i(P,Dx
¦Dd-X)^nr...-nd-eP(X)
may also be deduced algebraically from the formula of Lech A), stated in
the Notes and References to Chap. 7. Let Gr(A) = © m"/m"+l, and let
/¦ 6 m"'/m^'+] be_the residue class of/. Then equality holds in (*) precisely
when F(/i,... ,fd) = 0 in Proj (Gr(^))._If Gr(A) is Cohen-Macaulay - e.g. if A
is regular - the latter is equivalent to/, ...,fd being a regular sequence. Note
that the left side of (*) is the multiplicity eA(f,... ,fd) (Example 7.1.2).
When P is singular on X, however, larger «, may sometimes satisfy the
conditions of the theorem. On the other hand, the «, need not be chosen
maximally, in which case the i?, may contain the exceptional divisor E. Note
that for general divisors containing E, the degree of the intersection class may
well be negative.
Example 12.4.10. Let X, V be non-singular subvarieties of a non-singular
variety Y, meeting properly at a point P. Suppose that TPVc TPX. Then
d=dim(V). (Write X as an intersection of d divisors near P. These divisors
meet Fin divisors D, which are singular at P. Then
d
i(P, X- V; Y) = i(P,Dr ¦¦¦¦ Dd; V) S J~

234
Chapter 12. Positivity
Notes and References
The results of this chapter represent joint work with R. Lazarsfeld, cf. Fulton-
Lazarsfeld C), D). The refined Bezout theorem of § 12.3 developed from our
work with R. MacPherson. In addition, Examples 12.1.9, 12.1.10, 12.2.2,
12.2.10, 12.3.5 and 12.3.6 were proposed by Lazarsfeld, and the proof of
Theorem 12.1 (c) is his. The inequality (iii) of Example 12.3.7 was proved by
Patil and Vogel A) by other methods.
A special case of the corollaries in § 12.4 and Example 12.4.8 appears
frequently in the literature. Let Ht ,...,//„ be hypersurfaces in A" meeting
properly at P = 0, and let m, be the multiplicity of H, at P. Then
with equality if and only if the leading forms of the //, have no common non-
trivial solutions. Indeed, nearly every proposed definition of intersection
multiplicity has been tested by this inequality. For n = 3, see Berzolari A), and
for the general case: Zariski B), Perron A), B. Segre G), Kirby A) and
Northcott A), among others. For varieties of larger codimension, the in-
inequality was considered by Severi (9), Samuel BI1.6.2, and Griffiths and
Harris A) p. 393. Using Serre's definition of intersection multiplicity, Tennison
A) showed that equality holds if the projective tangent cones do not intersect
(cf. Example 20.4.3). Teissier A), B) has studied the inequality for divisors on
a singular variety.
For the intersection of two plane curves, M. Noether B) gave the formula
for the difference between the intersection number and the product of the
multiplicities as the sum of the intersection numbers in the first infinitesimal
neighborhood (cf. Example 12.4.2). In higher dimensions, the proper trans-
transforms can have excess intersection; in this case we have not found classical
precedents for the inequalities of § 12.4.
The definition of ample vector bundle used in § 12.1 is that of Hartshorne
A), cf. Griffiths A). Special cases of the positivity theorem of Example 12.1.7
were proved by Kleiman, Bloch and Gieseker, Griffiths, Usui and Tango,
among others; see Griffiths A), C), and Fulton-Lazarsfeld C) for references.
Chapter 13. Rationality
Summary
Refined intersection products can be used to prove the existence of rational
solutions of algebraic equations, either in the given ground field K, or in
extensions of restricted degrees.
Suppose V\,..., Vr are subvarieties of a complete nonsingular variety X,
with 2Zy= i coding, X) = dim(X). By our construction, the intersection cycle
Vx ¦...¦ Vr is represented by a 0-cycle on CY]=1 Vj. Therefore there are points
P, ,..., P, in flj=i Vj, and integers nl ,..., nt such that
(*)
For example, if K = R, and the right side is odd, Dj=i F, must contain real
points. If some part of the intersection class is known, similar conclusions are
valid for the rest of it. Each isolated point Poff]rJ=l Vj appears in (*), with coef-
coefficient the intersection multiplicity of the Vj at P. With suitable positivity
assumptions on the tangent bundle of X, the coefficients n, can all be taken to
be non-negative, even when the intersections are improper.
Notation. If W is a complete scheme, i.e. W'\s proper over the ground field
K, and a = ? nr[P] is a 0-cycle on W, the degree of a is the sum
where K(P) is the residue field of 0,iVf at P, and [K(P): K] is the degree of the
field extension. Rationally equivalent 0-cycles have the same degree (§ 1.4).
For example, if K = R is the field of real numbers (or an arbitrary real
closed field), then K(P) = R if P is a real point of W, K{P) = <C if P is
complex, and
deg(oc)= ? 71,+ 2 X "p-
freal ^complex
In particular, if deg (a) is odd, W must contain real points.
Let i:X-* Y be a regular imbedding of codimension d, V a pure d-
dimensional subscheme of Y, W=Xf)V Let N be the restriction of NXY to
W, C = Cw V the normal cone to W in V, [C] = ? m, [CJ its cycle on N. Recall

236
Chapter 13. Rationality
that a subvariety of W is distinguished if it is the support of an irreduc-
irreducible component C, of C. For each distinguished variety Z set
a(Z)= Z m,sl,[Ci\
supp(C,)-2
in Ao (Z); here sN: W -> N is the zero section. Thus
is the canonical decomposition of X ¦ V. If Wis complete,
Thus one may deduce the existence of points on the distinguished varieties
from the knowledge of deg(X-V). The results of Chaps. 7 and 12 can be used
to restrict the possibilities for the a (Z).
We apply this to the intersection of r pure-dimensional subschemes
F,,..., Ff of a smooth variety X over K, forming the intersection product
Vr ... ¦ Vr by intersecting the Cartesian product of the Vj by the diagonal:
r\Vjc+ vx x...x vr
I I
X c+ X x ... x X .
Proposition 13. Assume ?J=i dim Vj = (r - 1) «, n = dimX, a«d 0 F;- is
complete. Let
F, •...¦ Cf=X
fee ?/ie canonical decomposition of the intersection product, summed over the
distinguished varieties Z c W, <x(Z) 6 ^40(Z).
(a) If Z is a point which is isolated in f] Vh then
a{Z) = i(Z, V,-...- Vr;X)[Z].
(b) If the restriction of the tangent bundle Tx to Z is generated by its
section, then az is represented by a non-negative cycle.
Proof. Note that isolated points in f] Vt are proper components of the
intersection, (a) was proved more generally in §7.1. (b) is a special case of
Corollary 12.2. ?
The following corollary suffices for many applications.
Corollary 13.1. Suppose P,,...,P, are distinct points which are isolated in
0 Vj, and
mk=i(Pk, Vr...- K;X)
is the intersection multiplicity at Pk.
Rationality
237
(a) Among the other components of'[} Vj, there are points Qx ,..., Q, and integers
nx ,..., n, so that
(*) Z n,[K(Q,): K] = deg(F,•... ¦ Vr) - ? mk[K{Pk): K].
i- I k- I
(b) If Tx is generated by its sections, one may find such Q,, «,, with «,>0. D
The field K is called a p-field, for a prime number p, if every finite
extension of K has degree over K which is a power of p. For example, the field
of real numbers, or any real closed field, is a 2-field.
Corollary 13.2. Suppose Px ,...,PS are distinct K-rational points which are
isolated points off] VJt and K is a p-field. If
ti(Pk, Vt-...- Vr;X)mdeg(Vr...- K) (mod/*)
then fl Vj contains additional K-rational points.
In particular, f] V} must contain K-rational points whenever degfFj •... • Fr)
is not divisible by p. Corollary 13.2 follows from Corollary 13.1 (a) since, when
(*) is taken mod p, all points which are not K-rational are ignored. ?
Remark 13.1. Similar conclusions may be drawn in other contexts where
classes are constructed on loci of interest, for example, on residual schemes
(cf. Example 13.11), double point loci (§ 9.3), or degeneracy loci (§ 14).
Remark 13.2. A reader more comfortable with geometry over algebraically
closed fields may work directly with the variety (or scheme) Xg over an algebraic
closure K of K. The point then is that if subvarieties Vt of Xg are defined over K,
the intersection classes and the contributions a (Z) of distinguished varieties are
represented by zero-cycles which are rational over K. (A 0-cycle ? nP[P] on Xg
is rational over K if each conjugate of P over K occurs with the same coeffi-
coefficient as P; in addition, if K is not a perfect field, each coefficient must be di-
divisible by the corresponding degree of inseparability.)
Example 13.1. Let K be a /?-field.
(a) If W is a subscheme of P" which contains a fe-cycle a whose degree is
not divisible by p, then W has a K-rational point. (If k = 0 this is obvious. If
k > 0, Cj (tf(l))kn a is represented by a zero-cycle with the same degree.)
(b) If F,,..., Vr are pure-dimensional subschemes of V, with Z codim(V7-)
^ n, and FldegfVj) is not divisible by p, then Pi F} has a K-rational point.
(Apply (a) to a representative for Vx -...- Vr.)
(c) (Pfister A)). If Hx ,..., H, are homogeneous polynomials in K[X0 ,...,XJ
of degrees prime to p, and r S n, then there is a non-zero solution x =
(x0,..., xn) to the system of equations
with all Xj 6 K.
In particular, (cf. Behrend A), Lang A)) if K= R, or K is any real closed
field, r form of odd degrees in more than r variables must have common non-
nonzero real solutions.

238
Chapter 13. Rationality
Example 13.2. Assume K is a jc-field.
(a) If W is a subscheme of Pm x P" which contains a cycle a with some
bidegree not divisible by p, then W has a AT-rational point. (If a ~ ^ n^s' tJ,
with the notation of Example 8.4.2, then sm~"t"~ba is represented by a zero
cycle on Wof degree «ai).)
(b) If Vx,..., Vr are pure-dimensional subschemes of Pm x P", and some
bidegree of [FJ ¦... ¦ [Vr] is not divisible by p, then CWj has a K-rational
point.
The analogous assertions are valid on arbitrary multiprojective spaces.
Example 13.3 (Behrend A)). Let K=R, or a real closed field. Let
Hi,..., H, be bihomogeneous polynomials in K[X,,..., Xm, Y[y..., Yn]o[odd
bidegrees. If the binomial coefficient KJ is odd for some k with r -n <k<m,
then there are real solutions (x, y) to the system
with x 4= 0, y + 0. In particular, if m = n = r, and r is not a power of 2, then
there are always non-trivial solutions. (With coefficients mod 2, the intersec-
intersection class of the corresponding hypersurfaces in Pm~' x P"~' is congruent to
the sum over k < m, r - k < n. If m = n = r, and (s + t)' = s' + V mod p, then r
is a power of p.)
Example 13.4. Let K be a /J-field. Assume there is a bilinear mapping
<p: Ux
W
of finite dimensional vector spaces U, V, W over K, with no zero-divisors, i.e.,
<p(x, y) = 0 implies x = 0 or y = 0. \i m,n,r are the dimensions of [/, F, W,
then p divides (f) for all r — n < k < m. In particular, if the three dimensions are
\KJ
equal, they must be a power of p.
It follows that, if K is a real closed field, a bilinear form K" x K" -* K" must
have zero divisors unless n is a power of 2. (Cf. Behrend A); for K = R, earlier
topological proofs had been given by Stiefel and Whitney.) If K= R, in fact, n
must be 1,2, 4, or 8; the only proof known at present uses topology, cf. Milnor
A). (Taking bases, <p is given by r polynomials of bidegree A, 1). Argue as in
Example 13.3.)
Example 13.5. An arbitrary polynomial/has a unique expression/= X/
with f{m) homogeneous of degree m. The non-zero term of minimal degree is
called the initial form of /, its degree the initial degree; the non-zero term of
maximal degree (= deg (/)) may be called the final form off.
Let/I,...,/, be polynomials in K[X,,..., Xn], not all forms. Assume that
the n initial forms of these polynomials have no common non-zero solutions in
Rationality
239
an algebraic closure K of K, and likewise for the n final forms. Then the system
of equations
/, (x) = ... =/„(*) = 0
has non-zero solutions xa = (xal,..., xan) 6 K", with
where d, = deg(/), c, is the initial degree of/, and the na are positive integers.
(For a polynomial/= ?/(/B) of degree d, let F= ? Hf"/*"" be the homo-
homogenized form of/, a form of degree d in ^[Zo,..., X,]. The n hypersurfaces
defined by F, = 0 have no common points on the hyperplane Xo = 0. The
intersection multiplicity of the hypersurfaces at A :0:...: 0) is Y\Ui ct by
Theorem 12.5. Apply Corollary 13.1.)
For example, if K is real, and fl di ~ Fl C; is °dd, there must be non-zero
real solutions.
Example 13.6. It is possible for n real polynomials in n variables to have a
finite number of real solutions, whose number is larger than the product of the
degrees. Let n = 3,m^3, and
m m
A = Z (*.-02+ Z (*i-02, f2=h = xz.
i-l /-I
Then there are m2 real solutions, while JJ de§ C/l) = 2 m. Or one may set
f2=X[X3, fi=X2X}, to obtain a similar example with /,, f2, f} algebraically
independent. In any such example, some of the given real solutions must lie on
a positive-dimensional variety of complex solutions.
Are there such examples with all / of the same degree? There are upper
bounds on the sum of the Betti numbers of such varieties (cf. Milnor B)); the
related question for the sum of the Betti numbers is apparently still open.
Example 13.7. Let I be a complete /j-dimensional variety, Du...,Dn
Cartier divisors on X. Then there are points P\,..., P, in the intersection of
the supports of the divisors, and integers nu ..., n,, such that
Example 13.8. Let H\,...,Hr be real hypersurfaces in P\ deg/T/=«,.
Suppose a non-singular curve Z of degree d is a scheme-theoretic component
of fl Hi. Then f) Ht must contain real points outside Z, provided
(Use Example 9.1.1.)
Example 13.9. Algebraic Borsuk-Ulam (Arason-Pfister A), cf. Knebusch B))
(a) Let #|, ...,#„ 6 R[X], ...,Xn+l] be odd polynomials, i.e. #,(— x) =
- g,(x), with R a real closed field. Let
S"={x=(xu...,xn+l)\x2 + ...+x2n+l=\}.

240
Chapter 13. Rationality
Then there is a point x e S" with <?,(x) = 0, i = 1,..., n. (For an odd polynomial
q, write q - ? g0), with g0) a homogeneous polynomial of odd degree j. If
deg (g) = d, define a form g by
,,..., Xn+1) = ? {X} + ... + Ai+,)(rf-;)/2 • 9W .
By Example 13.1 (c), the forms qu...,qn must have a non-trivial common
solution (fl|,..., an+]). Set .*, = a,/a, where a2 = ? a;.)
(b) For any /|, ...,/„ 6 i?[Z|, ...,Zn+,], there are points x e S" with
ft (x) =/• (- x) for »= 1,..., w. (Apply (a) to the odd parts of the/.)
(c) If i? = R, the topological Borsuk-Ulam theorem follows. (Approximate
given continuous functions on S" by polynomials, and apply (b).)
Example 13.10. Let X be a non-singular variety over R such that Xc S P2;".
Then X must have real points. (Since
JC|GVJ"=Bh+1)"
is odd, X has odd 0-cycles.) Therefore X is isomorphic to P2" (cf. Serre F) p.
168).
Example 13.11. Consider a residual intersection situation as in Definition
9.2.2, assuming also k = d and W is complete. Then there are points Pt in the
residual set R, and integers n, so that
? rii[K(Pd : K] = deg(Z- V) - J c(N) n j(Z, V).
z
Example 13.12. Let L be a finite extension of AT. For a scheme Zover K, the
base extension
^l = X ® * L = L xSpecW Spec (Z.)
is a scheme over L. There is a canonical morphism n: XL -* X (of AT-schemes)
which is finite and flat of degree [L : K]. The composite
is multiplication by [L : K] (cf. Example 1.7.4).
Example 13.13 (Colliot-Thelene and Ischebeck A)). Let X be a complete R-
scheme, X<c its complexification. Assume that the space X(R) of real points of X
has ^ connected components.
(a) There is an exact sequence
A0(Xd ^A0(X) ? B/22Y - 0
where /? takes a 0-cycle ^ "/¦ [^] to a vector whose components are the sums,
mod 2, of the coefficients of the points P on the corresponding components of
Z(R). (It suffices to verify this for X a smooth curve over R, in which case the
exactness is a theorem of Witt, cf. Knebusch A) p. 70.)
(b) There are 0-cycles of degree s— 2 which are not rationally equivalent to
any positive 0-cycle on X. (Choose P, in the /lh component of X(R). Then, by (a),
is such a zero cycle.)
Notes and References
241
(c) If i ^ 1, n*(A0(X<[)) = 2A0(X) is the largest divisible subgroup of A0(X).
(Use Examples 13.12 and 1.6.6.)
(d) If any two points of X<c can be joined by a chain of rational curves (e.g. if
Xtc is a unirational variety), then
Two points of X(R) are rationally equivalent if and only if they belong to the
same connected component of X{R). (Use Example 10.1.6.)
Using results of M. Knebusch and H. Delfs, Colliot-Thelene and Ischebeck
A) prove analogous results for arbitrary real closed fields.
Notes and References
When all intersections are isolated (proper), results like those in this chapter are
routine, once one has an intersection product that preserves rationality over a
given ground field, as in Weil B). When sufficient moving is available, one may
deduce the existence of solutions in general from the proper case. Several cases
of this appear in the literature, when the ambient space is a product of projective
spaces:
(i) If K=]R, one may use compactness arguments (cf. Shafarevich
A)IV§2.2).
(ii) If K is real closed, Behrend A) used Hensel's lemma and specialization
arguments. See also Lang A).
(iii) If K is a/7-field, Pfister A) used valuations and specialization.
Even when such moving is available the present intersection theory gives a
simple, direct approach. For more general intersections, the results here are new.
The applications to /7-fields were suggested by J.-L. Colliot-Thelene. S.
Kleiman urged working directly over the ground field, rather than descending
from an algebraic closure as in Remark 13.2. Applications to real forms (cf.
Example 13.10) were suggested by J. Harris and A. Landman. Example 13.6
came from discussions with R. Lazarsfeld and D. Eisenbud. Generalizations of
Example 13.9 were given by Terjanian A).

Chapter 14. Degeneracy Loci and Grassmannians
Summary
Let a: E -* F be a homomorphism of vector bundles of ranks e and / on a
variety X, and let k S min (e,/). The degeneracy locus
Dk (a) = {x s X | rank (a (x)) =g ?}
has codimension at most (e - k) (f— k) in X, if it is not empty. We construct a
class
E>k(a)eAm(Dk(a)),
m = dim(Z) — (e — k)(f—k), whose image in Am(X) is given by the Thom-
Porteous formula:
Here A\p)(c) denotes the determinant of the p by p matrix (c,+;_,)is,JS;l. If
dim (Dk(a)) = m, and X is non-singular, or, more generally, if a suitable depth
condition is satisfied, then F)k(a) is the m-cycle determined by the natural
scheme structure on Dk (a). In general the formation of Dk (a) commutes with
other intersection operations. These properties determine JDk(o) in case
dim(Dk(a)) > m.
If A] a ... czAd a E is a flag of sub-bundles of E, the determinantal locus is
Q{A;c) = {xeX\dim{Ker(a(x))C)Ai(x))^i for l^i^d}.
Similarly, there are classes SI (A; a) in A*(Q(A; a)), whose images in A*(X)
are given by certain determinants in Chern classes. If c (E/A,) = 1, the formula
1S Sl(A;a)=di(c(F-E))r>[X],
where A = (Ai,..., Id), A,=/— rank(^4,) + i, and z(^ is the Schurpolynomial
A special case of degeneracy locus is the zero set of a section of a vector
bundle. In this case the degeneracy class localizes the top Chern class of the
bundle (§ 14.1). The construction of general degeneracy loci is reduced to the
case of sections of bundles on Grassmannians; proving the formulas requires
some Gysin computations (§ 14.2).
Summary
243
Formal identities among Schur polynomials determine formulas for inter-
intersecting determinantal classes. When applied to Grassmann bundles, these
formulas yield generalizations of classical formulas of Schubert calculus: the
basis theorem, duality, Pieri's formula, and Giambelli's formula.
Notation. The fibre of a vector bundle E over a scheme X at a point x e X
is denoted E(x); it is a vector space over x(x). If a: E -*¦ F is a vector bundle
homomorphism, A*cr denotes the induced homomorphism on k'h exterior
powers.
If a: E -+ F is a homomorphism of vector bundles on a scheme X, the zero
scheme of a will be denoted Z (a). On an affine open set U where E and F are
trivial, a is defined by a matrix of elements in the coordinate ring of U, which
generate the ideal of Z (a) on U. In particular, if s is a section of a bundle E,
i.e., a homomorphism from the trivial line bundle to E, its zero scheme is
denoted Z(s) (cf. Appendix B.3.2).
More generally, for a non-negative integer k, we have the kth degeneracy
locus
Dk(a) = {x e X | rank(a(x)) g k] = Z( A*+' (a)).
The second description determines the scheme structure on Dk(a): locally its
ideal is generated by (k + l)-minors of a matrix for a.
Let A. be a flag of subbundles of E:
Given a: E -> F, set
Q(A; a) = {x e X \ &\
d
= n
Here a, :A, -<¦ F is the restriction of cr to ^,, and a, is the rank of At. The second
description determines the scheme structure.
For vector bundles E, F on X, set
c(F-E) = c(F)/c(E)=l+(cl(F)-cl(E)) + ...
and let c, (F- ?) be the term of degree i in this expansion.
If A|,..., Xd are integers, and c(",..., c(</) are formal sums:
with cf1 of degree/, y e Z, set
If cA)= ... = c(i° = c, we denote this
If, in addition, A, = ... = ld = e, this is
^(c), or
^ c).
), i.e. Ai{c) = |cil+j_

244
Chapter 14. Degeneracy Loci and Grassmannians
14.1 Localized Top Chern Class
Let ? be a vector bundle of rank e on a purely H-dimensional scheme X, and
let i : X -> E be a section, with zero scheme Z (s). We construct a class
Z(s)eAH-,(Z(s)),
called the localized top Chern class of E with respect to s, whose image in
An-e(X) is ce(E) n [X]. In case s is a regular section, i.e. the e functions locally
defining 5 form a regular sequence, Z(j) will be the cycle [Z(j)] determined by
the natural scheme structure on Z (s).
Consider the fibre square
Z(s)-
'I
X-
¦X
Is
where sE is the zero section, i the inclusion. Since sE is a regular imbedding of
codimension e, the construction of § 6.1 gives a refined intersection product
sE([X\) in An-e(Z(s)). Define
Proposition 14.1. (a) /* (Z(j)) = c,(?) n [X].
(b) ?ac/i irreducible component of Z(s) has codimension at most e in X. If
codim (Z (s),X) = e, then Z (s) is a positive cycle whose support is Z (s).
(c) Ifs is a regular section, then
(d) Let f:X' -*X be a morphism, s'=f*s the induced section of f*E, g the
induced morphism from Z(s') to Z(s).
(i) /// is flat, then g* 1(s) = Z (/).
(ii) Iff is a local complete intersection morphism, then /!Z (s) = Z (s').
(ii) If f is proper, and X' and X are varieties, then g*TL(s') =
deg(X7X)-Z(s).
Proof, (a) follows from the self-intersection formula:
i« s'E [X] = sE s, [X]*>s* j» [X] = ce (E) n [X]
by Theorem 6.2(a), Corollary 6.5, and Corollary 6.3, respectively, (b) and (c)
are special cases of Proposition 7.1 (a) and (b). In the situation of (d) there is
a fibre diagram
Z(s')
"I
yr r
sf'E
'I
X
¦+X1
I'
¦f*E
1
from which it follows that l(s')=sE[X'] (Theorem 6.2(c)). (d) therefore
follows from the corresponding properties of intersection products: Theorem
6.2 (b), Theorem 6.4 and Theorem 6.6, and Theorem 6.2 (a) respectively. ?
14.1 Localized Top Chern Class
245
Example 14.1.1. If X is Cohen-Macaulay, and dim Z(.s) = «-e, then .s is
regular, and Z(s) = [Z(s)] (Proposition 7.1).
Example 14.1.2. Z(.s) can also be defined to be s'[X], this time using the
zero section to imbed X in E (Theorem 6.4).
Example 14.1.3. Let ?¦ = ?¦,© ?2, ^ = ^ 1 © ^2, -yj a section of E,, e, = rank Eh
Then the image of Z (s) in An-e (Z (s2)) is
nZ(j2).
(See Example 17.4.8 for a generalization.)
Example 14.1.4. Residual formula for top Chern classes. Let s be a section of
a rank e vector bundle ? on a purely rc-dimensional scheme X. Assume that
Z (s) contains D, an effective Cartier divisor on X. Then there is a section s' of
E®0(-D) such that the canonical homomorphism from E®tf(-D) to E
takes s' to 5. (Locally the functions defining 5 are divisible by an equation for
D, and the quotients define s'). In addition, Z(s') is the residual scheme to D
in Z (s), and e
1{s) = 1 (s') + X (- 1 )'¦"' c,-, (E) n D"' ¦ [D]
/-1
in An-e(Z(s)). In particular, if j' is a regular section, then Z(j') = [Z (/)],
which gives an explicit formula for Z (s). (Apply Theorem 9.2 in the situation
Z(s')
D-
¦Z(s)-
I
X
¦X
Is
• E
That Z(s') is the residual class follows from Example 9.2.2.)
Example 14.1.5. Euler characteristics and Milnor numbers. Let Ibea non-
singular, «-dimensional variety over an algebraically closed field K, f\X-* C
a proper morphism onto a non-singular curve. The induced mapping on
tangent bundles df:Tx^f*Tc
determines a section sf of /* Tc®T)(. Assume / is smooth except over a finite
set {/[,..., tm] in C. The localized top Chern class Z (s/) is a 0-cycle class on the
singularity locus off.
(a) Assume C is complete. Then
degZ(sf) = (- 1)"(x(X) -X(QXiX,)),
where X, is a general fibre of /, and x denotes the Euler characteristic (degree
of top Chern class of tangent bundle). (One has
degZty) = \cn{f*Tc® T$) = I (- \)'ct(Tx) + I (- I)" {ct(f*Tc) cK.x(rx).
X XX

246
Chapter 14. Degeneracy Loci and Grassmannians
k\x,) = \x,cn-\(TXt) for general teC, the
Since Sxf*[t]-cH
formula follows.
(b) In the complex case (K = C), this formula may be rewritten:
deg 1 (sf) = (- 1)" ? (x {X,,) - x (X,)) ,
i.e. %.{sj) measures the change in Euler characteristic contributed by the
singularities of/ (Choose small disks Z), surrounding the /,, and apply Mayer-
Vietoris to the covering of X by f~l(DY),... ,fl(Dm), and f'l(C- UT=i Dt);
cf. Beauville B) p. 95.)
(c) In the complex case, the part of Z(jy) supported on Xh has degree
(~')"ix(X.) - X (Xi))- More generally, if C is replaced by a neighborhood of r,
so that Tc is trivial, ZEy) is a localized top Chern class for the cotangent
bundle of A", whose degree is (— 1)" (^ (X(i) — x (^i))- (One may prove this by
using outward pointing vector fields, as in MacPherson A).) Note that if C is
not complete, the global Chern classes may vanish, while the local Chern
classes still carry useful information.
(d) If x e X is an isolated critical point of /, the equivalence of Z (s/) in
(}
lix CO = dim* (<Cx,x/(df/dz,,..., df/dz.))
where the partial derivatives are defined in terms of local coordinates
Zj,..., z, at x, and a local coordinate for C at f(x). The number nx(f) is the
Milnor number, cf. Milnor C). For example, fix(f) = \ precisely when the
Hessian (d2//dz, dzj) is non-singular at x. For a general discussion of Milnor
numbers, see Orlik A).
(e) (cf. Iversen B)). More generally, if the singularity locus of/ contains an
effective divisor D, let s'f be the section of /* Tc ® 7* ® 0(- D) induced by sf
(Example 14.1.4). Then
Z(s) = 1 (# + (- I)" Z O' ¦ c,.,{Tx).
/-I
(This follows from Example 14.1.4 and the fact that /*[/] ¦ D = 0 for / e C.)
Therefore
( -x(C) ¦ X(Xt) + t ID' ¦ cm.,(Tx)\ .
i-l X /
If x is an isolated point of Z (j}), Iversen calls x a moderate critical point. In
this case the equivalence of 1 (s'f) at x is
with g, = (df/dzi)/h, df/dz, as in (d), and /i the greatest common divisor of the
partial derivatives. Iversen B) gives explicit formulas for these numbers when
X is a surface, in characteristic zero.
14.2 Gysin Formulas
14.2 Gysin Formulas
247
There are many formulas for pushing forward classes from Grassmann or flag
bundles. In this section we prove two such formulas, for use later in the
chapter.
Let A be a flag of vector bundles on X, i.e, a chain
0 5 A15^2 5 ¦ ¦ ¦ 5 Ad=A
of subbundles. Let <p: Fl(A) -* X be the associated flag bundle, whose fibre
over x e X consists of flags of subspaces of A (x):
with dimZ),(x) = / and D,{x) czAj{x). On Fl(A) there is a universal flag
D, c ... <= Dd, with rank A= i and D,-c <p*Ah An inductive construction of
Fl(A) will be given in the proof of the following proposition. Let a, = rank (^4,)-
Proposition 14.2.1. Let M\, ...,Md be vector bundles on X. Let i\,..., id be
integers. Then for all a eA*X,
•... ¦ cit
p*Md-Dd) n <p*z)
= cj,(Mi-Al)-...- cJt (Md -Ad)ntx,
wthjp= ip- ap + p, 1 ^ p ^ d.
Proof. The inductive construction of Fl(A) is as follows. If d= 1, Fl(A) =
P(A\), with D\ = 0Al(— 1) the universal line bundle. Assume
F' = Fl(Aic:...czAd-l)
has been constructed, with projection y/:F'-+X, and universal flag
D\ c ... c D'd-\. Then F=Fl(A) may be constructed as a projective bundle
over F':
Let <p= <p' o e, Di=Q* D] for; < rf, and determine Drf so Dd/Dd-{ is the universal
sub-bundle of (p*Ad/Dd-\. We claim that
(i) e*(cj^Mj-DJ) n g*^) = ^.^(^Mj-f/MJ n 0
for PeA^F'. From the exact sequence defining ?>,/, and the Whitney sum
formula, one has
c {<p* Md-Dd) = c (<p* Md - <p*Ad) c(Q),
where Q is the universal quotient bundle of rank ad—d on the projective
bundle /"over F'. By the projection formula,
ft, (c,
1-0
Md - <p'*Ad) ¦ q, (c, (Q) n Q* 0).
Since g*(c, Q n q*0) is 0 if i + ad- d, and it equals P i{ i = ad— d (Example
3.3.3), (i) follows. The general formula then follows from (i) by induction
ond. U

248 Chapter 14. Degeneracy Loci and Grassmannians
Corollary 14.2. If Xi,..., Xdare integers, and a eA*X, then
(p*{dh ks{c(<p* M,- D,), ...,c((p* Md- Dd)) r^ (p* a)
= AM M(c(M]-Al),...,c(Md-Ad)) not
where ^, = A, — a, + i.
Proof. Apply the proposition to each monomial in the expansion for the
determinants. ?
Let ? be a vector bundle of rank n on X, d g, n, Gd(E) the Grassmann
bundle of d-planes in E, n the projection from Gd(E) to X. Let S be the uni-
universal subbundle of n* E, of rank d. Let k = n — d.
Proposition 14.2.2. Let F be a vector bundle of rank f on X. Then for all
a eA*X,
n* {cdj{Sw ® n* F) n n* a) = /)}-* (c (F-E))r,ot.
Proof. (See Example 14.2.1 for another proof.) By the splitting principle
(Remark 3.2.2) it suffices to prove this when E has a flag A of subbundles:
Aid ... <= Ad — E
with rank/4, = k+i. As usual we may assume that a = [X], X a variety. There
is a canonical morphism
H\Fl(A)-*Gd(E)
with /j*S = Dd. Over the open set consisting of d-dimensional spaces L in
fibres E(x) such that Lf)(/4;(x)) has dimension / for all i, n is an
isomorphism. The composite <p = n ° /x is the projection to X. Since p. is
birational,
cdl (Sv ® n* F) n [Gd(E)] = ^ {cif{Dwd ® <p* F) n
Therefore
7c*(QyE'v® 7c*F) n 7t*a) = <p^{cdf{Dwd ® <p*F) n <p*a).
If x, = ci (Di/Dj-t), and>>|,...,>y are Chern roots for (p*F, then
1=1 j-l
It follows from Lemma A.9.1 (ii) that
cdl{D),® <p*F)=Af{c(<p*F- Di),...,c{<p*F- Dd)).
By Corollary 14.2, since ap - p = k for all p,
By Lemma A.9.1 (i),
which concludes the proof. ?
14.3 Determinantal Formula
249
Example 14.2.1. (a) If Q is the universal quotient bundle on Gd(E), and
a eA* (X), then
(If / > fc, c,g = 0. If X (/ < dk, the left side vanishes for dimensional reasons.
To prove that 7c* (ck{Q)d r\ n* a) = a, one may assume E is trivial, X a point,
and one may induct on d, cf. Example 3.3.3.)
(b) From (a), one has another proof of Proposition 14.2.2. Indeed, since
cdf(Sv® n*F)=Af{c{n*F-S))=d^{c{n*F-n*E) ¦ c(Q)),
one may apply (a) to all the terms appearing in the expansion of this deter-
determinant.
Example 14.2.2. Jbzefiak, Lascoux and Pragacz A) prove the following
generalization of Proposition 14.2.2. For any integers Xu ..., Xd,
n*{dh Xd(c(n*F-S)) n n*a) =AMl Md(c(F-E)) n a ,
where ^, = A,— n + d.
14.3 Determinantal Formula
Let a: E -> F be a homomorphism of vector bundles of ranks e, f on a purely
n-dimensional scheme X. Let A be a flag of subbundles of ?:
0 At^...^Adc:E
and let F/^) be the associated flag bundle (§ 14.2), with universal flag
?>i cz ... cz Dd, rankZ>, = (, and projection <p from Fl(A) to X. The induced
homomorphism ,
Dd^(p*E H (p*F
determines a section st of ?>^ ® (p*F. Then p maps the zero set Z(sa) onto the
determinantal locus Q(A; cr). Let
be the morphism (of underlying reduced schemes) induced by <p. The construc-
construction of § 14.1 gives a localized top Chern class
m = dim Fl(A) — df. Define the determinantal class Sl(A; cr)e Am(Q(A; a)) by
the formula
t/iat
Theorem 14.3. Let at = rank(/4,), A,=/— a, + (, and /i =
m = n — h. Assume that f— ad + d^O. Then

250
Chapter 14. Degeneracy Loci and Grassmannians
(a) The image ofSl(A; a) in An~h (X) is
(b) Each irreducible component of O(A; cr) has codimension at most h in X. If
codim(fly; cr), X) = h, then $l(A; cr) is a positive cycle whose support is
Q{A;o).
(c) If codim (Q(A;o),X) = h, and X is Cohen-Macaulay, then Q(A; a) is
Cohen-Macaulay, and
Sl(A; a) = [fly; a)}.
(d) The formation of Sl{A; a) commutes with Gysin maps and proper push
forward.
Proof. Assertion (d) means that if /: X' -> X is a morphism, ?', F', A',, a'
the pull-backs to X', g the induced morphism from fly'; a') to fly; cr), then
formulas (i), (ii), (iii) of Proposition 14.1 (d) are valid, where Z(cr) (resp.
Z(s')) is replaced by fly; a) (resp. fly'; a')). To prove it, form the fibre
-^ Fl(A)
I*
—^X .
'1
X'
Then f*(sa) = v, so (d) follows from Proposition 14.1 (d) and the commuta-
tivity of pushing forward with Gysin maps (Proposition 1.7, Theorem 6.2 (a))
and the functoriality of push-forward (§ 1.4).
By Proposition 14.1 (a),
in Am(Fl{A)). By Lemma A.9.1 (ii),
cism ® <p*F) = Af f(c(cp*F - DJ,..., c(cp*F-Dd)).
By Corollary 14.2,
Dt),..., c{<p* F- Dd)) n,[Fl{A)\
which proves (a).
We next prove (b) and (c) in a "universal local case", namely: X = Aef is
affine space, with coordinate functions (x,-,-), 1 S i s/, I Sj S e; E and F are
trivial, E with basis vt,...,ve; A; is the trivial subbundle of E with basis
u, ,..., u0|; a is given by the matrix (xu). In this case Q = Q(A; a) is defined by
the vanishing of all (ap — p + l)-minors of the matrix consisting of the first ap
columns of (x^), for p= 1,..., d. This determinantal scheme is an irreducible
variety of codimension h in A'f and the morphism rj: Z (sa) -> Q is a bira-
tional morphism. Since Fl(A) is non-singular, and codim (Z(sa), Fl(A)) = df
Se is a regular section of Dwd®(p*F (Lemma A.7.1). In fact, Eagon and
Hochster have proved that Q is a Cohen-Macaulay variety (cf. Lemma A.7.2).
14.3 Determinantal Formula 251
By Proposition 14.1 (c),
Si (A; a) = r,a {sa) = r]* [Z (s.)] = [Q].
So (b) and (c) are proved in this case.
To prove (b) and (c) in general, it suffices to prove them locally on X, since
the construction of il is compatible with restriction to open subsets (by (d)).
Thus we may assume there is a morphism f:X-^lk'f, such that E, F, At and fl-
flare pull-backs from the universal case just considered. Let ?2 be the universal
determinantal subscheme of Aef, and let yf: X -> X x Aef be the graph of /.
Since Ae/ is non-singular, yf is a regular imbedding. By (d), and the case just
considered,
Now (b) and (c) follow from Lemma 7.1 and Proposition 7.1. ?
Remark 14.3. An alternative construction of the class fly; a) may be given
as follows. Let H = Hom(?, F), p the projection from H to X. On H there is a
canonical (tautological) homomorphism u from p*E to p*F. Then ii(p*A; u)
has pure codimension h in H; indeed, H is locally a product of X by the
universal case considered in the proof of the theorem. A given homomorphism
a: E -> F determines a section ta:X-* H such that t%u = a. Then
Indeed, (c) implies that ?l(p*A; u) = [Q(p*A; u)], and then (*) follows from (d).
This argument also proves the following corollary.
Corollary 14.3. The classes Sl(A; a) are uniquely determined by properties (c)
and (d) of the theorem. ?
Example 14.3.1. The general criterion for fly; a) to be the cycle deter-
determined by fly; a) depends on the notion of depth. For a closed subscheme Y
of a scheme X, depth (Y, X) is defined to be the largest integer d such that, for
all x 6 Y, the ideal of Y in the local ring of X at x contains a regular sequence
of length d.
Let fl = fly;a), Sl = Sl(A;a). If codim(fl, X) = h, and [Q] = '?lmi[f2i] is
the cycle of fl, then fl = ? eL [fl,], with 1 ^ e, ^ m; for all i. In general,
depth {Q,X)Sh,
with equality if and only if codim (fl, A") = h and SI = [fl]. For E a trivial line
bundle, and d=at = l, one recovers Proposition 14.1 (c). For X Cohen-
Macaulay, this contains Theorem 14.3 (c) and a converse to it. (The assertions
are local, so one may assume X is affine and one is in the situation at the end
of the proof of Theorem 14.3:
fl >X x fl > fl
i I
X x

252
Chapter 14. Degeneracy Loci and Grassmannians
There is a projective resolution 0 —* Fh —>...—> F\ —* I —> 0 over the coordinate
ring of Ik'f, where / is the ideal of the Schubert variety Q. The regular
sequence defining X in Xx Aef remains a regular sequence on Xx Q if and only
if the pull-back complex f*(F.) remains a resolution of the ideal of Q in the
coordinate ring of X; each condition is equivalent to the vanishing of higher
Tor of the coordinate rings of X and Q over the coordinate ring of Aef. These
conditions occur when depth (Q,X) = h (cf. Kempf-Laksov A) Corollary 8).
The conclusions then follow from Proposition 7.1.)
Example 14.3.2. Let ? be a vector bundle of rank r on an ^-dimensional
variety X. Let s{,..., s^ be sections of E, N ^ 2 r. For partitions
A: A| S ... S Xr ^ 0, set
Qx = {x e X\ dim Span(^,(x), ...,sr+i-Xl(x)) S r- A,- for »'= 1, ...,/•}.
(a) Codim {Qx, X) S h, where h = ? A,-, and there is a class S^ in /4,_a (?2^)
such that
inAn^h(X). If dim(Qx) = n-h, and A" is Cohen-Macaulay, or depth {Qx,X) = h,
then !Ii^= [flj, where the scheme structure on Qx is defined by vanishing of
appropriate determinants. (Let F be the trivial bundle with basis eu...,eN,
A, the subbundle of F spanned by e[t..., em, a, = r + i- A,. Apply Theorem
14.3 to the mapping from F to E given by (sf,..., sN).)
(b)IfA=(p, 0, ...,0), then
fl^= {x eX | dim Span ((¦$,(*),..., sr+,_,(*)) ^ r - p] ,
and ff2A represents cp (E) n [Z].
(c)IfA=(l,..., 1,0, ...,0), with p l's, then
Qx= {x eX | dim Span C?i (x), ...,5f+,_i (x)) ^ r- 1} ,
and S^^ represents (- 1)' sp (E) n [X].
(Use Lemma A.9.2 to calculate dt |.)
(d) If E is generated by its sections, and the ground field is infinite, then
for generic sections st,..., sn of E, codim (Qx,X) = h, and
(With F as in (a), H = Horn (F, E), let V be the corresponding degeneracy
locus for the canonical homomorphism on H. In light of Remark 14.3, it
suffices to apply Example 12.1.11 to the subscheme V of the bundle H.)
Example 14.3.3. Projective characters. Let I be a non-singular rf-dimen-
sional subvariety of P", and let
be a flag of subspaces of P", with a,= dimA,. Set
X(A) = {x e X | dim(Tx(X)f] A,) S «, 0 ^ i ^ <
14.3 Determinantal Formula
253
Here TXX is the tangent rf-plane to X in IP". Let A, = n -d - a, + i, \ X\ =
?f=0 A,. For general A XD) is a subscheme of X of codimension |A|, and
A)
where c« = A + h)"-"</c(Tx), A =
consider the diagram
I
. (Let P" =
o.
,), and
Then A1^) = Q {A ® 0X{\); a).)
The degree of X{A) is denoted X{a0,..., ad), or simply {a0,..., ad), and is
called a. projective character of A"c: P" (cf. Severi B)). From A), we have
B) X{a,,...,ad) = \h"-Wdx{c'-('\...,c^),
x
which gives the extrinsic invariants X{aQ,..., ad) in terms of the intrinsic
invariants c,GV) and the hyperplane class h.
Let L be a generic linear subspace of IP" of dimension m, Y= Xf)L. Then
the characters of Xa P" determine the characters of 7c: L. The character
Y{a0,..., ad-n+m) for YczL is the same as the character X{a0,..., ad) for
A"c= P", with a,¦= n - d + i for / > d — n + m. (This follows from the geometric
definition, or from B).) Similarly, if A" is a projection from a variety Y in a
larger projective space, the characters of X are the same as certain characters
of y(cf. Severi B)).
The (th class of X, 0 S i S d, denoted g,, is defined by
g, = X{n — d— \,n — d, ...,n — d+i — 2, n — d+i, ...,n).
The /"¦ rank of X, for 0 S / S min (rf, « - rf), is
a>i = X{n - rf— (', n - rf+ 1, ..., u).
Thus go = co0 - deg (A"), and
Qt = deg{x 6 X | dim T^XH^ ^ » - 1}
for a given general {n — d+i — 2)-plane A, while
a>, = deg {x e A" | TXX meets 5}
for a given general {n — d— (')-plane B. Thus g, corresponds to the partition
A = (l,..., 1,0, ...,0) with / l's, and a>, to its conjugate (/, 0, ...,0). From B)
and Lemmas A.9.1 and A.9.2 (cf. Examples 14.4.15 and 12.3.6)
/=o
tli
/-o \ »-J

254
Chapter 14. Degeneracy Loci and Grassmannians
14.4 Thom-Porteous Formula
Let a: E -> F be a homomorphism of vector bundles over a purely n-
dimensional scheme X; let e=rank?', /=rankF, and let k Smm(e,f). We
construct the ktk degeneracy class
Dk(c)eAm(Dk(c)),
with m = n — (e- k)(/- k), Dk(a) the klh degeneracy locus. Let d= e— k, and
let Gd(E) be the Grassmannian of rf-planes in E; let n be the projection from
Gd(E) to X, and let S1 be the universal subbundle of n*E. The composite
determines a section, denoted st, of 5V® tc*F. The zero set Z{sa) maps onto
Dk(a). Let 77 be the induced morphism from Z(sa) to Dk(o). The localized top
Chern class TL (sa) is in Am (Z (.?„)), since m = dim (G) - df. Set
Dt(ff) = 17* (!(*„))
in ,4m (/>*(*„)).
Theorem 14.4. (a) The image ofDk {&) in Am (X) is
(b) Each irreducible component of Dk(o) has codimension at most
(e-k)(f-k), in X. If codim (Dk(o), X) = (e - k)(f- k), then D*(ff) is a
positive cycle whose support is Dk (cr).
(c) If codim (Dk(ff),X) = (e - k)(f - k), and X is Cohen-Macaulay, then
Dk (cr) is also Cohen-Macaulay and
(d) The formation of Dk (cr) commutes with Gysin maps and proper push-
forward.
Proof. The interpretation and proof of (d) are the same as in Theorem 14.3.
Since
(a) follows from Proposition 14.2.2 (cf. Example 14.2.1).
To prove (b) and (c), we assume first that E contains a flag of the form
At c:... <= Ad — E, with rank Aj = k+ i. In this case
as schemes. We claim that
as well. To see this, let Fl(A) be the flag bundle,
ft: Fl(A) - Gd(E)
14.4 Thom-Porteous Formula
255
the canonical birational morphism constructed in the proof of Proposition
14.2.2, with n*(S) = Dd,<p=n°n, and n*{sa) the section of Dvd®<p*F used in
the construction of SI (A; cr). Then
by Proposition 14.1 (d)(iii). Now (b) and (c) follow from the corresponding
parts of Theorem 14.3.
In the general case, by the splitting principle there is a morphism
f:X' -*X, which is a succession of projective bundles, so that f*E contains a
flag of the type just considered. Since f (Dk(cr)) = Dk(f* cr) by (d), the asser-
assertions for/* cr imply the same for cr. ?
As in Remark 14.3, D^(cr) may also be defined as t'a[Dk(u)], ta the section
of Hom(?,F) determined by cr, and u the tautological homomorphism on
Horn (E, F). Similarly one has the following corollary.
Corollary 14.4. The classes Dk (cr) are uniquely determined by properties (c)
and (d) of the theorem. ?
Later we shall see that the classes D^(cr) actually live in certain bivariant
groups, which better expresses their nature as "cohomology" classes (cf.
Example 17.4.2). This will also provide a sharper formulation of the assertion
(d) that their formation commutes with other intersection operations.
Example 14.4.1. If e S f and k = e — 1,
Dk{<?) = {x 6X | a(x) is not injective}.
Then Dk(cr) has codimension at most f-e + ], and
Example 14.4.2. Localized Chern classes. Let ? be a vector bundle of rank/-
on an n-dimensional variety X, let p ^ r, and let Si,..., sr-p+i be sections of E.
Set
D(s) = {x 6X | st (x),..., sr-p+l (x) are dependent}.
Then D (s) has a natural scheme structure, all components have codimension at
most p in X, and there is a class D ($) in An-P (D ($)) such that
If codim(D(s\X) = p,JD(s) is a positive cycle supported on D(s). If the sec-
sections are nowhere dependent, then D($) = 0, so cp{E) = 0. If codim (?>($), X) = p,
and X is Cohen-Macaulay or depth (/)({), X) = p, then D(i) = [D(s)]. The
formation of D(,j) commutes with Gysin maps and proper push-forward.
(The sections determine a morphism cr from a trivial bundle of rank r — p + \
Example 14.4.3. Geometric construction of Chern classes. Let ? be a vector
bundle of rank r on a quasi-projective variety X over an algebraically closed
field.

256
Chapter 14. Degeneracy Loci and Grassmannians
(i) For a suitable line bundle L, there are sections Si,...,,y,+| of E ® L
such that for any p^r,
Dp = {x eX \s\ (x), ...,sr-p+\(x) are dependent}
has pure dimension/; in X, or is empty, and
(Choose L so E ® L is generated by its sections, and argue as in Example
14.3.2 (d).)
(ii) The classes cp(E) are determined from c,(?® L) and c, (L) by the
formula (Example 3.2.2)
cP(E) = Z (- I)'"'
, (L)'"' c,(
L).
Of course, if this is taken as the definition of Chern classes, work equivalent to
that in Chap. 3 must be done to prove that they are well-defined.
Example 14.4.4. If codim(Dk(o),X) = (e-k)(f-k), and [Dk{6)] =
Y, >«,-[A] is tne cycle of Dk{a), then Dk(c) = ? e,[A], with 1 ^ e, S m,. In
general,
depth (?*(*),*) 3 (*-*)(/¦-*)
with equality if and only if codim(Dk(a),X) = (e— k)(f— k) and T)k(a) =
[Dk(a)\ (The notation and proof are the same as in Example 14.3.1.)
Example 14.4.5. Special divisors (cf. Kempf A), Kleiman-Laksov B), C),
Arbarello-Cornalba-Griffiths-Harris A)). Let J be the Jacobian of a curve C,
and let w, = [ Wj-,-], with notation as in Example 4.3.3. Let Wd a J be the locus
of special divisor classes:
W'd = {L 6 J | dimH°(L ® 0(dPo)) Sr+l).
Let y be the Poincare bundle on J x C, normalized to be trivial on/xP0. Set
where p.JxC—>/, q: Jx C —* C are projections. SW m = t + d. From the
exact sequence
0 -*0(dPa) -+f{mPo) -*0,et^ 0
on C, applying p* (y ® q* (—)), one deduces an exact sequence
For t ^2g -I, Em and F, are locally free of ranks m + 1 — g and r respectively,
and
Wd=Dk{a), k = m-r-g.
Applying the Porteous formula to a, there is a class
^), e=g-(r+l)(g-d+r),
14.4 Thom-Porteous Formula
257
whose image in Ae(J) is z)?l+J!,(c(F,-?„,)) n [J]. Since F, is a successive ex-
extension of trivial bundles, c(F,) = 1. By Example 4.3.3, Si(Em) = w,. Therefore
in Ag(J). If dim f^= g, then Wd=[Wi\. Both WJ and Wrd are independent of
choice of / S 2g - 1.
Poincare's formula asserts that i! vf, is numerically (or homologically)
equivalent to 0', where 0=hv It follows (cf. Example A.9.3) that WJ is
numerically equivalent to
<>!• !!•...•/¦!
In particular, WJ #= 0, and so Wd #= 0, whenever p^O.
The non-emptiness of Wd also follows from the fact that Ev ® F, is an
ample vector bundle (cf. Example 12.1.6, and Fulton-Lazarsfeld B)).
If C is a curve of general moduli, Griffiths and Harris B) have shown that
the W'd are reduced, of dimension q. Arguing by specialization, as in Example
12.2.3 it follows that for any C, Wrd can be represented by a positive cycle, if
If C is smooth over a given ground field, and Po is a rational point, the
discussion of Chap. 13 can be applied to these classes. If gs 0, the locus Wd
always supports a zero-cycle of degree N(g, r, d), where
N(g,r,d)=-
r\q\
{g-d+r)\-...-{g-d+2r)\ '
(In fact, 6Q- Wrd is represented by such a cycle.) For example, if the ground
field is R, q S 0, and N (g, r, d) is odd, then Wd must contain real points.
Example 14.4.6 (Beauville C)). Continuing the notation of the previous
example, assume also that q = 0. Then
degWd= N(g,r,d).
If S1 :A0(J) -> J is the homomorphism of Example 1.4.4, Beauville has proved
that if pi ,...,/>„ are positive integers adding to g, then
S(wPl •... •
t t Pi(d-Pd) ¦ (K~Bg - 2) Po)
where K is the canonical divisor. Therefore
d
in J.
2g-2
N(g,r,d)-(K-Bg-2)P0)
If W'd is not infinite, Wd contains at most N (g, r, d) points. In fact [ W%\ is a
zero cycle of degree N(g,r,d). If C has general moduli, Wd consists of
N (g, r, d) distinct points.

258
Chapter 14. Degeneracy Loci and Grassmannians
Example 14.4.7. Excess Porteous formula. Let ff.E—*F, and let k be an
integer such that Dk_{ (cr) = 0. Then on D = Dk{a), there is an exact sequence
0 -> K-> ED-> Fo-C-^0
with K, Cvector bundles of ranks e—k,f—k. Then
(a)
in Am (Dk (cr)), m = n — (e — k) (f— k). If D is a local complete intersection in
X, of codimension d with normal bundle N, then
(b) 1Dk(<j) = cp(Kv® C-N)n[Dk(c)]
where p = {e — k)(f — k) — d. (Let H = Horn(E, F), u, ta as in the discussion
preceding Corollary 14.4, and let H c: H be the open subscheme consisting of
maps of rank at least k, and set D = Dt(u)f)H. On D there is an exact
sequence
and D is regularly imbedded in H with normal bundle Kw ® C (cf. Ronga
A)). Now /„ is assumed to map X into H. The class D^(cr) may be constructed
from the fibre square q in\ y v
k v /
by Theorem 6.4. Since K and C restrict to K and C on D, the conclusions
follow from the basic formulas for excess intersections (§ 6.1, 6.3).) See Harris-
Tu B) for a study of Chern numbers of K and C.
Let
Example 14.4.8. Let f:X"^> Ym be a morphism of non-singular varieties.
Sk(f) = {xeX\ rankdf(x) S k],
i.e., Sk(f) = Dk(df), with df: Tx ->/* ?Y the associated map of tangent
bundles. If S*(/) * 0, then dimSk(f)^N, N = n-(n- k)(m-k). The
scheme S^ (/) carries a cycle class §* (/) with
= ^S:|} (c (/* 7"r)/c (
n
in ,4* (Z). If dim Sk if) = N, then St (/) = [5, (/)].
In case m^n, k = n — \, one recovers the formula for the ramification class
(Example 9.3.12).
If m= 1, k = 0. §?(/) is theclass constructed in Example 14.1.5.
With appropriate transversality assumptions, one may use the Porteous
formula to study higher Thom-Boardman singularities (cf. Lascoux F), Ronga
(I)-)
When 7= Pm, / is given by a linear system without basepoint. For more
general linear systems, see Porteous C) and Piene C).
14.4 Thom-Porteous Formula
259
Example i4.4.9. If a: E -* F is a vector bundle homomorphism, one has the
dual homomorphism av: Fv -> ?v, with Dk (crv) = Dk (cr). Then
(i) D,t(o-V) = Dt(a)
in Am(Dk(ff)). (Dt(ffv) satisfies (c) and (d), so equals Dk(ff) by Corollary
14.4.) This corresponds to the formal identity (Lemma 14.5.1)
(ii)
W (c (E- - F")) = 4-"^ (.c (F- E)).
Replacing c(Ev — Fv) by c(E—F) in the left side of (ii) changes the sign by
(—l)('-*>v-*>, giving an alternative for Theorem 14.4(a):
(iii)
^) c (?- F) n [JT].
Example 14.4.10. Another construction of Dt(cr) may be given as follows.
Let Gk (F) be the Grassmann bundle of ^-planes in F, n' the projection to X, Q
the universal quotient of n'* F. The composite
determines a section s'a of rf*Ev
restriction 7' of if, and
Q. Then Z(si) maps onto Dk(o), by a
(Denote the right side by D?(cr). Under the canonical duality isomorphism of
G/cF with G/_t(Fv) s'a corresponds to the section sav constructed from the dual
homomorphism crv. Therefore D^(cr) = Dt(crv), and the assertion follows from
Example 14.4.9.)
Example 14.4.11. Symmetric and skew-symmetric degeneracy loci (cf. Harris-
Tu A), Jbzefiak-Lascoux-Pragacz A)). Let ? be a vector bundle of rank e on a
purely n-dimensional scheme X. A bundle map c: Ev -* E is symmetric if
crv= cr. Such cr correspond to sections sa of Sym2?. Let ks e, d= e- k. There
is a subcone D\(E) of Sym2?, of codimension ( ^ j, locally defined by the
vanishing of all (fc+l)-minors of the corresponding symmetric e by e
matrices.
Given a: Ev -> E symmetric, let Dk {a) be the locus where a has rank at
most k, and define Dk (cr) by
in ^4
cr)), m = n — f J ) • The analogues of Theorem 14.4 and Example 14.4.4
are valid for these classes, with (a) replaced by the formula
Similarly, if cr: ?v-» ? is skew-symmetric, i.e. crv= — cr, and Df(a) is the
locus where the rank of a is at most k, with k even, one constructs a class

260
Chapter 14. Degeneracy Loci and Grassmannians
Df (a) in Am{Dsks(a)\ with m = « -B). and
^(clop(Sym2Gr* ?)/Sym2(S)) n [GkE]).
(If 51 is the universal subbundle on GkE, Sym2(S) is a subbundle of
Sym2Gc*?), and
D|(ff) = 7r+(Clop(Sym2Gr*?)/Sym2(S)) n [G, E]) .
Similarly for D*, with A2 replacing Sym2. For the calculation of these Gysin
formulas we refer to the cited references; see also Example 14.5.1. A proof
more along the lines of § 14.4 has been given by J. Damon (unpublished).)
There are useful extensions of these formulas to symmetric and skew-
symmetric maps from Ev to E ® L, for L a line bundle, for which we refer the
reader to Harris-Tu A).
Example 14.4.12. For any positive integers a, b, the polynomial z)j,a)(c) is
characterized as the unique polynomial of weight ab such that for all vector
bundles A, B of ranks a, b,
If a is a vector bundle map from A to B on X, D0(cr) is represented by
d(p(c(B-A)) n[X]. Porteous' formula says that the same polynomial works
for higher degeneracy loci: if E, F have ranks k + a, k+b, and a: E -> F, then
Similarly, for any positive integer a, 2"da((c) is the unique polynomial of
weight f") such that for all vector bundles A of rank a
(see Example 14.5.1). If a is a symmetric map from Av to A, this says that
D0(cr) is represented by 2"da t(c(A)). The formula of the preceding
example states that for any bundle of rank k + a, and a: Ev -> E symmetric,
The analogous assertion is also true for alternating maps. Is there a proof of
these three formulas which uses only these characterizations of the poly-
polynomials involved?
Example 14.4.13. Positivity of degeneracy classes. Positivity assumptions on
the bundle ?v® F= Hom(?, F) force corresponding positivity on the classes
Dk{6). Let m = n - (e - k)(/- k) is 0.
(a) If Ev ® F is generated by its sections, then D^ (cr) is represented by a
non-negative m-cycle on Dk(a).
(b) If ?v® Fis ample, and L is an ample line bundle, then
14.4 Thom-Porteous Formula
261
In particular, if Ev® F is ample, and m S 0, then Dk(a)=?0 for any a
(Example 12.1.5). Corresponding assertions are also valid for symmetric and
skew-symmetric bundle maps. (Realize Dt(cr) as the intersection of a section
of Hom(?, F) with the universal degeneracy locus in Hom(?, F), and apply
Theorem 12.1.)
Example 14.4.14 (Giambelli B), C), Harris-Tu A)). Let m g n. The variety
of m by n matrices, modulo scalars, is a projective space P"-'. Let
Vk(m, ri) c: pm"-i be the sub variety of matrices of rank at most k. Then
(Apply Porteous to a: I
on P™". To calculate the resulting
determinant use Example A.9.4.) There are similar formulas for varieties of
symmetric and skew-symmetric matrices. For references to the classical litera-
literature see Baker B) p. 111.
Example 14.4.15. Classes, ranks, and polar loci, (a) Let A" be a non-singular
rf-dimensional subvariety of P", and let L be a linear subspace of IP" of
dimension n — d+ m. For p S 0, and —mSpSn — d, set
For general L, X(L, p) is a subscheme of X of codimension p(m + p + 1), and
(As in Example 14.3.3, if Pm = P(V), L = P(L), there is a canonical morphism
from L®0(\) to ./V^P", and X(L,p) is the locus where this morphism has
rank at most n — d—p.)
(b) The cases where p = 1 give formulas for the polar classes of X. Fix i,
0 S i S d, and let L be a subspace of dimension n — d+ i - 2. Set
For general L, codim(A'(L),A') = i, and X(L) is called an »th polar locus. For
general L,
j-o
The /"¦ class g, of X is defined to be the degree of X(L), for general L. Thus
the preceding formula refines the formula for q, given in Example 14.3.3. For
other approaches and generalizations to singular varieties see Example 4.4.5,
Todd (8), Pohl A), Porteous C), Piene C), and Kleiman (8).
(c) The cases where m + p +1 = 1 determine the ranks of X. Fix
0si§ min (d, n — d), let dimL = n — d—i, and set
X'(L)={xeX\ TXX meets L\.
Then for general L, codim (X' (L), X) = i, and
j-o \i-Jl

262
Chapter 14. Degeneracy Loci and Grassmannians
The (lh rank a>, of X is defined to be the degree of X'{L), for general L. This
refines the formula for a>, given in Example 14.3.3. Note that if pL: X-* Pd+'-'
is induced by projection from L, X'(L) is the ramification locus Sd_x(pL)
considered in Example 14.4.8.
For example, if A" is a curve in P", the (first) rank of X is the number of
tangents to X meeting a general («-2)-plane. If N = dtg(X), g = genus (X),
therankis2JV + 2#-2.
Example 14.4.16. Let ? be a vector bundle of rank e on an ^-dimensional
variety X, with a subbundle A of rank a. Let Gd(E) be the Grassmannian of d-
planes in E, and let Qk (A) be the subscheme of Gd(E):
\ x 6 X, dimLH A(x)^ k).
Qi(A) = {Lc E(x)\
Let p = e-d-a + k. Then Qk(A) has pure codimension pk in
&,t (/4) has a natural scheme structure so that
and
[0* 04)] = zJ<« (c (G - **/<)) n [Gd(E)],
where Q is the universal quotient bundle on GdE, and n the projection from
Gd(E) to A". (If a is the canonical map from n*A to g, then Qk(A) is Da^k{a).
The scheme determines the cycle, as in Example 14.3.1.)
Example 14.4.17. Linear systems on curves (cf. Schwarzenberger B), Arba-
rello-Cornalba-Griffiths-Harris A)). (a) Let C be a non-singular projective
curve, L a line bundle on C. An (s+ l)-dimensional subspace V of F(C,L)
determines an ^-dimensional linear system | V | on C. Set
VJ = {De Cw | dim Vf]r(C, L(- ?>)) ^ 5 + 1 - i + r},
i.e., Vd consists of those effective divisors of degree d which impose at most
d—r conditions on V. One may realize Vd as a degeneracy locus as follows. Let
jg><= Cx Cw be the universal divisor: @= {(P, D) \ P e D). Let p and q be the
projections from 3 to Cw and C. Set
Then ?L is a bundle of rank rf on Cw, whose fibre at D is T(C, L/L(-D)).
There is a canonical homomorphism of vector bundles
on Cw, and V'd = Dd-r(o). From Porteous's formula (Theorem 14.4) there is a
class V; in Am (V'd), with m = rf - r(s + 1 - d + r), so that
(b) The Chern class of EL, or the Chern polynomial c,{EL) may be
calculated as follows. For P e C, let ^""(P) be the image of the imbedding
of Od-[) in C"> by D -* D + P, and let xP eA[ (C^) be the class of
For any line bundle M on C set
= IIO
14.5 Schur Polynomials
263
where X! "/>[^>] is a divisor whose line bundle is M. Then
(i) c,(EL) = CK-aO • c,(Ot-) = cK-L(t) ¦ A - xt)"^+l/u*dw(-t/(\ -xt)),
where A^ is the canonical line bundle on C, x = xPll, and w(t) = X vf(f',
w, 6/4' G) as in Example 4.3.3. Since all xf are algebraically equivalent to x,
(n) c,{EL) = {\-x t)d^-"-'/u*dW{- //(I - x 0)
modulo algebraic equivalence, where n = deg (L). Modulo homological or
numerical equivalence, Poincare's formula gives
(iii)
c(E,) = (\-
-XI)
where 6 = u*d(w\). (If L = K, then EL = QxCn, and (i) follows from Example
4.3.3 (b). For P in C, there is an exact sequence
0 -> E?(_,
0 ,
so c,(EL) = c,{EL^P)) ¦ (\ - xPt). The validity of (i) for L or for L(-P) is
therefore equivalent, so the verification for one bundle K suffices for all.)
Formula (iii) is proved using the Grothendieck-Riemann-Roch theorem in
Arbarello-Cornalba-Griffiths-Harris A), where several enumerative applica-
applications may be found.
14.5 Schur Polynomials
Let c\, c2, c3, ... and s\, s2,
related by the identity
s3,
... be sequences of commuting variables,
A — S[t + S2t2 — S}t3 ...) = 1 .
For example, if Cj=c,{E) are the Chern classes of a vector bundle E, then
s, = s, (?v) are the Segre classes of Ev. If c, = c,(? — F), for vector bundles E, F,
then
A partition is a finite sequence A = (Xt,..., Xd) of non-negative integers,
arranged in non-increasing order. Set \X\ = Xt +... + Xd, the integer being
partitioned. To X corresponds a Young diagram, consisting of A, boxes in the ith
row from the top, lined up on the left. Thus
is the Young diagram for D,4,2,2,1). The Young diagram for the conjugate
partition is obtained by interchanging rows and columns in the Young

264
Chapter 14. Degeneracy Loci and Grassmannians
diagram. Thus E,4,2,2) is the conjugate to D,4,2,2,1). Denote the conjugate of
X by X.
Fora partition A=(Ai,..., Xj), set
the corresponding Schur polynomial (or S-function); dx(c) is isobaric of weight
|A| in the c,. Note that adding an arbitrary string of zeros to X does not change
Me).
Lemma 14.5.1. Let X and /j be conjugate partitions. Then
This is proved in Lemma A.9.2. We note several consequences. Corresponding
to X = A,..., 1), n = (d), one has
A)
sd = det
C\ C2 ... Cd
1 cx ... cd-\
0 1 ...
0 ... 1 c,
If X partitions de into d equal pieces of size e, the lemma gives
B)
(For the second equality see the last step in the proof of Lemma A.9.2.)
If r is an integer such that st = 0 for all / > r, and Xr+1 > 0, then
C) dk{c) = Q.
(Indeed, if p. is the conjugate to X, then p.x > r, so the first row of the matrix
whose determinant is dM(s) is zero.)
Lemma 14.5.2. Let X = (A|,..., Xj) be a partition, m a non-negative integer.
Then „
the sum over all partitions n = (p.u ..., Hd+\) such that
and | p. | = j X \ + m.
The Young diagrams for the partitions n appearing in the above formula
are those which one obtains by adding m boxes to the Young diagram for X, to
the right of the given rows, such that the new diagram still has rows of non-
increasing length, and such that no two of the new boxes appear in the same
column. For the proof see Lemma A.9.4.
The rule for multiplying general S-functions is somewhat more complicat-
complicated. A Young tableau is a Young diagram together with a collection of symbols
in the boxes. A simple m-expansion of a tableau is one obtained from a given
tableau by adding m new boxes according to the prescription of the preceding
paragraph, putting the same new symbol in each. A {p. x,..., ^-expansion of a
tableau is the result of e successive simple expansions, first adding //, boxes
14.5 Schur Polynomials
265
with the symbol a.\, then Hi boxes with the symbol 0B, etc. Given a (jxt,..., /ue)
expansion, write down the added symbols in order, proceeding from right to
left in the first row, then right to left in the second, and so on, obtaining a
monomial
where t = ^ + ... + /.ie. The expansion is called strict if for every 1 ^j S t, and
1 S k < e, the number of oe^'s occuring among the first j terms of this
monomial is no less than the number of a^+i's occuring in these firsty terms.
Lemma 14.5.3. Let X = (A|,...,
= {p. |, ...,pe) be partitions. Then
where the sum is over all partitions Q with \q\ = \X\ +\/x\, and NXtMiQ is the
number of ways a Young tableau on the Young diagram of q arises by a strict
(ft[,..., /xe)-expansion of the diagram for X.
This rule for computing the coefficients is known as the Littlewood-
Richardson rule. See Appendix A.9 for references.
Example 14.5.1 (Lascoux G); cf. Macdonald C) p. 31). Let ? be a vector
bundle of rank n. Let N = f")- Then, with the notation of Example A.9.1,
(a)
(b)
In particular, one has formulas of Giambelli:
(a') c/.J.n(S2?) = 2M».»_,.
(b#)
= 4,_, 0(c(E)).
(For (a) let x \,..., xn be Chern roots. Then
=n (i+x,+xj)=n (i+2x,
isj 1-1
= 2'Nse{\ + 2xu...,l + 2xn) =
the last two steps using Example A.9.1 (b) and (a). The proof of (b) is simi lar.)
Example 14.5.2 (Lascoux G); cf. Macdonald C) p. 37). Let E, F be vector
bundles of ranks n and m. Then
c(E ® F) = X d,Mdfi(c(E))dr(c (F))
the sum over partitions pc] with m S k\ § ... ^ Xn s 0; here
X'= (m — Xn, m — An_i,..., m — X\), and coefficients dXfl are as in Example

266
Chapter 14. Degeneracy Loci and Grassmannians
A.9.1. (The formula follows from ExampleA.9.2(c).) For m=\, the formula
specializes to that of Example 3.2.2.
Example 14.5.3. Let A, /j be partitions of length d, and assume s, = 0 for all
( > d. Then
This formula dates from Jacobi, and was rediscovered by Porteous, cf. Lascoux
C), Macdonald C) p. 30.
Example 14.5.4. Skew Schur functions (cf. Lascoux E), Stanley A), Mac-
Macdonald C) p. 39). If A, fi are partitions with n a A, define
Then
where NVtMii are the integers appearing in Lemma 14.5.3, and the sum over all
v with |v + u\ = I A I.
14.6 Grassmann Bundles
Let ? be a vector bundle of rank n on a scheme X, and let d be a positive
integer less than n. Let G = Gd(E) be the Grassmann bundle of rf-planes in E,
with projection n from G to X. There is a universal exact sequence
on G, with S1 the universal rank rfsubbundle, and Q the universal rank (n-d)
quotient bundle. Set
c(-=c,(e-**?).
The corresponding 5,- for the discussion of § 14.5 are
as follows from the Whitney sum formula.
In this section a partition A will be a sequence {X\,...,Xd) of d terms,
A, ^ ... s Xd^ 0. Note by consequence C) of Lemma 14.4.1 that dXl a,(c) = 0
if q > d and Xq > 0, so no non-zero Schur polynomials are lost by this restric-
restriction. Given X, set
dk = dk{c)=dh Xd(c(Q-n*E)).
Proposition 14.6.1 (Pieri's formula). For any partition X, and any m ^ 0,
the sum over all partitions p. with
and I u I = IAI + m.
14.6 Grassmann Bundles
267
This follows from Lemma 14.5.2 and the vanishing noted preceding the
proposition. Likewise, the following proposition follows from Lemma 14.5.3.
Proposition 14.6.2 (Product formula). For any partition A, n,
the sum over all partitions q with \q\ = \X\ + \/u\, and iV^j given by the Little-
Littlewood-Richardson rule. ?
Proposition 14.6.3 (Duality theorem). Let X and p. be partitions with
\X\ + \n\^d{n-d),andletaeA)f(X). Then
n*a) =
cc if A, + fii-nr i = n — d for 1 S i
0 otherwise.
Proof. By compatibility of it* and n* with proper inclusions, we may
assume a = [X], X a variety; therefore n*[X] = [G]. If | A | + | n \ < d(n - d) the
conclusion is clear for dimensional reasons. \f\X\ + \n\ = d(n — d), then
for some integer m. This equation remains valid if X is replaced by any open
subvariety, so we may assume E is trivial. In this case we claim that
@
if
otherwise.
= n- d for \SiS
Here z)'^ corresponds to the partition of d(n-d) into d equal parts. This
follows readily from the Littlewood-Richardson rule (or see Example 14.6.1).
Given A, the only strict (p.,,..., ^-extension of A to a tableau onarfby n — d
rectangle is obtained by adding, at each stage, a new symbol to every column
which has not been filled in previous stages.
To conclude the proof, we show that 71* (A$.d n [G]) = [X]. Choose a flag A
of trivial subbundles of E, with rank A,= i, and let c be the projection from
n* E to Q. By the determinantal formula,
Here Q(n*A; a) = {Le Gd(E) | dimLC\At ^ i} = Gd(Ad). Therefore n maps
Q{n*A; a) isomorphically onto X, so
as required. ?
Proposition 14.6.4 (Giambelli's formula). Let A be a flag of subbundles ofE:
Let Q(A) = Q (n*A; a), where a is the projection from n* E onto Q, i.e.,
Q(A) = {L e Gd(E) \ dimL 0 A, S i, 1 g i ^ d}.

268
Chapter 14. Degeneracy Loci and Grassmannians
Let X, = n-d+ / — rank (A/). Assume X is pure-dimensional. Then Q (A) has pure
codimension \k\ in Gd(E). If c-^E-Aj) = 0 for all />0 and all ] = 1, ...,d,
then
[Q(A)]=Axn[Gd(E)].
Proof. The determinantal formula (Theorem 14.3) gives
Sl(n*A; <j)=Ah Xd(c(Q - n*At),..., c(Q - n*Ad)) n [Gd(E)].
The assumption on Chern classes implies that
c (Q - n*Aj) = c(Q-n*E)-c(n*E- ji*A,) = c(Q- n*E).
To conclude, it must be verified that Sl(n*A; a) = [Q (A)]. This verification is
local, where the bundles are trivial and Gd(E) is a product XxGd(A"). On the
non-singular variety Gd(A") the equality is known by Theorem 14.3 (c), and it
pulls back by flat projection to Gd(E). ?
Proposition 14.6.5 (Basis theorem). For each i§0, there is a canonical
isomorphism
the sum over all partitions X = (X\,..., Xd) with
n-d^ l|i...glj?0.
Each element in Ak(Gd(E)) has a unique expression in the form
Y.dxr\ n*{ax),
ctx
+\lL\ (X), X as above.
Proof. (See also Example 14.6.2.) Let cp: ® xAk _ d{n _ d) + U| (X) -> Ak (Gd(?)) be the
homomorphism taking © ax toJ^Axr* n*txx. If ®ax is a non-zero element in
the kernel of (p, choose X with \X\ maximal such that ctx 4= 0. Define n to be the
partition with n, + Xd^i+l = n - d, 1 S i S d. By duality (Proposition 14.6.3),
0= ^(^-^z^n n*txx) = ax,
a contradiction.
To show that q> is surjective, as in the proof of Theorem 3.3 (b) — which is
the case d=\ of this proposition — it suffices to verify this when X is
irreducible and E is trivial. Choose a basis e,,..., en for E, which identifies
Gd(E) with XxG, G= Gd(A"). For each sequence a, 0 < a, < ... < ad^ n, let
Aj be the space spanned by et,..., eai, and
Q(a) = {Le G \ dim LHA^ i, \ ^i^d}.
Note that Q (b) c Q (a) if b ^ a (i.e., b, ^ at for all /). We have seen that Q (a)
is a subvariety of G, and that
14.6 Grassmann Bundles
269
with X, = n — d+ i — a,. To prove q> surjective, it therefore suffices to show that
for each a the homomorphism
taking © otj to Y, <*±x [® (t)], is surjective; here \b\ denotes Y, (d;— 1).
Let Fbe the span of eai, ea2,..., eai, W the span of the other basic vectors.
Let
Gy= {L e G\ L projects isomorphically to V].
Then Gv is an affine open subvariety of G, canonically isomorphic to
Hom(F, W). Any L in GKhas a unique basis projecting to the basis eQl, ..., eai
of V; L determines a d by n matrix
1
a2
* 0 *
1
0
ad
... * 0
0
0
1
whose 1 row gives the coefficients of the ith basic vector of L in terms of
e\,..., en. Conversely, any such matrix determines an L'mGy. Set
This is a linear subspace of Gv= Hom(F, W) of dimension Y, (ai~ 0i m the
above matrix description, it is given by the vanishing of all entries of all rows
to the right of the indicated l's. There is a union
Q(g) - Q(a)° = U
- 1,..., ad
where any sequence which is not a strictly increasing set of positive integers is
discarded. One concludes by induction on \a\ and the exact sequence (Proposi-
(Proposition 1.8)
Ak(Xx (Q (a) - Q (a) °)) -+Ak (Xx Q (a)) ^ Ak (Xx Q (a) °) ^ 0 . O
Example 14.6.1. The duality theorem may also be given a simple geometric
proof. One may assume E trivial, with basis eit...,en. Let A, be spanned by
the first a, basis elements, and let B, be spanned by the last bt elements. Let
X/= n — d+ i— a,, /u,= n — d+ i — bj. By Giambelli's formula,
Axn[G] = [Q(a)], A, n [G] = [Q (ft)].
Then Q(a) f] Q(b) = 0 unless a; + bd.i+, = n + 1 for all i, in which case Q(a) f]O(b)
is one (reduced) point. One concludes by Proposition 8.2.1.
Example 14.6.2. The basis theorem may also be proved by induction on d
and n. If E = F©1, with F trivial, one has an inclusion i of Gd_x (F) in

270
Chapter 14. Degeneracy Loci and Grassmannians
Gd(E); the complement U is an affine bundle over Gd(F), equipped with a
section s. One has a diagram
¦Ak(U)-
• 0
Computing how the classes Ax transport via the indicated maps, and using the
formula i# i* (a) = cn-d(Q) n a, the knowledge of generators for Ak(Gd-iF)
and Ak_d(GdF) determines generators for Ak(GdE). (For another proof based
on this geometric construction, see Laksov B).)
Example 14.6.3. If Q(a)° is the open affine subvariety of the Schubert
variety Q (a) constructed in the proof of the basis theorem, then
Q (a) ° c {L e G | dim (L f] A,) = <}.
This inclusion may be strict; the right side need not be an affine variety.
Example 14.6.4. The classes Ax (c @) may be used in place of
4t(cB~ n*E)) in the basis theorem. It can happen that Ax(c(Q)) + 0 for X a
partition with more than d non-zero terms, so Pieri's formula for these classes
must be stated in the modified form of Lemma 14.5.2. If c(E) = 1, there is no
distinction, and Ak = 0 if kx > n - dor kd+l > 0.
Example 14.6.5. There is a canonical duality isomorphism
such that cp*Sv is the universal quotient bundle on Gn_d(Ev), and (/>* Qv the
universal subbundle. Then
(p*Ax(c(Q - ji*E))=Aic(<p*Sv),
where Xis the conjugate partition to k. In particular, if c (E) = 1, then
Example 14.6.6 (cf. Grothendieck A), Jouanolou A)). If X is non-singular,
E a vector bundle of rank n on X, then A*(GdE) is the algebra over A*X
generated by elements a,,..., ad, b\,..., bn-d, modulo the relations
Z aibk-i= ck(E)
for k=\,...,n. (Take a, = c,E), bj=Cj(Q), with S and Q the universal
bundles.)
More generally, if F= F(d{, ...,dk) is a flag bundle of flags in E, with
universal flag Dtc ... c Dk = EF, rank(/),) = 4, then A*(F) is the A* (X)-
algebra generated by the Chern classes of the quotient bundles Z>,-/Z>,-_i, with
relations determined by the identity
14.7 Schubert Calculus
14.7 Schubert Calculus
271
The Grassmann variety of J-planes in P", denoted Gd{V"), is identified with
Grf+|(A"+I). The universal quotient bundle Q on G=Gd(V") has rank n - d.
The classes
om = cm{Q)eAmG, \Sm^n-d,
are called the special Schubert classes. For each X = (la, ...,ld), with
n- dg Ao^ ... ^ Xd^ 0, define the Schubert class {X} or {X0,...,Xd} by the
formula
{Xa,...,Xd}= Ak (a) = det (ff^+y-O
Thus am = {m, 0,..., 0}. By the results of the previous section, the Schubert
classes form a free Z-basis of A*G. In particular, Pic(G) =A{G= Z, generated
by <T|. The multiplication is determined by Pieri's formula:
the sum over /u with n — d^ /u0 § Xo S ... g ^a Xdt \n\ = \X\ + m. More
generally,
WM ZN{}
where the A^p are determined by the Littlewood-Richardson rule. If
\X\ + \jj.\ = (d+ \) (n — d), one has the duality theorem:
{n-d,...,n-d]
0
if kt +ftd-i = n-d for
otherwise.
If/lop /4|
P" is a flag of subspaces, with a,-= dim/4,-, let
4J = {Le Gd(P") | dim LD/4,- S i, 0 ^ i ^ d}.
Then 12(/40. •••. Ad) IS a subvariety of G of dimension S?=0(ai~0. i-e-.
2?=o at — \d(d + 1), called a Schubert variety. Its class in ^(G) depends only on
a0, • • •, ad, and it is denoted by (a0,..., ad); for 0 S aa < ax < ... < ad^ n,
(ao,...,ad) = [Q(Ao,...,Ad)].
Giambelli'sformula reads:
{Xo, ...,Xd}r\[G\ = (a0,..., ad)
with ki=n-d+i-a/. The notations {20, ...,-i</} and (aa,...,ad) originate
with Schubert; both remain in use, together with many others.
These results are all special cases of the formulas of the preceding section;
the shift in indexing from (ku..., kd+i) to (Xa,..., kd) compensates for the loss
of dimension in passing from affine to projective spaces.
The formulas of this section can also be proved by direct geometric
argument, as in Hodge-Pedoe A). The approach followed here, on the other
hand, shows how the formulas are consequences of the basic determinantal

272
Chapter 14. Degeneracy Loci and Grassmannians
formula, and of general polynomial identities, together with the vanishing of
certain higher Chern classes and inverse Chern classes that occurs on Grass-
Grassmannians.
Example 14.7.1. Let G=Gd(F"), aeAk(G). For each Schubert class
(&o> • • • > bj) of codimension k, let
a»0 b,= Ja-F0, ...,bd).
G
Then the expression of a in terms of the Schubert classes of dimension k is
<x = S ««-*¦ «-a.(flo»---. ad) .
(This is a formal consequence of duality.) If /? is a class of complementary
dimension, then
{(X • /? = X In-at R-ao/?ao,...,a< >
C
the sum over all Schubert classes of dimension k.
Example 14.7.2. For G, (P3), Schubert used a special notation:
1 =B,3) = {0,0} = [G,(P3)]
g = A,3) = {1,0} = a.
</, = @,2) = {2,1}
G = @, 1) = class of a point in G, (P3).
The products are: g2 = gp + ge, g ¦ gp = g ¦ ge= gs, g] = g\=G, gp-g€ = 0,
ggs=G.
It follows that g*=2G: there are two lines in P3 which meet 4 given lines
in general position.
Example 14.7.3. The special Schubert classes are
Om= (n — d— m, n — d+\, n — d+2, ...,n) .
If A is a subspace of P" of codimension d+ m, am is represented by the special
Schubert variety
Example 14.7.4. If (a0, ...,ad) and (b0, ..., bd) are Schubert cycles of com-
complementary dimensions, then
, II if a, + bd-,= n, O^i^d
l(aQ,...,ad)-(b0,...,bd)=\Q otherw.se
Example 14.7.5 (cf. Hodge-Pedoe A)XIV.2). Let ip: Gn_rf_|(P") -> Gd(V)
be the duality isomorphism of Example 14.6.5. Then
(p*((aQ,...,adj) = (ba, ...,bn-d-{) ,
where the integers bo,..-, bn-d-\ form the complement of n — ad,..., n — aa in
the set of integers from 0 to n. (This follows formally from Giambelli's
14.7 Schubert Calculus
273
formula and Example 14.6.5, using the combinatorial fact which begins the
proof of Lemma A.9.2.)
Example 14.7.6. (a) Let C be a reduced curve of degree din P3. Let
Vc = {I e G, (P3) j / meets C].
Then Vc is a hypersurface in G= G\ P3, and
[Vc] = d-g
with g = ff| as in Example 14.7.2.
(Consider the incidence correspondence 7cP3xG, consisting of pairs
(P, I), Pel, with projection/),: / ->• P3, p2:1 -> G. Then
Sincep2 mapspr1 (C) birationally onto Vc,
I Vc] = Pi* P* [Q = Pi*P* (d ¦ [/o]) = d-g.
where /0 is a fixed line in P3. Or one may simply use Example 14.7.1.)
(b) If C\,..., C4 are fixed curves in P3, in general position with respect to
the action of the projective linear group, then there are
2r[deg(C,)
i-1
lines meeting all four curves. (Use (a) and jG g* = 2.)
Example 14.7.7. (a) Let C be an irreducible non-planar curve in P3. Let Wc
be the surface in G = G\ (P3) which is the closure of the set of lines which meet
C in two or more points. Then, with the notation of Example 14.7.2,
where d is the apparent number of double points of C (the number of chords
to C through a general point), and n is the degree of C (so there are n(n— l)/2
chords to C in a general plane).
(b) Given curves C, C,, C2 in general position, with d, n as above for C,
and «, = degC,, there are
n]n2(d+ n(n- l)/2)
chords to C which meet C, and C2-
(c) If C is another such curve, with d' apparent double points, and degree
ri, in general position with respect to C, the number of common chords to C
and C is
Example 14.7.8. The rank of a curve C in P3 is the number r of tangents to
C which meet a given general line (cf. Example 14.4.15). The number of
tangents to C meeting a given curve C of degree «', in general position, is r n'.

274
Chapter 14. Degeneracy Loci and Grassmannians
For higher dimensional applications of Schubert calculus to tangency
problems, see Fulton-Kleiman-MacPherson A).
Example 14.7.9. Let X,/u be partitions, with /icl Set (see Example 14.5.4)
Au,, = | Oi,-,,,-,+j \osi,jsd
where a,= n — d+i — A,, b,= n — d+j — fiJt a<^6;. Then AXih is dual to the
class
(«- ad,...,n- a0) • (bo,...,bd) .
(If |v| = |>l|-|^|, by Example 14.5.4, Al/lt -A',= N,^x = A, -A,, ¦ A'x, where
A',, A\ are the classes dual to A,, Ax.)
Example 14.7.10. A general formula for multiplying Schubert classes is
given by Example 14.5.3:
{/lo, ••-, Id} ' {fiO> ¦¦¦>tid} = I ^li+iu.i-i+jloaijsd ¦
Example 14.7.11 (Schubert A), cf. Hodge-Pedoe A)XIV.7). Let G=Gd(P"),
N= (d+\)(n - d). For any Schubert variety (a0, •••, ad), of dimension A:, let
deg (a0>..., ad) = j a\ ¦ (a0,..., ad).
a
(The Pliicker imbedding of G is determined by the line bundle A"~d(Q), so
<jl = ct (Q) is the class of a hyperplane section.)
d
(i) deg (a0,..., ad) = X deg (a0)..., a,¦ - 1,..., ad)
i-O
where deg (b0,..., bd) is defined to be zero if the conditions 0 ^ b0 < b\ < ... <
bd^n are not all satisfied. (Use Pieri's formula.)
(ii)
dsg(ao,...,ad) = -
k\
¦TT (a -a,)
XX \ J •/
uO\ ... ad\ {<j
where k = ??=0 a, — d(d + l)/2. (Use (i) and induction on k.)
_ V.2\...d\N\
(Set ai= n — d+ i.) This number is the number of J-planes meeting N given
(n — d— l)-planes in general position in P". For d=\, this number is
1 Bn-2\
More generally, Schubert C) uses Pieri's formula and induction on k to
prove
(iv) (an,..., ad) ¦ of = X (*!) 11/(«* -bj)\\(b0,..., bd) ,
where the sum is over all OsAo<... <bd^n with X ^;= X a* ~ ^'. in tne
determinant, \/(a,— bj)\ is taken to be 0 if a, < bj. Equivalently,
(v) deg((a0, •••, ad) ¦ (ba, ...,bd)) = k\ \ l/(a, + bd-,- n)\ \ ,
where k — X (a,- — 0 + X (Pj ~j) ~ (d+\)(n — d).
14.7 Schubert Calculus
275
Example 14.7.12. Parameter spaces for many enumerative problems can be
formed by a finite number of projective, Grassmann, and flag bundle construc-
constructions. Let P" = />(K), dimF=« + l, G = GdW" = Gd+l (V), with universal sub-
bundle S.
(a) The variety of hypersurfaces of degree m in d-planes in P" may be
identified with the projective bundle P (Symm EV)) over G.
(b) If M= P(W) is a subspace of P(V), Wa V, the incidence variety
IM={(P,L)eMxG\PeL}
is a Grassmann bundle over M. If A is the universal rank 1 subbundle of W on
M, IM is Gd(VM/A). If dim M= n — d, the projection from IM to G is birational.
(c) The space X = P (Symm EV)) xc IM represents triples (H,L,P), H a
hypersurface in L as in (a), P a point of Lf]M. The subvariety D of triples with
P e H is given by the vanishing of the composite
with <?A) the universal line bundle on /)(Symm5'v). The image of D in
/>(SymmE'v)) represents hypersurfaces H c L which meet M
(d) For « = 3, d= 2, m = 2, one has the 8-dimensional variety P (Sym2 Sv)
of conies in P3. With X as in (c), one may calculate that
which corresponds to the fact that there are 92 conies meeting 8 lines in
general position (cf. Example 3.2.22).
Example 14.7.13. Fano schemes (cf. Altman-Kleiman B)). Let P" = P(E), E
a vector space of dimension n + \. Let G= Gd(P") = Gd+i (E), S the universal
subbundle of ?c.
A hypersurface X c P" of degree m is given by a section of Symm (?"v) on
P". Since Sv is a quotient of ?g, this determines a section of SymmEv) on G,
whose zero scheme is the Fano scheme F of J-planes in X. If codim (F, G) =
, then the class of [F] in A*G is
In case m = 3, d=\, [F] = c<(Sym3(Sv)) = \&s^s2+9si, where 5,-=Ci(Sv).
The degree of F is
27Bn-6)!
For n = 3, this gives 27, the number of lines on a cubic surface. For n = 4, it
gives 45, the degree of the Fano surface of lines on a cubic three-fold.

276 Chapter 14. Degeneracy Loci and Grassmannians
Example 14.7.14 (cf. Hodge-Pedoe A)XIV.7, Altman-Kleiman B)).
(a) The lines in P" which lie in a given non-singular quadric hypersurface
form a subvariety F (a Fano variety) of codimension 3 of G= G, (P"), with
In fact [F] ¦ @, 4) = 0 since no line in the quadric passes through a general
point, and \G [F] ¦ A, 3) = 4 since four lines in a general section by a 3-plane
meet a general line in the 3-plane. Or one may construct F as the zero scheme
of a section of Sym2Ev), as in the previous example, so (Example 14.5.1)
[F\ = c3(Sym2(Sv)) = 22zl2,, (c(Sv)) = 4zf2J (c@).
(b) There are 16 lines common to two general quadrics in P4.
(JOA,3J = 1.)
Example 14.7.15. If 2d S n - 1, the J-planes in P" which lie in a given non-
singular quadric hypersurface form a subscheme F of codimension
(d+ \)(d+2)/2 in G = Gd(Pn), whose class in A* G is
[F] = 2d+l{d+\, d, ...,\} = 2d+'(n-2d- 1, n- 2 d+ 1,...,«- 1) .
(As in the previous example, this is the top Chern class of Sym2(Sv).) If
n = 2d+ 2, two general quadrics in W2d+2 have 4d+] J-planes in common.
Example 14.7.16. Flag manifolds. Fix n, and fix points e0,..., en in P" which
span P". For any integer m between 0 and n, let [m] denote the ra-plane
spanned by eo,et,..., em. For any set a of integers between 0 and n, number
the elements of a in increasing order:
0 ^ aa< a, < ... < adS n , d-#a—\.
Let [a] denote the corresponding Schubert variety:
[a] = {Le Gd(F") | dim Lf]^] ^i,0^i^d}.
Denote by a* the dual sequence:
a* = {n — (id, n — ad-\,...,« — aa] ,
so that [a*] is the dual Schubert variety to [a].
If a and b are sequences, write b< a if b is a proper subset of a.
For integers 0 S dx < d2 < ... < dr < n, let F = F(d\,..., dr; n) denote the
flag manifold whose points are flags of linear subspaces
L, c: L2e ... c: Lr c P"
with dim L, = d,. Consider nests of r sets:
a1 < a2 < ... < ar, # d'= d,¦+ 1 .
For each such nest, let
14.7 Schubert Calculus 277
Then [a1;...; ar] is an irreducible subvariety of F, with
t d,
In this sum any term (a)—j) is omitted if a) appears in the previous a'.
Let (a';...;ar) denote the class of [a1;...; ar] in A*F. This class is inde-
independent of the choice of basis e0,..., en.
(i) (Basis) The classes (a1;...; ar) form a free basis for A* F.
(ii) (Duality) If dim [a1;...; ar] + dim [?'; ...;br] = dim (F), then
if b! = a'* for 1 ^ i S
otherwise .
Thus if a eAkF,
a = Z (a • la1*; • • •; a'*]) [a1; ¦¦¦;ar]
the sum over all [a1;...;a'] of dimension k. (Realizing the flag manifold as a
succession of Grassmann bundles, one verifies that A^F is a free abelian group
with the same number of generators as the number of nests a1 < ... < ar,
# a'= 4 + 1. It therefore suffices to prove (ii). In case
a} + b'dl-i = n for all \Si^r,0sj^ dt,
or a'j + b'd^j < n for one such i, j, the conclusion follows by using comple-
complementary flags, as in Example 14.6.1. In any other case, dim [a1,..., a'] +
dim[i>',..., br] > dimF; equivalently, if a)^ c) for all i,j, with some inequality
strict, then dim [a1,..., ar] < dim [c1,..., c']) See Ilori B) for a Pieri formula for
flag manifolds.
Example 14.7.17. Incidence varieties (Martinelli B)). Fix d S n, and let
a smooth variety of dimension d(n - d) + n. For 0 ^ a0 < ... < adS n, and
0 S k S d, let
[fl0,... A,... ,ad] = {(P, L)e / | dimLDk-]^ i,0 ^ i ^ d, and Pe[ak]}
with [a,,] as in the previous example. This is a variety of dimension ? (a,— i) + k.
The classes (a0, ¦¦¦, ak,..., ad) of these varieties form a basis of A+I. The class
(n — ad, ...,n — ak,...,« — aa)
is dual to (a0, ...,ak,..., ad). To know the coefficients of a class in A^I, it
suffices to compute the intersection number with all dual classes. This basis is
often more convenient than that obtained by realizing / as a Grassmann
bundle over P" (Example 14.7.12 (b)) and applying Proposition 14.6.5. The
appealing notation was used by Martinelli. (The assertions are special cases of
the preceding example.)
Example 14.7.18. Fix d, n and / as in the preceding example. Let V be a
subvariety of P" of codimension eS d+ 1. Let Vo be the non-singular locus of
V, and let Vc /be the closure of
{(/>, L) e / | P e Vo, dim(TP Vf]L) S d - e + 1}.

278
Chapter 14. Degeneracy Loci and Grassmannians
Then V is a subvariety of / of codimension d + 1. Let /u (V) be the class of [ V]
in Ad+lI. Let /uk = /u(M), where M is a linear subspace of codimension
ft(V) =
mQftd-e+i
where m{ is the 1th class of V (Example 14.4.15).
From this one may deduce the formula for the number of varieties in an
r-dimensional family tangent to r given varieties in general position, in terms
of the corresponding numbers for r given linear varieties (cf. § 10.4). (For the
proof, intersect both sides with the basis for Ad+](I) described in the
preceding example. For details see Fulton-Kleiman-MacPherson A).)
Notes and References
In topology, for suitably generic maps of vector bundles on an oriented
manifold, Thorn showed that the degeneracy loci must be Poincare dual to
some universal polynomials in the characteristic classes of the bundles. These
universal polynomials were determined by Porteous B).
In algebraic geometry, on a non-singular projective variety X, and when the
degeneracy loci have the expected dimensions, the formulas for their classes in
Aif(X) were proved by Kempf and Laksov A). The extension to arbitrary
singular varieties, with possibly excess degeneracy loci, as given in § 14.3 and
§ 14.4, is new here. In fact, once one has the machinery of Chaps. 1 - 7 to
handle excess intersections on possibly singular varieties, the proof of Kempf
and Laksov goes over without difficulty. The particular case of the localized
top Chern class was discussed in Fulton-MacPherson A).
The realization that Schubert calculus is essentially the same as the algebra
of Schur polynomials (resulting from representation theory of the symmetric
group) has occurred more than once. Giambelli B) was apparently the first to
express general determinantal loci by such polynomials. The polynomials
themselves go back to Jacobi. The connection was explicitly pointed out by
Lesieur A), after Ehresmann A) had worked out the cell structure and
cohomology ring of Grassmannians, cf. Horrocks A). Recently the importance
of Schur functions has been illuminated by Lascoux (l)-G). The seminars in
Strasbourg A976) and Torun A980) (cf. Stanley A) and Dieudonne B)) are
good sources for this. For combinatorial facts about Schur functions, the book
of Macdonald C) is also recommended. It should be pointed out that, although
much of this algebra goes back to Schubert, Pieri, Giambelli, and other
classical algebraic geometers, the general rule for intersecting Schubert
varieties was not given before the Littlewood-Richardson rule.
For symmetric and skew-symmetric degeneracy loci (cf. Example 14.4.11),
early formulas were also given by Giambelli C). Barth and Tjurin gave special
cases, with applications, for vector bundle maps. General formulas have been
Notes and References
279
given by Jozefiak, Lascoux, and Pragacz A), and by Harris and Tu A), to
which we refer for additional applications and references.
Intersection theory on Grassmann bundles, at least over a non-singular base,
has been studied by Grothendieck A), Kempf and Laksov A), Laksov B), and
Scott C).
An important generalization of Schubert calculus is from Grassmannians
and flag varieties to homogeneous varieties of the form G/P, for P a parabolic
subgroup of a semi-simple linear algebraic group G. For this we refer to
papers of Berstein-Gel'fand-Gel'fand A), Demazure A), Lakshmibai-Musili-
Seshadri A), Marlin A), and Hiller A), B), C). The basis for flag mani-
manifolds described in Example 14.7.16 was given by Ehresmann A); it agrees with
that arising from the description of a flag manifold as SL(« + I)/P.
An interesting calculation of the cohomology ring of Grassmannians via
zeros of vector fields is given by Carrell and Lieberman A).
The article by Kleiman C) contains general constructions and functorial
properties of Grassmann bundles, as well as applications to smoothing cycles.
Formulas for Gysin push-forward homomorphisms for Grassmann and flag
bundles are given by Damon A), Ilori A), J6zefiak-Lascoux-Pragacz A), and
Harris-Tu A).
The residual formula for Chern classes in Example 14.1.4 is new. The
excess Porteous formula of Example 14.4.7 is the result of conversations with
G. Ellingsrud, J. Harris, and R. Lazarsfeld. Examples 14.4.5 and 14.4.17 in-
include strengthening and simplification of known facts for linear systems on
curves.
Even to summarize the applications of Schubert calculus to enumerative
geometry could double the length of this book; it is hoped that the examples
give some hint of the possibilities. From the classical period there are the
books and papers of Schubert, Zeuthen, Pieri, Giambelli, and Severi. The
books of Semple and Roth A), Hodge and Pedoe A), and Baker A), B)
contain hundreds of these applications. For a modern survey the article of
Kleiman (8) is recommended. Other classical as well as new applications may
be found in Griffiths and Harris A) and Arbarello, Cornalba, Griffiths, and
Harris A).
For a sampling of recent enumerative applications of the Porteous formula
and related ideas, some references are: Harris and Mumford A), Laksov D),
E), Eisenbud and Harris A).

Chapter 15. Riemann-Roch for Non-singular
Varieties
Summary
The Grothendieck-Riemann-Roch theorem (GRR) states that for a proper
morphism/:!'-* Y of non-singular varieties,
ch(/,a) • td G» =/, (ch(a) ¦ td (Tx))
for all a in the Grothendieck group of vector bundles, or of coherent sheaves,
on X. When Y is a point, one recovers Hirzebruch's formula (HRR) for the
Euler characteristic of a vector bundle E on X:
X (- 1)' dim H'(X, E) = J ch (E) ¦ td (Tx).
x
The aim of this chapter is to show how the geometry of the deformation to
the normal cone leads to a simple proof of GRR when/ is a closed imbedding.
The same proof gives the corresponding theorem without denominators, which
in turn yields a simple proof of the formula for blowing up Chern classes.
The reader of this chapter is assumed to have some familiarity with the
cohomology of coherent sheaves, although the necessary facts are reviewed in
the first section. In addition, the proof of GRR when /is a projection is only
sketched briefly. The first nine sections of the article of Borel and Serre A) are
recommended for a detailed discussion of these points.
Although the theorem is stated here for arbitrary non-singular varieties,
the proof in this chapter makes an additional assumption of projectivity. The
general case will be considered, together with singular varieties, in Chap. 18.
15.1 Preliminaries
For any scheme X, K°X denotes the Grothendieck group of vector bundles
(locally free sheaves) on X. Each vector bundle E determines an element,
denoted [?], in K°X. K°X is the free abelian group on the set of isomorphism
classes of vector bundles, modulo the relations
15.1 Preliminaries
281
whenever E' is a subbundle of a vector bundle E, with quotient bundle
E" = E/E'. The tensor product makes K°X a ring: [E] ¦ [F] = [E<g)F]. For any
morphism /: Y—> X there is an induced homomorphism
taking [E] to [f*E], where f*E is the pull-back bundle; this makes K° a con-
travariant functor from schemes to commutative rings.
The Grothendieck group of coherent sheaves on X, denoted by KOX, is
defined to be the free abelian group on the isomorphism classes p?~\ of
coherent sheaves on X, modulo the relations
for each exact sequence
0 -> Jf' -> &'-» 3r" ->0
of coherent sheaves on X. Tensor product makes KOX a K^-module:
K°X®KOX->KOX,
For any proper morphism/: X -*Y, there is a homomorphism
which takes \&\ to ?jg0(- 1 )'[#/* ^ Here R'f* ^ is Grothendieck's higher
direct image sheaf, the sheaf associated to the presheaf
U ->•//'(/-' (U),^)
on Y. It is a basic fact that the R'fifSr are coherent when y is coherent and/is
proper ([EGA]III.3.2.1). The fact that this push-forward/,, is well-defined on
KaX results from the long exact cohomology sequence for the R' /„,. The fact
that Ko is a covariant functor for proper morphisms uses the spectral sequence
for composite morphisms.
The push-forward and pull-back are related by the usual projection
formula:
/•(/•« -/0 = «•/*/»
for f:X->Y proper, aeK°Y, peKoX. This follows from the formula
J?'/» (/* E ® F) = E ® K'/a-T for ? locally free on Y, .9~ coherent on X.
On any X there is a canonical "duality" homomorphism:
K°X -> KOX
which takes a vector bundle to its sheaf of sections. When X is non-singular,
this duality map is an isomorphism. The reason for this is that a coherent
sheaf T on a non-singular X has a finite resolution by locally free sheaves, i.e.,
there is an exact sequence
0 ->?„->¦?„_,->¦... ^?l^?0->Jr->0

282
Chapter 15. Riemann-Roch for Non-singular Varieties
with ?„,...,?„ locally free. The inverse homomorphism from KOX to K°X
takes [&] to Z?=0(-1)i[?il. for such a resolution. (See Appendix B.8.3 for
details.)
From now on we consider schemes X which are smooth over a given
ground field K. For such X we identify K°X and KaX, and write simply
K(X). For X= Spec (AT), K(X) = I, and we make this identification. We write
A(X) for the ring of cycles modulo rational equivalence on X, and A(X)ni for
There is a homomorphism, called the Chern character (cf. Examples 3.2.3,
determined by the following properties:
(i) ch is a homomorphism of rings;
(ii) if/: Y ->• X, ch °/* =/* ° ch;
(iii) if L is a line bundle on X,
.so
For a vector bundle E we write either ch? or ch [E].
The Chern character does not commute with proper push forward. A
fundamental insight of Grothendieck was to phrase the Riemann-Roch
problem as the problem of comparing ch °/* with /* ° ch, for a proper
morphism f:X-*Y. When Y is a point, and ? is a vector bundle on X,
eh/* [E] is v
*(*,?) = ? (-iydimKH'(X,E)
in/l(Spec(/r))Q=Q. ;t0
Model for closed imbeddings. To motivate the general formula, we work out a
special case of a closed imbedding f:X-* Y in which all the terms may be
calculated explicitly. In this model, X is arbitrary, and Y is P(N ©1), where N
is an arbitrary vector bundle of rank d on X. The imbedding / is the zero
section imbedding of X in N, followed by the canonical open imbedding of N
in P(N®\):
f:X-> NczP(N® 1)= Y.
Let p be the bundle projection from Yto X, and let Q be the universal quotient
bundle, of rank d, on Y. Let s be the section of Q determined by the projection
of the trivial factor in p* (N © 1) to Q. This section s vanishes precisely on X,
i.e., Z(s) = X. In particular, for any a e A (Y),
A) /•(/*«) = <* ¦/*[*] = Q(S)-oc
(Proposition 14.1 (a)). In addition, the Koszul complex determined by s:
0-> A'gv -»...-+ A2QV -*QvA(^-+/(i(!))t-»0
is a resolution of the sheaf f*0x (Lemma A.7.1). For any vector bundle E on X,
we therefore have an explicit resolution of/* ?:
0 -> Ad8v ® p*? - ... - 8V <g> p*? -+ p*?_>/,,?-> 0.
15.1 Preliminaries
Therefore
B)
283
ch/J?]= S (-l)pch(A'ev)-ch(p*?).
To write this as the image of a class on X, we want, by A), to divide the right
side by cd (Q). In fact, the Todd class td (Q) is determined by the identity
d
C) X (-l)pch(Apgv) = cd(g)- td(g).
(see Examples 3.2.4 and 3.2.5).
Combining A), B), and C),
ch/*? = cd{Q) td(Q)-1 • ch(/>*?)=/*(/*td@-1 -/*ch (/>*?)) ¦
Since/* Q= N, and/*/>*? = E, this can be rewritten
D) ch/,?=/,(td(A0H-ch(?)).
The Todd class, like the total Chern class, takes sums to products. Therefore
(cf. Appendix B.7.2). The right side of D) is therefore
/,(/• tdGY)-1 • tdG» ¦ ch?) = tdGY)-' •/*(td(^) • ch?) ,
and D) may be rewritten:
E) ch (/„?) • td (TY) =/, (ch (E) ¦ td G»).
The Grothendieck-Riemann-Roch theorem we shall prove is the assertion that
E) holds for an arbitrary proper morphism f:X-* Y.
Example 15.1.1. (a) The group Ko (Pm) is generated by the classes
ign|m. If p maps F" to a point, then
n + m
m
(b) For any X, Kthere is an exterior product
KOX ® KOY ± KO(X x Y)
with [J^] x y$\ = [pr* (&) ® pr* (^)], prf the projections. If Y = F", this product
KOX ® KJT -+ KO(X x F")
is surjective.
(The proofs are quite similar to the proofs of corresponding facts for rational
equivalence given in § 3.3. In fact, if ? is a vector bundle on X, the pull-back
KaX -> KOE is an isomorphism, and KOP (E) is a direct sum of e copies of
K0X. An elegant proof of this is given by Quillen B) § 8.)

284
Chapter 15. Riemann-Roch for Non-singular Varieties
Example 15.1.2. (a) Let c{ be the ilh elementary symmetric function in variables
xx,..., xe. Let pk = x* + ... + x\. Then (cf. Example 3.2.3 and Macdonald C) p. 20)
Pk =
2 c,
\'iCk Ck-l
Thus/>0 = e, />, = C|, pi = c\- 2c2, p3 = cf - 3c, c2 + 3c3)... .
(b) If ? is a vector bundle of rank e on X, c, = c, (?), and pk is defined as in
(a), then (cf. Example 3.2.3)
*=o
(c) If a = 0 for 0 < i < d, then pd={~ \)d'' d cd. If ? is a vector bundle of
rank e, and c, (?) = 0 for 0 < i < d, then
ch(?) =
'/(</- 1)!)
(d) If A' is any connected scheme, the function which assigns to each vector
bundle its rank is additive on exact sequences, and defines a homomorphism
rk:K°X->Z.
With this notion of rank, and the determinantal definition of pk in (a), the
formulas of (b) and (c) extend to arbitrary elements of K°X. Note that any
? e K°X has Chern classes c,-(<j;) by Example 3.2.7.
Example 15.1.3. For any partition X:Xx a ... a I, ^ 0 of e, and x,, c, as in
Example 15.1.2, define px(c) =Pi(cu ..., ce) to be the sum of all the distinct
monomials in x,,...,xf which are obtained from x\l-... ¦ x\- by permutation
of the variables xu...,xe. For example, if X =(k, 0,..., 0), pk is the poly-
polynomial pk of the preceding example.
If ? is a vector bundle, let px(c(E)) be the polynomial obtained by
substituting c,(?) for c,-. If
0 -> ?'->?->?"-+ 0
is an exact sequence of vector bundles, then a result of Thorn (cf. Milnor-
Stasheff(l)§ 16.2) states:
where the sum is over all pairs of partitions a, /? whose juxtaposition a A is X.
This generalizes the formula ch (E) = ch (?") + ch (?").
Example 15.1.4 (cf. Hirzebruch A) Lemma 1.7.1, or Borel-Serre A) Prop.
10). On Pm, letx = c, (<VA)). Then for all n is 0,
p-
n + m
n
15.1 Preliminaries
285
(The integrand is a power series in x, for which one wants the coefficient of
xm. Divide the integrand by xm+\ and compute the residue by changing
variables: y=l — e~\)
Example 15.1.5. On any scheme X, the topological filtration on KaX is
defined by letting FkKaX be the subgroup generated by coherent sheaves
whose support has dimension at most k. Equivalently, FkKoX is generated by
the classes [0y], as Franges over closed subvarieties of X of dimension at most
k. If/: X -> Yis a proper morphism, then
U(FkKoX) a FkKoY.
The associated graded groups GrkKaX = FkKaX/Fk_lKoX are covariant for
proper morphisms.
If J"" is a coherent sheaf whose support has dimension at most k, y
determines a A>cycle Zk (J"") on X:
where my^) is the length of the stalk of $~ at the generic point of V, as a
module over the local ring 0VtX of X along V.
There is a unique, surjective homomorphism
q>:ZkX->GrkKaX
which takes [V] to [#v], and Zk(!F) to [?"] for any coherent sheaf T whose
support has dimension at most k. This homomorphism is covariant for proper
morphisms, and passes to rational equivalence, determining
q>:AkX-+GvkKoX
which is surjective, and commutes with push-forward by proper morphisms.
(For the covariance, reduce to the case where/: X -> Y is a proper morphism
of varieties, and Jr = #x\ replacing Yby an open subvariety, one may assume/
is finite, and fix is free over <*V> m which case R'f*fix~0 for i> 0 and
ftlfix] = deg(X/Y) [fiY]- To show that q> passes to rational equivalence, use
Example 1.6.4: if/: X ->• P1 is dominant, X a variety, Da =/"' @), D^ =f~l (oo),
then
0 ->f*fi(- \)^fix^ fio, -» 0
for t = 0, oo; therefore [&Do] = [&DJ in KOX. Since [D] = Zk@D) for D an
effective Cartier divisor on a (k + l)-dimensional variety, cp[D0] = (p[Dx\, as
required.)
Example 15.1.6. If X is an arbitrary algebraic scheme, and a e KaX, there is
a quasi-projective scheme A", a projective morphism f:X'->X, and a class
a' e KOX' such that /„ a' = a. If the ground field has characteristic zero, one
may even take X' to be smooth (but not necessarily connected); for this one
uses Hironaka's resolution of singularities. See Lemma 18.3 or Fulton D) for a
proof.

286
Chapter 15. Riemann-Roch for Non-singular Varieties
Example 15.1.7 (see [SGA6] or Baum-Fulton-MacPherson B) App. 2).
(a) If X is a scheme, Xrti the associated reduced scheme, then the inclusion
of XKi in Xinduces an isomorphism
(b) If A* is a closed subscheme of a scheme Y, then Ka(X) is canonically
isomorphic to the Grothendieck group of coherent sheaves on Y whose support
is contained in X.
(c) If X is a closed subscheme of a non-singular scheme Y, then KO(X) is
canonically isomorphic to the Grothendieck group of bounded complexes of
locally free sheaves on Y which are exact off X, modulo the subgroup
generated by complexes which are exact on all of Y.
Example 15.1.8 (cf. [SGA 6]III). There is a class of morphisms/: X — Y, of
possibly singular schemes, called perfect morphisms, for which there are
functorial Gysin homomorphisms:
and
: KOY -> KOX.
For the first, / is assumed to be proper. For simplicity, we describe these
concepts for schemes which are quasi-projective over a fixed non-singular base
scheme.
If/is a closed imbedding,/is perfect if and only if/*^ may be resolved
by a finite complex E. of locally free sheaves on Y. It follows that every locally
free sheaf Fon X has a resolution G. on Y, and
For any coherent sheaf J*" on Y,
where Torf^, J*") is the Ith homology sheaf of the complex E. ®ey9~.
A general morphism/: X — Y of imbeddable schemes is perfect if, when/
is factored into a closed imbedding i: X -> P followed by a smooth morphism
p: P — Y, i is perfect in the above sense. If/is proper, one may take P = P(E),
E a vector bundle on Y, p the projection. One defines/,, = p* ° /*,/* = i* °p*.
Here p* [F] = [p*^] for a coherent sheaf J*" on Y. The homomorphism
Pm:K°(P(E))->K°Y
is uniquely determined by the projection formula p*(p*a • /?) = a ¦ p^fl for
a e K°Y, /? e K°(P(E)\ and the identity pt[OE(n)] = [Sym"?v].
15.2 Grothendieck-Riemann-Roch Theorem
Theorem 15.2. Let f: X —> Y be a proper morphism of non-singular varieties.
Then for all a. & K(X),
ch(/,a) • tdG» =/*(ch(a) • tdG»)
15.2 Grothendieck-Riemann-Roch Theorem
287
Proof We shall assume/factors into a composite
JT- yxT-i Y
where g is a closed imbedding, and p is the projection. Such a factorization
exists whenever X is quasi-projective, for if i:X-* Pm is a locally closed
imbedding, then g=(f,i) is a closed imbedding of X in YxPm. For the
general case, see Example 15.2.9 and Corollary 18.3.1 (c).
The theorem may be interpreted to say that the homomorphism
defined by
commutes with proper push-forward: /* ° ry= ?Y°f*- It follows that, if the
theorem is valid for g and forp, then it is valid for/= p ° g.
For the projection, consider more generally the projection/: YxZ-* Y,
for Z non-singular. There is a commutative diagram
K(Y)®K(Z)
"I
K(YxZ)
The point here is that td (TYx z) = td Gy) x td (Tz), as follows from the multi-
multiplicative property of the Todd class. Now if Z = Pm, the left vertical map is
surjective, and K (Pm) is generated by [#(n)], n§0 (Example 15.1.1). One is
therefore reduced to verifying the theorem for the mapping p from P to a
point, and a = [<#(n)], i.e., to verifying the formula
Since tdGV») = (x/\ - e-x)m+l, where x = c, (<VA)) (cf- Appendix B.5.8) this
is the content of Examples 15.1.1 (a) and 15.1.4.
Riemann-Roch for closed imbeddings. We turn now to the case where
f:X-+ Y is a closed imbedding. Let N be the normal bundle to X in Y. W e shall
use the deformation to the normal bundle to deform the imbedding/into the
imbedding f:X-* P(N® 1) discussed in the "model" at the end of the
previous section. From § 5.1, we construct a commutative diagram
where M is the blow-up of KxP1 along Xx{oo}. We may assume a =[E],
with E a vector bundle on X. Let E = p*E, where p is the projection from
Xx P1 to X. Choose a resolution G. off* (E) on M:
0
Go
0.

288
Chapter 15. Riemann-Roch for Non-singular Varieties
Since JfxP1 and M are flat over P1, it follows from Lemma A.4.2 that the
restrictions of this exact sequence to the fibres Ma and Mx remain exact.
Therefore j*G. is a resolution of y*F, (?), and j&G. is a resolution of
jZ, F,(?). Since;?F, ? =/,ij F =/, (F),
(i) y*G. resolves/, (F) onr=M0.
Similarly, j%,G. resolves/, (F) on Mo,. But /(A') is disjoint from Y. Therefore
(ii) k*G. resolves/,(E) on P(N®\), and
(iii) /* G. is acyclic.
For a complex F of vector bundles, we write ch(F) for ? (— l)'ch(F). We
compute the image of ch(/,F) m
M (ch (/, E)) =y0, (ch (/S G.)) by (i)
= ch (G.) ¦ y0, [ Y] (projection formula)
= ch(G.)-(?,[P(JV ©!)] + /,[?])
by the basic fact that [Mo] - [Mm] = [div(e)] = 0 in ^(JW), (cf. Example 5.1.1).
Continuing, with the projection formula again:
= A:, (ch (k* G.)) + /, (ch (/* G.))
= *,(ch(/,?)) + 0 by (ii) and (iii).
For / ch(/,F) was calculated in the model of the previous section
(formula E)). Therefore
(iv) Jo* (ch (/,?)) = M/.ftdW1' ch(?^ in A(Mh-
Let q:M~* Y be the composite of the blow-down map from Af to YxW1,
followed by the projection to Y. By construction of M, q°jo = idy and
q ° k °f=f. Applying q* to (iv), we deduce
ch(/,?)=/,(td(A0-'-ch(?)).
As in the transition from formula D) to E) in the previous section, this is
equivalent to the theorem. D
Corollary 15.2.1 (Hirzebruch-Riemann-Roch). Let E be a vector bundle on a
non-singular, complete variety X. Then
/(*,?) = | ch(?)tdGV).
x
Proof. This is the content of the theorem for the mapping of I to a
point. ?
Corollary 15.2.2. If X is a non-singular, complete, n-dimensional variety, then
Example 15.2.1. Let X be a non-singular complete curve of genus g, i.e.,
Since dim H° (X, 0X) = 1, this is equivalent to dim Hl (X, ffx) = g.
15.2 Grothendieck-Riemann-Roch Theorem
Let ? be a vector bundle of rank e on X. Then by HRR
289
In particular, if E = (9 (D) is a line bundle,
/(*,*(/>)) = deg(Z>)+1-ff.
If E, Fare bundles of ranks ejon X, then (cf. Weil A))
X(X, Horn (E, F)) = e\ c, (F) -f\ c, (E) + (ef)(l - g).
x x
Example 15.2.2. Let A" be a non-singular complete surface. Then
with c, = Ci{Tx). Therefore
If K = — c1(Tx) is a canonical divisor, X = $x C2(TX) is the topological Euler
characteristic, pg = dim H1 (X, 0X), and q = dim Hx (X, fix), this reads
If ? is a vector bundle of rank e on X, with dt = c,(E), then
2d2 + ct dx) + ex(X#x) ¦
In particular, if D is a divisor on X,
X (X HD)) = 12
If D is an effective divisor on X,
X (X HD)) = 12{(DD)-{K- D)) + x {X fix) ¦
(Use the exact sequence Q -* fi (- D) -* fix ^ fiD -> 0.) If D is an irreducible
curve on X, and pa{D) = dim H] (D, 0D) is its arithmetic genus, then
pa{D) = \{{DD)+{KD))+\.
For example, a plane curve of degree n has arithmetic genus j (n — 1) (n - 2).
A curve of bidegree (m, n) in P1 x P1 has arithmetic genus (m — 1) (n — 1).
Example 15.2.3. Let /: X1 -* Y2 be as in Example 9.3.2, with f(X) = X.
(By the double point formula, deg D (/) = X ¦ X+ KY ¦ X- degA:^. Then use
Examples 15.2.1 and 15.2.2.) Comparing with Example 9.3.2, this is the well-
known formula for the degree of the conductor (cf. Serre E) IV).
Example 15.2.4. Index theorem. Let A" be a non-singular projective surface,
H an ample divisor on X. Let D be a divisor on X, such that \ D ¦ H = 0. Then

290
Chapter 15. Riemann-Roch for Non-singular Varieties
j D ¦ D S 0, and the following are equivalent: (i) J D ¦ D = 0. (ii) \ D ¦ E = 0 for
all divisors E on X (iii) a non-zero multiple of D is algebraically equivalent to
zero.
Here we write J D ¦ E for the intersection number (Z) • E)x- One may replace
7/ by a multiple if necessary to assume // is represented by a smooth
hyperplane section of X. The proof is in several steps:
1) There is a c = c(X,H) so that H2(X,tf(D)) = 0 for all Z) with
\ D- H ^ c. This follows from the cohomology sequence of the exact sequence
0
- 1)//)
0
using Serre's vanishing theorem, and the vanishing of Hl (L) for line bundles
Lof degree larger than 2g - 2 on a curve of genus g (cf. Example 15.2.7).
2) If J Z) ¦ Z) > 0, then f Z) • // > 0 if and only if //° (<?(« Z))) =t= 0 for some
n > 0. For if n Z) ~ ? with ? effective, then E ¦ H > 0. Conversely, by Riemann-
Roch, if {/)¦ D> Othen;r(^(nZ))) ~* oo as n -> oo.
3) If JD ¦ D > 0, and J D ¦ H > 0 for one ample divisor H, then J Z) ¦ H > 0
for all ample divisors H.
4) If | Z) • Z) > 0 and J D ¦ H = 0, then J (n Z) - //J > 0, J (n Z) - H) ¦ H < 0,
and
for n > m > 0, contracting 3). This proves the first assertion.
5) If | Z) • D = \ D ¦ H= 0 but | D ¦ E 4= 0, replace ? by a? + bH so that
j?// = 0. By 4),
for all integers m, which is absurd.
6) To see (ii) => (iii), consider Dmn = mD + nH. By Riemann-Roch, for
n S n0>
dimH°(X, #{Dm,n)) + dimH2(X, #{Dm,n)) > 0 .
By 1), for n> n0, there are effective divisors Em linearly equivalent to Dmn.
All Em have the same Hilbert polynomial, and since such curves are
parametrized by an algebraic scheme (cf. Mumford B)), there are Emj and Em,
in the same algebraic family.)
A generalization appears in Example 19.3.1. For the singular case, see
Kleiman's Appendix to Expose XIII of [SGA 6]. The theorem was discovered
and proved by Hodge using De Rham theory, as in Griffiths-Harris A).
B. Segre A) gave a proof more like the one sketched here, cf. Mumford B).
Example 15.2.5 (a) If dim (A') = 3,
tdG»=l+yc,+ —
c{c2,
with Cj=Ci(Tx). If ? is a vector bundle of rank e on X, with Chern classes
d, = c, (?), then
X{X E) = \^{d]-
15.2 Grothendieck-Riemann-Roch Theorem 291
If E = fi(D), this simplifies to
X(X, 0{D)) = | j ZK + j c, ¦ ZJ + -L (cf + c2) ¦ D + ~ c, c2.
(b) If X=P\ and ? is a vector bundle on X, let c,(E) = n,hi, h a
hyperplane class, n, e TL. Then
3n3+ lln,
is divisible by 6. For example, if ? has rank 2, n, n2 must be even.
Example 15.2.6. Let X be an abelian variety of dimension n, E a vector
bundle on X. Then
If L is a line bundle, then
nl
In particular, for any divisor D and X, the self-intersection number \x D" is
divisible by n\.
Example 15.2.7. The Hirzebruch-Riemann-Roch theorem is most useful
when one has some control on some of the cohomology groups. Some basic
facts in this direction are:
(a) For Xcz FN, a theorem of Serre A) implies that H'{X, ?(«)) = 0 for
n P 0. Therefore
dim H° (X, E («)) = | ch (? («)) ¦ td (Tx)
x
for n sufficiently large.
(b) Serre duality gives isomorphisms
H'(X, E) S H"-'{X, K®EV).
Here n = dim (A'), and K = Q J = Tx is the canonical line bundle (cf. Serre B),
Grothendieck D), Hartshorne B)).
(c) In characteristic zero, the Kodaira vanishing theorem states that i f L is
an ample line bundle on X, then
H'(X,K®L) = 0 for / > 0.
More generally, Le Potier (cf. Verdier B)) has shown that, for ? an ample
vector bundle on X,
Hi(X,QJx®E) = 0
for / +; S dim (X) + rank (?), Q'x = A> Tx .
Example 15.2.8. Let /: X -* Y be a smooth proper morphism of non-
singular varieties, and let 7} = KerGV -¦/* TY) be the relative tangent bundle.
Then for all a e ? (X),
ch(/,a)=/,(ch(a)-tdG>)).
(Since tdG» = id(Tx) ¦f*td{TYyl, this is equivalent to GRR for/, as in the
equivalence of formulas D) and E) of § 15.1.) For example, if X= P(E), for ?

292
Chapter 15. Riemann-Roch for Non-singular Varieties
a vector bundle on Y, and/is the projection, then (cf. Appendix B.5.8)
ch(/,a) =/,(ch(a) td(/*? ® ^A))).
Similarly, if X ~* Y is a family of abelian varieties, Tf=f*F for a vector
bundle Fon Y, and
ch(/*oO=/*(ch(oO)-td(F).
Example 15.2.9. In characteristic zero, the proof of Theorem 15.2 can be
completed, in case X is not quasi-projective, as follows. Given a e K(X), there
is a non-singular, quasi-projective A", a proper morphism g : X' -* X, and an
element a' e K{X') with g*(a.') = a (Example 15.1.6). The theorem as proved
applies to g and to/° g. Therefore
and
»(ch (a') • td G») = ch (a) • td G1,),
') • tdG») =
td(TY).
Since (/#)* =/*#*, the theorem follows.
For a proof without resolution of singularities, valid in arbitrary charac-
characteristic, see Corollary 18.3.1.
Example 15.2.10. The arithmetic genus satisfies a fundamental "modular"
law:
If A", Y, and Z are non-singular divisors on a non-singular projective variety
M, such that X is linearly equivalent to Y + Z, and Y meets Z transversally in
a disjoint union of varieties V{,..., Vr, then
t
In fact, there is only one way to assign a number to every non-singular
projective variety, taking the value 1 on a point, and satisfying this modular
law. This is proved in Washnitzer B) and Fulton G).
If A" is a non-singular subvariety of a non-singular variety Y, deformation
to the normal bundle gives a linear equivalence between Y and P(N®1) + Y
with P(N®\) and f meeting transversally in P(N). Since
= x(X, #x) for any vector bundle E on X, it follows that
Example 15.2.11. Signatures. The signature (index) a {X) of a compact
oriented manifold is defined to be zero if its (real) dimension is not a multiple
of 4, and the index of the quadratic form given by cup product on H2k(X; R),
if dim (A") = 4k. If A" is a complex projective variety, Hodge theory gives the
formula
(cf. Hirzebruch AI5.8.2, or Griffiths and Harris A) p. 126.).
The geometry of deformation to the normal bundle can be used to prove a
formula for the signature of the blow-up Y of a complex manifold Y along a
15.2 Grothendieck-Riemann-Roch Theorem
293
submanifold X. Namely,
\a(Y) if codim(X, Y) is odd
v ' \a{Y)-a{X) if codim(X, Y) is even .
(With the notation of the previous example,
a{Y) = a(Y) + a{P(N © 1)).
This follows from Hirzebruch AI1.3.1, since Y does not meet P(N) in the
deformation space M, and, for any vector bundle E on X, a{P{E)) is zero if
rank(?) is odd, and is a{X) if rank(?) is even.) A similar method may be
used for Hirzebruch's general 7^-genus.
Example 15.2.12. Todd classes. If A" is a non-singular variety set
Then
(i)
(»)
In particular, x(Xx Y, 0X%r) = X(%• #x) x(Y,#y)-
(iii) If f:X~* Y is a closed imbedding of codimension d, and X is the
intersection of rfCartier divisors Du ..., ?>rfon Y, then
d
Td(A-)=/*(Td(y)-rj A -exp (-?)/?)),
so
/,(Td(JQ) = TdG) ¦ n A - exp(- D,)).
i=\
Example 15.2.13. Chern numbers. Each isobaric polynomial P= P(ct,..., cn)
of weight n determines a Chern number P{X) for any M-dimensional non-
singular complete variety X:
> = \P{c>{Tx),...,cn{Tx)).
For example, the nth Todd polynomial tdn gives the arithmetic genus
(a) If px are the polynomials defined in Example 15.1.3, then (cf. Milnor-
Stasheff(l)§ 16.4) for non-singular X, Y
Pi{Xx Y)= X P*{X)Pe(Y),
a result of Thorn. In particular, if X =(k, 0,..., 0), and neither X nor Y is a
point, then pk (X x Y) = 0.
(b) An isobaric polynomial P of weight n is determined by its values on X,
where X varies over all M-dimensional Cartesian products of projective spaces.
~ . form a basis for the polynomials of weight n, and
10 if X is not a refinement of /u

294
Chapter 15. Riemann-Roch for Non-singular Varieties
(c) (cf. Hirzebruch AH.3) The nth Todd polynomial is the only Chern
number whose value is 1 on all products of projective spaces. (Use (b) and
Example 15.2.10.) Hence multiples of tdn are the only Chern numbers which
are invariant under blow-ups along non-singular centers.
Example 15.2.14. In characteristic zero, at least, the arithmetic genus is a
birational invariant. If A'is a non-singular complex projective variety,
dim H' (X, 0X) = hOi = hi0 = dim H° (X, Q'x),
So x{X,0x) = Y,i- \y&raH°{X,Qx). The individual numbers ti° are bira-
birational invariants (cf. Hirzebruch A)).
Example 15.2.15. Let/: X -* Y be a regular imbedding of possibly singular
schemes, with normal bundle N. Assume that Y admits a closed imbedding
into a non-singular variety, so that any coherent sheaf on Y is quotient of a
locally free sheaf (Appendix B.8.1). There is a homomorphism
/„ : K°X -> K°Y
which takes [T] to ? (- 1)' [?/], if E. resolves/*jr. Then for all as K°X
ch(/,a) n [Y] =/,(td(A0H • ch(a) n [X])
in AzYq. (The proof is the same as that given for the non-singular case; see
§ 18.3 for generalizations.)
Example 15.2.16. Give A*X its natural filtration:
FkA,X='?AlX,
isk
so the associated graded group FkAJFk- {A* isAk.
(a) If A'is non-singular, and Kis a ^-dimensional subvariety of A", then
where/is the inclusion of Kin X, and a. e Fk-\A* V. (Let 5 be a proper closed
subscheme of V, so that V— S is regularly imbedded in X — S. Since
A,S -> A,X-*A,(X- S) -> 0
is exact, and the image of A*S is in /*(/*_] A* V), it suffices to prove the
assertion when 5 is empty. In this case
where N is the normal bundle to Kin X.)
(b) It follows from (a) that the homomorphism cp-.A+X^ Gr*KoX of
Example 15.1.5 becomes an isomorphism after tensoring with Q, and that the
Chern character determines an isomorphism
15.2 Grothendieck-Riemann-Roch Theorem
295
of <Q-algebras. (By (a) ch maps FkKoX into FkAt.(X)^. This induces a homo-
homomorphism
Gr KOX -> Gr/t,(A-)Q = A^X)^
such that the composite with q> is the natural inclusion of A* X in A^X^. Since
q> is surjective, both maps are isomorphisms after tensoring with Q. Since, after
tensoring with Q, ch determines an isomorphism on associated graded groups,
the same must hold on the original groups. For a more conceptual proof see
Corollary 18.3.2.)
Example 15.2.17. Let ? be a vector bundle on a scheme X, p: P(E) -* X
the associated vector bundle, <?A) the universal quotient bundle of p*Ev.
Then for all ae A*X, and all n ^ 0,
ch(Sym"(?v)) n a. =/>*(ch(<?(«))¦ td(/»
np*a).
(The equality in each degree amounts to a formal identity in the Chern classes
of a vector bundle. One may deduce it without calculation as follows: A) it
suffices to prove it when a = [X], X a smooth projective variety; indeed, one
may use any Grassmannian where monomials of given degree in Chern classes
of the universal quotient bundle are independent. B) Since Sym"?v =p*<?(«),
and R'p*#(n) = 0 for / > 0, the desired formula follows from Riemann-Roch
(Example 15.2.8). For later use we note that only the projective case of
Riemann-Roch is used in this argument.)
Example 15.2.18. Let G=G} (P"), / the incidence variety of (P, L) e TP"xG
with PeL (cf. Example 14.7.12); I=P(S) is a P1 bundle over G, with
projection p:I->G, and line bundle 0(\). Let X=IxGI, pup2 the two
projections from A" to /. For any d^ 1, set
where -f(A) is the ideal sheaf of the diagonal imbedding of / in X. Then Ed is
a vector bundle of rank 2n — 1 on /, and
c(Ed) = II A +{d- 20 c,
1=0
+ ip* («r,)),
with <r, as in § 14.7. (Apply GRR to p2:X-* I, and calculate ch(?«/). For
another approach, see Ran B).)
A hypersurface F e ^{^"^{d)) determines a section of Ed, whose zeros
consist of all (P, L) such that L has contact of order at least 2 n — 1 with F at P.
The weighted number of such (P, L) is therefore
2n-2
2.-1 (Ed) = | II ((</- 20 c,
+ i a,).
Since ptr(c1(e>(l))i+1) is the i'h Segre class of S, i.e., a,, the right side may be
evaluated by Schubert calculus. For example, if n = 2, one gets 3d(d — 2); and
if n = 3, it gives 5d(d - 4)Gd - 12).

296 Chapter 15. Riemann-Roch for Non-singular Varieties
15.3 Riemann-Roch Without Denominators
If ? is a vector bundle on a non-singular variety X, the total Chern class
c (?) = 1 + C] (?) + ... is an element of the multiplicative group
By the Whitney sum formula, the total Chern class defines a homomorphism
c:K(X) ->A*(X)
from the additive group K{X) to the multiplicative group A*(X). If E. is a
complex of vector bundles, or any indexed collection of vector bundles on X,
we write
nW1I
for the image of ? (- 1)' [?,] by c.
Let/: Jf-> y be a closed imbedding of non-singular varieties, of codimen-
codimension d, with normal bundle N, and let ? be a vector bundle on X. Then
/»[?] ?^(y), and the object is to find a formula for c (/* [?]) in ^x 7.
To solve this, we first consider the model situation, as in § 15.1:
Y= P(N ® \), X=Z(s), s the canonical section of the universal quotient
bundle Q, p the projection from Y to X. Then
SO • = «
c(f*[E]) = c(KQv®p* E) = Fl c(A'<2v ® p* E)'*1.
i = 0
Lemma 15.3. F/x positive integers d, e. There is a unique power series
P{T\,...,T,i,U\,...,Ue) with integer coefficients such that for all vector
bundles D, E of ranks d, e on any variety V,
c(A'Dv ® E) - 1 = cd(D) ¦ P(D, E)
where P(D, E) denotesP(c, (D),..., cd(D), c, (?), ...,ce(?)).
Proof. Let xu...,xd be Chern roots for D, y]t...,ye Chern roots for E.
Then by Remark 3.2.3
C(A-Dv®?)-i = nn n (\+yi-xii-...-xipr»p-\.
It suffices to verify that the right side of this equation is divisible by x{, for
then, by symmetry, it is divisible by all the x,, and hence by the product,
which is cd{D). If xx is set equal to zero, each term in the product with /, > 1
and given p cancels a term with i, = 1 and p replaced by p + 1. D
Returning to the model, let e = rank E, and define P as in the lemma for
this d, e. Then
c(f*[E])=l+cd(Q)-P(Q,p*E).
15.3 Riemann-Roch Without Denominators
Since cd(Q) =f*[X], as in § 15.1 this gives
297
Theorem 15.3. Let f: X -> Y be a closed imbedding of non-singular varieties,
of codimension d, with normal bundle N. Let E be a vector bundle of rank e on X.
Then
c(f*E)=\+MP(N,E)),
with P as defined in Lemma 15.3.
Proof. The proof is identical with the proof given for Theorem 15.2 for the
case of a closed imbedding, with the substitution of total Chern class c for the
Chern character ch throughout. ?
Example 15.3.1. Let P be defined as in Lemma 15.3, for fixed d, e, and let
Pq be the term of isobaric weight q of P. For a e K(Y), let c,(a) denote the
component of c (a) in ,4' Y. For/: X -> Y, ? as in Theorem 15.3, and q > 0,
cq{f*[E])=U{Pq_d{N,E)).
In particular, c, (/„[?]) = 0 for 0 < q < d. Since Pa{Tu...,Td,Uu...,Ue)
= (- l^-'Crf- l)!e(cf Example 15.1.2 (c)),
In particular,
Example 15.3.2 (Kleiman C)§ 5). Let X be non-singular over an algebra-
algebraically closed field, i: X-* P a locally closed imbedding, h = c, (/*<?( 1)). For
any a e AFX, there is a vector bundle ? on A'and integer n so that
(p- iy.0L = cp(E)-nh".
In addition one may assume c,(?) = n,/i' for i < p, n, some integers,
rank(?) ^ dim (A'), and ? ® <?(— 1) generated by its sections.
Kleiman uses this result, together with the geometry of Schubert cycles, to
show that (p — 1)! a is smoothable, i.e., rationally equivalent to a cycle ? n, [V,]
with all V, non-singular, provided p > (dim {X) - 2)/2. Earlier, in charac-
characteristic zero, Hironaka(l) had shown that a itself can be smoothed if
dim (a) s 3 and p > (dim (A") + 2)/2. The impossibility of smoothing cycles in
general was shown by Hartshorne, Rees, and Thomas A), although the
question of smoothability with rational coefficients remains open. For a
discussion of this problem see Hartshorne D).
Example 15.3.3. Let D, E be vector bundles of ranks d, e, on a scheme X.
Let Q be the universal quotient bundle on P(D © 1), p the projection from
P(D © 1) to X. Then for all a. e A*X,
P*(c(A'Qv ® p*E) n p*a) = P(D, E) n a
where P is defined in Lemma 15.3. This formula also determines P. (Assume
a = [X], Xz. variety. Let/be the canonical zero section. Then

298
Chapter 15. Riemann-Roch for Non-singular Varieties
as in § 15.1.) For other descriptions of P, bringing in A-rings, see [SGA6]
Exp. 0.II.5 and V.6, and Jouanolou B).
Example 15.3.4. When d=\, and D=LV, the polynomial P(D,E) of
Lemma 15.3 is determined by the identity
c(E®L)P (D, E)=(c(E® L)-c
, (L)
(Example 3.2.2). One may obtain a closed expression for P(D, E) by writing
out c(?®L)~' (cf. Example 3.1.1).
Example 15.3.5 (cf. Mumford G)). Let E = 1 be a trivial line bundle. If
d = 1 or 2, then the power series of Lemma 15.3 is
P(D, 1)=(- l)''c(Z)v)-1.
It follows that if/: Jf-» 7 is a closed imbedding of non-singular varieties of
codimension d = 1 or d= 2, with normal bundle N, then
If d= 3, P(A 1) = c(Z>v)-' B - c, (Z)))(l - c, (/)))-'.
Example 15.3.6. Let X be a non-singular n-dimensional variety, and set
F"X = Fn_p KOX (Example 15.1.5). If a e F"X, then c,(a) = 0 for 0 < i < p. The
mapping a h-> c,,(a) determines a homomorphism
The composites cp°q> and ^"c,, which are endomorphisms of APX and
fpX/Fp+iX respectively, are both multiplication by (- \y~l (p - 1)!. (Argue
as in Example 15.2.6, using Example 15.3.1. Note that Theorem 15.3 extends to
arbitrary regular imbeddings, as in Example 15.2.15.) In particular
rank : F3 X/F1 X^Z
det: F1 X/F2X ^ Pic(X) = A]X
Thus if X is a surface, A(X) ^ Gr K(X). For afiine varieties, there are some
stronger results, cf. Kumar-Murthy A).
15.4 Blowing up Chern Classes
For any non-singular variety X, write c{X) for c{Tx). Consider a blow-up
diagram (§ 6.7) xUf
15.4 Blowing up Chern Classes
299
with Y, X, and therefore Y and Xnon-singular. Our object is to compare c(Y)
with f*c(Y). Let N be the normal bundle to X in Y, of rank d, and identify X
with P(N), so Nx? is ff^ir ')¦ Let F be the universal quotient bundle on
P(N);
Lemma 15.4. There are exact sequences
0
(i) Q-*fi{- 1)
(ii) 0-*J^> g*N
(iii) 0 -¦ 7> ^;*
(iv) 0^rf^r,^,(fl^(l.
77ie first three are exact sequences of vector bundles on X, the fourth an exact
sequence of sheaves on Y.
Proof, (i) is the universal exact sequence on the projective bundle P{N).
(ii) follows from the exact sequence giving the relative tangent bundle of a
projective bundle (Appendix B.5.8):
and the identification of Tg with the kernel of dg:Tx-> g*Tx. (iii) is the
usual relation between tangent and normal bundles (Appendix B.7.2). For (iv),
note that
df:Tf->f*Ty
is an isomorphism off X, so is a monomorphism of sheaves. To prove (iv) it
suffices to map/*/* TY = g* i* TY to F so that
j*Tf-* g*i*TY~* F-* 0
is exact on X. Consider the diagram
0-> Tx -> j*TY -> NXY -»0
i i i
0- g*Tx -> g*i*TY~* g*N -0
with exact rows. The first vertical map is surjective by (ii), the third injective,
with cokernel F by (i). The desired map is the composite
g* i*TY-*g*N -* F,
and the required exactness is a simple diagram chase. D
We apply Riemann-Roch without denominators to the vector bundle F on
X, and the inclusion/: X -* Y:
By (iv) of the lemma,
c{j*F)=f*c{Y)/c{Y).
Therefore
A) f*c{Y) - c(Y) = c(Y) -MPP(- 1), F)) =j*(j*c(Y) P{0{- 1), F)).

300 Chapter 15. Riemann-Roch for Non-singular Varieties
By (iii) and (ii) of the lemma,
B) j*c(Y) = c(X) c@(- 1)) = g*c(X) c(g*N®0(\)) c@(- 1)).
From the definition of P@(- 1), F),
_c(F)
or
(F®0(\))
-\=cx@(-\))-P@(-\),F),
C)
c(F®0(\))-P@(- \),F) =
c(F®0(\))-c(F)
c,
where the right side denotes the result of formally dividing by c{@(l)) (cf.
Example 15.3.4). By (i) of the Lemma, and the result of tensoring (i) by<?(l),
c(F) = c(g*N)/c@(- 1)),
c(F® 0A)) = c(g*N ® 0(\)).
Combining B), C), and D), we have
E) j*c(?) P@(- 1), F) = g*c(X) c^~ '» ^f^ ~ '^
Theorem 15.4. With the above notation, and ? = c, I
c(Y)-f*c(Y)=j*(g*c(X)-a),
where
D)
C L/-0 ;-o
In this expression, the term in brackets is expanded as a polynomial in ?, and a is
the polynomial one obtains after formally dividing by (.
Proof. This follows from A) and E) and the identity
d
c(g*N®0(\)) = g c@(\))ig*cd-i(N)
of Remark 3.2.3 (b). D
Example 15.4.1. The formula for a may be written explicitly as follows:
where Kj is defined to be zero if p < q. With this formula for a, and
(- 1)' Ci(V) for the /* canonical class of a non-singular variety V, Theorem 15.4
takes on a form similar to that conjectured by Todd G).
Example 15.4.2. Suppose ck(N) = i*ck, for some classes ck e Ak Y. For
example, if N is the restriction of a vector bundle E on Y, then ck = ck (E) will
do. Let n = c, @?(X)). Then
c(Y)-f*c(Y)=j\(g*c(X))-fi
15.4 Blowing up Chern Classes
301
where
(Since j*rj= — (, it follows that j*f} = a with a as in the theorem. Apply the
projection formula.) Using c(X) = i*c(Y) (? ck)~\ andy^l) = r\, this can be
rewritten:
c(Y)=f*c(Y) [?f*ck
u-o
c(Y)=f*c(Y)(\ + r,)(\-r,)d.
3)
(a)
Ifc(AO=l,then
(b)
This holds in particular when A" is a point. Then f*a- i=j\(j*f*oi) =
/„ (g*i*a) = 0 for a. e A"X, p>Q,so
(c) c(Y)=f*c(Y)+(\ + rt)(\-rt)d-\.
Example 15.4.3. Equating terms of degree 1,
c[(Y)~f*cdY)=j*('i-d)=(\-d)[X],
which is the usual formula relating the canonical divisors on Y and f:
Ky=f*KY + (d- \)[X].
For terms of degree 2,
C2(Y)-f*c2(Y) = -jt((ct- l)g*Ct(X)+-
lfd=2,
c2(?) -f*c2(Y) = -j*g*c\ (X) - [X] ¦ [X] =/•/»[*] -f*c,(Y) ¦ [X].
(For the second equation, use Proposition 6.7.)
Example 15.4.4. Let A" be a non-singular projective surface, and suppose a
Lefschetz pencil of curves is constructed on X, i.e., all curves in the pencil are
non-singular curves, of genus g, except for d curves, which each have one
ordinary node; at each of the a base points the curves are assumed to be non-
singular and meet transversally. The Zeuthen-Segre invariant I is defined by
I = d- 4g - a .
Then x(x) = \ c2 (X) = / + 4. (If X is the blow-up of X at the base points, then
x(X) = x(X) + a by Example 15.4.2(c). The pencil is given by a morphism
from A" to P1, and </=/(f)-2B-2#) by Example 14.1.5.) In particular, / is
independent of the choice of pencil.
With the notation of Example 15.2.2, define pm = (K ¦ K) + 1, and pa =
pg— q. Then equation (*) of Example 15.2.2 is equivalent to Noether's formula
(cf. Noether(l)):
/><" + /= 12/>a+9.
Example 15.4.5. Atiyah and Hirzebruch C) proved GRR for closed im-
beddings of complex manifolds. The proof given here can also be used in the

302
Chapter 15. Riemann-Roch for Non-singular Varieties
analytic case. Similarly, the formula for blowing up Chern classes (Theorem
15.4) is also valid, with the same proof, for X and Y arbitrary complex
manifolds, and Chern classes in singular cohomology with integer coefficients.
Notes and References
For more than a century, the Riemann-Roch problem has stimulated the
development of intersection theory and the search for invariants of algebraic
varieties. This began with Riemann's inequality
dimHa(X,l?x(D)Mzdeg{D)+ 1 -g
for a divisor D on a non-singular projective curve of genus g, and an
identification of the error term as dim H° (X, 0X (K - D)) by Roch. For
surfaces, analogues of Riemann's inequality were developed primarily by
Noether, Castelnuovo, and Severi (cf. Severi A3), B0), Zariski D)) while
Zeuthen and Segre (cf. Example 15.4.4) generalized the notion of genus. For a
sketch of this early history see Zariski A). The arithmetic genus was studied
quite generally by Severi C), B0).
The notion of canonical divisor was generalized from curves to surfaces by
Noether. Severi F) was the first to find an analogue in codimension greater
than one, when he defined a canonical zero-cycle on a surface, as the zero set
of a holomorphic one-form, for surfaces which have such one-forms. There
followed a succession of papers by Segre, Todd, and Eger, constructing
canonical classes in all dimensions on arbitrary non-singular projective
varieties; the /th canonical class of these authors is
c,{Tx)={- \yCi{Tx)={- \)'Ci{X),
although this was not proved until the machinery of Chern classes was highly
developed (Nakano A)). By ingenious calculations, Todd D) found formulas
for the arithmetic genus in terms of canonical classes (Corollary 15.2.2), and
proved them in many cases.
There was also much work by Segre and Todd on the formula relating
canonical classes of a variety and its monoidal transforms. Formulas were
proved in many cases, and Todd G) conjectured a general formula. The article
of Todd (8) may be recommended for the early history of canonical classes.
Two developments were vital for the modern solution of these problems.
One was the introduction of the methods of sheaf theory, primarily by
Kodaira and Spencer for complex manifolds, and Serre for algebraic varieties.
The second was the development of characteristic classes of vector bundles in
topology, by Whitney, Stiefel, Pontrjagin, and Chern. (It is interesting that the
study of canonical classes in algebraic geometry and Stiefel-Whitney classes in
topology began almost simultaneously, with many common geometric con-
Notes and References
303
structions, involving singularities of projections in ambient projective or
Euclidean spaces; apparently it was some time before there was any contact
between these schools.)
The general formula (Corollary 15.2.1) for the Euler characteristic of a
vector bundle on a non-singular complex projective variety was given and
proved by Hirzebruch A), using Thorn's results in cobordism. The third
edition of Hirzebruch's book contains useful historical notes at the end of
chapters, and an appendix by Schwarzenberger with applications and more
recent history. For surfaces and threefolds, other proofs of Corollary 15.2.2
have been give by Piene E) and Piene and Ronga A).
Grothendieck's transformation of the Riemann-Roch problem (cf. Borel-
Serre A), [SGA6] Exp. 0) has influenced a wide spectrum of mathematics.
Grothendieck groups and ^-theory are two legacies of this work. The
generalization from giving a formula on individual varieties to a formula for
morphisms between varieties, and the construction of a natural transformation
of functors, has had profound consequences. At the same time, and using
similar sheaf-theoretic and geometric constructions, Washnitzer B) gave an
axiomatic characterization of the arithmetic genus (cf. Example 15.2.10). Some
recent applications of GRR may be found in Harris-Mumford A), and
Arbarello-Cornalba-Griffiths-Harris A).
For complex projective varieties, by GAGA theorems of Serre C),
Riemann-Roch for algebraic sheaves and morphisms is equivalent to cor-
corresponding formulas for analytic sheaves and morphisms. A generalization of
GRR for imbeddings of complex manifolds has been given by Atiyah and
Hirzebruch C), where the values are taken in topological A:-theory. Another
far-reaching generalization of the Hirzebruch formula is the index theorem of
Atiyah and Singer A). Obrian, Toledo and Tong C) have generalized GRR
to complex manifolds, with values in the cohomology H* (X, Q*x).
The proof of GRR given in this chapter is a simplification of ideas in
Baum-Fulton-MacPherson A), B), using the deformation to the normal
bundle. The generalization to non-projective varieties, over arbitrary ground
fields, depends first on having an adequate intersection theory in that
generality. In characteristic zero, one may then use Chow's lemma and
Hironaka's resolution of singularities to conclude the proof (cf. Example
15.2.9) — an observation I owe to Kleiman and Gillet. In all characteristics,
GRR for non-singular varieties can be deduced from the full singular
Riemann-Roch theorem (see Chap. 18).
Riemann-Roch without denominators (Theorem 15.3) was conjectured and
proved in characteristic zero by Grothendieck [SGA6] and proved in general by
Jouanolou B). The general formula for blowing up Chern classes was deduced
from GRR by Porteous A), after conceptual simplification by Van de Ven A);
Porteous's proof was only valid modulo torsion until Riemann-Roch without
denominators was proved. Lascu and Scott A), B) gave a proof without using
Riemann-Roch. Mumford, Jouanolou, Lascu and Scott used the space which we
constructed in Chap. 5 as the total space of the deformation to the normal bundle;
the further simplifications may be attributed to realizing the full potential of the
deformation itself.

304
Chapter 15. Riemann-Roch for Non-singular Varieties
It is possible to give more conceptual proofs of GRR for the case of a
projection YxWm -> Y, although with some loss in brevity. See Baum-Fulton-
MacPherson B) App. 3, or Obrian-Toledo-Tong C).
Hirzebruch pointed out the application of the geometry of the deformation
to the normal bundle to signatures in Example 15.2.9. Example 15.2.18 is J.
Harris's solution to a question of M. Green. Most of the other examples are
standard consequences of Riemann-Roch, at least in the case of projective
varieties over an algebraically closed field.
Chapter 16. Correspondences
Summary
A correspondence from X to Y, denoted a.: X \- Y, is a subvariety, cycle, or
equivalence class of cycles on Xx Y. The graph of a morphism, or the closure
of the graph of a rational map, are basic examples, but more general cor-
correspondences have played an important role in the development of algebraic
geometry. On complete non-singular varieties correspondences have a product
ft ° a, and a correspondence a : X \— Y determines homomorphisms a* from
A(X) to A{Y), and a* from A(Y) to A(X), these notions generalizing
composition, push-forward, and pull-back for morphisms. The basic algebra of
correspondences is deduced easily from the general theory of Chap. 8.
If X= Y has dimension n, and T is an n-dimensional correspondence, then
the degree of the intersection class T ¦ A of T with the diagonal is the virtual
number of fixed points of T. In case there are non-isolated fixed points, the
excess intersection formulas can be applied. (If T= VxW, with V, W sub-
varieties of X, T- A = V- W is the intersection class studied in Chap. 8.)
When one has explicit formulas for the equivalence class of [7] or of [A] on
Xx X, fixed point formulas for T¦ A can be deduced.
Notation. Unless otherwise stated, all ambient varieties X, Y, Z,... in this
chapter are assumed to be complete and non-singular, i.e., proper and smooth
over the given ground field.
16.1 Algebra of Correspondences
Definition 16.1.1. A correspondence from a variety A' to a variety Y is a cycle,
or an equivalence class of cycles, onJfx Y. We shall write a: X \- Y to denote
that a is a correspondence from X to Y.
If a.: X \- Y, P: Y h Z, with X, Y, Z non-singular, the product (or
composite) correspondence, denoted /? ° a, is the correspondence from X to Z
defined by the formula
P ° a = Pxz* (Pxy* a • P yz*P) ¦

306
Chapter 16. Correspondences
Here pXY, Pyz, Pxz denote the projections from XxYxZ to XxY, YxZ,
XxZ, respectively. The product pXY* a " Pyz*P is the intersection product on the
non-singular variety Xx YxZ. If a and /? are cycles and pXY*a. meets pYz*P
properly, then /? ° a is a well-defined cycle on X x Z; in general /? ° a. is defined
up to rational equivalence. This product defines a bilinear homomorphism
^(A'x Y)®A{YxZ) ^ A(XxZ).
A correspondence a : X f- 7 has a transpose a.': Y \- X defined by
a' = t* (a)
where i:IxN Yx X reverses the factors, i.e., x(P, Q)=(Q, P).
An irreducible correspondence from A' to Y is a subvariety V of Xx Y,
identified with its cycle [V\. Any morphism/: Jf-» Y determines an irreduc-
irreducible correspondence Ff from X to Y, given by the graph imbedding of X in
A'x Y.
Proposition 16.1.1. Let a.: X h Y, 0 : Y h Z.
(a) Ify.Zh W, theny°(P°a.)=(y°p)°ai.
(b) 08 ° a)' = a' ° P, and fa')' = a.
(c) (i) IfP=rg,thenl]°o.=(\xxg)*(o.).
(li) //a = />, r/>e« y8 o a = (/x lz)* (y8).
(iii) //a = />, y3 = r,, f/ien y8 ° a = />.
?roo/ (a) Denote by p/^ the projection from Xx Z x W to Xx W, and
similarly for other projections; for projections from Xx YxZx W, the super-
superscript XYZW is omitted. By Proposition 1.7, we have the formula
as homomorphisms from A (X x Y x Z) to A (X x Z x W). Then we have
= p?w%{Pxzw* (Pxyz* (pVvZ* a •
= p?w*{Pxzw*{{Pxy* <*-Pyz*P
= Pxw* ((Pxy* a • Pyz* P) ¦ Pzw* i)
• PXzZww* y)
*p?ww* y))
(P
XY*
- PZW* 7)).
These five equalities follow from: (i) the definition of products; (ii) formula
(*); (iii) the compatibility of pull-back with intersection product, functoriality
of pull-back, and the projection formula; (iv) functoriality of push-forward and
pull-back; (v) associativity of intersection products. Symmetrically, the last
expression is (y ° P) ° a which concludes the proof of (a).
(b) Let a: Xx Yx Z -* Z x Yx X map {P, Q, R) to (R, Q, P). For any S, T
let rsr denote the morphism from S x T to 7"xS that reverses the factors. By
Proposition 1.7,
16.1 Algebra of Correspondences
Arguing as in (a), one has
*'°ir=p?x\ (piV*
307
= PzzYxi'>*{pxxYYZ**-pVzz*P)
Vzl (p?yZ** ¦ pVzz*P) = (fi ° a)' •
The third equality uses the fact that o* is a ring homomorphism (Example
8.3.5); the fourth follows from the identity pffi o¦= tX2pW'¦ The other
identity (a')' = a follows from the identity xYX xXY = Ixxr-
(c) Let yg : Y -* YxZbe the graph of g. Then
r, °« = pVzl (pW* (y9* [ Y]) ¦ pVyz* («))
= pVzi
yg)* p?yz* («)))
= O*x0),(a).
The second equation uses Proposition 1.7, the third the projection formula,
and the fourth the identities
= \XxY.
Similarly for c (ii),
P" r, •=
?yz* (y/* [X]) ¦ pVzz* Pi
Forc(iii),r,orf
Corollary 16.1.1. For a non-singular variety X, the product a. x fS —» a ° fS
makes A{Xx X) into an associative ring with unit [Ax], and with an involution
a. -> a'. D
Definition 16.1.2. For a: X \- Y, define a homomorphism
by the formula a* (a) = pf I (a • /r^* (a)), and a homomorphism
by the formula a* (b) = pxx\ (a ¦ pxYY* (b)).
Proposition 16.1.2. (a) If a : X h Y, 0:Y\- Z, then
08 ° a)* = yS* ° a, anrf (yS ° a)* = a* » p* .
(b) Ifa.:X\- Y, then (a')* = a*.
(c) ///: X^Y, then (/», =/, and (/»• =/*.

308
Chapter 16. Correspondences
Proof. Let P=Spzc{K). If a e A(X) = A{Px X), then a* (a) may be
identified with a°a in A(Z)=A(PxZ). Similarly if b e A(Y) =A(Y x P),
a* (b), b°a. With these identifications, (a), (b) and (c) follow from cor-
corresponding parts of Proposition 16.1.1. D
Corollary 16.1.2. For a non-singular variety X, the homomorphism
A(X xX)->- End(A (X)), a t-* a, (resp. a i-* a*)
is a homomorphism (resp. anti-homomorphism) of rings. ?
Remark 16.1. Several modifications of A(X xX) have been useful, and have
been called the ring of correspondences on X:
(i) Let C(X) = A"(Xx X), where n = dim(X). This is a subring of the ring
of correspondences, closed under the involution and containing the identity
and graphs of morphisms, and a e C(X) induces homomorphisms of A*X
which preserve degrees (cf. Example 16.1.1).
(ii) One may consider A(XxX)/I, where / is an ideal of degenerate
correspondences (cf. Example 16.1.2).
(iii) One may replace rational equivalence by algebraic, numerical, or
homological equivalence (cf. Chap. 19).
The results of this section extend readily to arbitrary smooth, proper
schemes over a field, using the identity
A(Xx
= ®A(X;x
U
where Xh Yj are the connected components of X and Y.
In addition, the completeness of the ambient varieties is not always
necessary. For example, if a is a correspondence from X to Y, and the support
of a is proper over Y, then a induces a homomorphism a* from A (X) to A (Y).
Indeed, as in Definition 16.1.2, if a 6 A(X), then a ¦ p*/* (a) is represented by
aj by §8.1; the proper push-forward of this class
a well-defined class on
is a* (a).
Example 16.1.1. A correspondence a : X" f- Ym is homogeneous of degree
(or codimension) p if ae Am+p(Xx Y). If a: X \- Y has degree p, and
0: Y f— Z has degree q, then 0 ° a has degree p + q. In particular the correspon-
correspondences of degree zero are closed under composition. If a has degree p, then a*
maps AkX to Ak-P Y, and a* maps Ak Y to Ak+pX. Iff: X -> Y is a morphism,
then Ff has degree 0. If a : X \- Y has degree p, then a' has degree p + m — n;
in particular, Ff has degree m — n. (A dual definition is discussed in
Example 16.1.12.)
Example 16.1.2. Degenerate correspondences, (a) Let I(X, Y) be the sub-
subgroup of A(Xx Y) generated by correspondences of the form [Vx W], with V
(resp. W) a subvariety of X (resp. Y). If a 6 I(X, Y) and 0 e A (Y x Z), then
0°ae I(X,Z). It follows that I(X, X) is a two-sided homogeneous ideal in
A(XxX), with I(X,X)'=I(X,X). (If a=[VxlV], then p°a = [V]x
pYzl{[WxZ}-0).)
16.1 Algebra of Correspondences
309
The subgroup of A(XxX) generated by correspondences of the form
[PxX] and [Xx P] for points P e X, is likewise a two-sided ideal.
(b) Consider only ^-dimensional varieties X, Y, Z. Let J(X, Y) be the sub-
subgroup of A"(Xx Y) generated by irreducible correspondences [V], where Fis
a subvariety of Xx Fsuch that pY (V) * X or pV (V) * Y. lfaeJ(X, Y) and
0 s A"{Yx Z), then 0° a e J(X, Z). Hence J(X, X) is a two-sided, involutive
ideal in C(X) = A"(XxX).
(Let a = [V], 0 = [W], V, W irreducible. If pxxY\V) = ScJ, then 0 o a is
represented by a cycle on S x Z. Suppose pV(V) = T $ Y. If W <?T x Z, then
PzYZ{{V xZ)C\(Xx W))<z pYzz((TxZ)C)W)'~Z by a dimension count. Any
rc-cycle on Yx Z is rationally equivalent to a cycle none of whose components
are contained in Tx Z; indeed, on any smooth variety, any cycle can be moved
off any given proper subvariety, as a simple local argument shows.)
(c) If X is a curve, J (X, X) = / (X, X) f] C (X). The quotient ring C (X)/J {X, X)
of non-degenerate correspondences operates on the Jacobian
io(X) = Ker(/fo(X) ^ Z)
by the operations a* and a* of Definition 16.1.2.
Example 16.1.3. If a and 0 are correspondences from X to Y, then
inA{X), with/>: Xx K-> X the projection. If X, Y, a and 0 are ^-dimensional,
then J a • 0 is the virtual number of coincidences of a and /?. The virtual number
of coincidences of a and 0 is therefore equal to the virtual number of fixed
points of 0' ° a, or of a' ° 0. (Let
square XxY_
> = (\xx6rx\x)*(ax0'). Form the fibre
1 V V V
—» A X / X A
I'
->• XxX
?= #>*(y) and
° a = q* (y); and
1
X -
where q is the projection. Then a ¦
p* <p* by Theorem 6.2.)
Example 16.1.4. In this example, X, Y, Z are irreducible of the same dimen-
dimension n. The indices (or degrees) dl (a) and d2(a)of a correspondence a e A"(X x Y)
are defined by
pxxl (a) = d, (a) ¦ [X], /rj^ (a) = d2 (a) ¦ [ Y].
(a)If 0e A" (YxZ), then, for/ = 1 and 2,
(Use Proposition 16.1.2(a) and the identity a*[Y] = d, (a) ¦ [X].)
(b) For any a, rf, (a') = rf2(«) •
(c) For any rational point P on X, d\(tx) = \<x-[P x Y].
With a = di{a), b=d\ (a), a is called an (a, by correspondence.
Example 16.1.5. Valence. The varieties X, Y, Z are assumed to be n-
dimensional. According to Severi A0), a correspondence aeA"(Xx Y) has

310
Chapter 16. Correspondences
valence zero if a is in the group I(X, Y) of degenerate correspondences defined
in Example 16.1.2 (a). A correspondence a e A"(X x X) has valence v if a + v A
has valence zero, where A is the identity correspondence. These notions are
most useful if rational equivalence is replaced by algebraic or numerical
equivalence, as in Remark 16.1.
(a) If a and p are correspondences on X with valences u and v, then /?°<x
has valence - u v. (P ° a - u v A = {ft + v A) ° a — v (a + u A).)
(b) If a has valence v, so does a'.
(c) If a has valence v, and one knows a decomposition
a. + vA~Y,n,[ViX IV,]
with V,, W^ subvarieties of X such that dim F)+dim W/, =
where % = \ A ¦ A is the Euler characteristic of X.
(d) Every correspondence a on P1 has valence zero. If a is an (a, b)-cor-
respondence on P1, then
Jot- A =a + b.
This correspondence principle of Chasles A), D) - that an (a, b) correspon-
correspondence on P1 has a + b fixed points - was one of the primary tools of classical
enumerative geometry.
(e) (Chasles-Cayley-Brill-Hurwitz). If a is an (a, ^-correspondence with
valence v on a curve X of genus g, then
j<X ¦ A = a + b + 2v g .
If a = [7], T irreducible, T 4= A, then T has a + b + 2v g fixed points, counting
multiplicities. (Let a +vA = cx[X] + [X]xd. Then deg(c) = a + v, deg(</)
= b + v. Apply (c).)
(f) If X is a projective space, Grassmann variety, or any flag variety, then
every correspondence \nA"(XxX) has valence zero. (See Example 1.10.2.) On
a curve of general moduli, every correspondence has a valence (Hurwitz A)).
(g) If /: X -> P is a closed imbedding, and y is a purely m-dimensional
subscheme of Xx Pm, then
a = (]xxi)*\r]eA"(XxX)
is a correspondence of valence zero. If the projection from y to X is flat, with
fibre V(P) over P e X, then a* (P) is the intersection class of V(P) with X in
Pm. With some additional hypotheses, Severi A0) calls such a a generalized
Zeuthen correspondence.
Example 16.1.6 (cf. Severi E), p. 174). If a is a correspondence on a curve
of genus g which is a product of r correspondences a,, with a, an (a,, b,)-cor-
respondence of valence «,-, then the virtual number of fixed points is
I-1
I-1
16.1 Algebra of Correspondences
311
Example 16.1.7. Let <X| and <x2 be correspondences on a curve of genus g, a,
an (a,, ^-correspondence of valence vh Then the virtual number of coinci-
coincidences is given by
Jat ¦cc2 = alb2 + a2bl-2gvlv2.
(This follows from Examples 16.1.3 and 16.1.5 (e).)
For enumerative applications of this and the preceeding formulas for
correspondences on curves, see Severi E)VI§ 4.
Example 16.1.8 (cf. Zeuthen A), Severi E) § 60). Let T be an irreducible
(n, «')-correspondence between curves C and C. Let g be the genus of C, and
let d be the number (suitably weighted) of points of C which correspond to
fewer than n' points of C; define g' and d' symmetrically. Then
d-2n(g'-\) = d'-2n'{g-\).
(Apply the Riemann-Hurwitz formula (Example 3.2.20) to f -> C and f -» C,
fthe non-singular model of T.) More generally, if D (resp. /)') is the divisor of
double points, and K (resp. K') a canonical divisor on C (resp. C), then
. (Writer=?°a', a : f -> C, 0: T -> C.)
Example 16.1.9. Let /: X -> X be a morphism, P a fixed point of/, rational
over the ground field. Regard 1 — rf/as an endomorphism of the tangent bundle
(a) If det A - df)F 4= 0, then P is an isolated fixed point of/, and
i((P, P),rf-A;XxX)= 1 .
(b) If X is a curve, and the ground field has characteristic zero, then
i((P, P), rrA;XxX)=\+ ordP(l - df) .
Here ord,>(l - df) is the order of vanishing at P of 1 - df as a section of
T)c ® Tx. (If t is local coordinate at P, and /(?) = ? a, ?', the left side is the
order of vanishing of t-f(t) at ? = 0.)
Example 16.1.10. Castelnuovo-Severi inequaltiy (cf. Mattuck-Tate A)).
(a) Let Fbe an (n, «')-correspondence between curves C and C. Then
(Apply the index theorem (Example 15.2.4) to the divisor D=T- n{P xC)
-n'(CxP') on the surface CxC, with ample divisor H= CxP'+ P xC.)
Note that the equality occurs precisely when a multiple of T is algebraically
equivalent to a degenerate correspondence, or, modulo numerical equivalence,
Fhas valence 0.
(b) With T as above, and 5 an (m, m') correspondence between C and C,
(JS- T'- m n' - m' nJ ^ 4(n ri - \\T¦ T) ¦ (m rri - \ \ S ¦ S).
(Let q{a, b) = Q{a S + b T), where Q(U) = n(U) ¦ ri(U) - {J U ¦ U. Then q is
a non-negative quadratic form, so its discriminant is non-positive.)

312
Chapter 16. Correspondences
(c) With Fas in (a), and C = C a curve of genus g, then
(JF- A-n-n'J^4g(n n'-\\T-T).
Equality holds precisely when F has a valence, modulo numerical equivalence.
(d) If, in (b), 5 and F are graphs of morphisms e and/from C to C, then
N= j 5 • F is the weighted number of coincidences and
| N - deg (e) - deg (/j| ^ 2g' J/deg(e) ¦ deg(/) ,
where #' is the genus of C (By the self-intersection formula, \T-T =
lf*cx (Tc)=B - 2g') deg(/), so Q(T) = g' deg(/).)
(e) In (d), if C'= C, e= id., the weighted number N of fixed points of/
satisfies
(f) If C is defined over the field F, with q elements, then the number Nq of
F,-valued points of C satisfies the "Riemann hypothesis":
\N,-q-l\s2gf?.
(Apply (e) to the Frobenius morphism.)
Example 16.1.11 (cf. Colliot-Thelene and Coray A)§6). If X and Y are
birationally equivalent complete non-singular varieties over an algebraically
closed field, then A0X = Ao Y. (The closure of the graph of a birational map
determines a correspondence a from X to Y. To see that a* and <xi are inverse
isomorphisms on Ao, note that a' ° a is the sum of the identity correspondence
and correspondences whose projections are contained in proper subvarieties of
X; any point on X is rationally equivalent to a zero-cycle disjoint from a given
subvariety of X.)
Example 16.1.12. Grothendieck's motives (cf. Manin A), Kleiman E),
Deligne-Milne-Ogus-Shih A)). Let y be the category of smooth schemes (over
a fixed ground field). Define an additive category -ffV as follows. The objects
of &V are the objects of T; denote by X the object in 4Y" determined by the
scheme X. The morphisms in €V are defined by
ttom{X,Y) = A(Xx Y)
with composition defined by the product of correspondences. Define a direct
sum in 4T by X® Y= X 1LY, and a tensor product by X ® Y=Xx Y. For
each morphism/: Y -> X inX, define/: X -> Y'm^r by/= [/>]'; this gives a
contravariant functor from V to O^.
-, define X(f) = Hom(T,X). For p 6 Uom(X, f), define
by <PT(g) = <P°g- When r is a point, X(T) = A(X), and #>T is the homo-
morphim #>„, of Definition 16.1.2. If #>=/ #>r = (lr></)*, while if <p=f,
(pT = (\Txf)+.
Martin's identity principle. Let (p, i// e Hom(X, Y). Then the following are
equivalent: A) <p=i//. B) <pT=y/T for all F. C) <px=Vx- (Indeed, (px([^x])
= (p°\x=<p,soC)=> A).)
16.1 Algebra of Correspondences
313
This principle can be used to show that formulas known in rational
equivalence therory A* are also valid in a cohomology theory H*, when one
has a cycle map A* -» H*. For example, the key formula (Proposition 6.7(a)),
for non-singular varieties, can be expressed as an equality of correspondences
between Y and X; the validity of the formula, after crossing the diagram with
an arbitrary T, impHes its truth in H*.
Define Horn'(A", Y) = ® An>+'(XjX Y), where Xt are the connected com-
components of X, n,-= dirnX,. The degrees of morphisms add under composition.
Allowing only morphisms of degree zero determined a subcategory €T^ of€y.
Grothendieck defines a motif to be a pair (X, p), JeT, p e Hom(X, X),
with p°p=p. The motif of the variety X is the pair (X, ]x), denoted X. Define
Horn((*,/>), (Y, 9)) to be
(y6Hom°(I, Y)\q<p = <pp}/{<pe Horn0(X, ?)\q <p= <pp = 0].
With these morphisms, the category of motives, denoted €T^, is pseudo-
abelian, and the natural functor from €ya to -O^ is universal for functors to
pseudo-abelian categories. (An additive category.® is called pseudo-abelian if
every p.D^D in & with p°p= p has a kernel Ker(/>), and the canonical
map from Ker(/>) © Ker(lo-/>) to D is an isomorphism.) The category Jy°
inherits a tensor product: (X, p) ®(Y, q) =(Jx Y,pxq).
The Tate motive L is defined to be (P',/>), with p = [P1 x {0}] eAl(P' xP1).
Then P1 = 1 © L, where 1 = P, P = Spec(#)- If ? is a vector bundle of rank
c+lonJ, there is a canonical morphism of motives
P(E)=
(Manin A)§7). Similar formulas are valid for blow-ups (loc. cit. §9). For a
curve X, the essential part of the motif of X is determined by the Jacobian
variety of X(loc. cit. § 10).
Important variations are obtained by replacing rational equivalence by
algebraic, homological, or numerical equivalence, as in Remark 16.1.
Example 16.1.13. The theory of correspondences can be extended to
quotient varieties, provided rational coefficients are used. If X,= YJG,, Y,
non-singular, rc,: K,-> X, as in Example 8.3.12, then Cartesian products
X\ x ... xXr are identified with quotients of Ytx ... x Yrby Gt x ... x Gr. One
may construct a ° /?, a*, a* as in this section, using the intersection product for
quotient varieties described in Example 8.3.12. In particular, if/: X[ -> X2 is a
morphism, one may define the pull-back
to be /}*, where f; is the graph of/ This is a ring homomorphism, with the
usual functoriality, and projection formula for proper/; if/is the identity, so is
/*. (One way to verify this is to show that this /* agrees with that of
Example 17.4.10.)
In particular, the definition of intersection product in Example 8.3.12 is
independent of presentation of the variety as a quotient variety. Keeping track

314
Chapter 16. Correspondences
of supports of correspondences, one sees likewise that the same is true for the
refined intersection products, and hence for the rational intersection numbers.
Example 16.1.14. Let a: X f- Y be a correspondence, and let a be a cycle
on X, b a cycle on Y. Then
x y
(Both are equal to \x%Ya-(ax b)\ cf. Clemens-Griffiths A) p. 289.)
If a = [7] is an irreducible correspondence, and p : T -> X, q : T -> Y are the
projections, then
(ii) T+(a) = q*p*a , T*b = p*q*b .
Here p* and q* are the pullbacks of § 8.1. (If i is the imbedding of T in Xx Y,
then i+(p*a)= T-(ax Y); apply pf% to prove the first statement; apply the
involution 2"-» T to deduce the second.)
Iff:X'->X, g :Y' -> K are morphisms, then (cf. Lieberman B) p. 1168)
(iii) (f'*Q)*(a) — r'g°a.°rf.
Example 16.1.15. Lefschetz fixed point formula. The formalism of this sec-
section is also valid for homological correspondences a between compact oriented
manifolds X and Y. To avoid sign problems, consider only a e H"{Xx Y), with
n = dim(X) = d\m(Y) even; the cohomology groups are taken with rational
coefficients. Such a determines
by the formalism of Definition 16.1.2, using Poincare duality to define the
push-forward px*. By the Kiinneth isomorphism, this corresponds to an iso-
isomorphism
H" {Xx Y)^@ Horn (//' Y, H'X).
If X= Y, the diagonal class A corresponds to the identity map. The Lefschetz
fixed point formula states that for a e H" (XxX),
J a ¦ A = X (- 1)' trace (a* : H'X^ H'X).
XxX i
This gives a formula for the virtual number of fixed points of a (cf. Klei-
man B)).
If a corresponds to a morphism f:X^>X, one recovers a fixed point
theorem for / There are formulas for the intersection numbers occurring on
the left. For example, in the differentiable case, if P is an isolated fixed point
and dfp does not have 1 as an eigenvalue, then the intersection number of Ff
and A at (P, P) is the sign of the determinant of /— dfP.
When the manifolds are algebraic, and a is the class of an algebraic cycle, it
follows from the discussion in § 19.1 that the topological and algebraic calcula-
calculations of | a ¦ A agree, as do the notions of a*, a ° /?, etc., cf. Example 19.2.7.
Note however, that, in general, the right side of the fixed point formula
necessarily involves non-algebraic cycles.
16.2 Irregular Fixed Points 315
16.2 Irregular Fixed Points
Let T be an irreducible ^-dimensional correspondence on an n-dimensional
variety X, with T+ A. Let XxX be the blow-up of XxX along the diagonal
A. The exceptional divisor E is the projective tangent bundle P(TX); let p
denote the projection from E to X- A. Let f be the proper transform of T in
XxX, i.e., f is the blow-up of T along Tf]A. The points of TOE are_certain
tangent lines to X, called principal tangents. Since tf]E is a divisor in T, it has
pure dimension n — 1.
The fibres of tf]E over a point P in TO A can have any dimension from
0 to n - 1. If T f]E contains the full fibre P(TP\ P is called a perfect fixed
point. All isolated fixed points are perfect, but there may also be a finite
number of perfect fixed points lying on larger components of Tf] A.
The degree j T ¦ A of the intersection is the virtual number of fixed points of T.
The following proposition can be used to relate this number to the geometry of
Tf]A. Several classical formulas of this type are derived in the examples that
follow.
Proposition 16.2. Let i, be the universal quotient bundle on the projective
bundle P(TX). Then
T-A = {c(Tx) n s(T0A, T)}0 = P^.^) n [ff]E])
Proof. The class T ¦ A may be calculated by intersecting T <= X x X by
A <=. X x X, according to the prescription of §6.1. Since Tf]E is the projective
normal cone to Tf] A in T, the result follows from Proposition 6.1 (a) and Exam-
Example 6.1.8. ?
Example 16.2.1 (cf. Pieri A)). If X= P", there is a canonical morphism
which takes a projective tangent line to the corresponding imbedded tangent
line in P". In fact <p extends to a morphism from XxX to G which takes (P, Q),
P * Q, to the line between P and Q (cf. Kleiman (8) VB). The principal lines at
P are the images by <p of the principal tangents Tf]P{TrX).
Define the indices It,..., In+l of T by setting
Ii+, = * {(P, Q) e T\P s[i],Q e[n- i])
where [i] and [n - /] are general linear spaces in P" of the indicated
dimensions. These indices are simply the bidegrees of the n-cycle [7] on
P" x P".
Pieri defined the ranks R x,..., Rn of T by
R.+l = $ {p e t(] A | P e [n - i], and some principal line at P meets [»]}.
Thus J?t is the number of perfect fixed points. Since the scheme TOE may
not be reduced, these ranks should be interpreted with multiplicities, as
follows:

316
Chapter 16. Correspondences
Let am be the class in Am(G) represented by a Schubert variety of lines that
meet a linear space [n — 1 — m\ Then
Pieri's Theorem. With this notation,
Previously cases with n = 1, 2, and 3 had been given by Chasles, Zeuthen,
and Schubert. (Since [A] = ? ['] x [n - 0. the left side of (*) is J F- A. Since
ff»> = c<n@. where Q is the universal quotient bundle on G (§ 14.7), the right
side of (*) is
n [THE]).
By Proposition
q>*Q®p*0(\).
?,=
16.2 and Remark 3.2.3(b) it suffices to show that
() Indeed, the canonical surjections p*^(\)Bn+] -> p* Tx -> ?
realize ^ ® p*0{— 1) as a quotient of the trivial bundle of rank n + 1, and #> is
determined by the universal property that this quotient is <p* (Q).)
For example, if T is the closure of the graph of a projection from a
subspace A"~r~l to a complementary space B', the principal lines to P e B are
the lines joining P to A. In this case /,• = R,¦ = 1 for 1 § j s r + 1, and /, = i?, = 0
for j > r + 1.
As an application, take T= Vx. W. The left side of (*) is deg(K) deg(W).
Suppose F meets ff in JV isolated points, counted with multiplicities, and also
scheme-theoretically in a non-singular variety Z of dimension A: > 0. Then
where >>(j) is defined as follows (cf. Example 12.3.6). Fix a general i-plane L
and a general (n - i)-plane M. Then ym is the number of points P in LflZ
such that the span of the tangent spaces to V and W at P meet M. Previous
cases of this formula had been given by Salmon and Caporali. For applications
see Baker B) Ch. II, Pt. III.
Pieri's theorem can also be deduced from the formula for/4(P"*: P"), using
Proposition 6.7.
Example 16.2.2. Proposition 16.2 and its proof go through for an arbitrary
T of pure dimension m^n, giving a formula for T ¦ A in v4m_n(rTl A).
If X=F", the degree of T- A is equal to the sum of the indices of T
(bidegrees of [7]). In particular, every irreducible correspondence of dimen-
dimension m g n on P" must have fixed points. The degree of T ¦ A is also equal to
the sum of the ranks of T, as defined in the preceding example. If Z,,..., Z,
are the irreducible components of TO A, then (Theorem 12.2(b))
16.2 Irregular Fixed Points 317
Example 16.2.3 (cf. Severi A)). Let X= Gd(Wm), n = (d+ 1) (m - d), and let
T be an rt-dimensional correspondence on X. For 0 S a0 < <?i < ¦¦• < ad = m,
define an index I(aQ,..., ad) of T by
I(aQ, ...,ad)=#{(L,M)eT\Le [a0,..., ad], M e [m-ad,..., m - a0]}
with notation as in Example 14.7.16. Then
the sum over all such sequences (aQ,..., ad). (In fact,
[zl] = Z (ao,...,ad) x(m- ad,...,m-aQ)
in An(Xx.X), as one sees by intersecting both sides with (m - bj, ...,m — b0)
x(c0,..., cd).) A similar formula is valid for general flag manifolds.
Example 16.2.4. If X is a surface, and T contains a curve D of fixed points,
Severi (8) p. 874 gave a formula for the contribution of D to the virtual
number of fixed points, so that the difference is the number N of perfect fixed
points. His formula is:
lT-A-N=v+2-2n+Q-2oi,
where: v is the degree of D, i.e. J D ¦ D; n is the virtual (arithmetic) genus of D;
choosing a general one-dimensional linear system L of curves on X, a> is the
number of points of D f] C for general C in L; and if D is the lift of D to
P(TX), i.e. the closure of the tangents to smooth points of D, then q is the
number of points t e D which are tangent to any C in L passing through ^@-
This may be explained, including cases when multiplicities occur, as
follows. Write
where the P, are the perfect fixed points, and each component of D maps
finitely by p to a curve in X. Define D to be the curve p*D. Then
By the adjunction formula (Example 15.2.2)
on E= P{TX),
By Example 3.2.19, i
Seven's formula now follows from Proposition 16.2:
A-T={c(Tx)ns(Af]T,T)}0
= c^Tx) r, p, [T[)E] + p+(ct((P(l)) n [TOE]).
Combining the displayed equations gives Seven's formula in case D has no
multiple components, and shows how to interpret the formula in general.

318
Chapter 16. Correspondences
Notes and References
A glance at the long encyclopedia article of Berzolari C) impresses one with
the importance of correspondences in mathematics through the early part of
this century. In particular, many problems in enumerative geometry were solved
by constructing appropriate correspondences between curves, and using coin-
coincidence or fixed point formulas for such correspondences (cf. Examples
16.1.5(d), (e), 16.1.6,16.1.7).
There were many attempts to find higher dimensional analogues. Among
these Pieri's theorem for correspondences on P" (Example 16.2.1) stands out as
a precursor of modern excess intersection theory. Zeuthen and Severi also
devoted a number of papers to this (cf. Examples 16.1.5, 16.2.3, and 16.2.4).
Most of the success in higher dimensions was achieved for those correspon-
correspondences T for which one can write the class [7] of Ton Jxlas a sum of
exterior products of cycles on X, or on varieties X for which the diagonal has
such a decomposition. In general such Kiinneth decompositions are only
possible if one allows non-algebraic cycles, including odd-dimensional homo-
logy classes, on X. The problem of finding a general fixed point formula in this
context was solved by Lefschetz A), cf. Example 16.1.15.
For history and applications of the theory of correspondences — which we
have made no attempt to repeat here — we recommend the encyclopedia
article quoted above, Zariski A) Ch. VI and App. B, Severi E) § 6, Lefschetz
B)VIII, Conforto A), and Baker BI,11. Recently correspondences have
appeared in the guise of Hecke operators (cf. Shimura B) § 7, Deligne A)).
The material in §16.1 is a routine application of standard intersection
theory; for curves this appears in Weil D). The deduction of the formulas of
Pieri and Severi in Examples 16.2.1 and 16.2.4 from the excess intersection
formula are apparently the first modern proofs to appear.
Chapter 17. Bivariant Intersection Theory
Summary
Our basic intersection construction has assigned to a regular imbedding (or
l.c.i. morphism)/: X -* K of codimension rf a collection of homomorphisms
for all Y' -> Y, X' = Xxy Y', all k. In this chapter we formalize the study of
such operations. For any morphism/: X -> Y, a bivariant class c in AP(X -> Y)
is a collection of homomorphisms from Ak Y' to Ak-pX', for all J" -> Y, all A:,
compatible with push-forward, pull-back, and intersection products.
The group A~k(X -> pt.) is canonically isomorphic to Ak(X). The other
extreme Ak(X-^-*X) is defined to be the cohomology group AkX. The
bivariants groups have products
A'(X ? Y)(
which specialize to give a ring structure on A*X, and a cap product action of
A* X on A+X. If X \s non-singular, A*X = A*X. There are also a proper push-
forward and a pull-back operation for bivariant groups, generalizing the push-
forward on A* and defining a pull-back on A*. There are compatibilities
among these three operations which allow one to manipulate bivariant classes
symbolically with a freedom one is accustomed to with homology and cohomo-
cohomology in topology.
Many constructions of previous chapters actually produce classes in
appropriate bivariant groups. For example, Chern classes of vector bundles on
X live in A*X. Flat and l.c.i. morphisms f:X -> Y determine canonical ele-
elements \nA*{X -? Y), which are denoted [/]. An element c of AP(X -> Y) deter-
determines generalized Gysin homomorphisms
AkX
AkY$ Ak-pX
and
• Ak+P Y
(for the latter / is assumed to be proper). Intersection formulas such as the
excess and residual intersection formulas achieve their sharpest formulation in
the bivariant language.
There is a useful criterion which implies that an operation which produces
rational equivalence classes on X' from subvarieties of Y' (for all Y' -> Y),
passes to rational equivalence and defines a bivariant class (Theorem 17.1).
This will be used in the next chapter to deduce the important properties of
local Chern classes.

320 Chapter 17. Bivariant Intersection Theory
17.1 Bivariant Rational Equivalence Classes
Definition 17.1. Let/: X -> Y be a morphism. For each morphism g : Y' -> Y,
form the fibre square „
with induced morphisms as labelled. A bivariant class c in A"(X -4 Y) is a
collection* of homomorphisms
for all g : Y' -> Y, and all k, compatible with proper push-forward, flat pull-
back, and intersection products, i.e.:
(C,) If h : Y" -> Y' is proper, g : Y' -> Y arbitrary, and one forms the fibre
diagram
C)
X" -
X' -
'1
X -
Y,
then, for all a e Ak Y",
cgk) (A* (a)) = hi cfl (a)
inAk-pX'.
(C2) If h: Y" -> Y' is flat of relative dimension n, and g : Y' -> Y is
arbitrary, and one forms the fibre diagram (*,), then, for all a e Ak Y',
in A fY"\ ^Qfi \ CL) = n Cg (Ot)
111 "k + n-P\A )¦
(C3) If g : Y' -> Y, h : Y' -> Z' are morphisms, and /: Z" -> Z' is a regular
imbedding of codimension e, and one forms the fibre diagram
then, for all a e Ak Y',
X
'"[
X
i
X
f"'
f '
Y"
{"
Y' ¦
iS
Y
-iTz"
1'
->r z'
Notation. The group AP{X -* Y) may be denoted simply Ap(X -> Y) or
(/). The homomorphism cik):AkY' -> A.^A" determined by an element c
in Ap(X-> Y) will usually be denoted simply c, with an indication of where it
acts. Since these homomorphisms will be seen to generalize the cap products
of previous chapters, we may also write c n a in place of c(<x) = c(k)(a) for
* To avoid set-theoretical complications, one should work in a fixed universe, cf. Mac Lane A).
17.1 Bivariant Rational Equivalence Classes
Y) orA(X-
321
Y) the direct sum of all A"(X- Y\
aeAkr. Denote by A*(X
pel.
Proper push-forward and flat pull-back are defined on the cycle level. One
case of intersection product is defined on the cycle level, namely, when i is the
imbedding of a principal Carder divisor (Remark 2.3).
Theorem 17.1. Let f:X-* Y be given. Suppose for all g : Y' -» Y, and all k,
there are homomorphisms
cf:ZkY'^Ak_pX'
satisfying the formulas of (C|) and (C2) for all h : Y" -> Y' proper and flat
respectively, and satisfying the formula of (C3) whenever i: Z" —> Z' is the
imbedding of the point {0} in A'. Then the homomorphisms c(gk) pass to rational
equivalence, and the resulting homomorphisms from Ak Y' to Ak-pX' determine a
bivariant class in AP{X -> Y).
Proof. We show first that c passes to rational equivalence. Since c is
compatible with proper push-forward, it suffices (Proposition 1.6) to show that
if Fis a subvariety of Y' x P1, dominant over P1, of dimension k + 1, then
Since, for t = 0, 00, /, is a composite of an imbedding of {0} in A', followed by
an open imbedding of A' in P1 (which is flat),
with c ([ V\) ? Ak+1 _, (A" x P1). But 4 a = 4 a for any ae^fl'xF1) (Exam-
(Example 3.3.6), so c passes to rational equivalence.
It remains to show that c verifies (C3), when / is an arbitrary regular
imbedding. Let N be the normal bundle to Z" in Z', n: N -> Z" the projection,
and let M° — Mz"Z' be the deformation space for deforming Z" ^-> Z' into
Z"^JV(§5.1).Let yv^M^Z'xA'^Z'
be the canonical inclusions and projection. Let NY» be the pull-back of N to
Y", with projection nr: Nr -* Y", and let MT = M° xz, Y'. There is a unique
homomorphism ctso that the diagram
Ak + l(Ny.) —^-> Ak+l (Mr) —-—> Ak+l(Y' x A1) —»0
A) ^\'*l }"'
Ak(Nr) «—;¦ — Ak(Y')
commutes. Then i] = (nf:)'i ° a: AkY'^ Ak_eY". This follows easily from
the construction of i!, as in §6.2: if V <=. Y', j*[M^rF] = pr*[F], and
00 L KM y"' J — t ^ vf) Y"' 1 ~
One has a similar diagram for the base extension X' -> Z' in place of
Y' -> Z', and a similar description for /•': Ak_pX' -> Ak.p^eX". From the fibre
Nx» -> A'k" ~* °o
1 1 1'-
Mr -> M?. -» P1
1 i
x -> y

322 Chapter 17. Bivariant Intersection Theory
one sees as in the first step that c commutes with i^. Since c commutes with
!<»* (by (CJ), and withy* and pr* (by (C2)), c also commutes with a. Since c
commutes with n* (by (C2)), c commutes with i = (n*)~l ° a, as required. ?
Example 17.1.1. A pseudo-divisor D = (L, Z, s) on a scheme X determines
a bivariant class c(D) e A' (Z -> X). If g : X' -> X is a morphism, a 6 ^*(A"),
then c(D) (a) is defined to be g* (D) ¦ a (cf. Definitions 2.2.4 and 2.3). (The main
results of Chap. 2 verify the hypotheses of Theorem 17.1.)
17.2 Operations and Properties
There are three basic operations on the bivariant groups A* (X -> Y).
(Pi) Product. For all morphisms /: X -> Y, g :Y -> Z, and integers p, q,
there is a homomorphism
The image of c <g> d is denoted c • d. Given Z' -» Z, form the fibre diagram
If a e 4tZ', then rf(a) e Ak-q Y', and c(rf(a)) e Ak^q-pX'. We define c ¦ d by
(P2) Push-forward. If /: A' -> K is a proper morphism, g : Y -> Z any mor-
morphism, and/) an integer, there is a homomorphism
ft-.A^X-^Z) -^AP(Y-^ Z).
Given Z' -> Z, form the fibre diagram (**). If c e A"{gf), and a e Ak(Z'),
then c(a) e Ak^p(X'). Since/' is proper,/;(c(a)) e Ak-P(Y'). Define/*(c) by
the formula
(P3) Pull-back. Given/:
r, ^: K, -> K, form the fibre square
For each p there is a homomorphism
Given ce/ (/), Y'->YU a e Ak(Y'), then composing with # gives a mor-
morphism r-> Y. Therefore c(a) e Ak.p(X'), X' = Xxy Y' = Xt xKl r. Set
17.2 Operations and Properties
323
It is very easy to verify that, in these three cases, c ¦ d, /*(c), and g*{c)
determine bivariant classes in the appropriate groups, i.e. that (C,)-(C3) of
Definition 17.1 are verified. The following seven axioms satisfied by these
three operations are similarly straightforward to verify, using the basic
functorial properties of Chaps. 1 and 6.
(A,) Associativity of products. Ifc e A(X' -> Y), d e A(Y -> Z), e e A(Z -* W),
then
(c • d) ¦ e = c ¦ {d ¦ e) e A {X -> W) .
(A2) Functoriality of push-forwards. If f:X-* Y and g.Y^Z are proper,
Z -> W arbitrary, and c e A (X -> W), then -
Y, h : Y2
(A3) Functoriality of pull-backs. If c e A{X -> Y), g : Y,
then
(gh)*(c) = h*g*(c)
X2 = XxYY2.
(A,2) Product and push-forward commute. If /: X -> K is proper, K -» Z,
Z-> Ware arbitrary and c e ^ (X ->¦ Z), rf 6 A (Z ->¦ ff), then
/*(c) ¦</=/,,(c • </) e ^(r-- w).
(A13) Product and pull-back commute. H c e A(X ^ Y), d e A(Y -> Z), and
^ : Zj -> Z is a morphism, form the fibre diagram
Then
= g'*(c)-g*{d)eA(X1 -» Z,).
(A23) Push-forward and pull-back commute. If f: X -> Y is proper, K —> Z,
g : Z, -> Z, and c6^(I->Z) are given, then, with notation as in the preceding
diagram,
(A123) Projection formula. Given a diagram
X'^U Y'
'I I'
X -L+ Y-^Z
with g proper, the square a fibre square, and c e A{X -> Y), de A(Y' ->• Z),
then
c ¦ g*(d) = gi(g*(c) ¦ d) e A(X ^ Z) .

324
Chapter 17. Bivariant Intersection Theory
17.3 Homology and Cohomology
Let S = Spec(K), K the ground field. For each p, there is a canonical homo-
morphism
A-P(X S)A(X)
taking a bivariant class c to
Proposition 17.3.1. The above homomorphisms (p are isomorphisms: A~P(X^> S)
Proof. Given a e AP(X), define a bivariant class i//(a) e A~p(X -> S) as fol-
follows: for any morphism Y -* S, and a e Ak Y, define
y/(a) (a) = a x a e Ap+k(Xxs Y)
where ax a. is the exterior product (§1.19). Since exterior products are
compatible with proper push-forward, flat pull back, and intersections (Propo-
(Proposition 1.10, Example 6.5.2), y/(a) is a bivariant class. Clearly y/(a)([S]) = a, so
<p ° \fi is the identity. To show that i// ° <p is the identity, we must show that
c{a) =
e Ak+p{Xxs Y)
for all a e Ak Y. By compatibility with push-forward, we may assume a = [ V\,
V=Y a variety of dimension k. Then a=p*[S], where p : F-> 5 is the
morphism from Fto S. Since c commutes with flat pull-back,
as required. ?
Definition 17.3. For any scheme X, and any integer p, define the pih
cohomology group APX by
A"X=Ap{X ^ X).
Thus an element c e A'X is a collection of homomorphisms AkX' -> Ak-pX',
for all X' -» X and all k, compatible with proper push-forward, flat pull-back
and intersections ((C|)-(C3)). There is an element 1 e A°X, which acts as the
identity on all AkX', such that
\-c = c, d-\=d
for all c e A {X -> Y), d e A (W -> X).
The product from the composite X -^ X ^ X determines "cup"products
A"X®A"X^A"+iX
which make A*X into an associative, graded ring with unit 1. For any
g : Xx -> X, the pull-back g* : A*X -> A*X\ is a ring homomorphism; this is
functorial inj?.
There are canonical homomorphisms
17.3 Homology and Cohomology 325
taking c®atocna = c (a). If one identifies AqX with A~q(X -> S), this is the
bivariant product from the composite X-^—>X -» S. This "cap" product makes
^i a left^4*X-module. One has the projection formula
for /: Xt -> X, aeA+Xi, PsA*X. (All of the above assertions are formal
consequences of the seven axioms.)
Given a vector bundle ? on a scheme X, and an integer^, there is a Chern
class cp(E) in A"(X). The action of cp(E) on a 6 Ak(X'), g : X' -> X, is defined
where the right side is the class defined in § 3.2. Theorem 3.2 (c), (d) and
Proposition 6.3 amount to saying that cp(E) is a bivariant class. In addition, all
the formal identities proved for Chern classes in § 3.2 are valid for these
classes.
In fact, Chern classes commute with all bivariant classes, not just the three
stated in the definition. Put another way, any operation which commutes with
push-forward, pull-back, and intersections automatically commutes with Chern
classes:
Proposition 17.3.2. Let ceAq{X-^Y), Y'-*Y, a e Ak{Y'), E a vector
bundle on Y'. Then
c(cp(E)n a) = cp(f'*E) n c(a) e Ak.q.pX'.
where f : X' = XxY Y' —* Y' is the morphism induced by f.
Proof. Since Chern classes are polynomials in Segre classes, which come
from operations of the form a ->^»(cl (<^A))' np*(a)) (cf. §3.1), and since c
commutes with p+ and p*, one is reduced to showing that c commutes with
c\ (L), L a line bundle on Y'. We may also assume a = [V], V= Y' a variety, so
L=#(D), D a Carder divisor on V. After replacing Fby V, where V -* Fis
proper and birational, we may assume D=Dl — D2, DUD2 effective (cf.
Theorem 2.4, Case 3). Since c,(L) = C|(^(D,)) - c,(^(ZJ)), we assume D is
effective. Let / be the inclusion of D in V. Then ct (L) n a = i+ /'(a), and since c
commutes with (* and i\ c commutes with c, (L). CD
Example 17.3.1. For X closed in Y, one may define a analogue of local
cohomology by setting
APXY = A'(X-* Y).
If also Z is closed in Y, there are products
A'XY ® A\Y^ AfffziY)
bycu</=i* (c) ¦ d, where i is the inclusion of Z in Y. This product satisfies an
obvious associativity, and refines the cup product on^* Y.
Example 17.3.2. Let X be a scheme, n: X' -> X a proper morphism such
that every irreducible variety in X is the birational image of some subvariety
of A". Then

326
Chapter 17. Bivariant Intersection Theory
is injective. More generally for any/: Y-* X, n* injects A*(Y-*X) into
A*(X' xx Y-> X'). (For any h : V-* X, Fa variety, there is a proper birational
morphism V -> Fso that the composite F' -> F-> X factors through X'. Then
the action of A*X on [V] factors through the action of A*X', and the action of
A*Xon[V] determines the action on [ K] by (C,).)
Example 17.3.3. Let/: X-* Y be a morphism. Let m = dim(T), and let n
be the largest dimension of any fibre/ (y), y e Y. Then
Ap(X^Y) = 0 if p<-n or /; > m .
In particular, for any X,
ApX=0 if p<0 or />>dim(Jf).
(Let c 6 /*>(/), /> < - n or /> > w. It suffices to show c([V\) = 0, h : V-> Y, V
a variety. Restricting to the closure of h(V), we may assume Y is a variety
and h dominant. By the flattening theorem of Raynaud and Gruson AM.5.2,
there is a proper, birational morphism Y' -» Y, and a closed subscheme
V <= VxY Y' such that the induced map g : V -> Y' is flat, and F' -> F is
birational. It suffices by (C,) to show c([F']) = 0. By (C2), c([V']) =
g*(c([Y'])). But c([Y']) is in Am.p(Xxr Y'), which is zero since dim(XxK J")
=s m + n.)
17.4 Orientations
Certain morphisms/: X -> K have naturally determined elements in A(X -> F),
called canonical orientations, and denoted [/]:
A) Iff:X-> Y is flat of relative dimension n, then [/] e/r"(X-> Y) is
defined by flat pull-back. If g : Y' -> K is a morphism, the induced morphism
f':X'-> Y' is flat, and for «6^ Y', we set
[/] (a) =/'¦(«)
with/'* the flat pull-back of § 1.7.
B) If/:JT-> Y is a regular imbedding of codimension d, then [/]6^(X->- K)
is defined by the refined Gysin homomorphism. If g : Y' -> Y, and ae^ Y',
we constructed a class/' (a) inAk^dX' in §6.2. Set
C) More generally, if/: X -> Y\s a l.c.i. morphism which factors
with / a regular imbedding of codimension e, p smooth of relative dimension n,
set d= e - n, the codimension of/, and set
[/] = W • [/>] ¦
17.4 Orientations
327
Here [/] e Ae(X -> P) by B), [p] e A~"(P -> K) by A), so the product is in
Ad(X^Y). Equivalent^, if g : Y' - Y, a e AkY\ then [/](<x) =/!(a), where/1
is the refined Gysin homomorphism of § 6.6.
It follows from Proposition 6.6 that [/] is independent of the factorization,
and that the definitions of A) and C) agree if/ is both flat and l.c.i. If
f:X->Y and g : Y -> Z are both flat, or both regular imbeddings, or both l.c.i.
morphisms with compatible factorizations as in the proof of Proposition
6.6 (c), then .
lf][] lf] m A(X^Z).
Proposition 17.4.1 (Excess intersection formula). Let
X' -
X
Y'
1°
Y
be a fibre square, withfandf l.c.i. morphisms of codimensions dandd'. Then
g*[f} = ce{E)-{f'}eAd{X'^Y'),
where e = d— d', E the excess normal bundle.
If / is a closed imbedding, and N and N' are the normal bundles for / and
/', then E = g'* N/N'. The general definition, and the proof, are given in
Proposition 6.6 (c). ?
Several other compatibilities of these orientations are sketched in Example
17.4.6.
Proposition 17.4.2. Let g : Y -> Z be a smooth morphism of relative dimen-
dimension n, and let [g] e A-"{Y^> Z) be its orientation class. Then for any morphism
f:X-* Y, and any integer p,
is an isomorphism.
Proof. Form the fibre diagram
X —
1
X
Y
1'
X
where 5 is the diagonal imbedding, and p and q are the first and second
projections. Define the inverse homomorphism
L :
A"{X ^ Y)
by L(c) = [y] ¦ g*(c). Note that d and y are regular imbeddings of codimension
n, with f'*[5\ = [y\ The verification that L and multiplication by [g] are in-

328 Chapter 17. Bivariant Intersection Theory
verse isomorphisms is as follows: If c e Ap'"(gf), then
L(c)-[g] = [y]-(g*{c)-\g}) (A,)
= [y] •[/?']-c (C2)
= [/?' ° y] ¦ c = 1 ¦ c = c (A,).
Similarly, if ceAp(f), then
?(c -to]) =f'*[5] -p*(c) ¦ g*[g] (A,3), (A,)
= (/>°<5)*(c)-[<5W<7] (C3)
= c[^«] = rl=c (A,). D
Corollary 17.4 (Poincare duality). Let Y be a smooth, purely n-dimensional
scheme.
(a) The canonical homomorphisms
are isomorphisms.
(b) The ring structure on A* Y is compatible with that defined on At Y in
§8.3. More generally, if f:X -+Y is a morphism, 0 e A* Y, a e A*X, then the
class f*(fi) nie AtX coincides with that constructed in §8.3.
Proof, (a) Apply the proposition to /= 1Y, g : Y -» S, and identify
An-pY with Ap'"(g) (Proposition 17.3.1). (b) follows from the construction of
/*(/?) naasyf (ax/?) in §8.3, with yf the graph of/, and the construction of
the inverse isomorphism given in the proposition. ?
A bivariant class c in AP(X -^ Y) determines Gysin homomorphisms
(G,) c*:AkY->Ak.pX
and, if/is proper,
(G2) c*:AkX^Ak+pY.
Define e*(a) = e(a), <xeAkY, and c, (?)=/*(?¦ c), /3eAk(X); note that
/3-c e Ak+P(X^>Y), so /*(/?¦ c) eAk+p(Y^ Y). Formal properties satisfied
by Gysin homomorphisms are listed in Fulton-MacPherson C) § 2.5.
If/is flat or l.c.i., and [/] is its canonical orientation, we write/* for [/]*,
and/*for[/]*.
Example 17.4.1. Consider the situation of Proposition 17.4.1.
(a) If g is proper, then for all a e AkY'
(b) If/is proper, then for all /? e APX,
in Ap+llY'.
(These formulas, and other similar formulas when g has an orientation
class, follow formally from Proposition 17.4.1 using the axioms for a bivariant
theory (cf. Fulton-MacPherson C) § 9.2.1).)
17.4 Orientations
329
Example 17.4.2. Let a: E -* F be a vector bundle homomorphism on a
scheme X. Let e = rank E,f- rank F, fc^min(e,/). Let Dk(a) be the locus
where rank (a) ^ k. Then there is a class
dk(o) e Ai-W-kHDk(a) ^ X)
whose action on [V], h : K-» X, is
dk(a)([V]) = Dk(h*a),
with Dk(h*a) e At{h'{{Dk(a))) constructed in §14.4. (If s is a section of a
bundle E of rank e, define
where sE: X-* E is the zero section. Then with n:Gd(E) -+X, sa, rj as in
§ 14.4, define
Note that Theorem 14.4(d) is immediate from this description. The push-
forward of 5k(a) by the inclusion Dk(a) -» Xis the class
Example 17.4.3. If X is purely /j-dimensional, /: X -> Y a morphism, with
Y smooth and purely /n-dimensional, there is a class [/] e Am~"(X -» Y) which
corresponds to the element [X] in Am(X) by the isomorphisms of Proposition
17.4.2 (with Z= S) and 17.3.1.
(a) If/is flat, or /is a l.c.i. morphism, then this class [/] agrees with the
classes constructed in A) and C).
(b) If /: X -> Y, g : Y -» Z, all pure dimensional, y, Z smooth, then
[<?/] = [/]¦[<?]•
(c) The Gysin homomorphism
determined by [/] coincides with that defined in Chap. 8.
Example 17.4.4. Assume the ground field has characteristic zero.
(a) Let f,: Xl -> Y and f2: X2 -> Y be morphisms, c,- e /1P'(X,. ^> Y). Then
in ^P'+pj (jsf x x Yx2 -> Y). In particular, for all X, A* X is a commutative ring.
(b) Define /1P(X -> Y)^ as in § 17.1, but using cycle groups and Chow groups
with rational instead of integral coefficients. Show that the canonical map
A"(X -» Y) ® Q -> /1"(X -> Y)<p is an isomorphism.
(Use resolution of singularities, Example 17.3.2, and Corollary 17.4.) We do
not know a proof of these facts without resolution of singularities.
Example 17.4.5. (a) A diagram
Y^Z
y/,

330
Chapter 17. Bivariant Intersection Theory
with/flat of relative dimension n, p smooth of relative dimension m, g proper,
andp g =/ ps= \x, determines a class c in AdX, d=m — n. Indeed
so g* [f] e A'"(Z -^ X), and we may define
Explicitly, if X' -> X, c acts on AkX' as the composite
>
Y
where the primes denote fibre products of the given diagram with X' over X.
If two such diagrams, over the same base X, determine classes c, e Ad'X,
then the fibre product diagram determines the class c, • c2 in Adl+d'(X). (This
follows formally from the bivariant axioms, using the commutativity of the
orientation classes [s] with other bivariant classes.)
(b) Let E, F be vector bundles of ranks e, f on a scheme X, and let
Z = Hom(?, F), u the universal bundle map on Z (§14.4). Then Dk(u) is a
subcone of Z, flat over X. A bundle homomorphism a: E -> F determines a
section /„: X '-> Z. The class constructed from this data by the prescription of
(a) (for Y= Dk{u), g the inclusion,.? = /„) is the polynomial
Al'fZ${c{F-E)) e A^-
Similarly if A is a flag in E, the determinantal loci determine classes given by
the polynomials of § 14.3. (These assertions follow readily from the construc-
constructions, cf. Example 17.4.2. These classes were used in Fulton-Lazarsfeld
CI3 c.)
Example 17.4.6. Let /: X -> Y, g : Y -» Z be morphisms. Assume / and g
have compatible factorizations through smooth morphisms, as in § 6.6. Let
h= gf. Assume that each of/ g, h is either flat or a l.c.i. morphism. Then in
each of the following cases, [h] = [/] ¦ [g]:
i) / g, and therefore h l.c.i.
ii) / g, and therefore h flat.
iii) / h l.c.i., g flat. (When/is a regular imbedding there is a neighbor-
neighborhood o\f{X) on which g is a l.c.i. morphism, by [EGA] IV. 19.1.5.)
iv) / l.c.i., g, h flat. (Proposition 6.5 (a)
v) g l.c.i.,/ h flat. (Replacing Y by an open subscheme, g will be flat, by
[EGA] IV. 2.4.6, 2.2.13.)
vi) /flat, g and h l.c.i. (By Cor. 4 of Avramov A), / must also be l.c.i.)
Is there a class of morphisms, closed under composition, containing flat mor-
morphisms, and l.c.i. morphisms, with orientations [/], compatible with composition?
(See Example 18.3.17.)
Example 17.4.7. Let A <= Y, B <= Y be effective Cartier divisors on a
scheme Y, let D = A + B be the sum, and let /: D -» Y, a : A -> D, b : B -> D be
17.4 Orientations
331
Y).
the inclusions. Then
(i) [i\ = a*[ia] + b*[ib]
(ii) Pclpr(B))-a*lia]
(Vg: Y' -> Y is any morphism, and a is a cycle on Y', then
(i)' {g*D)-a={g*A)-a+{g*B)-cn
(ii)" (g*B)-(g*A) ¦ a =(g*A) -(g*B) ¦ a
by Proposition 2.3 (b) and Corollary 2.4.2. Here g* D, g* A, g*B are the pull-
back pseudo-divisors on Y'.)
Example 17.4.8. Localized top Chern classes are multiplicative, in the
following sense. Let s, be sections of vector bundles E, of ranks e, on X,
defining bivariant classes
(see Examples 17.3.1 and 17.4.2). Let E= ?, © E2,s = st ©52.Then
Example 17.4.9. The canonical homomorphism
L->Cl(L)
need not be injective. (Take X to be a singular curve, and use Example 17.3.2.)
In particular, if X is quasi-projective, the canonical homomorphism
hmA*Y->A*X
need not be injective; the limit is over all X -> Y, Y non-singular quasi-
projective (cf. Example 8.3.13). Mumford G) has considered the image of
lim^* Y in A* X as a cohomology theory with some of the concrete advantages
of the former, and formal properties of the latter.
Example 17.4.10. If X= Y/G is a quotient variety as in Example8.3.12,
then the canonical homomorphism
is an isomorphism of rings. This shows in particular that the ring structure on
A*Xiq is independent of Y, and constructs pull-back ring homomorphism for
arbitrary morphisms of such varieties. (Let n* : A^X^ -> (At y^H be the
isomorphism of Example 1.7.6. If c e A*X^, let c = c n [X]. For V a variety,
/: V->X a morphism, let rj: W-* V be a finite surjective morphism of
varieties, and/: W ->¦ Ka morphism so that nf=ftj. Then
Conversely, given c e A^X^, this formula defines a class in A+V^, indepen-
independent of choice of W\ one may use Theorem 17.1 to show that this construction
determines an element c of A*Xqi.)
Similarly for any /: X' -» X, A(X' -» X)^ ^A+X®. This may be used to
show that the refined intersection products (and hence intersection numbers)
of Example 8.3.12 are independent of the isomorphism X = Y/G.

332
Chapter 17. Bivariant Intersection Theory
17.5 Monoidal Transforms
Let X be a regularly imbedded closed subscheme of a scheme Y. Let/: Y -> Y
be the blow-up of Y along X, and form the fibre square
»1
X
Y
if
Y .
We assume that there is a surjection of a locally free sheaf E on Y onto the
ideal sheaf f{X) of X; such always exists, for example, if Y is imbeddable in
a smooth scheme. Then/factors into a regular closed imbedding in P(EV)
followed by the projection, so / is a factorable l.c.i. morphism, of relative
dimension 0. Therefore/has an orientation class [/] e A°( Y -> Y).
Proposition 17.5 (a) With the above notation
/*[/]=• eA°(Y).
(b) Let h : Y' -» Ybe any morphism, and form the fibre square
(*)
f-lf
n if
Then f*:Af(Y'-*Y)-*Al>(Y'-*Y) is a split monomorphism, with inverse
c^f(c[f])
Proof (cf. Proposition 6.7 (b) and Example 6.7.1).
(a) Let h : Y' -* Y, a e Ak Y'. Forming the fibre square (*) of (b), we must
show that f^ ([/] a) = a. We may assume a = [V], V a variety, and, by covariance,
V = r. Let X' = h~l(X), 2' = %~1{X), and let g' :X' -> X', / : ?' -> f' be
the induced morphisms.
If h(V) <=. X, then by Proposition 17.4.1,
[/] («) = c,.! (F) ¦ M (a) = c,.»(F) • 3'* (a)
with Fthe universal quotient bundle oni= />(W), 3'* the flat pull-back. And
by Example 3.3.3, which proves (a) in this case.
If h{V) * X, let Kc f' be the blow-up of K along X'. Then
for some p e Ak(X')\ an explicit formula for (S is given in Example 6.7.1.
Therefore
/; &{fi = [K],
since g*(S e Ak{X'), and dimX' < dim K= A:.
17.6 Residual Intersection Theorem
(b) This follows from (a) and the identity (Axiom A123)
f*(f*(c)-[f]) = c ¦/,[/]. D
333
Example 17.5.1. (a) If n: E -» Y is a vector bundle,/: X -» y any morphism,
then
n* : AP(X -> Y) ^ A'(f*E ^ E)
is an isomorphism.
(b) With E.fds in (a), r = rank(?),
A"(P(f*E)
In particular, A"(P(E)) ^ ©JlJ ^*"'(y).
(c) Let (*) be a blow-up diagram as in §6.7. Let Y' -» y be any morphism.
Then there are split exact sequences
0 -> /4"-rf(A" -> z) -»/4"-1 (r -»z) e A'(Y' -> Y) -> ^"(r -> ?) ->¦ o
where X' = XxY Y', etc. In particular,
0 -> /4"-rfJf ->¦ ^'-' X ® A" Y ->¦ /4" ? -» 0 .
(The maps and proofs are parallel to those in § 3.3 and § 6.7.)
17.6 Residual Intersection Theorem
Consider a diagram
with the square a fibre square, and i,j, a, b closed imbeddings. Assume
(i) j a imbeds D as a Cartier divisor on Y', and R is the residual scheme to
D in X'.
(ii) / and/ b are regular imbeddings of codimensions d and e, respectively.
Set
c(i/j a) = cd-, ((g a)* Nx Y- (/ a)*0r (D)) ,
c(i/jb; D) = cd-e((gb)*NxY® {jb)*#r{- D) - NR Y') ,
classes in Ad~' D and Ad~'R respectively.
Theorem 17.6. With the preceding notation,
f* [i] = fl, (c(i/ja) ¦ [/'a]) + bt (c (i/jb; D) ¦ \jb])
in Ad(X' -U Y').

334 Chapter 17. Bivariant Intersection Theory
Proof. We first assume that e= 1, so X' is the sum of Cartier divisors D
and R on Y'. Let 5 = c, (j*^r(D)), Q = c[(j*#r{R)) in Al(X'), and let
a = a*[/ a], 0=b*[j b] in A1 (X' i F). Then (Example 17.4.7)
(i) [/] = a + /ff
(ii) Q-ct = 5(S.
From (ii) it follows by induction on 9 that
^ /a+ll
(iii) F + qY ¦ (a + /?) = 8q ¦ a + >
Let cp = cpC*^7). By Proposition 17.4.1,
(iv) f*[i] = cd-l(g*NxY-j*^r.(D + R)) ¦ [/]
p+q=d-\
= S (-l)?c^-a+ Z (-'
Writing out c (i/j a), one has
(v) a*
Similarly, and using Example 3.2.2,
(vi) bt(c(i/jb;D)-yb])=
= I
s
Comparing (iv), (v), and (vi) gives the required equation.
If e > 1, let n: Y' -» Y' be the blow-up of y along ?. Put a ~ over the
symbols for subschemes of Y', and morphisms between them, to denote their
inverse images, and induced morphisms, in Y'. By Proposition 17.5, it suffices
to show that both sides of the equation of the theorem become equal after
applying it*. Thus the case e > 1 is reduced to the case e = 1, once it is verified
that the three terms in the equation pull back to the corresponding three terms
in Ad(j). This is obvious for the first two. For the third, let r\: R -» R be the
induced morphism. Let
Since NR Y' is a sub-bundle of E (Example 9.2.2), and Ng Y' is a sub-bundle of
rj*NR Y', the Whitney sum formula gives
(vii) q_ , (r,* E/Ng Y') = e,,-, G* El if NR Y') ce., (r,* NR Y'INg ?').
By Proposition 17.4.1,
(viii) n* [/' b] = c,_, G* NR Y'INg ?') ¦ \fb].
From (vii) and (viii),
n* (c (ilj b; D) ¦ [/ b]) = c (i/jb; D) \jb],
which concludes the proof, if e > 1.
17.6 Residual Intersection Theorem
335
Finally, if e = 0, i.e. R = X' = Y', the required equation amounts to the
identity
cd{N) = cd.x{N - L) c, (L) + cd{N ® Lv)
in AdX', N = g*NxY, L=0Y(B). This is a simple formal calculation, using
Example 3.2.2. ?
This residual intersection theorem implies many of the previous intersec-
intersection formulas. Besides those in Chap. 9, it also implies such basic facts as the
functoriality theorem of Chap. 6 (cf. Example 17.6.3). Generalizations are
given in the examples.
Example 17.6.1. Consider the situation of Theorem 17.6.
(a) If/is proper, and a e Ak Y', then
'¦*/• («) =(9 a)* (c(}lj a) (/ a)*<x)+ (g b)*{c(i/jb; D) (j b)*a)
(b) For
mA"+dY'.
a)* 0)
i/j b; D)(g b)*fi)
Example 17.6.2 (Kleiman A2K.6). There is a useful generalization of the
residual intersection theorem. Consider a diagram labelled as in this section,
with the square a fibre square, and a a closed imbedding. Assume
(i) R = Proj(Sym(j*")), where j*" is the ideal sheaf of D in X', and
b: R -» X' is the projection.
(ii) i and; b are l.c.i. morphisms of codimension d.
(iii) j a is a l.c.i. morphism of codimension d'.
Assume also that all schemes appearing can be imbedded in smooth schemes
(weaker assumptions suffice to factor appropriate maps through smooth
maps). Then
f*li]
mAd(X'^Y'), where
c(ilj a) = ctsdg a)*Ni- Nja),
Nj and Nja the virtual normal bundles to i and ja. (Factor / into a closed
imbedding X^ P followed by a smooth morphism P -> Y. Let P' = PxrY'.
Let F be the blow-up of F along D. Then R is the residual scheme to the
exceptional divisor in J5', and Theorem 17.6 applies.)
Example 17.6.3. The residual intersection theorem can be used to prove the
functoriality theorem of § 6.5: If/: X -> Y, g :Y -» Z are regular imbeddings,
then [gf] = [/] • [g]. (Let n:Z^Z be the blo_w-up_ of Z along X, with
exceptional divisor X. The residual scheme R to X in Y = it~*(Y) is the blow-
blowup of Y along X (Appendix B 6.10). It suffices to show that n*[gf] =
V*[f]' n*[g], where tj: Y -> Y is the induced morphism. Use Theorem 17.6 to
calculate n* [g], and Proposition 17.4.1 for7r*[^/].)

336
Chapter 17. Bivariant Intersection Theory
Example 17.6.4. There is a useful generalization of the orientation class of a
regular imbedding. If /: X -» Y is a closed imbedding, and
CxYcE
is a closed imbedding of the normal cone in a vector bundle E of rank e on X,
these data determine a bivariant class c in /le (AT -?• y). Given j:r-»l', let
X' = cT'pO. h:X' ^X induced. Then
CrY'<=X'xx(CxY)<=h*E.
Define cf to be the composite
Ak r -i ^(Cr y) -^i ^(/,* ?) -^t_,.r
where a is the specialization map (§ 5.2), sE the zero section of E. Alternatively,
(That (Ci) and (C2) are satisfied follows from Proposition 4.2; (C3) is deduced
as in Theorem 17.1.)
(a) If/is a regular imbedding of codimension d, then
c = c,-d(E/NxY)-[f].
(b) If g : y -» y is a morphism, then g* (c) is the class determined by the
canonical imbedding of Cx. Y' in h* E.
Example 17.6.5. Consider a residual intersection diagram as in this section,
but with no assumptions on the imbedding; b of R in Y'. With E = (gb)* NXY
® 0 b)*0r (— D), there is a canonical imbedding of CR Y' in E (Example 9.2.2).
By the preceding example, this determines a bivariant class
reAd(R^*Y')
Then
Wheny b is a regular imbedding, this is equivalent to Theorem 17.6. (To show
both sides have the same effect on [V\, h: V-* Y' any morphism, apply
Theorem 9.2 if h{V)c?D. If h{V)<tR, let V be the blow-up of V along
h~'(R), Y' the blow-up of Y' along R. It suffices to show both sides have the
same effect on [V\; but V maps to Y', and the case e= 1 of Theorem 17.6
applies. Finally if h (V) c: D fl i?, it is a formal calculation, as in the case e = 0 of
Theorem 17.6.)
Example 17.6.6. Consider a diagram
'1
X
Y'
I'
Y
with the square a fibre square, a a closed imbedding. Assume / and j a are
regular imbeddings of codimension d and d'. Assume Y' can be imbedded in a
smooth scheme. Let R be the residual scheme to D in X', i.e., the ideal sheaf of
Notes and References
337
R in X' is the annihilator of the ideal sheaf of D in X'. Let b be the inclusion of
R in X'. Define
where iV, and A^ are the normal bundles to i and/a. There is a canonically
defined class r in Ad(R -^ Y') with
in Ad(X' -» y'). (Let nj. Y' ->¦ y be the_ blow-up of y along D, with
exceptional divisor D, X' = n~](X'). Let i? be the residual scheme to D i
X', rf.R
ieA"{R-
in
> R the induced morphism. In the previous example a class
F') was constructed. Then [rc] e /1° (F' -> y'), and one may set
Notes and References
The source and reference for most of this chapter is Fulton-MacPherson C).
There the reader can find examples of the utility of bivariant theories in areas
other than intersection theory or algebraic geometry.
Previously a cohomology theory had been developed for quasi-projective
schemes (cf. Example 8.3.13), which was adequate for formulating and
extending Grothendieck's Riemann-Roch theorem to singular quasi-projective
(Baum-Fulton-MacPherson A)). However, this cohomology theory lacks many
of the formal properties one would like, such as a Gysin push-forward for
proper l.c.i. morphisms. The coarser cohomology theory associated with the
operational bivariant theory (§ 17.3) has these formal properties, and may be
used on arbitrary algebraic schemes. For a recent application see Mumford
G).
Motivation for developing a pair of theories, both "homology" and "co-
"cohomology" came primarily from topology. For a time most topologists had
regarded cohomology as the proper object of study on singular spaces, superior
to homology — just as most algebraic geometers regarded the "Cartier divisor"
as a replacement for the notion of a "Weil divisor". Since Sullivan, MacPher-
son and others discovered that important invariants of singular spaces could lie
in homology instead of cohomology, a more balanced view has been achieved.
More recently we have seen that these two theories are not rich enough for
the study of singular varieties. Bivariant theory is one extension of homology-
cohomology, particularly useful for functorial and formal properties. Another
theory, called intersection homology, has been developed by Goresky and
MacPherson A), B). This had led to a deep insight into the geometry of
singular spaces. The functorial properties of intersection homology, and the
place of algebraic cycles in the theory, are not yet clear, however.

338
Chapter 17. Bivariant Intersection Theory
I have had useful conversations with D. Gabber and S. Kleiman on some
topics in this chapter, particularly related to the still unsettled question of
which maps have orientations for rational equivalence theory (cf. Example
17.4.6). Kleiman A2) has developed some variations on bivariant theory, and
a generalization of the residual intersection theorem (cf. Example 17.6.2).
The proof of Proposition 17.5 corrects an error in Fulton-MacPherson
C)9.2.2.
Chapter 18. Riemann-Roch for Singular Varieties
Summary
The basic tool for a general Riemann-Roch theorem is MacPherson's graph
construction, applied to a complex E. of vector bundles on a scheme Y, exact
off a closed subset X. This produces a localized Chern character1 chx(E.)
which lives in the bivariant group A(X^> Y)q. For each class a eA*Y, this
gives a class
chx(E.) not eA*Xv
whose image in A* Yq is X(-l)''ch(/ij) n a. The properties needed for Rie-
Riemann-Roch, in particular the invariance under rational deformation, follow
from the bivariant nature of chx(E.).
The general Riemann-Roch theorem constructs homomorphisms
xx: KoX -* A+Xq,
covariant for proper morphisms, such that xx(fi ® a) = ch (/?) n xx(a) for
fi e K°X, aeKoX. If X is imbedded in a non-singular variety M, and a
coherent sheafT is resolved by a complex of vector bundles E. on M, then
where Td (M) = td (TM) n [M]. Such xx is constructed for quasi-projective
schemes in the second section. The extension to arbitrary algebraic schemes,
using Chow's lemma, is carried out in the last section. As a corollary one has
the GRR formula
/, (ch(oc) • tdG») = cht/ioc) • td G»,
for f:X-> Y proper. X, Y arbitrary non-singular varieties, aeK°X. In the
singular case, there are refinements for /: X -> Ya l.c.i. morphism.
1 In topology, the Chern character of such a complex lives in H*(Y, Y—X; <Q); capping with
this class determines homomorphisms from Hm(Y;<S)) to Ht(X\<Q). The bivariant class
chjf (E.) is an analogue for rational equivalence.

340 Chapter 18. Riemann-Roch for Singular Varieties
18.1 Graph Construction
Let X be a closed subscheme of a scheme Y, and let E. be a complex of vector
bundles on Y which is exact off X:
„ g.+ l
Em-\ — 0 ,
with rf, homomorphisms of locally free sheaves, d, ° di+\ = 0, and Im (di+l) =
rf,) on Y-X. Let e, = rank?,. Let G, = Grasse, (E, @ ?,_,), and set
A) G=GnXyGn-iXy...XyGm.
Let ?,• be the tautological bundle of rank e, on G,, and set
n
pr,:G -> G, the projection. For each y e Y, and each X in the ground field K.
the graph of Xdiiy) is an e,-dimensional subspace r{Xdt{y)) of ?,(>') ©.Ez-iCv).
This determines a morphism
C) yx A' -» G,¦, (>>, A) -¦ T (A rf, (>-))
such that the pull-back of ?,• is the graph of Arf((>") at (>», A) e yxA1. This
determines a closed imbedding
(To make this pointwise description scheme-theoretic, see Example 18.1.1.)
Define integers kt by setting k, = 0 if i: ^ «, and requiring that
E) fc,+ ?,•_, = e,
for all /. Thus /t, = e,+, - ei+2 +... ±en. We assume that these kt are non-
negative. This is the case if E. is exact at any point y e Y (e.g. if Y—X is non-
nonempty), for then k, = dim (Ker (rf, (>-))). Let H, = Grass*., (?,¦), and set
F) H=HnxrHn-lx...xrHm.
On Y-X, Ker(rf,) is a subbundle of E, of rank &,. This determines a section of
//, on y—Jf, and hence a closed imbedding
G) y-jr^//°,
where H" is the restriction of // over y—JT. There is a canonical closed
imbedding
(8) H+G
which takes a collection of A:,-planes Z,, in E, to the collection of e,-planes
Lj® /,,_, in ?,© Ej-\. We identify // as a closed subscheme of G via i. Note
that
(9)
= 0 in
18.1 Graph Construction 341
Indeed, if C is the tautological bundle on //,,
n
•*@ = X (- l)'(pr? (C) + prf-i (C-,)) = 0,
i=m
where pr, is the projection from H to //,.
The inclusions in D), G), and (8) combine to form a fundamental (non-
commutative!) diagram
yxA' v-—
A0) U
(Y-X)xAlc(Y-X)xPl
J.xl
¦//°xlPlc://xlPl
Let n : G -> Y be the projection, Gx = n 1 (X), and let »;: Gx -» X be the
restriction. Given a /c-dimensional subvariety V of y, the localized Chern character
A1) chK?.)n[K]e^XQ
may be defined as follows. Let Wbe the closure of <p(Vx A1). Let
A2) Z. =
where ico:G=6x{oo}c»GxPl. Then Zx is a A:-cycle on G. One shows
(Lemma 18.1) that one irreducible component of Z^ is a variety V projecting
birationally onto V, and that
A3) Z=ZCO-[K]
is a A:-cycle on Gx. Set
A4) chK?.)nm = 7«(ch@n[Z]).
It is useful, however, to have a definition with more flexibility. Given a
cycle a on y, let a ° denote the restriction of a to Y— X.
(i) Choose a cycle a'onGx P1 which restricts to ^ (a x [A1]) on G x A.1,
(ii) Choose a cycle a" on //xP1 which restricts to t//t(a°)x[P'] on
//"xlP1.
Then a' — a" is a cycle on G x P1. Set
A5) y = i*(o'-a")eZ,(G).
Note that y is well-defined as a cyc/e on G since the normal bundle to ix is
trivial (Remark 2.3). In other words, y is the specialization cycle (a' — a")<»
discussed in § 10.1.
Lemma 18.1. (a) y is a cycle on Gx-
(b) Another choice of a' does not change y.
(c) Another choice of a" changes y to y + P, where {1 is a cycle on
HX = GXC\H.
Proof. Another choice of a' is of the form a'-t-g.ga cycle onGx(oo). Then
'oo (s) = 0, which proves (b). Another choice of a" is of the form a" + q, q a
cycle on Hxx P1. Then zj, (g) is a cycle on 7/^-, which proves (c).

342
Chapter 18. Riemann-Roch for Singular Varieties
To prove (a), since constructions of y commute with restrictions to open
subschemes, we may assume X = 0, in which case we must show, for some
choices of a' and a", that y = 0. Thus we may assume E. is exact on all of Y.
Let K, = Ker(rf,). Define a rank e, subbundle L, of the pull-back of ?, © ?,_i
to Yx P1, whose fibre over (y, (Ao: X\)) is
{(v,, v,..,) e E,(y) © KHt(y) \ Xo i>,-_, = A, 4 t>,-}.
This determines morphisms 7x P1 -> G,, and hence a closed imbedding
p:yxP' -» GxP1.
Away from oo =@:1), L-, is the graph of Xdh so <j> extends the imbedding q> of
D). At oo, Li = K-, © Ki-i, so, over oo, tj> is the imbedding i° y/of G) and (8).
Now given a on Y, let a' = f,(tx [P1]), and let a" = y/* (a) x [P1]. Then
Definition 18.1. Given ?. on y, exact off X, and a cycle a on y, define
A6)
in ^JTq. Here y is a cycle on G* defined by A5) and Lemma 18.1 (a), 7
projects Gx to X, and ? is the virtual bundle defined in B). By Lemma 18.1,
y is unique up to adding a cycle in Hx. Since ?, restricts to 0 on H, ch (?,) n y is
independent of choice of y, so chx(E.) n a is well-defined.
More generally, if g : Y' -» y is an arbitrary morphism, set X' = g~l (X), and
define, for aeZ, y,
A7) chx(E.) n a = ch?@*?.) n a e,4*A^.
Let 4(X -» y)<p= ©^(X -> y),Q be the bivariant group defined in §17.1,
but using cycles and cycle classes with rational coefficients (see Example 17.4.4
Theorem 18.1. The operation tx -» chx(E.) n a determines a bivariant class,
denotedchx(E.), inA(X^ y)Q.
Proof. We show that the operation defined by A7) satisfies the conditions
of Theorem 17.1. To show the commutativity with proper push-forward, it
suffices to show that if g : y'-> Y is proper, and h :X'-* X is the induced
morphism, and a is a cycle on Y', then
h*(ch%(g*E.) na) = chx(E.) ng*a
in A^Xn). Note that the fundamental diagram for g*E. is obtained from the
fundamental diagram A0) for E. by the base extension g :Y'-* Y. To avoid
cumbersome notation, we use gt and g* for push-forward and pull-back for all
morphisms (except h) induced by this base extension. If a' and a" are choices
satisfying (i) and (ii) for a, then g* a' and g* a" are choices for g* a which
satisfy (i) and (ii), since push-forward commutes with restriction. Similarly
if y=/S,(a'—a"), since g* commutes with i% (Theorem 6.2(a), cf. Proposi-
Proposition 1.4),
() Z('")
18.1 Graph Construction
343
is a valid choice for the y-cycle for g* a. Therefore
chx(E.) nj,a=i;, (ch (?) ng*y) = 7* g* (ch (g* ?,) n y)
= K r,'* (ch (g*i)ny) = h* ch/-' (g* (E.) n a).
Here rf: GX' -> X' is the map induced by rj.
The proof that chx(E.) commutes with flat pullback is entirely analogous,
and left to the reader.
Finally, let D <= y be a principal divisor, i.e., D = /r1 @) for some
morphism h: Y-> A1. Let j be the imbedding of 0 in A1. To complete the
proof, we must show that
f(chx(E.) n a) = chgn^(?.|D) nfct
in A^DCiX)^. The proof is again similar. If a' and a" are choices for a, and
y=i*,(a'- a"), then;'a' and/a" are cycles on GDxP' and HDxP[ which
satisfy (i) and (ii) with respect to the cycle;'(a); again, this follows from the
fact that/ commutes with restriction to open subschemes. Therefore
4./V - «") = 4/(«') - 4/(«")
is a valid choice for the y-cycle for /(a). Now by the fundamental com-
commutativity for divisors (Theorem 6.4),
4/(«'-«")=/»»(«'-«") =/(}')•
Therefore*, iff;' denotes the projection from GDf)X to DP\X,
chEnx(E.\D) nf a = 1,; (ch«|Ooni) nf y)
= >? j'¦ (ch @ n y) = / n*(ch (Q ny)=/ (ch? (?.) n a). ?
In particular, the homomorphism a -» ch^f.) n a passes to rational
equivalence, defining
A8) chHE-)n-:AtYq^A,Xq.
In addition, if g : Y' ->¦ Y is a regular imbedding or a l.c.i. morphism, and
A" = g~' (X), then, for all aeA*Y,
A9) »'(chf (?.) n a) = chj?to*?.) ng*a
Corollary 18.1.1 (Homotopy). Let X<= Y, and let E. be a complex of vector
bundles on YxA\ exact offXxAK For each rational point t e A1, let E., be the
induced complex on Y=Yx {/}. Then for any a e At Y,
chjf (?".,) n a = chx(E.o) n a e A*XV.
Proof. chx(E.,) na = i*(chrxlAA>(E.) n (ax [A1])) by A9), where i, imbeds
* at /. Since all homomorphisms
are the same (Corollary 6.5), the corollary follows. D
• Theorem 6.4 gives an equation of cycle classes in the intersection of the support of a' — at" and
the supports of the divisors, which is only in GD(\X u HD. However, the two sides are cycles
on GDc\ x,so this implies that they differ, up to rational equivalence, by a cycle on HDf] x. Such
a cycle can be ignored in the following calculation, since ? is trivial on HD(]X.

344 Chapter 18. Riemann-Roch for Singular Varieties
Proposition 18.1. Let i be the inclusion of X in Y.
(a) For all a e A* Y,
i, (chf (?.) n«)=t(- l)'ch (?,¦) n a
(b) Let 0 -> Fi1' -» F.B> -» F.C> -» 0 be an exact sequence of complexes of
vector bundles on Y, each exact offX. Then
(c) Z,e/ F be a vector bundle on Y. Then
?.) = ch(/*F) ¦ ch?(?.).
Proof, (a) Since the localized Chern character commutes with /,, we may
assume X— Y. In the choice of a' and a", we may take a" = 0. Then
ch(?) n y=ch(?) n /&«' = ch(?) n /Ja',
where /o:G=Gx@j^GxPl (cf. Example 2.6.6). If po: y-> G is the restric-
restriction of cp to y=yx{0}, then /$ a'= #>0* (a) by condition (i) determining a'.
Since #>0 is the section corresponding to the complex with zero boundary maps,
the graph of Qd, is Et © 0, so ?,• pulls back to ?,; therefore pj ? = X (- 1)' [•?.¦]¦
Hence ch @ n iJ a' = ?0* (Z (- 1)'' ch (?,-) n a).
Applying 77* yields A), since 7j(po = idY.
(b) We first deform Ei2) into the direct sum ?.(l> © ?.<3>. By Corollary 18.1.1,
it will then suffice to consider this split case. Let /? be the given map from
?.B> to EP\ Define a family of vector bundle surjections
h,: ?<2> © ?.<3> -» ?.<3>
parametrized by / e A1, by h,(v2, v3) = 0(v2) - t v3. Then Ker(/!0) = ?}'> © ??V
andKer(/!,) = ?<2>.
In the split case, denote by superscripts @ the spaces, bundles, and maps
constructed for EP, i= 1, 2, 3. There is a canonical imbedding of GA> xr GC> in
G(T> such that ?,(T> restricts to p* ?A> + q* <^C>, where p and q are the projections
from G"> xr G0) to G(l> and GC>. The imbeddings <p(l~> factor
G"> x P1
y
'^ G<3> x P1.
There are analogous factorings for the H(l) and t//(l). Therefore one may choose
the cycles a', a" for Ei2) to be cycles on GA> xr GC) x P1. Then the images of a',
a" by (/>xl), and (<?xl), are legitimate choices for Ei[) and ?.C>. Note also
that nB) = ni{) ° p = nC) ° q. Therefore if y — /J, (a' — a"), then
n a = rf$ (ch (p
q* ?<3>) n y)
n J7» y)
n a + chf (?i3)) n a .
n
y)
18.1 Graph Construction
345
(c) If G is constructed for E., G. for F ® ?., there is a canonical imbedding
of G in G so that if restricts to F ® ?. The proof concludes as in (b) (cf. Baum-
Fulton-MacPherson AI1.2.3). D
Corollary 18.1.2. Let i:X-+Y,j:Y^>Zbe closed imbeddings, where j is a
regular imbedding with normal bundle N. Let y be a coherent sheaf on X, and
supposed has a resolution by a finite complex of locally free sheaves E. {resp. F.)
on Y(resp. Z). Then
Proof. Note that by (b) of the proposition, chj^.E.) and chf(.F.) are inde-
independent of the resolutions. We must show that both sides of the displayed
equation have the same effect on a, for any cycle a on any scheme V, g : V -* Z
any morphism.
Assume first that Z= P(N © 1), 7 is the imbedding by the zero section, and
V=Z. Let p be the bundle projection from Z to Y, and assume also that
a =/?*/? forsome/? eA*Y. Let gbe the universal quotient bundle on P(N © 1).
Then (cf. §15.1) A'Qv®p*E. is a resolution of & on Z. Let
X = p~[(X), q:X -»X the projection induced by p. By compatibility of local
Chern character and push-forward, since; imbeds X in X,
A)
chf (
a = <j,(chf( t\Q< ® p*E.) n a).
Since p*E. is exact off AT, KQV ®p*?. is homotopic, as a complex on Z exact
off X, to the complex with the same vector bundles, but using the zero
boundary map on A'QV. Let Q be the restriction of Q to X. By Corollary 18.1.1
and Proposition 18.1 (c),
B) chf (KQV <g>p*E.) na = ch(A'Sv) n (chf (p*?.) n a).
Let k be the zero section imbedding of X in R. Since ch(A"?>v) =
Cd(Q)-td(Q)-\ d= rankN, and cd(Q) ¦ C= k* (k* Q for all C, and k*Q=i*N,
B) can be rewritten:
C) chf(A'Sv ®p*E.)nix = kt(td(i*N)-1 n k* q* (ch?(?.) n /?)),
using the commutativity of local Chern character and flat pull-back. Applying
q*, and noting that q k = \dx, A) and C) yield
D) chf (A'SV ®p*E.) na = id(/*#)-' n (ch|(?.) n ^),
which is the required equation in this case.
For the general case, deform the imbedding j into the imbedding in the
normal bundle N, by the construction of § 5.1:
J:YxP'^> MyZ.
Let pr: Y x P1 -> Y be the projection, and let F. be a resolution of pr*J* on
MYZ (which exists since J is a regular imbedding). Let i'o and im be the
imbeddings of Z and P(N®1) in M,,Z at 0 and 00. Then jj R (resp. i* F.) is
a resolution of J5' on Z (resp. P(N® 1)).

346
Chapter 18. Riemann-Roch for Singular Varieties
We may assume V is a variety and a = [ V\. Let W' = g ' (Y), h.W -» Y the
induced morphism, and consider the induced morphism MWV^ MYZ of
deformation spaces. Then, since local Chern character commutes with special-
specialization,
E) chf (K)na = 4(ch#f.(F.) n[MwV}).
As in the proof of Corollary 18.1.1, we may replace ib by 4> on the right side of
E), which yields:
F) chf (F.) n a = ch?<"®'>(/* /?.) n [P(C^© 1)].
By construction, [/>(CH,J/© 1)] = p*/[F], where p is the projection from
P(h*N®l) to W. Since i*F. can be replaced by A'gv®?., the required
equation
chf (F.) n a = td (/*#)"' n (chf (?.) n / a)
follows from F) and D). D
Example 18.1.1. For any vector bundles E, F of ranks e, f on a scheme y,
there is an open imbedding
A)
Hom (E, F) -* Grass, (E © F)
which, over geometric points y e Y, takes a{y) : E (y) -> F(>») to the graph of
crCy). (Indeed, if Y is affine, Hom(?, F) is one of the basic sets in an affine
open covering of the Grassman bundle (cf. [EGA] 1.9.7).
If a: E -* Fis a vector bundle homomorphism, there is a homomorphism
B)
Yx A1 -> Hom (E, F)
which, on geometric points, takes (y, X) to Xa(y). (Let sa be the section of
Hom (E, F) corresponding to a. Then
yxA'^»Hom(?, F)xA' -* Hom (E, F),
where /i is scalar multiplication on the bundle Hom (E, F).) The composite of
B) and A):
C)
Grass, (E ® F)
takes the geometric point (y, X) to the graph of A o(y).
The other constructions of this section extend similarly to arbitrary
schemes.
Example 18.1.2. Let E. be a complex of vector bundles on Y, exact off X.
Let E.[p] be the complex obtained by translating E. p steps to the left:
(E[p]),¦ = ?,._,. Then
(a)
Let ?7 be the dual complex: (?/)„= ?_„, with «lh boundary map t/_vn+,. Let
ch,(?.) denote the component of chf(?.) in^A'-*- y)^. Then
(b) ch,(?.v) = (-l)'ch,(?.)-
18.1 Graph Construction
347
(For (a), the geometry of the construction is the same for E. and for E. [/»]; only
(, changes to (—l)p?,. For (b), using the duality of Grassmannians (Example
14.6.5), the Grassmann bundles may also be identified, and ? corresponds to
Example 18.1.3. With the same notation as in this section, define
and define
Then for all i > 0, c{ (?.) determines an element in A' (X -» Y), called the fh
localized Chern class of ?.. There are analogues of Proposition 18.1 and
Corollary 18.1.2, "without denominators", for these classes.
Example 18.1.4. Let E., F. be complexes of vector bundles on Y, both exact
off X. Let a: E. -> F. be a quasi-isomorphism, i.e., a is a homomorphism of
complexes which induces an isomorphism of homology sheaves. Then
and similarly for local Chern classes. (There is an exact sequence
where G. is the mapping cone, which is a complex exact on all of Y.)
Example 18.1.5. Let E. and F. be complexes of vector bundles on Y, exact off
X and Z respectively. Then E.®F. is exact off Xf]Z, and, with the notation of
Example 17.3.1,
chjnz(?.® F.) = chf (?.) u ch|(F.) .
(It would be interesting to find a direct proof of this multiplicativity property,
as in Proposition 18.1; in topology this has been done by Iversen C). The
formula may be deduced from the formula of Example 18.3.12, the surjectivity
of the Riemann-Roch map xY, and the spectral sequence
3fp+q(E.® F. ®a?)
Sfp(E. ®#"„(F.)
for a coherent sheaf 3 on Y.)
Example 18.1.6. MacPherson's graph construction for vector bundle homo-
morphisms. Let a: E -> F be a homomorphism of vector bundles of ranks e, f
on a variety Y. Let G = Grass, (E © F)» with universal bundle ? of rank e on G.
There is a canonical imbedding
be the closure of the image of
taking (y, X) to (graph of X o(y), A:X)). Let
<p, and let
an rt-cycle on G, « = dim (Y). Let 7,:
tion.
y be the maps induced by projec-

projec348 Chapter 18. Riemann-Roch for Singular Varieties
(a) For any characteristic class cl (i.e., polynomial in Chern classes),
(b) If a has rank k off a proper closed subset X of Y, then there is one
component [Y] appearing in Zx with multiplicity one, such that Y projects
birationally to Y, and Y lies in
//= Grass,_ * (?)xy Grass* (F) ^ G.
The other components Vt all project into X.
(c) If k = e =/, then Y = Y, and
cl(E)n[Y]-cl(F)n[Y]= ? n, %(cl(Q n [Vt]).
K-> X
This is a special case of the construction of this section for the two-term
complex E -> F.
(d) If E is a trivial line bundle, so a is a section of F, and X = Z(s), then
),//=P(F),and
where Y is the blow-up of Y along A', and C is the normal cone to X in K.
(e) If a is sufficiently generic, then Zx = X [K], where V-, projects onto the
locus where rank a S i. In general, Zx contains much of the usable information
about how a degenerates.
@ If /: Y\ ->• K2 is a morphism of non-singular varieties, the graph con-
construction may be applied to df:TYl->f*TY2. The components of Zx will lie
over various singularity loci of/ For example, if K, is the blow-up of Y2 along
a smooth subvariety X2, and X{ is the exceptional divisor, then
where L=0N{\) on Xl = P(N), N=NXiY2. The formula for blowing up
Chern classes (§ 15.4) may also be derived from this.
On the other hand, if / is a branched covering, the graph construction
produces a decomposition
for a class /?on the ramification locus of/
Example 18.1.7. In the situation of Corollary 18.1.1, the classes ch|(?.,) e
7)^ are independent of / e A1. (See Example 17.5.1.)
18.2 Riemann-Roch for Quasi-projective Schemes
In this section we work in the category of schemes X which are quasi-projec-
quasi-projective over a fixed non-singular base variety S. Such schemes are equipped with
a morphism to S, which factors
X-U. ?/-i p(E) ^ S,
18.2 Riemann-Roch for Quasi-Projective Schemes
349
with / a closed imbedding, j an open imbedding, E a vector bundle on S, p the
projection. The case S = Spec (K), K the ground field is paramount; the extra
generality will be used in the extension to the non-projective case (§ 18.3).
If M is smooth over S, let 7V/s denote the relative tangent bundle of M
over 5, and set
For any closed imbedding i:X->M, M smooth over S, and any coherent
sheaf y on X, set
where E. is any resolution of ;'* (j*") by locally free sheaves ?; on M. It follows
from Proposition 18.1 (b) that this class is independent of choice of resol ution,
and that chjf determines a homomorphism
Define a homomorphism
by the formula
where Td (M) = td (TM/S) n [M]. Identifying A(X-*M)tl with A.X^, this
may also be written r#(a) = td {Tws) n (ch5f (?.) n [M]).
Theorem 18.2. The homomorphism
M. Denote by x or xx,
is independent of the imbedding ofX in
t • K Y —^ A Y
the homomorphism xx for any such imbedding. Then the following properties
hold:
A) (Covariance). Iff:X-* Y is proper, then
V Y *' . A V
commutes.
B) (Module). IfaeKoX,/]EKoX, then
C) (Local complete intersections). If f:X-> Y is a l.c.i. morphism, with
virtual tangent bundle Tf, then the diagrams
.l |
/.<tdG»-J
''I I
td(T,)f*
commute. For the first diagram, / is assumed proper. The Gysin homomor-
phisms/,, /* are those constructed in § 6.2 and Example 15.1.8. The virtual
bundle 7}is defined in Appendix B.7.6.

350
Chapter 18. Riemann-Roch for Singular Varieties
Proof. The proof is divided into ten steps.
Step 1. If i: X -+ M is a closed imbedding, and j : M -> P an open or closed
imbedding, with Af and P smooth over S, then xx = rf. This follows from the
definition if 7 is an open imbedding, and from Corollary 18.1.2 if 7 is a closed
imbedding.
Step 2. If i: X -> M is a closed imbedding, 0 e K°M, a e KOM, then
This follows from Proposition 18.1 (c).
Step3. If f:X-*Y is a closed imbedding, and j:Y->M is a closed
imbedding, M smooth over S, then /„, x% (a) = x" (/»a) for all aeKoX. This
is a consequence of the construction of rjf.
Step 4. Let /: K-> M be a closed imbedding, M smooth over S, and let
/>: P -> Mbe a smooth morphism. Form the fibre square
Thenforallae/JToK,
X-
'I
Y-
I'
¦ M
xxf*a = td(Tf)-f*xMY(a).
This follows from the commutativity of local Chern character with flat
pullback (Theorem 18.1).
Step 5. With notation as in Step 4, assume P = P (E), E a vector bundle on
M, p the projection. Then for all a. e KOX,
W* «)=/**?(«)•
It suffices to prove this for a = fi(n) ® /* /?, /? e Ko Y, since any class in KOX is
a sum of such classes (cf. Example 15.1.1). Then
/* rjf (a) = /* (ch (*(«)) • tdG» n /*
by Steps 2 and 4. By Example 15.2.17, since td Q}) = td (/* (E\ Y) ® ^ A)),
this may be rewritten:
/„ Tjf(a) = i* ch (Sym"?v) n
On the other hand, /* a = Sym" (;* Ev) ® yS, so
^(Z, a) = /* ch (Sym"?v) n
by Step 2 again.
Step 6. tx is independent of M By Step 1, it suffices to consider closed
imbeddings i:X-* U, U open in P (E), E a vector bundle on S. Suppose
j:X-*V<=P(F) is another such imbedding. Let P = P(E), Q = P(F). There
are induced imbeddings
x—Uixs2
1 I
U*SV >U*SQ
18.2 Riemann-Roch for Quasi-Projective Schemes 351
with k=(\J), d=(i,j). Let Z = XxsQ, p the projection from Z to X.
Consider the diagram
KaZ
"t
K0X
The upper square commutes by Steps 1 and 3, the lower square by Step 5.
Sincep*k* = {p k)* = id, xux= xux*sV; and x*i*sV= xxby symmetry.
Step 7. The covariance of r follows from Steps 3 and 5. Indeed, if
diagram
S, E a bundle, and Y<^> M -* S, with q smooth, one has a
M
and one may use Step 3 for ;', Step 5 (or p.
Step 8. The module property follows from Step 2 and the following lemma.
Lemma 18.2. Let E be a vector bundle on a scheme X. Then there is a closed
imbedding i:X->M, with M smooth over S, and a vector bundle E on M such
that i* E^E.
Proof. Choose a closed imbedding j of X in U, with U smooth over S.
Choose a vector bundle F on U and a surjection of sheaves F -* _/, (E) (Appen-
(Appendix B.8.1). Let e = rank(?T), and let Mbe the Grassmannian of ranke quotients
of F, n:M—* U the projection. Then there is a morphism i:X—>M with
n ° i=j, such that E is the pull-back of the universal quotient bundle on
M ?
Step 9. To verify that the first diagram in C) commutes, when/ is a proper
l.c.i. morphism, factor / as in Step 7. For the projection, the calculation is the
same formal calculation done in Step 5. For the case of a regular imbedding,
the proof is essentially the same as that given in § 15.2 in the non-singular case.
Indeed, a stronger result is given in Corollary 18.1.2.
Step 10. For the second diagram in C), again factor/ as in Step 7:
X^> UxsY^> P(q*E\Y) -^ Y.
For p, the assertion was proved in Step 4, and for open imbeddings it follows
from Step 1. It therefore suffices to prove that
for/: X-* Ka (closed) regular imbedding, with normal bundle N.
We first verify (*) when Y= N, and /is the zero section imbedding of X in N.
Let p be the projection of N to X. Since N is the restriction of a bundle on a

352
Chapter 18. Riemann-Roch for Singular Varieties
smooth variety containing X, it follows from Step 4 that
(*') xN (p* 0) = td(j>*N)np* xx @) = p* (td (N) n xx @))
for all 0 eK0X. Since p*:KaX-^KoN is surjective, we may write a = p* 0,
0 = /* a. Since /* and p* are inverse isomorphisms between A*X and A* N, (*)
follows from (*')•
We next verify (*) when X=div(t) is a principal Cartier divisor on Y.
Then / determines a morphism p : Y—> A]s, with X= p~l @). Choose an imbed-
imbedding ;': Y-* M, M smooth over S. Then j = (i, p) imbeds Y in A/xsA|= AJ^,
and one has a fibre square v / v
where g is the zero section, and k = ;' ° / It suffices to prove (*) for a = [0y], V
a subvariety of Y. If F<= A', both sides of (*) are zero, since /* »/„ is zero on
KOX and on A*X, and r,-/^/,-!,. If 7 4: X, let W = VTlX, so
/* a = \ffw\ Let E. be a resolution ofj*#v by locally free sheaves on A[M. Then
#* ?. resolves k^fiw on M (Lemma A.4.2). Now
f*xY(fiv)=f*
nTd (A^)) = ch%(g*E.) n
by formula A9) of § 18.1. But this last term is tx(<?w), which is the required
equation (*) since the normal bundle to A' in Y is trivial.
The general case of (*) follows from the preceding two cases and a
deformation to the normal bundle. As for rational equivalence (§ 5.2) there is a
specialization homomorphism
a:KoY^KoN
defined to make the diagram
KON
ipr*
commute. In addition,
/¦ = /¦ o a:KoY— KOX
where / is the zero-section imbedding of X in N. (To see this, consider the
deformation diagram
I 1'
— X
F
— N .
Let a e KOY, pr* a =/* P,p&KoM°. Then
f* a* (a) = f* i* 0 = i*,F* 0 = i*0 F* 0 = f* k* 0 = f* a .)
Now t commutes with the specialization maps a, since r commutes with pr*
(Step 4),j* (Step 1), and /* (since / imbeds A' as a principal Cartier divisor on
18.3 Riemann-Roch for Algebraic Schemes
M°). Therefore
353
f*Xy{a)=f*azY(a)=f*xN(a(a))
= td (N) n xx(J*a{a)) = td (N) n xx(f*z),
which concludes the proof. D
Example 18.2.1. Let /: X-* Y be a regular imbedding of codimension d,
with normal bundle N, and let ? be a vector bundle of rank e on X. D efine
P(N,E) e^*A"by the prescription of Lemma 15.3. Then
c(f,E)=l+f*(P(N,E))
in^4*K. (The proof is the same as in Corollary 18.1.2, cf. Example 18.1.3.)
Example 18.2.2. Lefschetz-Riemann-Roch. There are equivariant analogues
of the Riemann-Roch theorems, at least for automorphisms of finite order
prime to the characteristic. For a singular version see Baum-Fulton-Quart A).
The formalism and use of deformation to normal bundle are completely
analogous. There is an analogous formula for the Frobenius endomorphism in
characteristic p (cf. Fulton E)).
18.3 Riemann-Roch for Algebraic Schemes
In this section we work in the category of algebraic schemes (locally of finite
type and separated) over a fixed base field.
Theorem 18.3. For all schemes X there is a homomorphism
satisfying:
A) (Covariance). Iff: X -> Y is proper, a s KOX, then /, xx (<x) =
B) (Module). If a e KOX, 0 6 K°X, then
ft (a).
C) If i:X-y M is a closed imbedding in a smooth M, and E. is a resolution
i*y by locally free sheaves on M, then
xx{5T) = ch?(?.) nTd (M) = td (i* Tu) n (ch?(?.) n [M]).
D) Let f:X—>Ybe a l.c.i. morphism. Assume that there are closed imbed-
dings IcM, Kc= P in smooth schemes M, P. Then
xxf*a = td(Tf)-f*xY(a)
forallaeKaY.
E) (Top term). If Vis a closed subvariety ofX, with dim V= n, then
xx@v) = [V] + terms of dimension < n .

354
Chapter 18. Riemann-Roch for Singular Varieties
In addition, x is uniquely determined by properties A), D) (for open imbed-
dings of quasi-projective schemes), and E)(for V= X = IP").
The proof will be given at the end of this section. We first record some
corollaries. For any scheme X, we define the Todd class of X, denoted Td (X),
by the formula
Corollary 18.3.1. (a) If X is complete, then for any vector bundle E on X,
X(X,E) = jch(?)nTd(A').
In particular, x
(b) IfX is a l.c.i. scheme which is imbeddable in a smooth scheme, then
= td(Tx)n[X].
Here Tx is the virtual tangent bundle to X. This holds, in particular, for all
smooth schemes X.
(c) Let f:X^>Y be a proper morphism. Let /? e K°X. Assume there is an
element fa (p) in K°Ysuch that
For example, if f is a l.c.i. morphism, or more generally, any perfect morphism
(cf. Example 15.1.8), there is a canonical such element j* (/?). Then
/¦ (ch {p) n Td (A-)) = ch (/, fi) n Td (Y).
In particular, if X and Y are smooth, then
U (ch (fi) • td (Tx)) = ch (/„ /?) • td G» .
Proof, (a) follows from the covariance for the mapping from A' to a point,
and the module property B). If X is smooth, (b) follows from property C) of
the theorem, imbedding X in itself to calculate xx(fix). If X is a l.c.i. scheme,
and/: X-* Kan imbedding of A" in a smooth Y, then by property D),
x(fix) = td (#/)-'/* (td G» n [ Y]) = td (Tx) n [X].
(c) follows from the covariance and module properties A) and B):
/* (ch {fi) n xx(fix)) = /* (rxifi ® fix))
= zr{f* (P ® fix)) = rr(/¦ (A) ® fir)
= ch(ft(fi))nxr(fir). D
Corollary 18.3.2. For all X, the homomorphism xx induces an isomorphism
where KOX^ = (K0X) ®z Q .
Proof. In fact, if
KOX
18.3 Riemann-Roch for Algebraic Schemes
355
is the canonical surjection (Example 15.1.5) the composite ip^ ° x induces an
isomorphism on the associated graded groups Gr KoX^, as follows from
Property E). ?
In particular, if X is non-singular, the Chern character determines an iso-
isomorphism
hK°X
The proof of Theorem 18.3 will depend on the following two propositions.
If /: X -* M is a closed imbedding of a scheme A' in a smooth scheme M, and
s a coherent sheaf on X, we define ch#Er) eA* X^ by
?.) n [M]
where E. is a resolution of i^ on M. This determines a homomorphism
Proposition 18.3.1. Let X <^> M, X <^> P be closed imbeddings of a scheme X in
smooth schemes M and P. Then
td GV) n ch^(a) = td GV) n chj(a)
in A^X^for all a sKoX.
Proof. It suffices to show that
(i) td (Tp) n ch?(T) = td (TmP) n chf"'^),
where A"c= Mx P by the diagonal, and T is a coherent sheaf on A". Consider
the diagram
X -*-> M x X -^ M x P
1 I'
X , > P
with /; = (;', 1^-), k=\Mxj, p, q the projections. Note that h is a regular
imbedding with normal bundle ;'* (TM). Let E. be a resolution oijifSr by locally
free sheaves on P. Then p*E. is a resolution of q*3~ on MxP, since /? is flat;
therefore by Theorem 18.1,
Now apply Theorem 18.2C) to the regular imbedding h, regarded as a
morphism of schemes which are projective over the base scheme S = AfxP.
This gives
(iii) chf *p(h*/]) = td (/¦ 7V)-' n h* ch^(/3)
for any /3eKo(MxX). Apply this formula to p=[q*$-\. Then h*0 =
(q h)*[k\ = [?-], so by (ii) and (iii),
(iv) ch3fx/> (ST) = td (;* 7V)H n h* q* ch?(J^) = td (;* TM)-] n chjf (T).
Multiplying by td (TMxP), one achieves (i), as required. ?

356
Proposition 18.3.2. Let
Chapter 18. Riemann-Roch for Singular Varieties
r-
be a fibre square, with i a closed imbedding, and p projective, such that p induces
an isomorphism ofX'— Y' onto X-Y. Then the sequence
KOY' A KOY® KaX' -i KOX -* 0
is exact, where a (a) = (q* a, — _/, a), b (a, /?) = i»a+ p* /?.
Proof. Let U= X— Y, U' = X' — Y\ r the induced isomorphism from V to
U. Then there is a commutative diagram with exact rows
A KOW >0
KM
sb
KM
> K
y *
o ¦*
I"
y
OJ ;
KOX'
I"
KOX
KOU
0,
where K{ is Quillen's first higher AT-group (Quillen B)§ 7 Prop. 3.2, Fulton-
Gillet A)). Then the conclusion follows from a simple diagram chase. ?
The homomorphism tx:KoX-*A^X^ has been constructed in § 18.2 for all
schemes X which are quasi-projective over S = Spec (AT), K the ground field.
To extend this to arbitrary schemes X, we introduce the notion of a Chow
envelope.
Definition 18.3. An envelope of a scheme A' is a proper morphism p : X' -> X
such that for every subvariety Vo( X, there is a subvariety V of A" such that/?
maps V birationally onto V. If X' is quasi-projective over Spec (K), we call
p : X' -* X a Chow envelope.
We will use the following elementary lemma.
Lemma 18.3. A) If p:X'-^X and q:X"^>X' are envelopes, then
pq :X" -*X is an envelope.
B) Ifp:X'->X is an envelope, and f:Y-*X is an arbitrary morphism, then
the fibre product
pxxY : X'xxY-^Y
is an envelope.
C) For any scheme X, there is a Chow envelope p : X' -* X, and a closed sub-
scheme Y<=X, with X- Y dense in X, such that p maps X'-p~*(Y) isomorphi-
cally onto X- Y.
D) If pi:X!-*X are envelopes, i= 1,2, then there is a Chow envelope
p : X' -> X, with morphisms qt: X' -> X\ such that p = pt ° q, for i= 1,2.
E) For any morphism f: Y-* X, and any Chow envelope p:X'-+X, there is a
Chow envelope q:Y'->Y and a morphism f':Y'-* X' such that pf'=fq. Iff is
proper, one may achieve this with f proper.
F) Ifp : X' -* X is an envelope, then the homomorphisms
are surjective.
18.3 Riemann-Roch for Algebraic Schemes
357
Proof. A) is immediate. For B), if Va Y, let W be the closure of f(V),
and let p map W'czX' birationally onto W. There are open sets U a W,
U' c: W such that p maps U' isomorphically onto U. Then the closure of
U' x WV in W x WV is a subvariety of X' *XY which maps birationally onto V.
For C), by Chow's lemma (cf. [EGAJII.5.6), there is a proper morphism
P\\X\ -*X, with X\ quasi-projective over S, and a closed Kcl with X- Y
dense in X, so that p\ restricts to an isomorphism over X—Y. By Noetherian
induction, there is a Chow envelope pi :X'2^>Y. Then
is the desired Chow envelope for X.
In the situation of D), choose a Chow envelope
X'-+X\xxXi.
Then by A) and B), the composite X'-* X is also a Chow envelope. In E),
choose any Chow envelope
Y'-+YxxX'.
Finally, if F,KaX denotes the subgroup of KOX generated by sheaves
whose support has dimension at most ;', then FjKoX is generated by \fiy\,
where V ranges over subvarieties of dimension at most ;. If p maps V
birationally onto V, then
P* tyr] = [fy] + a
with a e F,-, KOX, and F) follows by Noetherian induction. The case for A* is
trivial since p* is surjective on the cycle level. ?
We can now complete the proof of the Riemann-Roch theorem. For all
quasi-projective schemes X over S= Spec (AT), we denote by xx the homo-
homomorphism which was constructed in § 18.2. The extension to general X will be
carried out in several steps.
Step 1. If p : X' -> X is a Chow envelope, there is at most one homomor-
homomorphism t : KoX^-A* A'iq so that the diagram
KOX'-
(*)
I'
1<P
commutes. Thus there can be at most one way to extend from quasi-projective
to general schemes, preserving the covariance property. This is a consequence
of Lemma 18.3F). If (*) commutes, we say ris compatible with p.
Step 2. Suppose r:KOX—^A^X^ is a homomorphism, compatible with a
Chow envelope p:X'->X. Then for any proper morphism /: Y-*X, with Y
quasi-projective, the diagram
K Y T> > A Y
4 7 *?

358
Chapter 18. Riemann-Roch for Singular Varieties
commutes. In particular, the homomorphism r, if it exists, does not depend on
the Chow envelope used to construct it. To see this, choose
Y
X
as in Lemma 18.3E). The covariance with respect to / then follows from
the known covariance with respect to q and /' (§ 18.2), the assumed covari-
covariance with respect to p, and the surjectivity of q*. For if aeKoY, let
a = q* a.', a' e KOY'\ then
/* rY(q* a') = /* q* xr (a') = p*f* tr («')
= P* XX' if*°0 = TP*f*'X' = Xf*i<i*'X') ¦
Step 3. We prove the existence of r by induction on the dimension of X.
Given A", choose a Chow envelope p:X' -*X, with Kc A' as in Lemma 18.3C).
Form the fibre square ) j
Y tA" .
Since; is a closed imbedding, Y' is quasi-projective. By Noetherian induction,
there is a homomorphism ry:KOY->¦ A^Y^, compatible with the Chow
envelope/i :Y'-*Y. Consider the diagram
KOY' -2-> KOY®KOX' -^KOX >0
t,,|
0
where a and b are defined as in Proposition 18.3.2. By assumption, and the
known covariance with respect to the morphism y of quasi-projective schemes,
the square in this diagram commutes. By Proposition 18.3.2 the top row is
exact; in fact, the bottom row is also exact (Example 1.8.1), but this is not
needed. Therefore there is a unique homomorphism r: KOX-^A^X^ making
the diagram commute. In particular, r is compatible with p.
Step 4. By steps 2 and 3, we have, for all X, a homomorphism
tx' KOX —* A* A'iq
compatible with any Chow envelope X'-*X. We prove the covariance
property. If f:Y-*X is proper, choose p, q, f as in Lemma 18.3E). The
equality /* ry(a) = xxf* (°0 follows exactly as in Step 2.
Step 5. For the module property, if peK°X, asK0X, choose a Chow
envelope p : X' -* X, and choose a' e KaX' with p* a' = a. Then
xx{fi ® a) = xxp*(p*/]® «')=/>* zr(p*P® «')
= p* ch (p* p) n rx. (a') = ch (fi) n p* xx, (a') = ch (fi) n xx (a)
by the known result for A" (Theorem 18.2), and the projection formula.
Step 6. To prove Property C), let i:X-*M be a closed imbedding, M
smooth, and let p : X' -> X be a Chow envelope. Let j: X' -» [/ be a closed
18.3 Riemann-Roch for Algebraic Schemes
359
imbedding, U open in some projective space. Then k = (ip, j) is a closed
imbedding of A'' in M x [/. Let a' e AToA"', a. = p* a.', and set
fjr (of) = td (k* TMKU)n ch^a'),
By Theorem 18.2, applied to the category of schemes which are quasi-projec-
quasi-projective over M,
P* fr (<*') = fyO»* «') = ^(a) •
(To be precise, Theorem 18.2 gives this equality before multiplying both sides
by td(;'* Tu).) But by Proposition 18.3.1, fr(a') agrees with the class obtained
by imbedding X' in U, i.e.
Therefore fx = p* ° i>, and by Step l,fx= rx, as required.
Step 7. Let f:X-> Y be a l.c.i. morphism as in Property D). One obtains a
commutative diagram
For the projection q, the proof of D) is the same as in Step 4 of the proof of
Theorem 18.2. For the regular imbedding h, D) follows from Theorem 18.2C),
regarding X and MxY as schemes quasi-projective over the smooth base
MxP.
Step 8. To prove Property E), by covariance one may assume V=X, and
by taking a Chow envelope, that X is quasi-projective. Since r is compatible
with restriction to open subvarieties, it suffices to prove it when X is projec-
projective. Choose a finite surjective morphism f:X-* P" of degree d. Then
a e /"„_, KOP". Then by covariance
/¦ tx(<Px) = </rP.(^
TP.(a) =
where /? is a class of dimension < «; this uses the truth of E) for P", which
follows from Property C), and the covariance of r with respect to inclusions of
subvarieties of P". If m [X] is the top term of xx (tfx), then
with dim (/?") < n. Comparing, one has the required conclusion that m=\.
As for uniqueness, suppose t' also satisfies the indicated properties. Since
Tq is an isomorphism (Corollary 18.3.2), one may consider the composite
T= Tq ° tip. Then T is a transformation from A^ to itself, which is covariant
for proper morphisms, contravariant for open imbeddings of quasi-projective
schemes, and T[W"] = [P"] + /?„ for some /?„ of dimension < n. Therefore by
Example 1.9.6, Tis the identity, as required.

360
Chapter 18. Riemann-Roch for Singular Varieties
Example 18.3.1. Cartesian products. Let X, Y be algebraic schemes. Then
the diagram
KOY
Ko(XxY)
A*Xq ® A,
commutes. In particular,
Td(A-xK)=Td(A-)xTd(K).
(If X and Y are quasi-projective, this is proved in Baum-Fulton-MacPherson
AI11.2. The general case follows by taking Chow envelopes X' ->¦ X, Y'-* Y.
This is equivalent to the multiplicativity of the localized Chern character
(Example 18.1.5).)
Example 18.3.2. Uniqueness. On the category of complete schemes, r is
uniquely determined by the covariance property A), the module property B),
and the normalization that r (fiP) = [P] when P = Spec (K) is a point. (See
Baum-Fulton-MacPherson AI11.2.) On the category of all algebraic schemes,
adding the property that r is compatible with open imbeddings determines r
uniquely.
Example 18.3.3. If n : X -* X is a proper birational morphism, isomorphic
off Z<=X, then
Td (X) = n* Td (X) + a
where a is a class supported in Z. In particular
for k > dim Z.
For example, if X is an ^-dimensional variety with a finite number of
singularities, and X is a resolution of singularities,
Tdk(X) = nt(td,-k(Tx) n[X])
for k > 0, and Tdo(A') = 7i*(td,,G» n [.?]) + X «/¦[•?], with the sum over all
singular points P in X, and
n-l
where /(?*")/> denotes the length of the stalk of the sheaf J*" at P. When X is
normal the last term is zero.
Example 18.3.4. (a) If A' is a projective curve, and n : X-> X is a resolution
of singularities, then
Td (X) = [X\ + \ Me, G» n [*]) - Z 5p[P],
where 5/.= l(n*#x/#x)p, the sum over singular points P of X. In particular,
Z(X, d)x) = J^- Tdo(X) = 1 - g(X) - S 5P. If ? is a vector bundle of rank r on X,
then
18.3 Riemann-Roch for Algebraic Schemes
361
(b) If A' is a normal surface, and n : X —> X a resolution of singularities, then
Td (X) = [X] + y 7t, (C.) + -jy E
where c, = c,G» n [^f], «/. = l(R] n^
+ C2) + E nP[P] ,
- In particular,
If ? is a vector bundle of rank r on X, then
Example 18.3.5. Let f:X-* Y be a regular imbedding of codimension d,
with normal bundle N. Assume Y can be imbedded in a non-singular variety.
Then
Td (X) = td (A') n /* Td (K).
If A" is a complete intersection of hypersurfaces Dt,..., Dd on K, then
where x, =
'/)), and
d
X (X,0X) = j II A - exp (- D,)) n Td (Y).
Y i-\
In particular, if A' is a Cartier divisor on Y, then
ldk(Y).
When Kis smooth ofdimension«,Td/i(r) = tdn-tG'y) n [K].
There are singular projective varieties X such that Td (A") does not lift to
cohomology. In Baum-Fulton-MacPherson A)IV.6 a normal projective 3-fold
X is constructed with one singular point, such that the image of Td2(A") in
//4 (A"; Q) cannot be written in the form a n [X], for any a e H2 (X; Q).
Example 18.3.6. Let X be an ^-dimensional complete scheme, 9~ a coherent
sheaf on X, Lx,..., L, line bundles on X, with x, = c, (Lf). Then
where rXik(^~) is the component of zx(^~) in
degree n in d\,..., d, whose top degree term is
. This is a polynomial of
This gives a concrete form to polynomials considered by Snapper A); for
related results see Kleiman A) and Mumford F).

362
Chapter 18. Riemann-Roch for Singular Varieties
Example 18.3.7. (a) Let ? be a vector bundle of rank n on a complete
scheme A'of dimension ^n. Then
(By Hirzebruch-Riemann-Roch and Example 3.2.5,
The conclusion follows since r(X) and [A'] agree in dimension n.)
(b) If E has a regular section s, with a finite zero scheme Z(s), the
equation in (a) reduces to
(cf. § 14.1). If E ® L has a regular section, for some line bundle L — for
example, if A" is a projective variety - then, as R. Lazarsfeld points out, (a) can
be deduced directly from (b).
(c) If A' is a non-singular complete variety, then
the topological Euler characteristic of X.
Example 18.3.8. Flat families. Let T be non-singular, X-* T a quasi-projec-
quasi-projective morphism. For t s T, let X, be the fibre over t, and let
be the specialization map (§ 10.1). Let y be a coherent sheaf on A" which is flat
over T. Then
where -T, is the fibre of y over f. (Use Theorem 18.1 and the fact that the
normal bundle to t in T is trivial). This generalizes (and reproves) the fact that
X(X,,y,) is a locally constant function of t. In particular, if X is flat over T,
then
This is a special case of the theorem discussed in Example 18.3.16.
Example 18.3.9 Let/:X-* Kbe a finite etale morphism. Then
ta-°/*=/* °ty-
(Taking a Chow envelope K' -+ Y, one is reduced to the case where Y and X
are quasi-projective. This is special case of Theorem 18.2C).) It follows that
Therefore /* Td (X) = deg (/) • Td (Y). In particular, if X and Y are complete,
18.3 Riemann-Roch for Algebraic Schemes
363
(S. Kleiman has given an elementary proof of the above equation for the
arithmetic genus.)
Example 18.3.10. Let f:X-* Kbe a proper l.c.i. morphism. Assume that/
factors into a regular closed imbedding X^P followed by a smooth proper
morphism P-*Y. Then for all /3eK°X,
ch(/,/?)=/*(tdG))-ch(/?))
in A*Yq. (It suffices to prove this when / is a regular imbedding or / is
smooth. The former case is covered by Theorem 18.2. If / is smooth, we must
show both sides have the same effect on a eA+Yq, Y'-* Y, Y' quasi-pro-
quasi-projective. Write a = xr(y), y s KOY^. The desired equation then follows formally
by applying Theorem 18.3D) to the morphism XxYY' —> Y'.)
Example 18.3.11. Let y be a coherent
dim Supp E*") ^ n. Define the H-cyde of$~,
sheaf on a scheme X with
dimV-n
where the sum is over all ^-dimensional subvarieties V in the support of y,
and ly{^~) is the length of the stalk of y over the local ring of A" at V. Then
txCr) = Z, {y) + terms of dimension < n .
If FnKoX is the subgroup of K0X generated by sheaves whose support has
dimension ^j«, then assigning Zn(y) toy determines a well-defined homo-
morphism
(In fact, modFn_|K0X, \\i is the associated graded homomorphism of xx.)
Since generators of FnKoX may be related by sheaves of higher dimensions,
this fact is not obvious from the definitions. Note that one must tensor by Q
for this to be well-defined.
Example 18.3.12. A Riemann-Roch formula. Let E. be a complex of vector
bundles on a scheme Y, which is exact off a closed subscheme X. Let y be a
coherent sheaf on Y. Then the homology sheaves 3^(?. ® y) of the complex
E. ®#yy are supported on X, so define classes [^(E. ®y)\ in KOX. Then one
has
(*) Z (-l)'rjrpr/(?.®.r)] = chJF(?.) n xY(y)
in A* A"iq. If Y is smooth, y = 0Y, and E. resolves a sheaf on X, this is Property
C) of Theorem 18.3. Property D) may also be deduced from (*) (cf. Example
18.3.10). Conversely, with resolution of singularities one may deduce (*) from
the Riemann-Roch theorem. Note that if X= Y (*) follows from the module
property B). We sketch the general proof of (*).
For fixed E., both sides of (*) are additive inJ*", so define homomorphisms
from KOX to A* X^. Let rx(sr(E. ® a)) denote the value of the left side on the
element a e K0Y. Then (*) is equivalent to
(**) ^xCT(E. ® a)) = chx(E.) n tr(a)
for all a eKoY. The proof will be given in four steps.

364
Chapter 18. Riemann-Roch for Singular Varieties
1. Suppose f:Y'-*Y is proper, X'=f~](X), g:X'-^X induced, and
a' sKoY'. Suppose
TX,(sr(f*E.®a')) = chj: (/*?.) n Tr(a')
in A+Xfa. Then (**) holds for a=/*a'. (Apply g* to both sides, and use the
covariance of r. One also needs the formula
g* CT(f*E. ® a')) =3T{E. ® /* a')
coming from a spectral sequence W*(E. ® R" f*y') => /?'+«/* (/* ?• ® •?"')¦)
2. If 0 -+ E'. -+?!.-» ?" -+ 0 is an exact sequence of complexes of vector
bundles on Y exact off A', and (**) holds for K and ?.", then (**) holds for E.
(Proposition 18. l(b)).
3. Suppose Y is a quasi-projective variety, and E. is an elementary com-
complex, i.e. there is an n so that E,¦ = 0 for ;' 4= «, « — 1, and En and ?„_, are line
bundles. Suppose also that y = #Y- Then (*) is true. (After translating and
tensoring by a line bundle, one may assume n = 1, Eo = 0Y, E\=fi(— D), D an
effective divisor on Y, Et -> ?0 the natural inclusion. Then (*) follows from
Theorem 18.2C) and Corollary 18.1.2.)
4. For the general proof, by Step 1 and Chow's lemma, we may assume Y is
a quasi-projective variety. We induct on the sum of the ranks of the bundles
Ej\ let E. be the complex
with En 4= 0. We may assume X 4= Y. Since the projection maps Ko (P (?„))
onto Ko Y, by Step 1 we may assume ?„ contains a sub-line-bundle L. In this
situation, we induct on the dimension of the support of y. We may assume
y=fiY. By Steps 1 and 2 and the inductive assumptions, it suffices to find a
proper birational f:Y'-*Y so that f*E. contains an elementary subcomplex
K, exact off/ (X).
Let IP = /¦(?„-1), p:P -* Y the projection, 0 (- 1) and 0 the universal sub
and quotient bundles of p*En-t. Let P' <= IP be the scheme-theoretic intersec-
intersection of the loci where the bundle maps
p*l p-^i p*?„_,-> Q and #(- 1) -> /?*?¦„_, ^p*En-2
vanish. The restriction of p*E. to IP' has an elementary subcomplex F., with Fn
and /"„_, the restrictions of p*L and <^(- 1) to IP'. On K— X, dn is injective, so
dn(L{y)) is a line in En-t(y) for _y e K— X. This determines a section
s:Y-X-*V such that j*F. is exact. The closure Y' of s(Y-X) in P' is the
required variety, f:Y'-*Yis induced by projection, and E'. = F. \ Y'.
Example 18.3.13. (a) Let /: X-* Y be a regular imbedding of codimension
d, with normal bundle N, and assume Kcan be imbedded in a smooth scheme.
Then for any coherent sheafs on Y,
S (- 1)' TX(ToT?@x,r)) = td (AT1 n /* ry(JO .
If dim Supp {y) = n, equating top terms gives
I (- lyZ^CTor/V^JO) =/*Zn(lr) •
(Apply Theorem 18.3D) or Example 18.3.12.)
18.3 Riemann-Roch for Algebraic Schemes
365
(b) Let A' be a non-singular variety, let V, W be closed subvarieties of X.
Then
[ V\ ¦ [ W\ = X (- 1)' Zm (Torf (*K, ^^))
in Am(Vf] W){>, where m = dim V + dim W — dim X. (Apply (a) to the diagonal
imbedding of X in X x X, !F = &v>, w.)
For a discussion of the role of Tor in defining intersection products, see
§20.4.
Example 18.3.14. Suppose E. is a complex of vector bundles on a scheme Y,
exact off a point X. For a coherent sheaf y on Y,
3f{E. ® J*~) = Z (~ !)' length (#*, (?. ® y))
in /ToA^Z. Let ch,(?V) sA'(X-* Y)^ denote the term of degree / of chj(?.).
Let n = dim K, and assume that
(*) ch,(?V) = 0 for i<n.
Let ?? be the dual complex of vector bundles. Then
(a) sr(E.®y) =0 if dim (Suppy) <n
(b) 3f(EY®y) = (-\ysr(E.®y).
(Use Examples 18.3.12 and 18.1.2.) The condition (*) holds whenever E. is the
pull-back of some complex of vector bundles on a non-singular variety.
However, Dutta, Hochster and McLaughlin A) have recently produced a
complex E. on a three-dimensional variety, resolving a module of finite length,
so that 3f(E. ® y) takes on both positive and negative values, for S~ the
structure sheaf of appropriate surfaces in Y; thus ch2 (E.) 4= 0. For a discussion
of relations between Riemann-Roch and local algebra, see Szpiro A).
Example 18.3.15. Let f:X-* Y be a closed imbedding such that fix can be
resolved by a complex of vector bundles on Y. Therefore / is perfect (cf.
Example 15.1.8), and Jf(?.®a)=/*a, where /* : KOY -* KOX is the Gysin
map. In this case Example 18.3.12 reads
In particular, if/ is a regular imbedding, then (Corollary 18.1.2)
M/*a)= td(Af)-|/*iy(a),
with N the normal bundle to X in Y. This reproves Theorem 18.3D). (Gillet
reports that, using twisted complexes as in Toledo and Tong A), the existence
of global resolutions E. is not needed for this conclusion.)
Example 18.3.16. Bivariant Riemann-Roch. All schemes are assumed to be
quasi-projective over a fixed non-singular base S. For a morphism f:X-* Y,
define ,
to be the Grothendieck group of f-perfect complexes on X; if one factors / into
a closed imbedding /: X -> M followed by a smooth p : M -* Y, a complex s/' of
sheaves on X is /-perfect if ;*(a/') is quasi-isomorphic to a bounded complex

366
Chapter 18. Riemann-Roch for Singular Varieties
E' of locally free sheaves, called a resolution of s/' on M. There is a natural
notion of push-forward (for proper morphisms), pull-back (for Tor-indepen-
Tor-independent squares), and product, making K into a bivariant theory (cf. Fulton-
MacPherson C)). If/ is factored as above, then K (X -4 Y) = K (X -+ M).
In particular, if Y is smooth, K(X
K(X -> X) = K°X.
Define a homomorphism
Y) = KOX, while if Y=X, / = id,
as follows. Factor / as above. Given an /-perfect complex sf on X, resolve it
by E' on M, and set
Theorem. 7Vie homomorphism x is independent of factorization, and compat-
compatible with push-forward, pull-back, and product.
(Except for the compatibility with products, the proof is much the same as
for Theorem 18.2. For products, the essential point is the following. Let
be closed imbeddings, sf /-perfect, SB' ^-perfect. Let F' resolve s/' on Y; let G'
resolve SBf on Z. Assume SB' is a complex of locally free sheaves on Y (not
necessarily bounded above). Then the product [a/'] ¦ [SS'] is represented by the
complex s/" ®0xf*SB'. Let E' be a resolution of s/' ®ftf*3)' on Z. Compatibility
of t with products asserts that for all Z'-* Z, a eA^Z^,
(*) chf (?•) n a = chJ(F) n (ch? (G1) n a).
At first glance, this appears to be difficult, since we do not know how E'
relates algebraically to F and G"; the essence of Fulton-MacPherson C)PRR
was to prove such a topological relation, for complex varieties. There is a simple
subterfuge, however. An element in K(X-^* Y) determines a homomorphism
from KOY' to Ko(XxY Y') for any h:Y' -*Y: if .s/'is an /-perfect complex on X,
and &~ is a coherent sheaf on K', then [s/-] takes pf] to
If / is a closed imbedding, and E' a resolution of stf' on 7, then
It follows from Example 18.3.12 that
The analogue of (*) for T in /sToZ', Z' -+ Z:
follows from the Tor spectral sequence. Then (*) follows, since rz.® Q is
surjective.)
18.3 Riemann-Roch for Algebraic Schemes
367
Example 18.3.17. Let/: X-* Kbe a perfect morphism of pure-dimensional
quasi-projective schemes, and let d= dim Y— dimX. Define
to be the term of degree d in xf @X), with xf as in the preceding example.
(a) If/ is flat, l.c.i., or Y is smooth, then [/] is the image of the integral
class in Ad(X-> Y) defined in § 17.4. The terms of degree less than d in xf{fix)
are all zero.
(b) If f:X-* Y, g : Y-* Z are morphisms such that / and g can each be
factored into a sequence of morphisms which are flat, l.c.i., or morphisms to
smooth schemes. Then
It is not known if this holds with integer coefficients (cf. Example 17.4.6). For
a general perfect morphism / it is not known if the terms of degree less than d
in if@x) vanish, or if the term of degree d comes from a class with integer
coefficients. For / a closed imbedding, and E. a resolution of fix on Y, this
asks for vanishing and integrality for the local classes ch,-(?.). which seems
doubtful, cf. Example 18.3.14.
Example 18.3.18. For complex algebraic schemes, there are homomorph-
lsms
Ko{X)-*i?V{X)
where AToop denotes the homology theory (with locally closed supports) of
topological AT-theory. This homomorphism satisfies covariance, module and
other properties analogous to those in Theorem 18.3; if A' is non-singular, the
image of0x is the orientation class of X. If
ch : K^X - H+ (X; Q)
is the Chern character in homology, the composite
KOX
(X; Q)
satisfies the same properties and formulas as x. (In the quasi-projective case,
the map is constructed in Baum-Fulton-MacPherson B). The extension to non-
projective schemes proceeds as in this section.)
Example 18.3.19. Riemann-Roch and Grothendieck duality. Let s/' be a
bounded complex of sheaves on a scheme X such that the cohomology sheaves
are coherent. Assume that A'can be imbedded in a smooth scheme. Let
= RHom(jf,co'x)
with co'x a dualizing complex on X (cf. Hartshorne B)). Then for all k,
I (- 1)' rt(*"(«O) = (~ 0* Z (- 1)' r*CT (#O)
. If A' is Cohen-Macaulay, with dualizing sheaf cox, and E is locally
xk (E ® ffx) = (- I)""* xk (Ev ® cox) ,
in ^
free, then

368
n = dimX. When k = 0, this yields
Chapter 18. Riemann-Roch for Singular Varieties
which also follows from Serre-Grothendieck duality. (If X a M, M smooth,
and E' resolves sf on M, then E'v resolves D (s/'), cf. Fulton-MacPherson
C)§ 7.)
Example 18.3.20. For an algebraic scheme X, let Kl X be the Grothendieck
group of sheaves on X whose supports are complete. There is a homo-
morphism
x-.KiX^AiX^,
with A*X as in Example 10.2.8, covariant for arbitrary morphisms, and
satisfying analogues of the properties of Theorem 18.3. In particular, x induces
isomorphisms KiXq S/l? Xq.
Notes and References
The localized Chern character of §18.1 was developed in Baum-Fulton-
MacPherson A), modelled on an earlier graph construction of MacPherson (cf.
Example 18.1.6). The fact that chx(E.) determines an element in the bivariant
group A(X^> F)n, is new here. This strengthens the properties proved in (be.
cit) as well as simplifying the proofs; in addition, no separate argument is
needed for base fields which are not algebraically closed. A topological
construction of local Chern classes, with values in H\Y = H* (Y, Y— X), has
been given by Iversen C). An important precedent had been given by Atiyah
and Hirzebruch B), C). A version of Riemann-Roch for singular curves may
be found in Serre E), and for normal surfaces, in Zariski C).
The Riemann-Roch theorem of § 18.2 was first given in Baum-Fulton-
MacPherson A), with the exception of Theorem 18.2C). The first part of C) is
analogous to the RR theorem for l.c.i. morphisms proved earlier in [SGA 6]. In
the SGA 6 version, values were taken in graded .K-groups. Using deformation
to the normal bundle, the proof of the SGA6 theorem may also be simplified
as in Chap. 15. The second formula of Theorem 18.2C) was conjectured in
Baum-Fulton-MacPherson A), and proved by Verdier E). The proofs in § 18.2
follow these sources, with modifications for use in the non-projective case.
The extension to arbitrary algebraic schemes (§ 18.3) follows Fulton-Gillet
(D-
In the complex case one also wants a Riemann-Roch theorem to give a
transformation from algebraic geometry to topology. The most natural such
transformation is from algebraic A'-theory to topological ^-theory. Such was
constructed in the non-singular case in Atiyah-Hirzebruch C), and in the
quasi-projective case in Baum-Fulton-MacPherson B). Using the methods of
§ 18.3, this extends to arbitrary algebraic C-schemes (cf. Example 18.3.18).
Notes and References 369
Composing with the homology Chern character gives a homomorphism
satisfying the properties of Theorem 18.3.
For arbitrary complex analytic spaces such homomorphisms have not yet
been constructed. For complex manifolds X, Obrian, Toledo and Tong C) have
constructed a homomorphism with values in H*(X,Q*), and proved the GRR
formula in this context.
The Riemann-Roch formula of Example 18.3.12 has not appeared before,
although special cases were known from the singular Riemann-Roch theorem.
This was motivated by joint speculation with MacPherson, Peskine, and Szpiro
on the relations of Riemann-Roch with conjectures in local algebra. The note
of Peskine-Szpiro B) provided initial evidence that such relations exist.

Chapter 19. Algebraic, Homological, and Numerical
Equivalence
Summary
Each fe-dimensional complex variety V has a cycle class cl(V) in H2kV, where
//* denotes homology with locally finite supports (Borel-Moore homology). If
V is a subvariety of an n-dimensional complex manifold X, then H2k(V) =
H2"~2k(X,X—V). The resulting homomorphism from cycles to homology
passes to algebraic equivalence. There results in particular a cycle map
for complex schemes X, which is covariant for proper morphisms, and
compatible with Chern classes of vector bundles.
If V and W are subvarieties of dimensions k and / of a non-singular «-di-
mensional variety X, a refined topological intersection product cl(V)-cl(W)
is constructed in H2m(Vf]W), m = k + l-n. If cl"V is the class in
Hln~2k{X,X- V) dual to cl(V), and similarly for clx(W), then cl{V)-cl(W)
is defined to be the class dual to
clx(V) u clx(W) e H2"-2m(X, X-VC\W).
We show that the cycle map takes the refined intersection V• We Am(Vf]W)
of Chap. 8 to the class cl(V) ¦ cl(W). In particular, cl is a ring homomorph-
homomorphism from A*X to H*X. More generally, if /: X-* Y is a regular imbedding of
codimension d, the cycle classes of the refined products /' (a) of Chap. 6 are
given by cap product with an orientation class uXiy in H2i(Y, Y—X).
In the final section we discuss what is known about algebraic, homological,
and numerical equivalence on non-singular projective varieties. Only a few
salient facts are mentioned which relate most directly to other chapters, and
few proofs are included. Together with the examples, this may serve as an
introduction to the literature on the transcendental theory of algebraic cycles.
Notation. Unless otherwise stated, all schemes in this chapter are assumed
to be complex algebraic schemes which admit a closed imbedding into some
non-singular complex variety. All topological spaces will be locally compact
Hausdorff spaces which admit a closed imbedding into some Euclidean space.
As in preceding chapters, a A>cycle on I is a formal sum of algebraic
subvarieties of X.
19.1 Cycle Map
19.1 Cycle Map
371
Even when one is primarily interested in projective varieties, it is useful, as we
have seen, to have rational equivalence groups defined for varieties which may
not be complete, so that every variety has a fundamental class, with restriction
homomorphisms for open imbeddings, etc. The analogous homology theory
for locally compact topological spaces is the Borel-Moore homology, which we
denote by //*. Geometrically, Borel-Moore homology may be thought of as
constructed from possibly infinite singular chains with locally finite support.
We sketch here the main features of Borel-Moore homology needed for
present purposes; for details, see Example 19.1.1 and the references cited there.
Borel-Moore homology can be defined using singular cohomology. If a
space X is imbedded as a closed subspace of R", then
A) ff,Is//""'(R*,R"-Ar)
where the group on the right is relative singular cohomology with integer
coefficients. More generally, if A' is a closed subspace of a space Y, there are
cap products
C^\ f-lJ (V V Y\ (&i f-f V t 11 y
If Yis an oriented, connected, real n-manifold, then HnY is freely generated by
a fundamental class fi >-, and capping with fi >- determines an isomorphism
C) H"-!(Y,Y-X) ^ H,X.
When Y= R", this is the isomorphism of A). When X= Y, C) is the Poincare
duality isomorphism of H"~'Y with //, Y.
If/: X -> Y is a proper morphism, there are covariant homomorphisms
(A\ f ¦ H Y -* H Y
If j: U -> Y is an open imbedding, there are contravariant restriction
homomorphisms
E)
H,U.
If X is the complement of U in Y, i: X-* Y the closed imbedding, there is a
long exact sequence
F) ... -* //,+i U -* H,X-> HjY-* HiU -> //,_i X ->....
If X is a disjoint union of a finite number of spaces Xt,..., Xm, then
G) H,X=H,Xi@...@HiXm.
There are evident compatibilities among these homomorphisms, which will
be used without comment. One such is the equation
(8) (u u v) n a=j* (u) n (v n a)
in //„, (X fl W), where X c, Y, j:W -» Y are closed imbeddings,
ueH*(Y, Y-X), v<=H*(Y, Y- W), ae H^Y.

372
Chapter 19. Algebraic, Homological, and Numerical Equivalence
Lemma 19.1.1. If X is an n-dimensional complex scheme, then HjX= 0 for
i >2n, and H2nX is a free abelian group with one generator for each irreducible
component of X.
Proof. Let S be the singular locus of X, U-X~ S. For U this follows from
G) and the Poincare isomorphism C), since a complex manifold has a
canonical orientation. Assuming the assertion for S by induction on the
dimension, one concludes by the exact sequence F). ?
The generator of H2nX corresponding to an n-dimensional irreducible
component X, will be denoted cl(X,); it is determined by the fact that it
restricts to the fundamental class fiUt for any connected open subset ?/, of the
non-singular locus of Xh More generally, if V is a A>dimensional closed sub-
variety of a scheme X, define the cycle class clx(V) by
clx(V) = iifcl(V)eH2kX,
where / is the inclusion of V in X. When no confusion is likely, we may write cl (V)
in place of clx(V).
Lemma 19.1.2. Let /: F-> W be a proper, surjective morphism of varieties.
Then
)
Proof. Let n = dim V. If dim W < n, the lemma is obvious since H2n W = 0.
If dim W= n, let U be a small open topological ball contained in the smooth
part of W, such that f~'(U) is the disjoint union of d=deg(V/W) balls
U\,...,Ud, each of which is mapped homeomorphically by / to U. The
lemma follows from the commutativity of the diagram
and the fact that the restrictions H2n W -> H2n U and H2n V -> H2n G, are iso-
isomorphisms, as are the induced maps from H2n U-, to H2n U.
For any X, there is a homomorphism denoted cl, or clx,
cl:ZkX -> H2kX
from the algebraic A:-cycles to the Borel-Moore homology, which takes
X Ki[V] to 21 «(C h{Vi). It follows from Lemma 19.1.2 and the definitions in
Chap. 1 that c I commutes with push-forward for proper morphisms, and with
restriction to open subschemes.
If MaP is a closed imbedding of complex manifolds of codimension d,
there is a canonical orientation class uM P,
uMiPeH2d(P,P-M).
In case P is a complex vector bundle over M with M imbedded in P by the
zero section, uMP is the Thom class of the bundle. If/: P' -* P is transversal to
M, and M'=f~xM, then f*uMP= uMF. In particular, if N is the normal
bundle to A/in P, and /: N-> P is a tubular neighborhood, uMP is determined
19.1 Cycle Map 373
by the fact that/* uMtP is the Thorn class of N. If M and P are oriented, then
(9)
Lemma 19.1.3. Let X be an n-dimensional variety, T a non-singular curve,
X-* Ta surjective morphism, t eT,X, =/"' (/). Then
inH2..2(X,).
Proof. Since cl and specialization commute with proper push-forward
(Proposition 10.1), we may replace X by its normalization, i.e., we may assume
X is normal. It suffices to show that both sides of the required equation restrict
to the same class in H2n^2{Uf\Xt), where U is any open subset of X which
meets X, inside a given irreducible component of X,. Thus we may assume T is
the unit disc in C, / = 0, and that X is non-singular, with /= gm, where g = 0
defines a non-singular hypersurface Y^X. Consider the commutative
diagram
, {T, T- {0})
(X,X-X,)^ i,
r*(T, T- {0})
where n(z) = zm. Let u = u,tT, the canonical generator of H2(T, T- {0}). Then
n* u = m ¦ u, and
/,,(«) n cl(X) = m • g*(u) n cl(X) = m ¦ uY x n fix
= cl[X,}. ?
Proposition 19.1.1. If a cycle a on a complex scheme X is algebraically
equivalent to zero, then c/(a) = 0 in H^X.
Proof. It suffices to show that
in //* V where V is a variety, /: V-* T dominant morphism to a non-singular
curve T, t0, /, e T(cf. Example 10.3.2). By Lemma 19.1.3,
in H»F, where u, T is the image of u, T in H2 T. Since T is connected, ut T is
independent of t. Therefore cl(Vt) is independent of t, as required. ?
It follows that the cycle map passes to algebraic equivalence, defining
homomorphisms
cl: BkX = ZkX/A\gkX -> H2kX
which are covariant for proper morphisms. The composite
is also called the cycle map, and denoted cl or clx.
A complex vector bundle E on a space X has Chern classes Cj(E) e//2' (X),
satisfying formulas analogous to those proved in § 3.2.

374
Chapter 19. Algebraic, Homological, and Numerical Equivalence
Proposition 19.1.2. If E is an algebraic vector bundle on a scheme X, a a
k-cycle on X, then
cl(ci(E)na) = c,(E)ncl(a)
in H2k-vX,for all i.
Proof. By the projection formula, it suffices to find a proper morphism
f:X' -*X and a' e A* X' with/* a' = a, so that the formula holds for the bundle
f*E and the cycle a' on a. Taking X' -* X to be a composite of projective
bundles (cf. Example 3.3.3), one may assume E is filtered, with line bundle
quotients. By the Whitney formula, one may therefore assume E is a line
bundle, and i=\. As usual, we may assume a = [J/], V= X a variety,
E — 0{U), D a Cartier divisor. Blowing up denominators as in Case 3 of the
proof of Theorem 2.4, and using the additivity of the first Chern class, we may
assume D is effective. Normalizing, X may be assumed to be normal. Let s be
the section of E whose zero is D. If u is the Thorn class of E in H2(E, E — {0}),
then s*(u) e H2(X, X - D) is a localized first Chern class of E, i.e., the image
of.s*(w) in//2 A" is ci(E). The same argument as in Lemma 19.1.3 shows that
s*(u)ncl(X) = cl[D]
in //* (/)). Since [D] = c{ (E) n [X], the conclusion follows. ?
Definition 19.1. Define Hom^A" to be the group of ^-cycles whose homol-
ogy class is zero, i.e.
Hom^AT = Ker(c/: ZkX -> H2kX).
Denote by HomJA' the cycles for which some non-trivial multiple is in
Hom^A", i.e., HomJA' is the kernel of the cycle map from ZkX to H2k(X; Q).
Two cycles are homologous if their difference is in HomiA'(or in Hom^A",
according to some authors).
For any complete scheme X, we say that a &-cycle a is numerically
equivalent to zero if
for all polynomials P in Chern classes of vector bundles on X. When X is non-
singular, this is equivalent to requiring \x /? • a = 0 for all peAkX (cf.
Example 19.1.5). Let Num^A" be the group of ^-cycles numerically equivalent
to zero on X, and let NkX= Zk AVNunuX.
Let Algi X (resp. Rat? A") be the group of cycles some non-zero multiple of
which is in Alg^A" (resp. Rat* A"). By the results of this section, for any
complete complex scheme X, there are inclusions
RaUA-o Alg*A'c=Hom*A'
n n n
Rather Alg*A"c= Hom*A'c= Num*A"c= Z*A\
Example 19.1.1. On Borel-Moore homology. For a space X which admits a
closed imbedding into a Euclidean space R", one may define the Borel-Moore
19.1 Cycle Map 375
homology groups //, (A") by
H,(X) = H"-'(Rn, R"-X)
where the groups on the right are relative singular, or Cech (cf. Dold
A) VIII.6.12) cohomology groups. Most of the basic properties needed in § 19.1
were verified more generally in Fulton-MacPherson C), which we cite here as
[BT]. For example the fact that this definition is independent of the imbedding
is proved in [BT] 3.1.5, the covariance for proper maps in [BT] 3.1.8, the cap
product (equation B) of § 19.1) in [BT] 3.1.7, the duality isomorphism (equa-
(equation C)) in [BT] 4.1.3.
If /: X -> Y is a closed imbedding, j: U -> Y the complementary open set,
the long exact sequence (b) of § 19.1 is the cohomology exact sequence of the
triple (R", R" - X, R" - Y), for some closed imbedding of Y in R". Equation
G) follows from the definition and excision; equation (8) is a special case of
[BT] 2.2(A1); equation (9) follows from [BT] 4.1.3.
If the one-point compatification Xc = X u {*} of A' is a CW complex, one
may imbed Xc in the n-sphere S", and then
H,(X) = H"-'(S"- {*}, S" - Xc) s
*}),
which identifies H,X with homology with closed or locally finite supports, i.e.
Borel-Moore homology (cf. Borel-Moore A) Thm. 3.8 and §5). Borel and
Moore give a sheaf-theoretic construction of homology groups H,X for any
locally compact space, from which the basic properties can also be derived.
For the analogous /-adic theory for algebraic varieties, see Deligne B) and
Laumon A).
Example 19.1.2 (cf. Bloch-Ogus A)). Let A" be a complete scheme over an
algebraically closed field. Then
Alg*A'/Rat*A'
is a divisible group. (The group is generated by classes in A*X of the form
P*q*{P) for V a subvariety of Xx C, C a complete non-singular curve, /? a
zero-cycle of degree zero on C, p, q the projections from Fto X and C. Since
Alg0C/RatoC = y(Q is divisible (Example 1.6.6), for any ./V we may write
/? = Nfi', so a = Np^q*/?', as required.)
Example 19.1.3. For any complex algebraic scheme X, the groups //„ X are
finitely generated. It follows that Z,,A'/Hom*A' is finitely generated. If X is
complete, the quotient group Z*A'/Num*A' is a finitely generated free abelian
group, and HomiA'/Hom*A' is finite. (Using the long exact sequence F) and
Noetherian induction, one is reduced to the case where X is projective. For a
simple proof of the triangulability of projective varieties, see Hironaka B).)
Example 19.1.4. For any complete scheme A" over any algebraically closed
field, Ar*A'=Z*A7Num*A' is a finitely generated free abelian group. (There
is a pairing N' ® Z*A"-» Z for an abelian group N', so that Num*A" is the
annihilator of N'. Thus N= N*X is torsion-free. Using the cycle map to etale
homology, it follows that N ® Q; is finitely generated. Let x,,..., xn e N give

376
Chapter 19. Algebraic, Homological, and Numerical Equivalence
a Q-basis for N ® Q. Choose y, e N' so that y, • x, = w, 4= 0, and yt ¦ x)¦ = 0 for
/ =#/ Then iV <= ? Z(x,7w;) c jV ® Q, so JV is finitely generated.)
Example 19.1.5. (a) If X is a non-singular, complete, n-dimensional variety
over a field, then a ?-cycle a on A' is numerically equivalent to zero in the sense
of this section if and only if \x a ¦ /? = 0 for all (n - fe)-cycles /? on X. (By
Riemann-Roch (§ 18.3), the Chern character determines an isomorphism form
(b) If X is a complete scheme which can be imbedded in a non-singular
variety, then the following are equivalent:
(i) a is numerically equivalent to zero,
(ii) J
x
(iii) $f*fi n a = J a yfi= 0 for all/: X -
x x
(Use Lemma 18.2 with the argument in (a).)
7, 7 non-singular,
Example 19.1.6. Let/: X -» 7 be a morphism of complete schemes over an
arbitrary field. Then/*(Num#A') c= Num* 7, so/induces a homomorphism
ft-.N^X -> N* 7, making N* a covariant functor.
If X and 7 are non-singular, /*(Num*7) <= Num,^ and / induces a
homomorphism f*:N*Y^N*X of graded rings. (Both follow from the
projection formula, using Example 19.1.5) in the second case.) More generally,
for any correspondence a : X Y, a* and a* induce homomorphisms between
N,X and N,Y.
Example 19.1.7. MacPherson Chern classes (MacPherson A)). For any
complex scheme X, and any subvariety Fez X, the function which assigns to P
the local Euler obstruction Eu/>F (cf. Example 4.2.9 (a)) is a constructable
function Eu_ Kon X. The mapping
defines an isomorphism 0 from the group of cycles Z*X to the group of
constructable functions F* X.
Define cM (? «,- [K]) to be X «; cM(V,) with cM(V,) the Mather Chern class
of Vi (Example 4.2.9 (b)). Define
cm:F,X-+AmX
by c, = cM° 0~\ The theorem of MacPherson A) is that the composite
commutes with push-forward for proper morphisms; for/: X-+Y, \? the
characteristic function of a subvariety Fof X, f*(l v)(y) is defined to be the
topological Euler characteristic of V(\f~l(y). The MacPherson Chern class
c*X of X is defined to be the class c*(l^-). For X complete, mapping X to a
point shows that MacPherson's Chern class satisfies
the topological Euler characteristic of X.
19.1 Cycle Map
377
In fact, the proof of MacPherson A) shows that c* :F* ->/4* commutes
with proper push-forward. Verdier F) has shown that these Chern classes are
compatible with specialization. Relations with other intersection operations
are discussed by Sabbah A).
Note: Three other natural definitions of Chern classes of singular varieties,
all agreeing with c(Tx) n [X] for X non-singular, have been discussed
(Examples 4.2.6 (a), (c), and 4.2.9 (b)). Few functorial properties are known for
these other classes, however.
Example 19.1.8. Cohomology operations. Let S' ://*-> H* be a cohomol-
ogy operation, where H* is cohomology with "E/p 1 coefficients, and let w be
the corresponding characteristic class:
where tp: H* X -> H* (E, E - {0}) is the Thorn isomorphism for an //-oriented
bundle E. For example, if S' is the Steenrod square, w is the Stiefel-Whitney
class (cf. Milnor-Stasheff A), Atiyah-Hirzebruch A).) Assume that for any
non-singular projective algebraic (resp. complex analytic) manifold M,
w(TM) nfiM is represented by an algebraic (resp. analytic) cycle. For example,
if S' is the Steenrod square, w is the reduction of the total Chern class mod 2,
so this is clear. Then S' preserves algebraic (resp. analytic) cycles.
More generally, for any complete algebraic scheme (resp. compact analytic
space) X, the corresponding homology operation
preserves algebraic (resp. analytic) cycles. (If V is a subvariety of X, by
Hironaka there is a proper morphism n: M ^> V, M non-singular, with
%*i^u) = cl[V]. Therefore
S.(d(V)) = TE.S.to*) = MwGV)-' n fiM)
is algebraic.)
The case of Steenrod squares and reduced power operations on non-
singular varieties was proved by Kawai A), by a more complicated ar-
argument. Note in particular that Sq' vanishes on algebraic cycles if / is odd.
Related results were given by Atiyah and Hirzebruch A).
We do not know if Sq lifts to an operation on the group ^^AT®(Z/2Z). The
above proof shows what Sq must be. The question is whether nt(c(Tuyl n [M])
is independent of the resolution.
Example 19.1.9. There are exterior products
for Borel-Moore homology. If a and /? are algebraic cycles on X and Y, then
(Since the products are compatible with restriction, this follows from the fact
that hm*n = Hm x Hn for manifolds M, N.)

378
Chapter 19. Algebraic, Homological, and Numerical Equivalence
Example 19.1.10. Refined topological intersections. For X a closed subspace of
Y, ceH*{Y,Y- X), and f:Y'->Y, ae //*(F), define cna in H,(X'),
X' =f~\X), by setting
c n fl=/*(c) n a ,
where the cap product on the right is the cap product B).
Let M be an oriented real n-manifold, Acz M a. closed subspace, a e HkA.
Define clM(a) e H"'k(M, M - A) to be the class dual to a, i.e.,
n
If also B a M,beH,B, define
ab = (clM(a) u
in Hk + i_n(A OB). If ^ e H"(M x M, M x M - A) is the orientation class of the
diagonal, then
a- b = usr\(ax b).
In addition, b-a = (- l)<"-*>('I-'>a • />.
Example 19.1.11. There is an important class of schemes X for which the
cycle map
clx:AtX->HtX
is an isomorpism. Such varieties can have no odd-dimensional homology, so
the only curves for which this holds are P1 and A1.
(a) If Y <= X is a closed subscheme, U = X — Y, and clY and clv are isomor-
isomorphisms, then clx is an isomorphism. (Use the compatible exact sequences of § 1.8
and §19.1 F).)
(b) If X has a cellular decomposition in the sense of Example 1.9.1, then
c lx is an isomorphism.
(c) If X= GIB, with B a Borel subgroup of a reductive group G, the Bruhat
decomposition (cf. Borel A)IV.14) is cellular in this sense, so clx is an isomor-
isomorphism.
(d) If AT is a projective space, Grassmannian, or arbitrary flag manifold, then
clx is an isomorphism. More generally, if AT is a flag bundle over a scheme Y, then
clx is an isomorphism if and only if clY is an isomorphism.
Example 19.1.12. If H+X denotes singular homology (homology with
compact supports), and Ai X is the group defined in Example 10.2.8, there are
homomorphisms
cl:AckX -> Hc2kX,
covariant for arbitrary morphisms.
19.2 Algebraic and Topological Intersections
Every regular (closed) imbedding i: X -> Y of codimension d has an orienta-
orientation class
Ui=uXjYe H2d(Y, Y-X).
19.2 Algebraic and Topological Intersections
379
We describe this class in three important cases, which suffice for many
applications, and then give the general construction:
(i) If Y is a vector bundle on X, and / is the zero-section imbedding, then
ux, y is the Thom class of the bundle.
(ii) If X and Y are non-singular, ux Y is the class defined in § 19.1.
(iii) If there is a vector bundle E of rank d on Y and a section s of E so that
X is the zero-scheme of s, then ux Y = s* (u), where u is the Thom class of E.
(iv) In general, choose a vector bundle E of rank e S d on Y and a section 5
of E so that X is the zero-scheme of s (Appendix B.8.2). The normal bundle ./V
to X in Y is a sub-bundle of the restriction ?"|^- of the E to X. Choose a
(classical) neighborhood Fof Xin Y, and a complex topological sub-bundle C
of .E|Kso that C\x® N= E\x. Let Q be the quotient bundle of E\vby C, and
let s be the section of Q induced by s. Shrinking V if necessary, s maps the pair
(V,V-X) to (Q, Q - {0}), with {0} the zero-section of Q. Let u be the Thom
class of Q. Then set
uxr = s*(u)eH2d(V, V-X) = H2d(Y, Y-X).
It may be verified (Baum-Fulton-MacPherson A)IV.4) that this class is
independent of choices, and agrees with the classes considered in (i) - (iii).
Lemma 19.2 (a) Consider a fibre square
X' ^ Y'
n, if
X -> Y
with i and i' regular imbeddings of codimension d. Then
/¦(«,) = «/'? Hld(Y', Y' -A").
(b) If, in (a),fandf are also regular imbeddings of codimension e, then
Ut \J Uf= Uf\J Uj = Ufj- = Ufi
inHM+2e(Y,Y-X').
(c) Let T be a non-singular, connected curve, X a scheme, i,: X -> X x T the
imbedding i, (x) = (x, /). Then for alia e Hk{XxT),
uhnae Hk-2(X)
is independent oft in T.
(d) Let i: X ^ N be the zero-section of a vector bundle N of rank d on X.
Then for all k-cycles a on N,
in H2k.2d(X).
Proof. Properties (a), (b) and (c) follow readily from the constructions of
the orientation classes. For (d), by Theorem 3.3 (a) one may assume a = [tt V\,
V a subvariety of X, n the projection from ./V to X. Using (a), we may assume
V= X. The required formula is then w, n ixn = Mx\ this is verified by
restricting to a non-singular open subset of X, in which case equation (9) of
§19.1 suffices. ?

380 Chapter 19. Algebraic, Homological, and Numerical Equivalence
Theorem 19.2. Consider a fibre square
'I
X
Y'
I9
with fa regular imbedding of codimension d. Then for all k-cycles a on Y',
clx, (/•' a) = g* (uXi r) n clr. (a)
in Hik-ldX'. Here f 'a is the class in A^-d (A") constructed in §6.2.
Proof. We may assume a = [V\, with V= Y' a variety. Construct, from §5.1,
the diagram for deformation to the normal bundle N of X in Y:
X —+ N >{oo}
4 h I
Here M° = M%Y is the blow-up of 7xP' along A'xjoo}, with the proper
transform Y of Yx {oo} omitted. Factoring # into a closed imbedding followed
by a smooth morphism, one is reduced to the case where g is a closed
imbedding. Then M°' = M^Y' is a subvariety of M°, imbedded by a
morphism G: M°' -> M°. By Lemma 19.1.3,
ncl(M")
where C' = (A/°')oo is the normal cone to X' in 7'. By Lemma 19.2(b) and
equation (8) of § 19.1,
=(9'xl)*(uJn(G*(uF)ncl(M°')).
By Lemma 19.2(c), we may replace w,0 by «,-„, in the last expression. Reversing
the argument at oo,
(g'x 1)*(«,J n (G*(uF) n cl(M°')) = g*(uf) n cl[C],
where g is the imbedding of C in iV. By Lemma 19.2(d),
This concludes the proof, since/' [ Y'] is defined to be/* [C]. ?
Let y be an /-cycle on an n-dimensional non-singular variety Y, with
support \y\ c 7. Define
to be the class such that clY(y)n nY - cl(y) in H2l(\y\) (cf. equation C) of
§19.1). If /: X -> Y is a morphism, AT any scheme, x a fe-cycle on X, then
19.2 Algebraic and Topological Intersections
381
we have constructed a refined intersection class x -fy in Am(Z), where
Z=\x\ r\f-\\y\),m = k + l- n (§8.1).
Proposition 19.2. Letf be the morphism from \x\to Y induced byf. Then
inH2m(Z).
Proof. Note that/'*c lr(y) e H*(\x\, \x\ — Z), so the cap product on the
right lands in H*Z. As usual, we may assume x = [V\, V= \x\ = X a variety,
so/' =/ Apply Theorem 19.2 to the fibre square
Z >Xx\y\
X >Xx Y
y
with y the graph of /, g the inclusion. This yields:
clz{x-ty) = g*(u1)ncl(Xxy).
Now if p is the projection from X x Y to Y,
cl(X xy) = p*(clY(y))ncl(X x Y).
By the commutativity of even dimensional cohomology classes, and equation (8)
of§19.1,
g*(uy) n (p*(dY(y)) ncl(Xx Y)) = f*(clY(y)) n («, ncl(Xx Y))
= f*(clY(y))ncl(X),
the last step by Theorem 19.2 again. Combining the displayed equations
concludes the proof. ?
Let AT be a fixed non-singular n-dimensional variety, V, W subvarieties of X
of dimensions k, I. We have constructed V- W in Am(V f]W),m = k + I - n,
in §8.1. A corresponding refined class is defined topologically as follows. Since
clx(V)eH2"-2k(X,X-V), clx(W)eH2n-2i(X,X- W),
we have dx(V) u clx(W)e H*"-2"-2'(X, X-Vf)W). Set
cl(V) ¦ cl(W) = (clx(V) u clx{W)) n nx
in H2m(Vf) W) (cf. Example 19.1.10).
Corollary 19.2. (a) With the above notation,
d(V- W) = cl(V)-cl(W)
inH2m(Vf]W).
(b) The mapping cl:A*X^>H*X is a homomorphism of graded rings,
contravariant for morphisms of non-singular varieties.

382 Chapter 19. Algebraic, Homological, and Numerical Equivalence
Proof. For (a), apply Proposition 19.2 to the inclusion/of Fin A", x = [V\,
y =[W}. This yields
W) = f*clx(W)nclv(V)
= f*clx(W)n(clx(V)nnx)
using equation (8) of §19.1, as required. The fact that cl commutes with /* for
/ : X -> Y, is a special case of Proposition 19.2, with x = [X]. ?
For X quasi-projective and non-singular, the fact that cl is a ring
homomorphism may also be deduced from the moving lemma, using the
strong form stated in Example 11.4.2.
Analogous results hold for quasi-projective schemes over an arbitrary
algebraically closed field, with values in etale homology and cohomology. For
details, as well as a sheaf-theoretic approach in the classical case, see Exposes
VI-IX in the seminar of Douady-Verdier A). This approach yields several
important generalizations, including: (i) a flat pull-back homomorphism in
homology, compatible with that defined in § 1.7 for cycles; (ii) cycle classes for
varieties not necessarily imbeddable in smooth varieties, and for general
complex analytic spaces. Gillet C) constructs the classes uXY for a regular
imbedding X ?-> Y without assuming Y can be imbedded in a smooth variety.
Example 19.2.1 (cf. Baum-Fulton-MacPherson A)IV.4, and Fulton-Mac-
Pherson C)). For any l.c.i. morphism /: X -> Y of relative dimension d there
are Gysin maps
P.HiY^H^X.
These are functorial, and compatible with the cycle class, i.e. f*clr(a)
= clxf*a for any cycle a on Y. (Imbed AT in a smooth M, so / factors into an
imbedding / into Yx M followed by the projection p to Y. Define/* = i*p*,
where p* (a) = a x fiM, and /* (/J) = w,n /?.) If/is also proper, there are also
functorial Gysin maps/*: H'X -> Hi+2dY.
Example 19.2.2. Excess intersection formula (cf. Fulton-MacPherson C)).
Given a fibre square
A"-> Y'
el if
X -fY
with / (resp. ;') a regular imbedding of codimension d (resp. d'), with excess
bundle ?(§6.3), then
in H2d(Y', Y' - A"). e = d-d'. The product on the right is defined as follows. If
c e H* A", extend c to a class c e H* U' for some topological neighborhood U'
of A" in Y'. For u in H* (Y', Y' - A"), identified with H*(U', U' - A") by
excision, c ¦ u is defined to be c u u.
Theorem 19.2 follows from this formula, since by blowing up we may
assume a = [ K], V= Y', and either A" is a divisor on Y' or A" = Y'.
19.2 Algebraic and Topological Intersections
383
Example 19.2.3. Let /: X -> Y be a l.c.i. morphism of complete schemes
which are imbeddable in non-singular varieties, over an arbitrary field. Then
/* (Num* Y) c Num* X, so A** is a contravariant functor for l.c.i. morphisms.
(If aeNum,!; write the image of a in A^Y^ as xrF), 9eKoY, and use
Theorem 18.3 D).)
Example 19.2.4. Linking numbers. Let V, Wb be subvarieties of a non-
singular complex variety X", of the indicated dimensions, meeting properly at
a point P in X; so n = a + b. Identify (complex analytically) a neighborhood of
P in A" with a neighborhood of 0 in C". If S is a small Bn - l)-sphere about P,
then S meets Fin a real topological Ba — l)-cycle, which we denoted by A. In
fact, if D is the closed ball around P with boundary S, then /) Pi F is
homeomorphic to the cone over A with vertex P. Similarly, S meets W in a
topological B6 - l)-cycle 5.
The linking number L(A, B) of A and B in 5 may be defined as follows.
Choose a 2/>-chain B' in 5 whose boundary is B, so B' determines a class {5'}
in H2b(S, B). Let cls(ANfl2l(S,S-^) be the class dual to the class of A in
H2a-i(A) (cf- Example 19.1.10). Then the relative cap product cls(A) n {B1} lies in
Ho(A, AC\B) = H0A, since ^f]B = 0, and we may set
L(A,B) = deg(cls(A)n{B'}).
Proposition. L(A, B) = i(P, F- FF; X).
(Since c/*G) e tf "(AT, X - V) restricts to
(i) L(A,B) = deg(clx(V)n{B'}).
Let WD = Wfl/X Then WD determines a chain on D whose boundary is B, so a
class {%} in H2h(WD,B), and
(ii) J"(P, F- W; X) = deg(c/x(F) n {WD})
where c/*G) n {%} eH0(VC\ WD, VC\B) = H0(P). This follows from Corol-
Corollary 19.2 and the fact that {WD} = clx{W) n {?»}, where {?>} is the canonical
generator of H2n(D, S).
Now B' and WD both have boundary B, so B' — WD is & cycle on ?>. Since
H2bD = 0, {B1} and {%} have the same image in H2b(D,B). The zero cycles
appearing in (i) and (ii) therefore have the same image in H0(VC\D, Vf]B)
- H0(Vf]D), so they have the same degree.)
The case of two curves on a smooth surface is discussed by Martinelli A) and
Reeve A).
Example 19.2.5. Analytic spaces. Much of the intersection theory developed
in this text for algebraic schemes can be extended to analytic spaces. (One
should use projective bundles and cones, rather than affine cones, to avoid
arguments using closures of locally closed subvarieties.) For spaces imbedd-
imbeddable in complex manifolds, the methods of this chapter apply. For example, if
V and W are irreducible analytic subspaces of a complex manifold X, the
intersection class cl\V)-cl(W) can be constructed in H^VCWV); if V and
W meet properly, this determines intersection numbers, and therefore V- Was

384
Chapter 19. Algebraic, Homological, and Numerical Equivalence
an analytic cycle on X. By the methods of Chap. 6, cl(V)cl(W) is always repre-
represented by an analytic cycle on Vf] W (cf. King C)).
For divisors, the results of Chap. 2 extend without change. Such a refined
intersection theory is useful in studying families of compact analytic (or
algebraic) families parametrized by a disk (cf. Persson A)).
The multiplicity of an analytic variety at a point, as in Chap. 4, is the
degree of the projective tangent cone to the variety at the point. This multiplic-
multiplicity can also be realized analytically as the Lelong number of the variety at the
point (cf. Harvey A), Griffiths-Harris AK.2). More generally, for any closed
imbedding X a Y of analytic spaces, there is a Segre class s (X, Y) in H* (X).
If ?>,,..., ?>„ are divisors meeting properly at a point P in an n-
dimensional manifold X, and/ is a local equation for ?>, at P, then (Griffiths-
Harris A) 5.2, cf. Picard A))
,-(P,/V .-¦ A,; *) = Res0(^ a ... a4^) •
\J\ hi
For an n-dimensional subvariety A'of P, with the Fubini-Study metric,
Volume(X) = deg(A-) • Volume (L)
with L an ^-dimensional linear subspace of P. For this and other relations
between volumes and intersection theory, see the discussion of Wirtinger's
theorem inMumford E) and Griffiths-Harris A).
Example 19.2.6. If a vector bundle E of rank r on a scheme X has a section
s, and Z is the zero-scheme of s, then the top Chern class cr(E) in H2r(X)
comes from a well-determined class in H2r(X, X— Z), namely the class s*(u),
where u e H2r(E, E — {0}) is the Thorn class of E. More generally, if Z is
closed in X, a nowhere vanishing section of E on X — Z determines such a
localization of cr(E) (cf. Atiyah-Bott-Shapiro AI1).
A complex E. of vector bundles on a space Y exact off a closed subspace X,
determines a local Chern character chYx(E.) in H*(Y, Y-X; Q) (cf. Atiyah-
Hirzebruch B), Iversen C)). Similarly, it is only necessary to have the bundles
E, on Y, and the exact complex on Y-X, to construct such a class (Atiyah-
Bott-Shapiro AI1, cf. Baum-Fulton-MacPherson B) App. 1). This Chern
character is compatible with that of § 18.1, i.e.,
clx(chrx(E.) n a) = chrx(E.) n c/r(a)
in //* (X; Q), for any a eA*Y.
Example 19.2.7 A topological correspondence 0 : X \— Y between compact
oriented manifolds X and Y is a class 0 e H* (X x Y). The formalism of
Chap. 16 goes through without change. For example, 0 determines homo-
morphisms 0*:HifX^> H*Y by 0*(a) = q*(a • p*a), where p and q are the
projections from XxY lo X and Y, and similarly for 0*, and for composites of
correspondences.
19.3 Equivalence on Non-singular Varieties
385
If X and Y are non-singular complete varieties, and a e A*(X x Y) is an
algebraic correspondence, then cl(a) is a topological correspondence, and if
aeA^ (X), then
(This follows from Lemma 19.1.2 and Corollary 19.2.)
19.3 Equivalence on Non-singular Varieties
In this section X will be a non-singular, complex projective variety of
dimension n. The groups of cycles will be indexed by superscripts denoting
codimension. Our main purpose is to sketch what is known about relations
among the groups defined generally in § 19.1:
Rat"X c Alg'X c= Hom/'A'
n
Hom?A' cz Num'A' <= Z"X.
19.3.1. Divisors. For p=\, the situation is quite well understood. In this
case:
(ii) Alg{A'=HomiA'=Num1A'.
(iii) The Neron-Severi group NS(X) = Z'X/A^X is finitely generated,
(iv) The Picard variety Pic0A"= Alg1 AVRat1 X has the structure of an
abelian variety whose dimension is dim H' (X, fix).
(v) AlgtJST/Alg1A'= H2(X, Z)lore, a finite group.
The basis for these facts is the cohomology sequence of the exact sequence of
analytic sheaves
0 -> Z -*0X" -+ fix"* -* 0 ,
where e(f) = expBre if), and the GAGA theorems that H'(X,W) =
//' (X, fix), Hx (X, fix" *) = Hl (X, fi*x) = Pic (A"). This gives an exact sequence
W (X, 1) -> W (X,
Pic (X) -i H2 (X, TL) - H2 (X, 0X).
The boundary map S is the first Chern class, i.e. S(O(D)) = c,(<P(D)) = cl [D].
The Picard variety Pic°(JQ is the quotient of Hl(X,0x) by the image of
H\X, Z), which is a lattice in H] (X,fix)- Assertions (i)-(v) follow from these
facts.
19.3.2. Z*A-/Hom*A-.
In general, since H* X is finitely generated (for example, from the fact that
Xis a compact manifold), Z* A'/Hom*A'is finitely generated. Therefore
(i) Z/)A7Num''A'is a finitely generated free abelian group,
(ii) Horn? A'/Hom^A' is a finite group.

386 Chapter 19. Algebraic, Homological, and Numerical Equivalence
It is not known if homological and numerical equivalence coincide for all p,
i.e.
(iii) IsHom?Ar=Num'IAr?
For p = 0, n, this is trivially true, and for p = 1 it is true by 19.3.1. Lieberman A)
has verified the equality for p = 2, and for all p when X is an abelian variety; in
general the question is closely related to Grothendieck's "standard conjectures"
(cf. Example 19.3.14, Grothendieck F), Kleiman B)).
19.3.3. Hom*AVAlg*Ar.
Griffiths B) showed that Alg?X can be unequal to Hom?A' if 1 < p < n.
Griffiths constructed a 3-fold with
HomjAVAlg?* infinite.
Ceresa A) has shown that if X is the Jacobian of a general curve C of genus
n ^ 3, <pthe endomorphism of X which is multiplication by — 1, and wp eApXis
the class of the image of C1""'0 in X, then
is not algebraically equivalent to zero, if 1 < p < n, although it is homologous
to zero, since q>* is the identity on H*X. Other naturally occurring examples
have been given by Ceresa and Collino A).
Recently Clemens A) has shown that there are 3-folds X with Hom?A7Alg?A'
not finitely generated.
It is not known if Alg^X/Mg^ is always finite.
19.3.4. Mg*X/Rat*X.
For p > 1, the structure of the groups Alg? X/Raf X has been determined
in only a few cases. In several cases this group has been given the structure of
an abelian variety, usually by relating these groups to Jacobians of certain
curves associated to X. For examples of this see Murre B), Bloch-Murre A),
and Beauville A).
In general, however, for/? ^ 2, the groups AlgpX/Raf X can be larger than
any abelian variety, as shown by examples of Mumford, discussed next.
19.3.5. Zero-cycles. Forp = n, Alg" X= Num" X, and Z"X/Alg"Z = Z. The
interesting group is the group
i0A'=Alg"A'/Rat"A'
of zero-cycles of degree zero, modulo rational equivalence.
Mumford C) showed that when X is a surface with geometric genus pq > 0,
then Aq X cannot be given a structure of an abelian variety, if one requires that
the canonical maps
for Pa a base point, be morphisms of algebraic varieties. Indeed, the fibres of
this map act as if A0Xis infinite dimensional.
Bloch has conjectured that A0X is finite dimensional when pg=0- For a
discussion of evidence for this, see Bloch-Kas-Lieberman A) and Bloch D).
For relations with vector bundles, see Murthy-Swan A).
19.3 Equivalence on Non-singular Varieties
387
For a general X, Roitman B) has shown that the canonical map (Example
1.4.4)
9:A0X->Mb(X)
to the Albanese variety induces an isomorphism on the torsion points.
19.3.6. Hodge theory. With complex coefficients, the cohomology of X has a
canonical Hodge decomposition
Hk(X;C) = © H"'i(X)
p + q'k
with H^(X) = W(X) s H"(X, *2?). If a is an algebraic cycle of codimen-
sion p on X, then the image of cl(a) in H2p{X; C) lies in H"-P(X). This follows
from the fact that, if zu ..., zn are local holomorphic coordinates on X, a form
with more than n — p dz's or dz's must vanish on the non-singular locus of a
subvariety of codimension p.
For p = 1, the Lefschetz-Hodge theorem asserts that every class in //'•' (X)
which is in the image of H2(X,Z) is algebraic. This is proved by a direct
calculation (cf. Griffiths-Harris A) p. 163). For p > 1, the celebrated Hodge
conjecture asks if every class in Hp-p(X)r\H2p(X; Q) is a rational combination
of algebraic cycles.
A principal aim of the theory of intermediate jacobians has been to study
algebraic cycles by means of a complex torus P (X) and an Abel-Jacobi map
generalizing the map A0X -> Alb(A') for 0-cycles. Griffiths, modifying a con-
construction of Weil, achieves this by defining
J"X
1 (X;
+' (X;
where Hi (resp. H?) denotes the image of H'(X;Z) (resp. Hi(X;Z)) in
H(X; R) (resp. H,(X; R)), and the * denotes the dual of a real vector space
Set
Then
where V denotes the complex dual. These descriptions put a complex structure
onJ"X.
Given a cycle a e Hom^, write a as the boundary of a Bn
chain (or current) r, and define 9 (a) to be the functional
Je»,
r
co e
These complex tori vary holomorphically if X varies in a holomorphic family
ST-* T. If a is a cycle of codimension p on the total space %~, whose specializa-
specializations a, are homologous to zero on X,, one may study how 9(a) varies with /.
For some of the successful applications of this theme, see Griffiths B), C),
Clemens-Griffiths A), Zucker A), Carlson-Green-Griffiths-Harris A).

388
Chapter 19. Algebraic, Homological, and Numerical Equivalence
The Hodge conjecture is also closely related to Grothendieck's "standard
conjectures" for algebraic cycles (Grothendieck F), Kleiman B)). For some
recent progress on the Hodge conjecture see Deligne-Milne-Ogus-Shih A).
Example 19.3.1. Let X be a non-singular projective surface. The group
N(X) = Z'AVNum'A^ Z'AVAlgU
is a free abelian group of rank q = rank NS (X) = dim H' •' (X) 0 H2 (X; % This
group plays an important role in the study of curves on X (cf. Mumford B),
Beauville B)).
(a) By the Hodge index theorem (Example 15.2.4), the intersection pairing
N(X)®
given by C, D -» \C ¦ D, is a perfect pairing over Q, of index 2 — q.
Equivalently, if C, D are divisors on X, with J C2 > 0 and j C ¦ D < 0, then
j D1 S 0, with equality if and only if a multiple of D is algebraically equivalent
to zero.
(b) If X is the blow-up of X at a point P, then
with ee = —1, and e perpendicular to N(X). (See Proposition 6.7; e is the
class of the exceptional divisor.)
(c) If X = P (E), where ? is a vector bundle of rank 2 on a curve Y, then
g=2 and N(X) has a basis x, y with x2 = 0, x-y= 1, y*= — c, where c = jc! (?).
(d) (Segre B)). For any divisor D on X,
\ D2 ^ (deg(D)J/deg(X)
where deg (D) = j D • //, deg (X) = j //2, for any ample divisor H on X.
Equality holds when D and H are dependent in N (X). (Apply (a) to
deg(X)-D- deg(D)H.)
Example 19.3.2. The index of X" is the index of the symmetric bilinear
form on H"(X\W) given by the cup product (intersection pairing). When
n = 2, the index theorem of Example 15.2.11 reads
Index (X) = 2 + 2h°-2 -//>¦',
where hM= dim//A'. Equivalently, with c, = c(G»,
(I c2 = ?(- l)p+« //"•* and J c2 + c2 = 12A - /zoa + /i0'2) by Examples 18.3.7 and
15.2.2.) Since the pairing puts H2'° and H°'2 into duality, this index theorem is
equivalent to the assertion, that the induced pairing on H11 has index 1 — h1'1,
which implies the index theorem of Example 15.2.4.
The Hodge index theorem also has implications for algebraic cycles if
n > 2. For example, if n = 2p is even, and a is a p-cyde on X such that H ¦ a is
homologous to zero for an ample divisor H, then
19.3 Equivalence on Non-singular Varieties
389
with equality if and only if a is homologous to zero. For n > 2, only the
transcendental proof is known. For generalizations to p<2n, see Kleiman
B)-
Example 19.3.3. Let X be a projective variety of dimension n fe 2 over an
algebraically closed field, H an ample divisor on X. For a Cartier divisor D on
X, the following are equivalent:
(i) \ D ¦ a = 0 for all 1-cycles a on X.
x
(ii) fi(m D) is algebraically equivalent to zero in Pic (X), for some m =t= 0.
(Hi) \D-H"-{ = \D2Hn~2 = 0.
x x
(iv)
= X(x>$~) for a11 coherent sheaves^on
In particular, D is numerically equivalent to zero if and only if the restric-
restriction of D to a hyperplane section of X is numerically equivalent to zero. (For
non-singular complex varieties, (iii) => (i) follows from the Hard Lefshetz
theorem and the index theorem on surfaces, (i) <=> (iv) also follows from the
singular Riemann-Roch theorem. For details see Matsusaka A) and Kleiman
[SGA6]XIII.4.6.)
If X is normal and n §= 3, Weil E) (cf. [SGA6]XIII.3) shows that D is
linearly equivalent to zero if and only if the restriction of D to a generic
hyperplane section is linearly equivalent to zero. A similar result holds for
DeRat[A'if/7 = 2.
There are few results of this kind known for cycles of codimension greater
than 1. Note that by Griffiths' example a 1-cycle a on a 3-fold X can be
homologous to zero, so the restriction of a. to all surfaces in X will be
algebraically equivalent to zero, without any multiple of a. being algebraically
equivalent to zero.
Example 19.3.4. If D is a Cartier divisor on a complex manifold X, corres-
corresponding to a line bundle L with a sections, then cl(D) = cl(L)eHl'l(X) is
represented by the form — TTt 9, where 9 is the curvature form of any
connection for L; equivalently, 0 = <^3 log|s|2, where the norm is taken with
respect to a Hermitian metric on L. (Cf. Griffiths-Harris A) p. 141.)
Weil C) showed that, if X is projective, D is homologous to zero if and
only if D is the residue of a closed meromorhpic 1-form on X. For extensions
of this result to 0-cycles and cycles of arbitrary dimension see Griffiths E) and
Coleff-Herrera-Lieberman A).
Example 19.3.5. For p > 1, using a construction of Serre, Atiyah and
Hirzebruch B) give examples of classes of finite order in H2p(X, 1) which are
not algebraic.
Example 19.3.6. If at" is the sheaf associated to the pre-sheaf U -> H"(U, 2),
in the Zariski topology, on a non-singular complex variety X, Bloch and Ogus
A) show that there is a natural isomorphism
Z'(X)/Mg"X

390 Chapter 19. Algebraic, Homological, and Numerical Equivalence
Example 19.3.7. The canonical decomposition
where r = min (j>, m) (Theorem 3.3) induces corresponding isomorphisms on
the subgroups G*/Rat*, where G* denotes any of the groups Alg*, Horn*,
Num*, Algf, or Horn*:. In particular, examples of X and p where any of the
quotient groups is large can be crossed with Pm to find similar examples with
larger codimensions and dimensions.
Example 19.3.8 (Roltman BL.2). If an ^-dimensional variety X is an
irreducible component of the intersection of r hypersurfaces of degrees
du...,d, in P"+r, with Y, di ^ n + r> then A0X=0. If, however, X is a non-
singular complete intersection of r hypersurfaces, with ^ d, > n + r, n S 2,
then A0X is infinite dimensional.
Example 19.3.9. If X is non-singular, the images of A\g*X, Hom*A\
Num**, Mg*X, and Horn** in A*X are all ideals. The quotient of Z*X
modulo each of these subgroups is a graded ring, contravariant for morphisms
of non-singular varieties.
Example 19.3.10. If X is a non-singular complete intersection in Pm, then
HiX=HiFm for i < n, so Z" X mom" X = "E forp < nil, Hartshorne D) asks if
Hom"A'= Alg'* = RafXforp < nil.
Example 19.3.11. Several equivalence relations lying between Rat*X and
Alg*X have been defined, usually involving families of cycles parametrized by
abelian varieties. For discussions of some of these, we refer to Samuel D),
Weil E), Griffiths C), and Kleiman D).
Example 19.3.12. If X is a complex scheme which is imbeddable in a non-
singular scheme, and one defines
A*X=hmA*Y
the direct limit taken over all f:X-* Y, Y non-singular (cf. Example 8.3.13),
there is an induced homomorphism of graded rings
cl:A*X-+H*(X).
If A*X is the bivariant cohomology ring defined in § 17, however, we do not
know if there is such a map.
Example 19.3.13. If _/": JT —> Y is holomorphic, then the Abel-Jacobi map 0
commutes with /* and /*. Equivalently, if T.YV-X is an algebraic cor-
correspondence, then for all cycles a on Y
(cf. Example 19.2.7). See Griffiths C) for a discussion of this fact, and King
A) for the technique involved.
The fact that 0 vanishes on Rat*X may be seen as a special case of this
formula (cf. Lieberman B)). Indeed, a cycle in Rat*X is of the form T* (a), for
Notes and References
391
T: F[\-X, a a cycle of degree zero on P1. Then d(T* a) = 7*0 (a) =0 since
0(a) e/1Pl = Pic°(P1) = 0.
Example 19.3.14. If H"-p'"'p(X)f]H2n-2l'(X; Q) consists of rational linear
combinations of algebraic cycles, (i.e., the Hodge conjecture holds for p-cycles
on X), then Hom?X = Num"X.
Notes and References
Several references for the transcendetal theory of algebraic cycles have been
included in the text and examples. For general surveys, we recommend Grif-
Griffiths C), D), Griffiths-Harris A), Carlson-Green-Griffiths-Harris A), Hart-
Hartshorne D), and Kleiman B).
Although the use of topology and homology on algebraic varieties can be
traced back to Riemann and Poincare, it was Lefschetz who developed a
simplicial theory for intersecting homology classes on an ordinary manifold,
and applied it to the classes of algebraic subvarieties of complex projective
manifold (cf. Lefschetz B)). This theory depended on the triangulability of
complex varieties (cf. Hironaka B)). Borel and Haefliger A) used relative
cohomology and Borel-Moore homology to construct the cycle map and give a
modern proof that algebraic and topological intersections agree. Refinements
were given by Atiyah and Hirzebruch B) and Douady A).
Sections 19.1 and 19.2 extend these results to refined intersections on non-
singular varieties, and to local complete intersections on possibly singular
varieties. These results stem from Baum-Fulton-MacPherson A), Fulton-
MacPherson A), C), Deligne B), and the sources cited at the end of § 19.2.
Many of the above sources construct intersections on complex analytic as
well as algebraic varieties. Others have used the machinery of differential
forms, residues, currents and kernel functions to give more or less explicit
analytic formulas for cycle classes. In particular, King C) gave an analytic
version of the excess intersection formula, some time before the formula
appeared in modern algebraic geometry. Other references for the analytic
approach are Draper A), King A), B), Raisonnier A), and Toledo and Tong
B).
Borel-Moore homology is the natural homology theory for use on non-
compact algebraic or analytic varieties. In addition to the original source,
Borel-Moore A), one may refer to Verdier A), Douady-Verdier A), or Iversen
D) for the general sheaf-theoretic development. Since this theory also arises
naturally in the intersection homology theory of Goresky and MacPherson A),
B), we look forward to its becoming more familiar — perhaps appearing in
standard topology texts.
Many of the results discussed in this chapter are valid for varieties over
arbitrary algebraically closed fields, using etale in place of classical homology
and cohomology. For this we refer to Deligne B), and to articles by Laumon

392
Chapter 19. Algebraic, Homological, and Numerical Equivalence
and Verdier in the seminar of Douady-Verdier A). Grothendieck and Katz
have extended Griffiths' example (§ 19.3.3) to characteristic p (Deligne-Katz
A)XX). The basic facts on divisors (§ 19.3.1) were proved in the abstract case
by Matsusaka and Weil (see Example 19.3.3 for references). Bloch C) and
Milne A) have extended Roitman's theorem (§ 19.3.5) to characteristic p
For discussion of relations of algebraic cycles to A^-theory and number
theory, as well as much more on zero-cycles, we recommend the lectures of
Bloch D).
Chapter 20. Generalizations
Summary
Much of the intersection theory developed in this text is valid for more general
schemes than algebraic schemes over a field. A convenient category, sufficient
for applications envisaged at present, is the category of schemes X of finite
type over a regular base scheme S. Using an appropriate definition of relative
dimension, one has a notion of A:-cycle on X, and a graded group A* (X) of
rational equivalence classes, satisfying the main functorial properties of Chaps.
1 — 6. The Riemann-Roch theorem also holds; in particular
KO(X) ® Q .
The main missing ingredient in such generality is an exterior product
Ak(X)
+l(XxsY) .
If S is one-dimensional, however, there is such a product. In particular, i f X is
smooth over S, then A* (X) has a natural ring structure.
When 5 = Spec(/?), R a discrete valuation ring, and X is a scheme over S,
with general fibre X ° and special fibre X, there are specialization maps
which are compatible with all our intersection operations. If X is smooth over
S, a is a homomorphism of rings.
For proper intersections on a regular scheme, Serre has defined intersec-
intersection numbers using Tor. For smooth schemes over a field, these numbers agree
with those in § 8.2. Indeed, even for improper intersections, a Riemann-Roch
construction shows how to recover intersection classes from Tor, at least with
rational coefficients.
Although higher A^-theory is outside the scope of this work, the chapter
concludes with a brief discussion of Bloch's formula
20.1 Schemes Over a Regular Base Scheme
In this section S denotes an arbitrary regular scheme, i.e., S is a Noetherian
scheme all of whose local rings are regular local rings. All schemes X will be of

394
Chapter 20. Generalizations
finite type and separated over S. No assumption of a ground field is made; in
particular, S could be Spec (A), with A: (i) the ring of integers in a number
field on any Dedekind domain, (ii) a p-adic ring of integers or any discrete
valuation ring; (iii) any regular ring.
We introduce a notion of relative dimension for schemes over S, which
shares many properties with absolute dimension over a ground field. For
p: X-* S a scheme over S, and Vcz X a closed integral subscheme of X, define
dimsV=tr.deg.(R(V)/R(T))-codim(T,S)
where T is the closure of p(V) in S, and R(V),R(T) are the function fields. If
v eX is the generic point of V, and t = p(v), then
dims V= tr.deg. (x(v)/x(t)) - dim (f,iS).
Note that in case 5, is an algebraic scheme over a field, dims V=
dim (V) - dim E).
Lemma 20.1. A) If V° is a non-empty open subscheme of V, then dims V° =
dims V-
B)IfWa V is a closed integral subscheme of V, then
dims V= dims W + codim (W,V).
C) //"/: K-> W is a dominant morphism of integral schemes over S, then
dims V= dims W + tr. deg. («(V)/R (W)) .
In particular, dims (-F^ dimsK, with equality if and only if R(V) is a finite
extension ofR(W).
Proof. A) and C) are immediate from the definition. For B), let T and U
be the closures of the images of V and W in S. Since S is universally catenary,
we have the dimension formula [EGAJIV.5.6.5:
tr. deg. (R (V)/R (T)) + codim (U,T) = tr. deg. (R (W)/R(U)) + codim (W,V),
which is equivalent to B). ?
Using this relative notion of dimension, a A:-cycle on a scheme X can be
defined, as in § 1.3, to be a formal sum X "/[^L with V, closed integral sub-
schemes of X such that d\msVj=k. The notion of rational equivalence is
defined as in § 1.3. Denote by AkX or Ak{X/S) the group of rational
equivalence classes of A:-cycle. Note that the group
A,X=@Ak(X/S)
k
does not depend on S\ only the grading depends on the dimension relative to
S.
One has the covariance of Ak for proper morphisms, contravariance for flat
morphisms (of some relative dimension), and the exact sequence of § 1.8. (For
Proposition 1.4(b) the proof appealing to [EGAJIII must be used.) The
alternate definition of A*X (§ 1.6) is also valid provided one uses XxsPs in
20.1 Schemes Over a Regular Base Scheme
395
place of XxF1. However, one does not in general have exterior products as in
§ 1.10, since subvarieties of X need not be flat over S.
With these foundations, the rest of § 2 - § 6 carries over without essential
change. As above, appearances of XxP" or Xx A" should be replaced by
XxsFs or XxsAs, and any other mention of exterior products should be
omitted. Section 7.1 goes over as stated, but §7.2 (where finiteness of a
normalization is appealed to) does not.
In particular, one has Chern class operators
for E a vector bundle on X, and refined Gysin homomorphisms:
f:AkY'-+Ak-d(XxYY')
for f:X^> Fa (factorable) l.c.i. morphism of codimension d, F'-> Farbitrary,
satisfying the fundamental properties of § 3 and § 6. If F is a scheme which is
smooth over S of relative dimension n, the diagonal is a regular imbedding, so
one has Gysin homomorphisms
&*:Ak(YxsY)-+Ak.t{Y).
Lacking exterior products AirY®AirY-^Air{YxsY), however, this does not
determine products for arbitrary cycle classes on F (cf. Example 20.1.1).
The residual intersection theorem (§ 9) generalizes without change. Many
of the other results of § 9 — § 13 have analogues in this generality; we leave this
to the interested reader. The formulas for degeneracy loci in § 14 generalize
without change. In addition, the Riemann-Roch theorems in § 15 and § 1 8 are
valid, with the same proofs, in this generality. In particular, this implies that
Jf V '""*¦' A V '""*¦' (~* r Jf V
where GrKoX is constructed from the filtration of KOX using relative
dimension over S.
The definition and formalism (§17) of bivariant theories A(X-^Y)
extends to schemes over S; only Proposition 17.3.1 and Corollary 17.4, where
exterior products were used, must be omitted. In particular, one has a contra-
variant functor A* from schemes over S to rings.
Example 20.1.1. (a) If F-> S is smooth of relative dimension n, then, with
the notation of Chap. 17,
Ap-"(J^ S).
"(X-S).
(b) If Vc Y is a closed subscheme of F which is flat over S of relative
dimension m, then Vdetermines an element cv in A~m(Y-+ S), and hence a
class inA"~mYif Fis as in (a). In particular, (^determines homomorphisms
f*(cy)n_:AkX-+Ak+m-nX
for any /: X-* Y. (For proofs of (a) and (b) see Proposition 17.4.2.)

396
Chapter 20. Generalizations
(c) If S is smooth over a ground field, then the canonical homomorphism
from A* (Y—* S) to A* Y is an isomorphism. On the other hand, if one knows
that this homomorphism is an isomorphism, and Y is smooth over S, then
A*Y = A* Y has an intersection product (cf. Example 20.2.3).
Example 20.1.2. Iff-.X-* Yis a morphism of smooth S-schemes, of relative
dimensions n and m over S, there is a pull-back
f:Ak(Y/S)->Ak+m-m(X/S)
defined as the composite yj ° p*, yf:X-+XxsY the graph of /, p : X xs Y-* Y
the (flat) projection (for this X need only be flat over S). The construction of
the double point class D (/), and the proof of the double point formula in
§ 9.3, generalize without essential change to this setting.
Example 20.1.3. For an arbitrary Noetherian scheme X, let Z.X be the free
abelian group on the subvarieties (closed integral subschemes) V of X. For
r e R(V)*, let [div(/•)] = ? ord»,(r)[ W], the sum over all subvarieties W of V
of codimension one. Define A.X to be Z.X/~, where ~ is the subgroup
generated by all such [div (/•)]. If /: X -> Y is proper, define /„ : Z.X -> Z. F by
where deg (K//(K)) = [/? (K): «(/(K))] if this number is finite, and
deg(V/f(V)) = 0 otherwise. Then/* passes to rational equivalence, making A.
a covariant functor for proper morphisms. Similarly, A. is contravariant for flat
morphisms.
Using infinite cycles Y, nv[V\ which are locally finite (i.e., each point has a
neighborhood U such that only a finite number of V with nv =t= 0 meet U), the
definition extends to arbitrary locally Noetherian schemes.
For X locally of finite type over a universally catenary base scheme S, the
definition of relative dimension given in this section determines a grading for
A.(X). We do not know if the Riemann-Roch theorem extends to this
generality.
Example 20.1.4. The preceding construction applies in particular to the case
X = Spec (R), R a local Noetherian ring. If R is the local ring of a variety Y at
y, A^X has been called the local Chow group of Y at y. Equivalently, AlfX =
lim A*(U), where the limit is over all open subschemes U of Y which contain
y. The case when Y is the cone over a smooth projective variety, with vertex y,
is discussed by Grothendieck in [SGA6] Exp. XIV.8.
If R is regular, it follows from Riemann-Roch that A* (X)q = Q, generated
by[Jf].
A related, but different, construction of rational equivalence for locally
Cohen-Macaulay rings has been studied by Claborn and Fossum A).
Example 20.1.5. If the ordinary dimension of integral schemes is used, none
of the three assertions of Lemma 20.1 is true. However, the reader who wishes
may replace each dims ^by dims V+ dimE), with only notational changes; in
§ 20.3 specialization will then change dimension of cycle classes. When X is proper
over S, this notion of relative dimension is studied in [SGA 6] Exp. V.
20.2 Schemes Over a Dedekind Domain
20.2 Schemes Over a Dedekind Domain
397
In this section S is a one-dimensional regular scheme. The important case is
when 5 = Spec (/I), A a Dedekind domain. All schemes are of finite type and
separated over S. Any variety (integral scheme) V is either flat over S, or
maps to a closed point P in S. Let X and Y be schemes over S, V<=.X, We Y
subvarieties (closed integral subschemes). Define a.product cycle [V] xs [W] on
XxsYby
[ Vxs W] if V or W is flat over S
0 otherwise.
[V\xs[W] =
If dim5 V= k, d\ms W=l, then [K]xs\W\ is a (k + /)-cycle.
Proposition 20.2. The above product passes to rational equivalence, defining
an exterior product
Ak(X/S)®Ai(Y/S)^Ak+,(XxsY/S).
Proof. If a is rationally equivalent to 0 on I, W <=. Y a subvariety, we must
show that a xs [ W\ ~ 0. If W is flat over S, the induced map p : X xs W-* X is
flat, and
since rational equivalence pulls back by flat maps. If W maps to P in S, let ir
be the imbedding of P in S. Then
a.xs[W}= i'p{a.)xP[W}~ 0
since i'p preserves rational equivalence, as does exterior product over P
(§1.10). ?
From the construction one sees that this product satisfies the usual com-
commutative and associative properties for exterior products. For schemes which
are smooth over S, the material of Chap. 8 extends without change. If Y is
smooth of relative dimension n over S, set
There is an internal product
A"(Y/S) ®A"(Y/S)
defined by a. ¦ 0= <5*(<x xsP), with 5:Y-+YxsY the diagonal - a regular
imbedding of codimension n. This makes A* (Y/S) into a commutative, graded
ring with unit 1 = [F]. Given f:X-+ Y,A*(X/S) is a module over A*(YlS) by
Y/:X-+ XxsY the graph of/ In particular, if X is smooth, defining /* (/?) =
yj ([X] xs /?) defines a pull-back
f*:AP(Y/S)-^A"(X/S),
making A* a contravariant functor from smooth schemes over S to rings.

398
Chapter 20. Generalizations
Example 20.2.1. Refined products. Let f:X-*Y, Y smooth of relative
dimension n over S, and let X'-*X, Y' -* Y be given morphisms. If
a eAk(X'/S), peA,(Y'/S), define a refined product
by a sp = Yj(a xs j}). For example, if X' and Y' are the supports of cycles a
and /? on X and Y, then a.-sp is a class in /4#(|a| n/^d/?!)) which maps to
/*(/?) n a in A^X. The basic properties are given in Proposition 8.1.1.
Example 20.2.2. Intersection multiplicity. If K, W are subvarieties of a
scheme Y which is smooth over S, and f is a proper component of Ff) W, i.e.
dimi/> = dimsK+ dim.? W- dim^F,
then the coefficient of [P] in [ V\ ¦ [ W\ is a positive integer, denoted
i(P,VW;Y). The discussion of Chap. 7 carries over to this intersection
multiplicity. (If neither V nor W is flat over S, no component of Vf) W can be
proper in this sense. Otherwise [Vx$ W\ is a positive cycle on YxsY, and the
intersection with the diagonal proceeds as in § 8.2.)
Example 20.2.3. For any X over S, the canonical homomorphism
A~k(X-+ S)-+Ak(X/S),
c-*c([S\), is an isomorphism. (An element a in Ak(X/S) determines homo-
morphisms
A1(Y/S)-+A1+k(XxsY/S)
by P -> a xs p. Clearly this operator takes [S] to a. To show that the maps are
isomorphisms, it suffices to show an element c in A~k(X-* S) vanishes if
c([S]) = 0. If V is integral, and V-+ S is flat, then c([V]) = 0 by property (C2)
of § 17.1. If V maps to P e S, iP the inclusion of P in S, then c acts on [V]
through ;?(c) eA~k(Xf -> />) =^(JT/.); but i*P(c)[P] = «?(c[5]) = 0, so ifc = 0
by Proposition 17.3.1.)
If Y is smooth of relative dimension n over S, then (Example 20.1.1)
A'-"(Y-+ S) =An-p(Y/S).
20.3 Specialization
Let S, S be regular schemes, /: S -> S a closed regular imbedding of codi-
mension d. Assume that the normal bundle to S in S in trivial, or at least that
its top Chern class is zero. Let S° = S — E, j the inclusion of 5° in S.
For any scheme X over S, set X = X x s S, X° = X x s S°. Note that
sequence
l = Ak-d(x/s) and At(X°/S°) = Ak(X°/S)
by our conventions on relative dimensions. From § 1.8 we have an exact
Ak (X/S) -S Ak (X/S) ± Ak(X°/S°) —> 0,
20.3 Specialization
399
where !„, and 7* denote push-forward and pull-back induced by inclusions of X
and X° in X. From § 6.2 there is a Gysin homomorphism
i':Ak(X/S) -^A
=Ak(X/S).
Since r f* = 0 (Theorem 6.3), there is a unique homomorphism
a: Ak(X°/S°) ^ Ak(X/S)
such that a{j*a) = /!(a) for all a eAk(X/S). This homomorphism <r is called
the specialization map.
For any morphism f:X -> F of schemes over S, let /: X -> P, /° : X° -> r°
denote the induced morphisms.
Proposition 20.3. (a)///:I->Fij proper, and iei,X°, J//e« J+ a (a) =
(b) If f:X-* Y is flat, or a regular imbedding, and aeA^Y0, then
J*a(a) = a f°* (a) in A^X.
Proof. These follow from the commutativity of i! and j* with proper push-
forward, flat pull-back, and intersection products (§ 6). Q
Suppose S = Spec (R), with R a discrete valuation ring, with residue field x
and quotient field K, and S = Spec (x), S° = Spec(X). For a scheme X over S,
X and X° are algebraic schemes over x and K, and Ak (X) = Ak (X/S),
Ak (X°) = Ak (X°/S°). In this case a has a simple description on the cycle level:
If V° is a subvariety of 1°, and V is the closure of V° in X, then K is flat over
S and
[V\ [V\
Corollary 20.3. If X is smooth over S, S as above, then the specialization map
is a ring homomorphism, i.e., a preserves the intersection product.
Proof. Given subvarieties V°, W° of X°, let V, W be their closures in X.
Then Kxs W is a flat subscheme of X x.sX, restricting to V° xso W° over S°.
Applying (b) of the proposition to the regular imbedding S : X -> X x SX,
= a 5°
= 5*a [
as required. D
In a similar way, all our other intersection operations are compatible with
specialization.
Example 20.3.1. If Y is smooth over S= Spec(R), R a discrete valuation
ring, then the homomorphisms in the commutative diagram
A*(Y/S)
¦A*(Y)
are all homomorphisms of graded rings.

400
Chapter 20. Generalizations
Example20.3.2. Let S = Spec(R), R a discrete valuation ring, f:X-+ Y a
morphism of S-schemes, with Y smooth over S. Let Z be a closed subscheme
of X. Suppose a° and 0° are cycles on X° and Y°, with |<x| f](f°)'l\ /3°| c Z°.
in /4, (Z). (This is a refinement of Corollary 20.3, with a similar proof.)
Example20.3.3. Let X be a scheme over 5, D{, ...,Dn Cartier divisors, or
more generally, pseudo-divisors (cf. § 2.2) on X. Let K be a subscheme of X,
flat over S of relative dimension n. Assume that C\D,C\V is proper over S.
Let ?>?, ?>, denote the restrictions of Dt to X°, X. Then
f D?-...-D°-[K°] = J
Indeed, for any pseudo-divisor D and any cycle a on X°, a(D° ¦ a) = /> • a (a).
Example 20.3.4. The Riemann-Roch homomorphism commutes with spe-
specialization map a : Ko (X°) -> Ko (X), defined as with rational equivalence, pro-
provided the normal bundle to S in S is trivial. Then the diagram
commutes. In particular, if $~ is a coherent sheaf on X which is flat over S,
then a(tx'(^'a))j= 1x(&)\ this generalizes the equality of Euler characteristics
X (X °,3^°) = x (X,T), when X is proper over S.
Example 20.3.5 (cf. [SGA6]XApp. and Fulton B)§4). Let R be a
complete discrete valuation ring with quotient field K and residue field x. Let
K and x be algebraic closures of K and x. For a scheme X over R there are
specialization homomorphisms
o:A*(X®RK)->Ai,(X®Rx)
compatible with our intersection operations. (For all finite extensions R' of R
in K, with quotient field K' and residue field xf, we have specialization maps
Alf(X®RK')^Alf(X®Rx'),
constructed from the /{'-scheme X ®R R'. Passing to the direct limit over R'
gives a.)
If A'is smooth over R, one has a commutative diagram
A'{X®RK)
I"
-A'{X®Rx)
H2i(X ®R K, Z,@) = H2'{X ®R x, Z,(/)) •
where / * char(x), and H2i{ , Z,(j)) is /-adic etale cohomology.
Example 20.3.6. Let X be smooth and proper over Spec (R), with R, K, x as
in the preceding example. Then
rank NS (X <g>R R) ^ rank NS (X <g>R x).
20.4 Tor and Intersection Products
401
(If a,,..., aQ is a basis for divisors modulo numerical equivalence on X <S>R K,
there are I-cycles fa,..., fiQ on X ®R K with det (a, • fa) =t= 0. Then
so <j(ai),..., ff(ae) are independent.)
20.4 Tor and Intersection Products
Let X be a regular scheme, V and W subvarieties (closed integral subschemes)
of X. The sheaves loxf'((9v,(9w) are supported on Vf)W, so determine ele-
elements of the Grothendieck group Ko(Vf]W) of coherent sheaves on VC\W
(cf. Example 15.1.7 (b)). Set
dim (X)
Tor*(K,^)= S (~ I)'[Tor,'*(*„,*„)]
inXo(KDW).
Let f be an irreducible component of Vf] W, with its reduced structure. Any
element a. in Ko(Vf] W) may be written in the form
a = trip (a) \fir] + a'
where mP{a) is a well-defined integer, and a' is a linear combination of
sheaves whose supports do not contain P. Serre D)V.6 has proved that
codim(P,X) =s codim(V,X) + codim(W,X) .
If P is a proper component, i.e., equality holds in this inequality, Serre defines
an intersection number i(P, VW; X) by the formula
i(P, VW; X) = mP(Torx(V, W)).
This intersection number is obviously commutative in V and W, and the
expected associativity formula follows from a Tor spectral sequence. Serre's
conjecture that this intersection number is always positive is one of the
outstanding open questions in commutative algebra (cf. Example 20.4.2).
If X is smooth over a field, this definition agrees with that given in
Chap. 8. For then one may reduce to the diagonal, where both terms agree with
the multiplicity obtained from a Koszul complex (cf. Example 7.1.2 and Serre
D)V.C).
More generally, with no assumption on the properness of the intersection,
the refined intersection class, at least with rational coefficients, is determined
by Tor. Let X be smooth over a field, V, Wsubvarieties of X. Set
m = dim V+ dim W — dim X ,
so the refined intersection class VW\% in Am(Vf) W) (§ 8.1). Let

402 Chapter 20. Generalizations
be the Riemann-Roch map (§ 18). Then (Example 18.3.13 (b))
r(Torjr(K, W)) = V- W+ terms of dim < m .
Example 20.4.1 (Fulton-MacPherson A)§6). With notation as in the last
paragraph of this section,
Tor*(V, W) e Fm Ko(Vf] W).
This is not known for X an arbitrary regular scheme. Indeed, Serre also
conjectured that
mP(ToTx(V, W)) = 0
for P any irreducible component of Vf] W of dimension greater than m; this
conjecture too remains open.
Example 20.4.2. Let X be a regular scheme, P an irreducible component of
Vf]W. Let A be the local ring of X at P. Serre D) p. V-15 proved the
positivity of mP(ToTx(V, W)) in the case of proper intersections, and its
vanishing in the improper case, under the assumptions that all localizations of
A at prime ideals are either equicharacteristic or unramified. This problem is
also discussed in Nastold A) and Malliavin A).
For A a regular local ring, Serre's question is equivalent to the positivity or
vanishing of
Z(-l)'7,(Tor?(M,iV))
for finitely generated /1-modules M, N with lA(M®AN) < oo. The hope that
this might hold for arbitrary local ring A, if M or N had a finite free
resolution, has been dashed by the recent example of Dutta-Hochster-
McLaughlin A). Note that, since the completion A of A is flat over A, it
suffices to consider the question for complete regular local rings.
Example 20.4.3. Let X be regular, P a proper component of Vf] W.
Assume (with no loss of generality, replacing X by an open subscheme) that P
is regular, and V f]W= P (set-theoretically). Let n : X -> X be the blow-up of
X along P, E the exceptional divisor, rj: E -> P the induced morphism. Let V,
Wbe the proper transforms of Kand Win X. Then
(*) i(P, VW; X) = eP(V) eP(W) + mP(^ (Tor* (V, W))).
It is natural to conjecture that, as in the geometric case (Theorem 12.4), the
second term is always non-negative, so that
i(P, V- W;X) ^ er{V) eP{W).
(The multiplicities ep(V), eP(W) are defined as in the geometric case. The
proof of (*) given in § 12.5 works equally well with no base field. Note that the
proof of the non-negativity of the second term used reduction to the diagonal,
so does not work on general schemes).
A particular case of (*) is the formula
i(P, V- W;X) = eP(V)eP(W)
20.5 Higher AMTieory 403
in case Vf] Wf]E = P(CP V)DP(CP W) is empty. This had been proved by
Tennison A).
Serre's definition extends to the proper intersection of more than two
varieties, and (*) and the other results of this section extend readily to this
generality.
Example 20.4.4. For V, W subvarieties of an arbitrary scheme X, Pa compo-
component of Vf] W, one may form the formal power series
F(r) = ?m,(Torf(K W))T
in 1 \T\ \fX is regular, and P proper, then F(— 1) is the intersection number
i(P, V- W;X). Can F{— 1) be defined more generally, say by analytic
continuation? Is this related to fractional intersection multiplicities (cf.
Example 7.1.16)?
20.5 Higher A:-theory
This section sketches the relation of Quillen's higher A^-theory to cycles and
rational equivalence. For details we refer to the basic paper of Quillen B), and
to Grayson A), B) and Gillet A).
For a commutative ring R, Quillen B) has constructed a sequence of
higher ^-groups KP(R), with KQ(R) the Grothendieck group of projective R-
modules (locally free sheaves on Spec(/?). If R is local, K,(R) is the group R*
of units in R. For a scheme X, the presheaf
determines a sheaf, denoted $ff, on X. The sheaf 3C0 is the constant sheaf TL,
If X is regular, there are complexes
C),
with
i= ©
Here the sum is over closed integral subschemes Kof X, <pv is the inclusion of
Kin X, and Ki(R(V)) is the/h higher ?-group of the function field R(V),
regarded as a constant sheaf on V. In particular, F (X, $pp) is the group of cycles
of codimension p on X, and
)= © R(V)* .
F(X,^pp~l)
takes
The boundary map from the complex (*),
r e R(V)* to [div (r)]. The cokernel of this map is therefore the group of rational
equivalence classes of codimension p on X, as defined in § 1.3.

404
Chapter 20. Generalizations
Gersten has conjectured that (*% is exact for any regular scheme X. Since
the sheaves^ are flasque, Gersten's conjecture implies that H'(X,y?p) is the
<lh cohomology group of the complex F{X,^*). Quillen B) has proved
Gersten's conjecture when X is of finite type over a field. This gives the
following theorem, first discovered and verified in some cases by S. Bloch.
Theorem. IfX is a regular scheme of finite type over afield, of pure dimension
n, then
This theorem generalizes the isomorphisms H"(X, 2) = AnX,
= Pic(Jf) = An-\ (X). There are products
which make H* (X, JfJ into a ring. When X is smooth over a field, Grayson A)
has shown that this product agrees, up to a sign, with the product defined in
Chap. 8; for the general construction see Gillet A).
For possibility singular X, Gillet C) has constructed Chern classes of
vector bundles in Hp(X,Cfp). More generally, he defines, for Y a closed
subscheme of X,
where the subscript Y on the right denotes cohomology with supports in Y.
There are cap products
APYX ® AqX -+ Aq-PY.
A complex of vector bundles on X, exact off Y, has local Chern classes in
A*YX. This capacity for localization is one significant advantage of the higher
A>theory realization of cycle classes.
Higher AMheory has also led to insight into the group A0(X) of classes of
0-cycles of degree zero on a non-singular projective variety X over an
algebraically closed field. Bloch C) has used higher A%theory to prove a
theorem of Roitman B) that the canonical map from AQ(X) to the Albanese of
X is an isomorphism on torsion prime to the characteristic. Using W2, Bloch
has found evidence to support the conjecture that A0(X) is finite dimensional
when X is a surface with pg = 0. Higher A>theory has also been used to suggest
a theory of infinitesimal variation of Chow groups, cf. Bloch B) and Stienstra
A). For other relations between cycles and ANtheory the lectures of Bloch D)
are recommended. Some calculations on singular varieties have been made by
Collino B) and Srinivas A).
Notes and References
Specialization of cycles and proper intersections was studied by Samuel C) in
the geometric case, and by Shimura A) over arbitrary discrete valuation rings.
The general problem of showing that rational equivalence is preserved under
Notes and References
405
specialization was discussed by Grothendieck [SGA6]X App., and solved in
Fulton BL. Grothendieck also asked for the construction of a product on
A^X for X smooth over a discrete valuation ring; this is a special case of
Proposition 20.2, which is new here. For arithmetic surfaces see Lichtenbaurn A).
The construction of rational equivalence groups of cycles on Noetherian
schemes, covariant for proper morphisms, appears first in Fulton B); previous
theories using graded ^-groups were discussed in [SGA6], The definition
given in §20.1, with the notion of relative dimension, is new here; it enables
the usual constructions and proofs over a field to extend to schemes over a
regular base. Kleiman A2) is one who has advocated doing intersection theory
over general base schemes whenever possible. Until interesting applications
are in sight, however, the gain in generality must be weighed against the loss
of simplicity.
For schemes smooth over a ring of integers in a number field. Arakelov A)
has proposed a definition of intersection product which brings in the infinite
primes, as one would like for applications to number theory, cf. Neron A).
Such a ring should surject onto the ring constructed in § 20.2. Arakelov even
conjectured a Riemann-Roch theorem in this context. Recent progress on this
has been reported by G. Faltings and L. Szpiro.
The definition of intersection product using Tor, discussed in §20.4, was
originated by Serre D), for the case of proper intersections. The generalization
to improper intersections of algebraic schemes using Riemann-Roch is from
Fulton-MacPherson A); the numerical consequence of this formula, in case the
expected dimension is zero, had been proved by de Boer A).
The Riemann-Roch theorem for schemes of finite type over a regular base,
and the consequence that A+Xq^ KoXq, is new here. P. Wagreich (un-
(unpublished, 1966) had initiated a study of relative cycles and Riemann-Roch
over a base scheme, although his results were quite different from ours.
The formula relating rational equivalence to higher A^-theory was dis-
discovered by Bloch A), and established by Quillen B) when he proved a
conjecture of Gersten. The fact that Hp(X,2fp) makes sense for an arbitrary
scheme X was another early indication that one could construct a rational
equivalence ring for schemes other than smooth quasi-projective varieties over
a field. The material in §20.5 is based on lectures of Bloch, Gillet, and
Grayson at the Institute for Advanced Study in 1981-82.

Appendix A. Algebra
Summary
The main concepts needed from commutative algebra are set forth in this
appendix, and the facts needed for the basic theorems are proved. Some
further useful results are stated, with references to the literature for proofs.
Notation. All rings will be commutative, Noetherian rings with unit. The
localization of a ring A (resp. module M) at a prime ideal p is denoted Ap
(resp. Mp). If L is a finite extension of a field K, [L : K] denotes the degree of
the field extension.
A.1 Length
Definition A.I. For any finitely generated /1-module M there is a chain of
submodules
with M^/Mi^A/p^pi a prime ideal in A (Bourbaki A)IV. § 1 Th. 1). If the
Pi which occur in such a chain are all maximal ideals, M is said to have finite
length. Equivalently, the localization Mp is non-zero for only a finite number of
prime ideals p which are all maximal. In this case, the length r of a chain (*) is
independent of the chain, and is the length of M, denoted lA (M). Note that
for any ideal / such that IM = 0.
Lemma A.1.1. I/O —> M —* M —* M" —* 0 is an exact sequence of A-modules,
and two of the modules have finite length, then the third module also has finite
length, and
More generally, if
0 -> M, -> M,_,
Mo -> 0
is an exact sequence of modules of finite length, then
,) = o.
A.2 Herbrand Quotients
407
Proof. Chains (*) for M and M" of length r1 and r" determine one of
length r1 + r" for M. The second assertion follows by breaking the long exact
sequence into consecutive short exact sequences. ?
Lemma A.I.2. If M has finite length, then
the sum taken over the prime ideals p in A.
Proof. By localizing a chain (*) at a maximal ideal p, one sees that Alp
occurs as a factor Iav(Mp) times. ?
Lemma A.1.3. Let A-* B be a local homomorphism of local rings. Let d be
the degree of the residue field extension. A non-zero B-module M has finite length
over A if and only ifd < oo and M has finite length over B, in which case
lA{M) = d-lB{M).
Proof. Since both sides are additive on exact sequences, one is reduced to
the case M= B/q, q the maximal ideal of B. Up is the maximal ideal of A,
then
since length and vector space dimension agree over a field. D
Example A.1.1. Let A be a local ring with a subring k which maps
isomorphically onto the residue field of A. Then
for any /1-module M.
A.2 Herbrand Quotients
Definition A.2. Let fi:M-»Mbe an /1-linear homomorphism of a finitely
generated /1-module. Let
Mv = Coker(ip) = M/Image(ip)
pA/= Kernel (ip).
We say that eA(cp, M) is defined if Mv and ^M both have finite length over A.
Equivalently, the localization
cpp : Mp -> Mp
is an isomorphism for all but a finite number of prime ideals which are all maxi-
maximal. Then set
ll{

408
Appendix A. Algebra
In case ip is multiplication by an element a in A, we write eA(a, M) for
eA{tp,M). Note that
eA{<p, M) = eA,,(<p, A/)
for any ideal / with IM= 0.
For applications in the text, only the case where a is a non-zero-divisor on
M is needed, in which case eA(a, M) = lA{M/aM). As the following lemmas
show, the general case is more natural, from an algebraic point of view.
Lemma A.2.1. // lA{M) < oo, then
eA{tp,M) = 0
for all ip : M -> M.
Proof. This follows from Lemma A. 1.1 and the exact sequence
Lemma A.2.2. IfeA {<p, M) is defined, then
the sum taken over all prime ideals p in A.
Proof This follows from Lemma A. 1.2 and the fact that the construction of
kernels and cokernels commutes with localization. ?
Lemma A.2.3. Let A -* B be a local homomorphism of local rings, d the
degree of the residue field extension. Let ip : M -> M be a B-linear endomorphism
of a B-module M. If d < oo, then eA (ip, M) is defined if and only if eB {<p, M) is
defined, and
{M) = d-eB{<p,M).
Proof. This follows from Lemma A. 1.3. ?
Lemma A.2.4. Given a commutative diagram
0
M ^ M"
with exact rows, if two ofeA(ip't M'), eA{ip, M), and eA{<p", M") are defined, then
so is the third, and
eA (<p, M) = eA (<p', M') + eA {<p", M").
Proof. By the snake lemma, the above diagram determines an exact
sequence
0
A/;- -> M9
0 ,
and the result follows from Lemma A. 1.1. ?
Lemma A.2.5. Let ip and y/ be A-linear endomorphisms of M. If two of
eA(ip, M), eA(y/, M), andeA{ip yj, M) are defined, so is the third, and
eA{q> if/,M) = eA{q>, M) + eA(y/, M).
A.2 Herbrand Quotients
Proof. This follows from Lemma A. 1.1 and the exact sequence
409
0 -> yM^ VVM^ fM-* M^ M^^ M,-> 0.
The maps labelled y/ and <p are induced by y/ and <p, the others by the identity
map on M. D
Lemma A.2.6. If M is a finitely generated free A-module, <p: M —> M an A-
linear endomorphism, then eA{<p, M) is defined if and only if eA(de\.(q>), A) is
defined, and
¦ 4) = eA{de\.{<p),A).
Proof. (See also Example A.2.5.) Let M=A", and regard <p as an n x n
matrix. Note that <pp is an isomorphism if and only if det(p) ^ p, which proves
that both sides of the equation are simultaneously defined.
In case ip is a triangular matrix, the formula follows by induction on n from
Lemma A.2.4. Indeed, there are direct summands M' of M mapped to
themselves by ip, and the induced maps <p' on M', <p" on MIM' are given by
smaller triangular matrices. Another easy case is when <p is an isomorphism, in
which case both sides of the formula are zero.
We prove the lemma first under the assumption that A is a one-
dimensional domain — the only case needed in the book. In this case the terms
are defined whenever det(p) =t= 0, in which case <p is one-to-one, as follows
from the identity
Adjoint {<p) ¦ <p = det {<p) • I,
where / is the identity matrix. The required equation is then
Let K be the quotient field of A. Define a homomorphism
by the formula
h(y/) = eA(a yi, A") - eA(det(a v), A),
where a is any non-zero element in A such that a y/ has entries in A. To see that
h is well-defined, let b be another such element in A. Then
eA (a b yi, A") = eA (a I, A") + eA (b y,, A")
eA (a b yj, A") = eA (b I, A") + eA (a y,, A")
by Lemma A.2.5. Similarly
eA (det (a b y/), A) = eA (a", A) + eA (det (b y/), A)
eA (det (a b y/), A) = eA (b", A) + eA (det (a y,), A).
Since the formula of the lemma is obvious for scalar matrices, the four
displayed equations combine to give
eA {a yj, A") - eA (det {a y/), A) = eA {b yj, A") - eA (det {b v), A)
as required. Another application of Lemma A.2.5 shows that h is a homo-
homomorphism.

410
Appendix A. Algebra
To complete the proof, we must show that h is identically zero. Since any
element in GL (n, K) is a product of elementary matrices, it suffices to see that h
vanishes on these. Indeed, from the definition of h and the case considered at
the beginning of the proof, h vanishes on any triangular matrix in GLn{K), and
any permutation matrix is an isomorphism over A, which was also considered
initially.
For the general case of the lemma, by Lemma A.2.2, we may assume A is
local, with maximal ideal m. We may assume det(p) is not a unit, and that
A/det {ip) A has finite length, and therefore that A is one-dimensional. (If A is
zero-dimensional, both sides are zero). Let / be the ideal of elements in A
which are annihilated by some power of the maximal ideal. Since <p induces
endomorphisms of IM and MUM, and IM has finite length, we are reduced
by Lemma A.2.4 to the case where 7 = 0. Hence m does not consist of zero
divisors. Let K be the total ring of fractions of A. Define h from GL{n, K) to TL
by the same formula as before, but requiring a to be a non-zero-divisor in A.
Since K is Artinian, it is a product of local Artinian rings. It follows, by
standard elementary row operations, that every element in Gh{n,K) is a
product of triangular matrices, and the proof concludes as before. D
Lemma A.2.7. Assume A is a one-dimensional local ring, and let p{, ...,p, be
the minimal prime ideals of A. Let M be a finitely generated A-module, and let a
be an element of A not in any p,. Then
eA{a, M) = Z LPI(MP) ¦ eA(a, A/p,) = ? W
¦ lA {A/p, + a A)
Proof. The second equality follows from the fact that a $ pt and pt is prime.
For the first, since both sides are additive for exact sequences of ^-modules,
we may assume M = A/p, with p a prime ideal in A (cf. Example A.2.2 (iii)).
If p is maximal, then MPi — 0, and eA{a, M) = 0 by Lemma A.2.1. If/) is minimal,
then lA,(Mp) ~ 1> and the localizations of M at the other minimal primes are
zero, from which the required formula is clear. ?
Lemma A.2.8. Let tp and y be two commuting endomorphisms of M, both
assumed to be injective. Let q> (resp. ij/) be the endomorphism of Mv (resp. M^)
induced by <p (resp. y/). Then eA (<p, Mv) is defined if and only if eA {y, M,,) is
defined, and
eA{(p, Mv) = eA(ij/, MJ .
Proof. There are canonical isomorphisms of (Mw)^ with (M,,)^ and of
p(Mr) with p(My), from which the assertion is obvious. (See Example A.5.2
for a generalization to the case when <p and i// are not injective.) ?
Example A.2.1. Let T be an indeterminate, F, G polynomials in A[T], with
F monic of degree n. Let B = A[T]/(F), a free ^-module of rank n. Let ipa be
the endomorphism of B induced by multiplication by G. Let R = det {ipa) e A.
(i) R is the resultant of F and G. (See Van der Waerden D) for properties
of resultants. The assertion in (i) is formal. It therefore suffices to verify it
when G factors completely. The case where deg (G) S 1 is straightforward.)
A.3 Order Functions
(ii) If A is a domain, it follows from Lemma A.2.6 that
lA(A[T]/(F,G)) =
411
Example A.2.2 Let M be a finitely generated ^-module, <p an ^-linear
endomorphism of M.
(i) If ip is surjective, then ip is one-to-one,
(ii) If Coker (q>) has finite length, then eA (ip, M) is defined, and
OSeife AQS A,(Coker(p)).
(Let Nr = {meM\(pr{m) = 0}. Take n so iVn = U Nr. Then eA(q>,M) =
lA{M/Nn+<p(M)) by Lemmas A.2.4 and A.2.1.) 'i0
(iii) With the notation of Lemma A.2.4, if eA {<p, M) is defined, then
eA (ip', M) and eA (ip", M") are defined.
Example A.2.3. Let M be a finitely generated free A-modu\e, tp an ^4-linear
endomorphism of M. Then tp is one-to-one if and only if det(p) is a non-zero-
divisor in A, and in this case,
Example A.2.4. For any homomorphism <p : M-* N of finitely generated A-
modules, define
eA {<p) = lA (Coker {<p)) - lA (Ker (p))
where these lengths are finite. Lemmas A.2.1-2.4 have generalizations to this
setting. If M and N are projective A -modules of rank n, then eA{cp) = eA( A" q>),
where A" q> is the induced homomorphism on top exterior powers.
Example A.2.5. If A is a principal ideal domain, Lemma A.2.6 follows from
the theory of elementary divisors. If A is a one-dimensional domain whose
integral closure A' is a finite ^-module, the assertion for A follows from that for
A' and Lemma A.2.3. This case suffices for the application in Chap. 1.
A.3 Order Functions
Definition A.3. Let A be a one-dimensional domain, with quotient field A^. For
any non-zero element a in A, set
For any non-zero element r in K, set
oxdA{r) = oxdA(a) - oxdA{b)
for any a, bin A with r = alb. If also r = a'/b', then a' b — a b', so
ord^(a') + ovdA{b) = ordA{a) + ordA{b')

412
Appendix A. Algebra
by Lemma A.2.5, so ordA(r) is well-defined. Another application of Lemma
A.2.5 shows that
ordA:K* ^TL
is a homomorphism.
Lemma A.3. Let A be a one-dimensional domain -with quotient field K. Let
if. M-* M be an endomorphism of a finitely generated A-module, and let <pK be the
inducedendomorphism ofMK = M®AK. 7/det {<Pk) =1= 0, then
Proof. By Lemmas A.2.1 and 2.4, we may replace M by M/M', where M' is
the torsion submodule of M; i.e., we may assume M imbeds in M®AK.
Choosing a basis for M®AK from elements in M, one constructs a free
submodule F of M with F®AK= M®AK. Choose a common denominator a
for a matrix for ip, so that a ip{F) <=¦ F. Since (M/F) ®AK = 0, MIF has finite
length,so
eA {a<p,M) = eA (a ip, F)
by Lemmas A.2.1 and 2.4 again. Similarly eA(a, M) = eA{a, F). Then, using
Lemma A.2.5,
eA(a ip,M) = eA{a, M) + eA{(p,M) = eA{a, F) + eA{<p, M) = ord^(a") + eA{</>, M),
where n is the rank of F. By Lemma A.2.6,
eA(a <?,F) = OTdA(det(a ip)) = ord^(a") + ord^det (?>*)).
Comparing these equations gives the lemma. D
Example A.3.1. Let A be a one-dimensional local domain, with quotient
field K. Assume that the integral closure of A in K is a finitely generated
^-module. Then for any r e K*,
ordA(r) =
• [R/mR : A/mA]
where the sum is over all discrete valuation rings R of K which dominate A,
i.e. R =>A and the maximal ideal mR of R contains the maximal ideal mA of A.
(If B is the integral closure of A in K, then lA(B/A) < oo, so lA(A/aA) =
lA{B/a B) for a e A. The discrete valuation rings R are the localizations of B at
maximal ideals, so Lemmas A. 1.2 and 1.3 apply.)
Example A.3.2. If A is a discrete valuation ring, and r e A, then the order of
a is the maximum integer n such that asm", m the maximal ideal of A. For
general A such a maximality definition gives a non-additive order function (cf.
Example 1.2.4).
Example A.3.3. Let A be a discrete valuation ring with fraction field K, Ld.
finite extension of K of degree n, B the integral closure of A in L. If B is a
finitely generated ^-module, then (Lemma A.3) for a e A,
IA {B/a B) = n-
(a).
If a is a uniformizing parameter in A, this yields the familiar formula
"E e<fi=»"¦
A.4 Flatness
A.4 Flatness
413
Definition A.4. A homomorphism A -> B of rings is flat if every exact
sequence of ^-modules remains exact after tensoring over A with B. In
particular, for any ^-algebra A', the base extension A' -> A' ®AB is then flat.
Lemma A.4.1. Assume that A —> B is a flat, local homomorphism. Then the
induced mapping from Spec (B) to Spec {A) is surjective. If A and B are zero-
dimensional (Artinian), then
lA{A)lB(B/mB)
where m is the maximal ideal in A.
Proof. For the first statement, we must show that a prime ideal p of A is
the restriction of a prime ideal in B. Since B/pB is flat over Alp, we may
assume p = 0. All non-zero elements a in A are then non-zero-divisors; by
flatness, multiplication by a remains injective on B. A prime ideal in B not
containing the images of any non-zero elements in A then restricts to the zero
ideal in A, as required.
For the second statement, take a chain
A = /(, 3 /, Z> . . . 3 /, = 0
of ideals in A with /,_i /I, = Aim. Then
fl = /ofl =>/,?=>...=> 7,5 = 0
with
B ^
so /BE) = r • lA(B/mB) by Lemma A.1.1. D
Lemma A.4.2. Let 0 —* Mn —> Mn_i —>...—> M, —> 0
flat A-modutes. Then for any A-module N, the sequence
exact sequence of
n ®AN
n_, ®AN
M, ®AN -> 0
is exact.
Proof. Let M. be the given complex. We show //,(M. ®AN) = 0 by induc-
induction on i, the assertion being obvious for i ^ 0. Map a free ^-module F onto N,
and let N' be the kernel. There arises an exact sequence of complexes
0 -> M. ®AN' ~* M. ®AF -* M. ®AN -> 0 .
Since M. ®AFis exact, the long exact homology sequence gives isomorphisms
Hi(M. ®AN) S Hi-i{M. ®AN').
The inductive assumption, applied to N', concludes the proof. ?

414
Appendix A. Algebra
A. 5 Koszul Complexes
Definition A.5. Let ? be a finitely generated ^-module, and let s e Ev, i.e.,
s:E^A
is an /4-linear homomorphism. Let I(s) be the image of s, an ideal in A. Define
dk: AkE^ A1"^
by the formula
dk(elA...Aek) = 'Z (~ \)'+is{e,)elA...AeiA... Aek.
/-I
One checks that dkdk + 1 =0, so one has a complex of ^-modules, called a
Koszul complex; we denote it A'(s). Let Hk(s) be the felh homology group of this
complex:
Hk{s)
In particular,
In case E is free, we say that s is a regular section of ?v, or 5 is ^-regular, if
77* CO = 0 for all A: >0.
If all Hk (s) have finite length, we say that xa CO is defined, and set
Lemma A.5.1. Le/ ? = F© /f, 5 = / © u, t e Fv, u e Av = A. Then there is a
long exact sequence
... - Hk+](s) - 7/,@ -i tft(/) - tft(j) - 7/,_,(/) - ....
Proo/ Each A*? splits into a direct sum of AkF and A1^ which deter-
determines a short exact sequence of Koszul complexes
I
AkF —
1
1
-> A'E —
1
1
-> A
1
F-
where the complexes are written vertically in the diagram. The required long
exact sequence is the long exact homology sequence arising from this short
exact sequence of complexes. One checks that, up to sign, the boundary map
from Hk-1 (/) to Hk-, (/) is multiplication by u. ?
Lemma A.5.21 Let E = F@G, with F, Gfree, and lets = t®u,te Fv± u e Gv.
Let A = A/I(t), G = G/I(t)G, and let u be the homomorphism from G to A induced
by u. If t is A-regular and u is A-regular, then s is A-regular. The converse is true
if A is local with maximal ideal m, and u e mGv.
A.5 Koszul Complexes
415
Proof. By induction on the rank of G, one is reduced to the case G = A.
From the exact sequence of Lemma A.5.1, Hk(s) vanishes for all k > 0 if and
only if multiplication by u on Hk(t) is an isomorphism for all k> 0 and
injective for k = 0. This proves the first statement. For the converse, since
u e m, multiplication by u on a finitely generated ^-module is surjective only if
the module is zero (Nakayama's Lemma), from which the assertion
follows. ?
Lemma A.5.3. Let A -» B be a flat ring homomorphism. Let E be a free B-
module, s e Ev. Assume s is B-regular, and the induced homomorphism
A -> B/I(s) is flat. Then for any ring homomorphism A -> A', the induced section
ofEv ®A A' is B ®A A'-regular.
Proof. The complex
0- A"Cs).
A0CO-»7J/7CO->0
is an exact sequence of flat ^-modules. It follows from Lemma A.4.2 that this
complex remains exact after tensoring with A' over A, which proves the
claim. ?
Example A.5.1. Let E = F® A, s = t © u. Then xa(s) is defined if and only
if eA (u, Hk{t)) is defined for all k, and then
(From Lemma A.5.1 there are short exact sequences
0 -> Hk(t)u -> Hk(s) -> u7/t_, (/) - 0 .
Then apply Lemma A. 1.1.)
Example A.5.2. Let B be a commutative ^4-algebra, E a fi-module, .s a
section of ?v, so one has the Koszul complex A" (s) of 5-modules. For any JJ-mo-
dule M, let Hk(s, M) be the k'b homology group of the complex A'(s)® BM. If
these homology groups have finite length as ^-modules, define Xa(s> M) by
If ? is a free 5-module, %a(s>-) 's additive for exact sequences of 5-modules,
and Lemmas A.5.1, A.5.2, and Example A.5.1 generalize to this context. If
E= Bd,s is given by a sequence <px,..., <pd of elements of B. In this case set
eA((P\,...,(Pd,M) = xa {s, M) .
Note that to give an ^4-module M together with d commuting endo-
morphisms is the same as to give M the structure of a 5-module, for
B = A[T\,..., Td\ For d= 1, <p an endomorphism of the /4-module M, and
eA (tp, M) is the same as the multiplicity discussed in § A.2. For d= 2, let ip and

416 Appendix A. Algebra
i// be commuting endomorphisms of the ^-module M. Let y/9 (resp. 9y/) denote
the endomorphism of M9 (resp. ^M) induced by i//. Then, by Example A.5.1,
eA {<p, y/,M) = eA (<//„, M9) - eA („<//, ^M),
the two sides being simultaneously defined. In particular, the right side is
symmetric in if and i//, which generalizes Lemma A.2.8.
The positivity assertion of Example A.2.2 may also be generalized (see
Serre D) IV. App. 2). If M is finitely generated over A, and eA(c/>i,..., <pd, M)
is defined, then
where M is the cokernel of the map from Md to M given by (p,,..., ipd).
When B = A, M an /(-module, E = Ad, s e ?v is given by a sequence
(au...,ad)of elements of A, and eA {at,..., ad, M) is the multiplicity of M with
respectXoau ...,ad.
Example A.5.3. If s : E -> A is regular, and <r is an /(-linear automorphism
of E, then 5 ° a: E -> A is also regular.
Example A.5.4. If A is a discrete valuation ring, an /(-module M is flat if
and only if Mis torsion-free.
Example A.5.5. If s is an ,4-regular section of Ev, and A -> 5 is a flat
homomorphism, then the induced section of ?v ®,,5 is 5-regular.
A.6 Regular Sequences
Definition A.6. A sequence of elements a,,..., ad of a ring /( is called a
sequence if the ideal / generated by a,,..., ad is a proper ideal of /( and the
image of a, in A/(at,..., aM) is a non-zero-divisor, for i=\,...,d. Let
? = /*'', 5 the section of Ey determined by at,..., ad. If ax,..., ad is a regular
sequence, then 5 is a regular section of Ev; the converse is true if A is local and
all a, belong to the maximal ideal of A (Lemma A.5.2). In this local case, it
follows that the regularity of a sequence is independent of the order (cf.
Example A.6.1).
There is a canonical epimorphism of graded rings
a:A/I[Xu...,Xd]^ © 7V/"+1
nis.0
which takes X, to the image of a, in I/I2.
If a, is a non-zero-divisor in A, then A[a2/au ...,adla{\ is a subring of the
total ring of fractions of A, and there is a canonical epimorphism of rings
0:A[T2,..., Td]/J -> /I [a2/a,,..., ad/a,]
which takes 7", to a,/a,; J is the ideal generated by L2,...,Ld, with L,• =
A.6 Regular Sequences
417
Lemma A.6.1. If a\,...,ad is a regular sequence, then a. and (S are iso-
isomorphisms, and the sequence L2,..., Ldis a regular sequence in A[T2,..., Td\.
Proof (from Davis A)). First one shows by induction on d, that p is an
isomorphism, and L2,..., Ld is a regular sequence. If d= 2, and F(T2) eA[T2]
with F{a2/ax) = 0, dividing by L2 gives an equation
for some m > 0 and some G(T2) e A [T2]. Since of, a2 is a regular sequence,
one deduces from this equation that all coefficients of G{T2) are divisible by
of, so FeJ. Similarly if G(T2) ¦ L2 = 0, the coefficients of G must all vanish,
so L2 is a non-zero-divisor.
For rf > 2, consider the composite
A[T2, ...,Td]^A'[T3,...,Td]^A[aJau...,ad/ai]
where A' = A[a2/a]]. Since /('/a,/!' = /('/(a,, a2) A' S ^/(a,, a2) A [T2],
ait a3,..., ad is a regular sequence in A'. By the case d = 2, L2 is a non-zero-
divisor generating the kernel of the first displayed homomorphism; by
induction, Lit..., Ld is a regular sequence generating the kernel of the second.
Therefore L2,...,Ld is a regular sequence generating the kernel of the
composite, as stated.
To see that a is an isomorphism, it suffices to show that if F is a
homogeneous polynomial in A [X{,..., Xd] with F{a{, ...,ad) = 0, then F is in
IA[XU..., Xd\ But if F(a, ,...,ad) = 0, then F{\, a2lax,..., ad/a{) = 0; since 0
is an isomorphism, F{\, T2,..., Td) is in /. Hence all the coefficients of F are
in I, as required. ?
Remark A.6. Micali A) (cf. Bourbaki A) p. 160) has shown that
A[YU ...,Yd\/K=* SymA(I) =* @ I*,
where Y, -* a, e 7, and K is generated by a, Yj - aj Yh This also implies that a
is an isomorphism.
Lemma A.6.2. Let A be a d-dimensional local ring with maximal ideal m,
k — Aim. The following are equivalent:
(i) dim^m/m2) = d;
(ii) m has d generators;
(iii) m is generated by a regular sequence (of d elements);
(iv) © m"/m"+l ^ k[Xu ..., Xd]asgradedk-algebras.
Proof. By Nakayama's lemma, dim^m/m2) is the minimum number of
generators of m. Let au...,ae be minimal generators of m, and map
k[Xt,...,Xe] onto @ m"/mn+l by sending X, to a,. This gives a closed
imbedding of the projective cone P(Q to Spec (A:) in Spec (A), in PJ (see
AppendixB.5). But P(Q is always at least (d- l)-dimensional: if Spec(/t)c:
F, <= ... c Frf= Spec04) is a chain of subvarieties, then the blow-ups ^ of Vt
along Spec (?) give a chain of subvarieties of X which meet the divisor P (Q.
Therefore <? ^ d, with equality if and only if P(C) = PJ- The latter holds

418
Appendix A. Algebra
when the map from k[X\, ...,Xe] to © m"/m"+> is an isomorphism in large
degrees; since the polynomial ring has no zero divisors, the kernel must be
zero. From this one has the equivalence of (i), (ii), and (iv).
From the fact that © m"/m"+l is a domain it follows readily that A is a
domain. If pt= (a,,..., a,), it follows by descending induction on d that A/pt
has dimension d—i and satisfies the condition of (ii). In particular, p, is a
prime ideal, so (au...,ad) is a regular sequence, and (ii) => (iii). And
(iii) => (iv) by Lemma A.6.1. (See Serre D)IV.D for a purely algebraic proof of
Lemma A.6.2.) D
A ring satisfying the conditions of Lemma A.6.2 is a regular local ring.
100). Let A = k[X,Y,Z], ax=X,
is a regular sequence, but a2, a},
Example A.6.1. (i) (Matsumura A) p.
a2= Y{\— X), a} = Z{\ —X). Then a\, a2,
ax is not.
(ii) Let A = <E[X,Y,Z], a, = X A - 2X YZ) A - 3 X Y Z), a2 =
Y{\-XYZ){\-3XYZ), ai=Z{\-XYZ){\~2XYZ). Then {aua2,a3)
is a regular section of A3, but no permutation of a,, a2, a3 is a regular sequence.
Note that a sequence a,,..., arf in a ring A determine a regular section of Ad
if and only if (a,,..., ad) is a regular sequence in Ap, for each prime ideal p of
A which contains a,,..., ad.
A.7 Depth
Definition A.7. The depth of a local ring A is the maximal length of a
regular sequence in its maximal ideal. Every maximal regular sequence in the
maximal ideal has depth (A) elements (for a particularly elementary proof of
this fact, see Northcott-Rees A).) The depth of A is no larger than the
dimension of A; when depths = dim^, A is called Cohen-Macaulay. Every
regular local ring is Cohen-Macaulay (Lemma A.6.2). The following lemma
gives a useful criterion for the regularity of a sequence.
Lemma A.7.1. Let A be a local ring, with maximal ideal m, and let s e mEw,
Efree of rank d. Then
(*)
dim {All (s)) § dim {A) - d.
If A is Cohen-Macaulay (for example regular), then equality holds in (*) if and
only ifs is A-regular.
For homological proofs of the preceding facts, see Serre D)IV.B or Matsu-
Matsumura A) Ch. 6. For a simple elementary account, see Kunz A)VI. ?
A Noetherian ring A is called Cohen-Macaulay if all its localizations are
Cohen-Macaulay. The following result has been proved by Hochster. We refer
to Hochster A) Thm. 3.1*, or to Laksov A) for a proof; cf. also Arbarello-
Cornalba-Griffiths-Harris A), or De Concini-Eisenbud-Procesi A); the latter
discusses analogues for symmetric and skew-symmetric matrices.
A.9 Determinantal Identities
419
Lemma A.7.2. Let A be a polynomial ring in indeterminates xih 1 ^ i Sf
1 Sj S e, over afield K. Let 1 g: a\ < ... < ad S e be a sequence of integers, and
let I be the ideal of A generated by all (ak - k + 1) by (ak — k + 1) minors of
(XU ¦¦¦
Xf\...
for k = l,...,d. Assume that f-ad+ d^O. Then All is a Cohen-Macaulay,
normal domain, of dimension
ef-?L{f-ak+k).
yt-l
A. 8 Normal Domains
A normal domain is an integral domain which is integrally closed in its
fraction field.
Lemma A.8.1. Let A be a normal domain, a eA, a =t= 0. Then all prime ideals
associated to Ala A are minimal primes containing a.
Proof. Localizing at an associated prime, we may assume A is local and the
maximal ideal m is associated to A/a A. Then m = {a A: b) for some b eA.
Therefore {bla) m czA. If {bla) mam, then, since m is a finitely generated A-
module, bla is integral over A (cf. Zariski-Samuel A) Ch.V. § 1); then bla eA
since A is integrally closed, so b e aA and m =A. Therefore {bla) m =A, and
m = (alb) A is a principal ideal, which implies the minimality of m. ?
A.9 Determinantal Identities
Given formal power series c(l) = ^ cj1 V, with coefficients c^\ 1 S- i S h,
— oo <j < oo, in some commutative ring, and a partition X = (A|,..., !„),
11 § X2 S ... s: Xn S 0, set (cf. § 14 Notation)
the determinant of the n by n matrix whose ij entry is cj'+y.,. If cA) = ... =
c*n) = c, denote this simply Ax(c).
Lemma A.9.1. Let k S 0, m s 0 be integers. Let e = n + k, /= m + k, and let
X = (m,..., m) with m repeated n times. Let fiu..., jif, a.\,...,a.e be variables.
S
Set
/
no

420
and let c = d"\ Then:
Appendix A. Algebra
For (i), note that c(/) + at+1- / c(l) = c(l'"", i.e.,
By elementary row operations it follows that
Ax(c*'-",..., 6'~", c'",..., c<">) = zJ, (c<°,..., c(i), c<",..., c(">),
from which (i) follows inductively. Both sides of (ii) are homogeneous of
degree nm in the variables a.u ..., &„, /?i,..., jim. Setting oti = . .. = an = 0, both
sides give AX7=1 PjY- ^ therefore suffices to show that the left side vanishes
when a,• = /?/, by symmetry we may assume i=j= 1. But this is clear from (i),
since cA) is then a polynomial of degree less than m, forcing the top row of the
matrix whose determinant is Ax (cA),..., c(n)) to vanish. ?
Lemma A.9.2. Let c ~ Y.?=o c> tl, s = ??„ s, f be related by the identity
c(t) ¦ s( — t) = 1, c0 = s0 = 1. For an_v partition X, if X « //ie conjugate to X (cf.
§14.5), t/ien
Pwo/ Let n = l, and write A = (A,,..., An), Xn> 0, fi = (jiU ¦¦¦, tfm), Hm > 0.
Note that A, = m, fi\ = n. The fact that A and /^ are conjugate implies that the
sets of integers
{m-
and {m + nt -j + 1 | 1 Sj ^ m
form complementary sets in the set {\,2,...,n + m}. Indeed, the first set is
increasing, the second decreasing. If the i member of the first set equals the
/h member of the second, then
But conjugacy implies that if X, S j (resp. X, < j) then pij S i (resp. pij < i).
Letc,= (—l)'c,. Consider the product of two m + n by m + n matrices:
1
0.
1.
¦Si • •
..0
''I
0
1..
0
•¦«i,-l
• ^-2
¦Si,—
•1
¦Si,
¦Si,-1
•SA,+ 1 ¦ • •
Si,
0
1 c,...
0 1 c,
1 c,
A.9 Determinantal Identities
421
The determinant of the left (resp. right) matrix is (- \)m"Ax(s) (resp. 1).
Multiplying these matrices, one obtains the matrix
1 0
0 ... 0
1 0 ...
0 ...
1 ... 0
1
0 ... 1 ... cn
where, for i=\,...,n, the ilh row has a 1 in the (m — A, + 0th place, and zeros
elsewhere. The determinant is, up to sign, the determinant of the m by m
matrix obtained by removing the rows and columns containing these ones.
Using the complementarity of the two sets of integers obtained at beginning
of the proof, one obtains the matrix
The sign in front of this determinant is (- 1)?, with
n
e = ^?J {m — Xj) = m n — \X\= m n— \[i\ ,
where \X\= J%= 1X,,\n\ = 2Z™= i /^j • Equating the determinant of the product with
the product of determinants, and replacing the above matrix by a reflection in a
diagonal, one has
= (- II I (-!)"'-'(-
where ?' = E™= i (t*i — 0 + 2™= i j = I ^ I» which concludes the proof. ?
Let x,,..., xn be commuting variables, and set
: = S^ = II<
For a partition A set
= sx(xi,...,xn) =

422
Appendix A. Algebra
Lemma A.9.3 (Jacobi, Trudi). With this notation,
Proof. In the identity U"=i {t - xd = t" - ct f~x + ... + (-1)" cn, set t = xj
and multiply by xf~", obtaining
(i) xf-c, xf-'+ ... + (-l)"c»xf-" = 0
for all m S n. From the relation between s and c, for any integer q with
(ii)
„_, - c, jm_?_, + ... + (- 1)" cn sm-q-n = 0 .
The recursion equations (i) and (ii) are the same. Solving inductively, this
means there are, for all m § 0, and 1 § k S n, universal polynomials a(m,k)
in the variables c\,..., cn such that
-q = S a (
k=\
Kj k for all l^jSn,
For any non-negative integers X{, ...,Xa, this gives matrix identities
(x/'+"-')y = („ (A, + « - «, *)),* • (xJ-%
(sXi+J-dij = (« (*i +n-i, k))ik ¦ (Sj-k)kJ.
(The matrix subscript ij indicates how the ij entry of the matrix is formed.)
The proof concludes by taking determinants in these two equations, noting that
\Sj-k\ = 1 since (Sj-k)jk is a triangular matrix with ones on the diagonal. Note
also that
I x"~k I = II (xj ~ xk) (Vandermonde)
is not zero. ?
Lemma A.9.4. Let d =
for all m ^ 0,
a partition. Then
the sum over all n = (nl ,...,nn+1) with fi1^:Xi^ ... ^^n^Xn ^nn+1 ^0, and
Proof. Since Ax{d) does not change if an arbitrary string of zeros is added
to X, we may assume n S i for any d{ which appears in the identity to be
proved, and Xn = 0. We may therefore assume rf, = ^,, 5, as in Lemma A.9.3. Set
a, = Xj + n- i, bi = //,- + n - i, and set
A.9 Determinantal Identities 423
By Lemma A.9.3, and adding over m, we are reduced to proving the identity
(ii) 5(ai,...,fl,)-n(l-xi)-| = E^,-,y,
1-1
the sum over all (bt,..., bn) with bt S a, > b2 § a2 > ... > bn § an = 0. For any
integers e2,...,en, let Sk(e2, ¦¦¦, en) be the determinant formed as in (i), but
using integers e2,...,en and variables x,,..., xk,..., xn. Thus, expanding
along the top row,
n
S{au...,an)=y?(-l)MxfSk(a2,...,an).
k-\
Therefore
f
- x,)-1 5k(a2,..., an) Y[{\- x,)"'.
Assuming the result for n - 1, this expression becomes
the sum over all F,,..., bn) with b] ^ a, and b2 ^ a2 > ... ^ an. To conclude
the proof, one must verify that the sum of all terms with b2 S a\ is zero.
Together with any such {bu b2,..., bn), with b\ =t= 62, occurs also (b2, bu ...),
and
5{bl,b2,...,bn) + S(b2,b[,...,bn) = 0
by the alternating property of determinants. Similarly S(bu b2,..., bn)
vanishes if 6, = b2. ?
The Littlewood-Richardson rule (cf. § 14.5)
for multiplying general 5-functions was given in Littlewood and Richardson
A), although complete proofs have only recently appeared. For a proof along
the original lines we recommend Macdonald CI.9. Other proofs may be found
in Schiitzenberger A) or Akin-Buchsbaum-Weyman A).
Example A.9.1 (cf. Lascoux G), Macdonald C) p. 30, 31). For partitions
X = {XU ...,Xn),fi = (jiu ...,//„), write// c X if n,S A,-for lsis«. Set
+ n-i
Let x,,..., xn be commuting variables. Then
(a) sx A + Xi,..., 1 + xn) = X dXfl ^ (x,,..., xn).
(By Lemma A.9.3, sx{\ + x,,..., 1 + xn) = |A + x;)^+"-'|/l(l + x,-)""']. The
denominator equals |x,*~'|. Expand the numerator and compute coefficients of

Appendix A. Algebra
Lete=(n,n-l,...,l), 6= (n - 1, n-2,..., 0). Then
(b) s, (x,,..., *„) = x, ¦... ¦ xn ¦ JQ (x<+ xj)
ExampleA.9.2 (cf. Macdonald C) p. 35, 37). Let x,,...,xn, y\,...,ym be
variables. Then
n m
(a) liii x.yj i sx
the sum over all partitions X : X] S ... S Xn § 0 with A, S m; I is the conjugate
ofl
n m
the sum over the same X, with X' = (n - Im,...,«- ^1).
(C) n n o+*-+ya=srf^ ^ w jf w'
the sum over partitions // c A, A as above, dltl as in the preceding example.
Example A.9.3. (a) Let ao,...,ad be non-negative integers. Then
d
\l(fli +;)! |os/js</ = II (a< ~ tf/VlI (fl/ + rf)! ¦
(Multiply by Fl?=o (ai + dY- and use elementary row operations.)
(b) If c (/) = e' = Z A//!)''.then (cf- § 14 Notation)
i-0
^ w"/!-...-(/+e-l)! •
Example A.9.4 (cf. Harris-Tu A)). Let aa,...,ad and n be integers, with
n S a,: - d S 0 for all i. Then
Example A.9.5. Let f = \ + at t+ ... +ant", g = \ + bt t+... +bmtm. Then
^ (g/f) = R(J,g),
where /{(/g) is the resultant of/ and g. (This follows from Lemma A.9.1
Notes and References
Notes and References
425
We make no attempt to trace the development of multiplicity theory in com-
commutative and homological algebra, other than mentioning the names of a few
of the important contributors: Auslander, Buchsbaum, Nagata, Northcott,
Rees, Samuel, and Serre. The results of this appendix on length, Koszul
complexes, regular sequences, and flatness have been known for some time.
More general results, from a more sophisticated point of view, may be found
in Serre D). Other references for this material are Atiyah-MacDonald A),
Bourbaki A), B), [EGA]IV, Kunz A), Matsumara A), Nagata B), Northcott
B), and Zariski-Samuel A).
For the case of a domain, the important Lemma A.2.6 was proved in
Chevalley BI.2.3 and [EGA]IV. 21.10.17.3 by reducing to the case of a
discrete valuation ring. The elegant proof given here, motivated by A^-theory,
was shown to me by B. Iversen. The extension to general rings is apparently
new. The relation of this lemma to resultants (Example A.2.1) I learned from
L. Gruson.
The determinantal identities in §A.9 have an even longer history, going
back at least to Jacobi. For details, see Ledermann A) and Macdonald C).

Appendix B. Algebraic Geometry (Glossary)
The aim of this glossary is to fix terminology and notation from algebraic
geometry which are used in the text, and to give proofs or references for some
of the basic facts which are needed. Although a few scheme-theoretic construc-
constructions are needed, we have favored a more classical geometric language. For
example, "points" are always closed points, and, for a vector bundle E, P{E) is
the bundle of lines in E. The basic references for facts not proved here will be:
Grothendieck-Dieudonne A), denoted [EGA], and Hartshorne E), denoted
[H], and Berthelot-Grothendieck-Illusie, et al., denoted [SGA6]; other useful
references are Shafarevich A), Altman-Kleiman A), and Mumford B).
B.I Algebraic Schemes
B.I.I. An algebraic scheme over a field K is a scheme X, together with a
morphism of finite type from X to Spec (K). In other words, X has a finite
covering by affine open sets whose coordinate rings are finitely generated K-
algebras ([EGA]I.6.5, [H]II.3). The coordinate ring of an affine open set U may
be denoted A{U).
In Chaps. 1 — 19, the word scheme means an algebraic scheme over some
field.
A closed subscheme Y of a scheme X is defined by an ideal sheaf J'(Y) in
the structure sheaf ffx of X; for an affine open covering of X, Y corresponds to
an ideal in each coordinate ring of X. A closed subscheme Y of X comes
equipped with a closed imbedding Y^>X. In general, an imbedding, or a
subscheme is assumed to be closed, unless prefixed by "open", or "locally
closed". The notation Y^-* X is used to indicate that Y is a closed subscheme of
X.
If Y is a closed subscheme of a scheme X, X~Y denotes the open sub-
subscheme of X which is the complement of the support of Y.
B.I.2. A variety is a reduced and irreducible (integral) algebraic scheme.
A subvariety V of a scheme X is a reduced and irreducible closed sub-
subscheme of X; a subvariety V corresponds to a prime ideal in the coordinate
ring of any affine open set meeting V. The local ring of X along V, denoted
^v,x, is the localization of such a coordinate ring at the corresponding prime
ideal ([H]I.3.13); its maximal ideal is denoted Jfy.x- In Grothendieck's
B.2 Morphisms
427
language, 0VX is the stalk of the structure sheaf ^j of X at the generic point of
V.
The function fie Id of a variety V is denoted R{V). If V is a subvariety olX,
R(V)i% the residue field ^v.x^v.x-
B.1.3. The dimension of a scheme X, denoted dimA', is the maximum
length n of a chain
0* FoS^iS-.-S VKcX
of subvarieties of X. If X is a variety, dimA1 is the transcendence degree of
R (X) over the ground field. The notation X" may be used to signify that X is
an n-dimensional variety. A scheme X is pure-dimensional if all irreducible
components of X have the same dimension.
If V is a subvariety of a scheme X, the codimension of V in X, denoted
codim(F, X), is the maximum length dot a chain of subvarieties
V= Vn
VdaX.
A point on a scheme X is a O-dimensional subvariety of X. A point P is
rational over the ground field K if R(P) = K. We often write x(P) or K(P) in
place of R (P). A point P is a regular point of X if ffPX is a regular local ring
(Appendix A6). The open set of regular points in X may be denoted A^g.
B.1.4. Affine n-space, denoted A", or A?, is the affine variety whose
coordinate ring is the polynomial ring K[x\,...,xn\ For n= 1,2,3,4 co-
coordinates (/), (x, y), (x, y, z), (w, x, y, z) are often used. Projective n-space is
denoted P", or IPJ. Unless otherwise labelled, x0,..., xn will be homogeneous
coordinates on P", (x0:...: xn) the point in P" with homogeneous coordinates
Xo,...,xn. We identify A" as the open subscheme of P" where xo=t=0. The
point A : 0) in P1 is called zero point and denoted 0, while the point @ : 1) is
the point at infinity, and denoted oo.
The subscheme of A" defined by an ideal / = if],... ,/m) in K[x\,..., xn] is
denoted V{I), or F(/,, ...,/m). Similarly, if / is a homogeneous ideal in
K[xo,...,xn], generated by forms /,...,/„, V(I) or V{f,...,fm) may
denote the subscheme of P" determined by /.
B.2 Morphisms
B.2.1. A morphism f: X -> Y of algebraic schemes is assumed to be compatible
with the structure morphism to Spec (AT), K the ground field. If/maps an
affine open subset V of X into an affine open subset U of Y, then /
corresponds to a homomorphism/*:^([/) -* A(U') of/^-algebras.
The identity morphism of a scheme X is denoted idj, or 1*, or simply 1.
The composite of morphisms/: X -> Y and g : Y -> Z is denoted g °f or gf.
B.2.2. A morphism/: V—>Wo{ varieties is dominant if the image of/is
dense in W\ equivalently, the induced homomorphisms /*, described above,

428
Appendix B. Algebraic Geometry (Glossary)
are injective. If /: X -* Y is a morphism, and V is a subvariety of X, there is a
unique subvariety W of Y such that / maps V dominantly to W; f induces a local
homomorphism
/* :0WtY
The induced homomorphism on residue fields is an imbedding f*:R( W)
-*R(V). If dim V=dimW, then R(V) is a finite field extension of R(W)
([EGA]IV.5.5.6).
A morphism /: X —> Y of varieties is birational if it is dominant and /*
maps R{Y) isomorphically onto R{X). More generally, a morphism f:X -* Y
of schemes is birational if X and Y have the same number of irreducible
components, each component X, of X is mapped dominantly to a distinct
component Y, of Y, and the induced morphisms from <$YuY to <&x,,x are
isomorphisms.
B.2.3. If/: X -> 5, # : 7 -> 5 are morphisms, the fibre product of Z and Y
over 5 is denoted X x$ Y. If A', Y, and 5 are affine, with coordinate rings A, B
and A, then A'xi Y is the affine scheme with coordinate ring A (8)^5; in general
XxsY is constructed by patching together such affine schemes ([HJII.3,
[EGA]I.3). The fibre product comes equipped with projections p: Xxs Y -> X
and q : X xx Y -> Y. For any scheme Z with morphisms u : Z -+ X, v : Z -> Y
such that/0 u = g ° v, there is a unique morphism, denoted (w, u), from Z to
Jxj Ysuch thatp ° (u, v) = u,q° (u, v) = v. This universal property:
Hom5(Z, X) x Hom5(Z, Y) = Homs(Z, X xs Y)
characterizes Xxs Y up to canonical isomorphism.
A commutative square of morphisms
Xy S
is called a fibre square if Z is the fibre product of X and Y, and /> and g are the
canonical projections. In a fibre diagram, all squares appearing in the diagram
are required to be fibre squares.
When S= Spec (AT), K the ground field, we write Xx Y in place of X xs Y,
and call it the Cartesian product of X and Y.
Iff: X -> Y is a morphism, and Z is a closed subscheme of Y, the i'nvewe
image scheme, denoted f'l{Z), may be identified with the fibre product
XxYZ. \{J~ is the ideal sheaf of Z in Y, then/~'(Z) is defined by the ideal
sheaf f~](S) -fix. In case / is a closed imbedding, XxYZ =/"' (Z) is the
intersection scheme Xf]Z, the closed subscheme of Y defined by the sum of
the ideal sheaves of X and Z. If V1,... ,V, are closed subschemes of Y, we
write D V, or Ft fl... D Kr for the subscheme of Y defined by the sum of the ideal
sheaves of the Vt.
A morphism/: X-* Y is separated if the diagonal morphism from X to
A'xyA' is a closed imbedding. For the valuative criterion of separatedness, see
[EGA]II.7.2 or [H]II.4.3. An algebraic scheme is separated if its structural
B.2 Morphisms
429
morphism to Spec (AT) is separated. The reader who so wishes may assume all
schemes and morphisms are separated.
B.2.4. A morphism /: X -> Y is proper if it is separated, and universally
closed, i.e., for all Y' -> Y, the induced morphism from XxYY' to ,Y' takes
closed sets to closed sets. For the valuative criterion of properness see
[EGA]IL7.3or[H]IL4.7.
A scheme is complete if the structural morphism to Spec (K) is proper.
A morphism /: X -> Y is finite if for each affine open subset U c Y, the
inverse image ?/'=/"'(?/) is affine, and the induced homomorphism of
coordinate rings make A(U') a finitely generated A(U)-modu\e. A finite
morphism is proper with finite fibres (and conversely, cf. [EGAJIII.4.4.2).
If/: X -> Y is a proper surjective morphism of varieties, then/factors into
g °f, where f':X-*Y' has connected fibres, and g: Y' -> Y is finite. If
dim Z= dim Y, there is a non-empty open set of X which maps isomorphically
onto an open set of Y'. (These facts, proved in [EGAJIII.4.3.1, 4.4.1, are used
only in an alternative proof in § 1.4, and in § 20.)
B.2.5. A morphism /: X -> Y is flat if for UcY, U' <= X affine open sets
with /(?/') c U, the induced map /* :A(U) -* A(U') makes A(U') a flat
^4(?/)-module. Equivalently, for all subvarieties V of X, with W= f{V) ,?>v,x
is a flat^R/ y-module.
A morphism/: Z -> Yhas relative dimension n if for all subvarieties Kof Y,
and all irreducible components V of/~'(F), dim V = dim V+ n. If/is flat, Y
is irreducible, and Z has pure dimension equal to dimY+n, then / has
relative dimension n, and all base extensions ^xj-Y'-^Y' have relative
dimension n (cf. [H]III.9.6, [EGA]IV. 14.2).
Convention. In the text, unless otherwise stated, a flat morphism is assumed
to have a relative dimension.
B.2.6. A variety is normal if the coordinate ring of any affine open subset
is integrally closed in its function field. Any variety X has a normalization,
which is a normal variety X together with a finite, birational morphism from
X to X. If X is affine with coordinate ring A, then X is affine with coordinate
ring the integral closure of A in R(X); in general X is constructed by gluing
together these affine normalizations ([EGA]II.6.3.8).
B.2.7. If f:X-*Y is a morphism, the sheaf of relative differentials is
denoted Qx/Y. When Y= Spec(A^, we write Qx. If g : Y -> S is a morphism,
there is an exact sequence of sheaves on X:
f*alr/s-
0
([EGA]IV.16.4.19, [HJII.8.11).
We say that a morphism /: X -* Y is smooth if / is flat of some relative
dimension n, and fij^y is a locally free sheaf of rank n. It follows that for any
Y' -+ Y the base change X x Y Y' -+ Y' is also smooth of relative dimension n.
A scheme X is called non-singular, or smooth, if it is smooth over Spec(K). When
K is perfect, this is equivalent to its local rings being regular ([EGA] IV. 17.15.1).

430
Appendix B. Algebraic Geometry (Glossary)
Such a scheme is a disjoint union of irreducible n-dimensional varieties, for
some n. A simple point of a scheme X is a point in the open subscheme of X
which is smooth over Spec (K).
Convention. Unless otherwise stated, a smooth morphism will be assumed
to be separated. A non-singular scheme will be assumed to be irreducible.
If/: X -> Y is smooth of relative dimension n, the relative tangent bundle,
denoted TX/y, is the vector bundle whose sheaf of sections is the dual bundle
to Qxiy (§B.3.2). When Y= Spec (K), we write simply Tx, the tangent bundle
o(X.
B.3 Vector Bundles
B.3.1. A vector bundle ? of rank r on a scheme X is a scheme E equipped with
a morphism n: E -> X, satisfying the following condition. There must be an
open covering {?/,} of X and isomorphisms <pt of 7r~'(?/,) with t/,x Ar over [/,,
such that over U, f) C/,- the composites <pt o cpj1 are linear, i.e., given by a
morphism
These transitions functions satisfy: gik = g^g^, g7/ = gji, ga= 1- Conversely,
any such transition functions determine a vector bundle. Data (U1,, (p'i)
determine an isomorphic bundle if all composites <p| o q>j are linear on l][ f] Uj.
B.3.2. A section of ? is a morphism s:X-+E such that 7r°^ = idjj-. If E is
determined by transition functions gtj, a section of E is determined by a
collection of morphisms s-t: [/, —> Ar, such that
on UjDUj. The sheaf of sections of ? is a locally free sheaf <? of ©^-modules of
rank r. Conversely, a locally free sheafs (of constant rank) comes from a vector
bundle E, unique up to isomorphism. This may be seen by using transition
functions. For an affine open set Uc X with coordinate ring A, n~l(U) is an
affine open set in E, with coordinate ring the symmetric algebra
where ?v = Horrv,^, 0X), and F(U, ?") = H°(U, «?v) is the space of sections.
If 5 is a section of E, the zero scheme of s, denoted Z(s), is defined as
follows. Let .?,¦:?/,-> Ar determine ^ on ?/,-, s,—(sn,..., sir), sim in the
coordinate ring of I/,-; then Z(s) is defined in ?/, by the ideal generated by
si I > • • • , Sir ¦
B3.3. Several basic operations are defined for vector bundles, compatibly
with the corresponding notions for sheaves: direct sum E ® F, tensor
product ?®F, exterior product A'?, symmetric product S'E or Sym'?, dual
bundle ?v, pull-back /*? = X' xxE foif:X'-+X a morphism. The trivial
bundle of rank one on X is often denoted simply 1 or 0.
A homomorphism of vector bundles ? -* F corresponds to a homo-
morphism ?^>S?~ of corresponding locally free sheaves. To give such a
B.4 Cartier Divisors
431
homomorphism is equivalent to giving a section of the bundle Hom(?, F) =
?v ® F. A sequence
0
?„_,
0
of vector bundle homomorphisms is a complex if d,_i ° d,¦ = 0 for i = 1, ...,«.
The complex is exact at x e X if the corresponding complex of sheaves is exact
at x\ equivalently, the induced complex of vector spaces E.(x) over the
residue field x(x) is exact.
If F is a finite-dimensional space of sections of a vector bundle ? on X,
there is a canonical homomorphism from the trivial bundle Ixfto ?. The
sections F generate E if this homomorphism is surjective.
B.3.4 If 5 is a section of ?, there is a Koszul complex A" E):
0- Ar?v -> Ap?v ->...-> A2?v ->?v -> 1
0
which is exact on A'-Z(i). For the corresponding sheaf S, the image of
the ideal sheaf of Z (.?), and one has a complex of sheaves on X
(*)
0 -> A'
0,
obtained by globalizing the construction of Appendix A5.
The section ^ is called a regular section of ? if this last sequence is exact. If
x e Z(s), and ? is trivialized near x, so ^ is given by a sequence E,,..., jr),
5j e C, y, then (*) is exact at x if and only if s,,..., sp is a regular sequence of
elements in^jj- (Appendix A5, A.6).
We usually identify a vector bundle with its locally free sheaf of sections,
unless there is a particular reason to distinguish between them.
B.3.5. A line bundle is a vector bundle L of rank one. For n e Z, the line
bundle L®" is defined to be the n-fold tensor product of L if n > 0, the (- «)-
fold tensor product of Lv if n < 0, and the trivial line bundle 1 if n = 0.
B.3.6. In studying vector bundles, there is no loss of generality in assuming
that the base space X is connected. Indeed, prescribing a vector bundle of rank
r on X is equivalent to prescribing a vector bundle of rank r on each connected
component of X. It is sometimes convenient to allow the rank of a vector
bundle or locally free sheaf to vary on different connected components; for the
same reason, this makes no essential difference.
B.4 Cartier Divisors
References for this section are [EGA]IV.2O, Mumford B), Kleiman (9).
B.4.1. Let X be an algebraic scheme. For each affine open set U of X, let
K(U) be the total quotient ring of the coordinate ring A(U), i.e. the localiza-
localization of A (?/) at the multiplicative system of elements which are not zero
divisors. This determines a presheaf on X, whose associated sheaf is denoted

432
Appendix B. Algebraic Geometry (Glossary)
Of. LetX* denote the (multiplicative) sheaf of invertible elements in X, and ^*
the sheaf of invertible elements in<^=<^.
A Cartier divisor D on X is a section of the sheaf W*I0*. A Cartier divisor
is determined by a collection of affine open sets ?/, which cover X, and
elements / in K(C/;), such that ///• is a section of (9* over C/jfl ?/,-. Such / are
called local equations for D. The Cartier divisors on X form a group Div(X),
which is written additively.
B.4.2. The support of D, denoted Supp(?>), or sometimes |D|, is the
subset of X consisting of points x such that a local equation for D is not in
0%x. The support of D, like the support of the section of any sheaf, is a closed
subset of X.
B.4.3. A Cartier divisor is principal if the corresponding section of X*I0*
is the image of a global section of X*.
If X is a variety, W is the constant sheaf R (X). The principal divisor of
rsR(X)* is denoted div(r). Since the support of div(r) is a proper closed
subset of X, there are only a finite number of subvarieties V of codimension
one in X such that r$(9px.
BAA. A Cartier divisor Dona scheme X determines a line bundle on X,
denoted ^(i)), or <<7(D). The sheaf of sections of 0{ft) may be defined to be
the ^jj-subsheaf of Xgenerated on ?/, as above by/. Equivalently, transition
functions for^(?>), with respect to the covering U,, are g0 =f/f.
A canonical divisor Kx on a non-singular n-dimensional variety X is a
divisor whose line bundle <9(KX) is QJ = A"(T/).
B.4.5. A Cartier divisor /) is effective if local equations/; are sections of 0
on I/,-. In this case there is a canonical section ofrf(D), which we denote by sD.
Regarding <0{D) as a subsheaf of X, sD corresponds to the section 1; with
respect to the covering ?/,, sD is given by the collection of functions/, which
clearly satisfies / = gV] fi on Ut 0 U}. The section sD vanishes only on the support
of?>.
For an arbitrary Cartier divisor Dona scheme X, if U is the complement
of the support of D, there is a canonical, nowhere vanishing, section of 0{D)
over U, which we also denote by sD. (This section extends canonically to a
"meromorphic" section of#(D) on X, with poles on the locus where D is not
effective, cf. [EGA]IV.21.2.11.)
B.5 Projective Cones and Bundles
References for this section are [EGA]II.8, [H]II.7, and Lascu-Scott A).
B.5.1. Let S' = S° © S1 © ... be a graded sheaf of ^-algebras on a scheme
X, such that the canonical map from 0X to 5° is an isomorphism, and S' is
(locally) generated as an ^algebra by S\ To S' we associate two schemes
B.5 Projective Cones and Bundles
433
over X: the cone of S'
C=SpecE"), n:
and the projective cone of S', Proj (S'), with projection p to X. The latter is also
called the projective cone of C, and denoted P(C):
= Proj (ST),
The
On P(C) there is a canonical line bundle, denoted
morphismp is proper ([EGA]II.5.5.3, [H]II.7.10).
If X is affine, with coordinate ring A, then S' is determined by a graded
4-algebra, which we denote also by S'. If x0, ...,xn are generators for S\ then
5" = ^4[x0, ...,xn]/I for a homogeneous ideal /. In this case C is the affine
subscheme of Xx A"+1 defined by the ideal /, and P(C) is the subscheme of
XxP defined by /; the bundle 0CA) is the pull-back of the standard line
bundle on P". In general Proj E") is constructed by gluing together this local
construction.
If S' -> S'1 is a surjective, graded homomorphism of such graded sheaves
of ^-algebras, and C=SpecE'), C = Spec(Sv), then there are closed
imbeddings C c-> C, and P(C') <^ P(Q, such that^c(l) restricts to^c(l)-
The zero section imbedding of X in C is determined by the augmentation
homomorphism from S1' to ffx, which vanishes on S' for ; > 0, and is the
canonical isomorphism of 5° with fix.
If C= Spec(S') is a cone on X, and/: Z -> X is a morphism, the pull-back
f*C=CxxZ is the cone on Z defined by the sheaf of <^2-algebras /* S'. If
Zc A" we write C|2.
Each section of the sheaf Sl on X determines a section of the line bundle
). Let^(n) or^c(«) denote the line bundle^c(l)®"-
B.5.2. Let z be a variable, S' [z] the graded algebra whose nth graded piece
5"
5'z" © S°z" .
The corresponding cone is denoted C© 1. The cone P(C(B 1) is called the
projective completion of C. The element z in E'[z])' determines a regular
section of ^cffii(l) on P(C® 1) whose zero-scheme is canonically isomorphic
to P(C). The complement to P(C) in P(C © 1) is canonically isomorphic to C.
With this imbedding in P(C © 1), P(C) is called the hyperplane at infinity.
B.5.3. If C is a cone over X, each irreducible component D of C is a cone
over an irreducible subvariety Kof X, which is called the support of D. If X is
affine with coordinate ring A, then Z> is defined by a homogeneous prime ideal
P' in S" (cf. Zariski-Samuel A)VII.2); then Kis the subvariety of X defined by
the prime ideal P°, and D = Spec(SVP'), with S'/P' an algebra over A/P° =
5°/P°. This construction patches together to define the cone structure and
support of D when X is not affine. Since 5° = 0X, the union of the supports of
the irreducible components of C is X.
Since C is dense in P(C©1) ([EGA]II.8.3.2), there is a one-to-one
correspondence between the irreducible components of C and of P(C(B 1),

434
Appendix B. Algebraic Geometry (Glossary)
and the geometric multiplicities of corresponding components coincide. If D is
an irreducible component of C, there is a canonical closed imbedding of
P(D®\) in P(C©1), which identifies P(D® 1) with the corresponding
irreducible component of P(C©1); the support of D is the image of
P(D © 1) by the projection of P(C © 1) to X.
B.5.4. More generally, if S' is a graded sheaf of ^-algebras, generated by
S\ such that the canonical map from fix to 5° is surjective, then S' determines
a cone C, with a morphism from C to X. If X is the subscheme of X defined by
the kernel of the homomorphism from fix to 5°, then by the previous
considerations, S' defines a cone C over X; we call X the support of C. It is
convenient to call such C also a cone over X. Of course, in this case, the zero
section imbedding of Xin Cmay not extend to a morphism from Zto C.
B.5.5. A vector bundle E on X is the cone associated to the graded sheaf
Sym(<?v), where S is the sheaf of sections of E. The projective bundle of <?is
P(?) = Proj(Sym<?v)-
There is a canonical surjection p*Ev
imbedding
fiE(l) on P(E), which gives an
Thus P(E) is the projective bundle of lines in E, and fiE(— 1) is the universal,
or tautological line sub-bundle. More generally, given a morphism f:T-*X,
to factor / into p °f is equivalent to specifying a line sub-bundle (namely,
f*fiE(-l))off*E.
If ? is a vector bundle on X, L a line bundle, there is a canonical
isomorphism <p:P(?)-> P(? ® Z,), commuting with projections to X, with
We have adopted the "old-fashioned" geometric notation for P(E).
With,?as above, our P(E) is the P(«?v) of [EGAJII.8.
B.5.6. If ? is a sub-bundle of a vector bundle F, with quotient bundle
G=F/E, there is a canonical imbedding of P(E) in P(F)- If p:P(F)^> X is
the projection, the composite of the canonical maps fif(— 1) —* p* F and
p*F -* p*G corresponds to a section of p* G ®fiF{\). This section is regular,
and its zero-scheme is P(E).
B.5.7. If ? is a vector bundle of rank r on X, and 0 < d < r, there is a
Grassmann bundle, denoted Grassrf(?) or Gd(E), of d-planes in E, with a
projection p : Gd(E) -* X, and a universal rank d sub-bundle S of p* E; S is also
called the tautological bundle on G. The bundle Q = p*E/S is called the
universal quotient bundle, and
the universal exact sequence. This is characterized by the universal property
that for/: T -* X, factoring /through GdE is equivalent to specifying a rank d
sub-bundle of f*E. There is a canonical Pliicker imbedding of GdE in P(AdE)
corresponding to the line sub-bundle AdS of p* AdE. For the construction of
GdE, see [EGA] 1.9.7 or Kleiman C). Note that GXE = P(E).
B.6 Normal Cones and Blowing Up
435
B.5.8. With p : P(E) ->Ia projective bundle, the imbedding fiE(- 1) c p* E
corresponds to an imbedding of fif(E) in p* E ® <^-(l). The cokernel of this
imbedding is the relative tangent bundle of P(E) over X:
0 -> ^,(?) ->p*E® fiE(l) -> 7>(?)/Jr - 0
(cf. Manin B), Lascu-Scott A)). More generally, if G = GdE, with universal
sub and quotient bundles S and Q, then
rc/jr = Horn(S, Q) = Sv ® Q .
Indeed, the universal exact sequence on G determines a "second fundamental
form" homomorphism (cf. Altman-Kleiman AI.3) from S to QlG/x ® Q.
Dualizing, this gives a homomorphism from Tc/X to Sv ® Q, which is checked
by local coordinates to be an isomorphism.
B.6 Normal Cones and Blowing Up
The reference for this section is [EGA]II.8.
B.6.1. Let X be a closed subscheme of a scheme Y, defined by an ideal
sheaf ,f. The normal cone CXY to X in Y is the cone over X defined by the
graded sheaf of^ algebras
CxY=Sv,ec{®Jr"/Jrn+l).
If f: Y'-* Y is a morphism, X' =/ '(A'), and g is the induced morphism
from X' to X, there is a canonical closed imbedding
Cx.Y'^g*CxY=CxYxxX'.
Indeed, there is a canonical surjection of f*J~ onto the ideal sheaf J~' of X' in
Y', which gives a surjection of© ^*(t/'"/Jz'"+l) onto
B.6.2. If the imbedding of X in Y is a regular imbedding of codimensi on d,
then CxY is a vector bundle of rank d on X, and is denoted also NXY
(cf. § B.7.1); the sheaf of sections of Nx Y is (j^/j^2)v.
In particular, if /: X -* Y imbeds X as a Cartier divisor on Y, then
CxY=NxY=i*fiY(X).
B.6.3. The blow-up of Y along X, denoted BIXY, is the projective cone
over Y of the sheaf of fiValgebras @J":
2Hxy=Prqj ($./•).
Let i = BIXY, and let n denote the projection from Y to Y. The canonical
invertible sheaf (line bundle) fi(\) on the projective cone Y is the ideal sheaf
of n~l(X), which is therefore a Cartier divisor on Y, called the exceptional

436 Appendix B. Algebraic Geometry (Glossary)
divisor. Let E = n~'(X). By construction ? is the projective cone of
/.f+\ so
E = P(CXY)
is the projective normal cone to X in Y. From this description one sees that
where C=CXY. Let r\ be the projection from E = P(Q to X. If the imbedding
of J in Y is regular, then the canonical imbedding of normal cones
Ne Yc: r\*NxY is the imbedding of the universal line bundle fiN(- 1) in rj*N,
N=NXY.
In general, ninduces an isomorphism from Y— E onto Y — X.
B.6.4. If Y is a variety, X <z Y a closed subscheme, then BIXY is also a
variety. This follows from the fact that if / is any ideal in a domain A, then
© /" is also a domain.
B.6.5. If X is nowhere dense in Y, then n: f -> Y is birational. Indeed,
since ? is a Cartier divisor on Y, no irreducible component of Y can be
contained in E, and since X contains no irreducible components of Y by
assumption, all irreducible components of Y and Y meet the open sets Y — E
and Y— Xrespectively, which are isomorphic by n.
B.6.6. If 7 has pure dimension k, then CXY also has pure dimension k, for
any closed subscheme X cr Y. To see this, consider the imbedding of X in
YxA1 which is the composite of the given imbedding of A' in Y and the
imbedding of Y in Yx A1 at 0 e A1. The normal cone to X in Yx A1 is C © 1,
where C = CXY. Since X is nowhere dense in yx A\Blx(Yx A1) is birational to
YxA1, so has pure dimension k + 1. Since the exceptional divisor P(C®\)
is a Cartier divisor on Blx(YxA1), it must have pure dimension k, and C is an
open subscheme of P(C © 1).
B.6.7. If y is flat over a non-singular curve T, then Blx Y is also flat over T, for
any X a Y. Indeed, all ideal sheaves in (9Y are torsion free over &T (cf. Example
A.5.4), so © J" is flat over <BT; hence BIXY, whose local rings are localizations of
© J", is flat over T.
B.6.8. If X is Cartier divisor on Y, then Y=Y. More generally, suppose
Xa Y is defined by an ideal sheafs, and D is an effective Cartier divisor on Y,
defined by an ideal sheaf/". If Z is the subscheme of Y defined by the ideal
sheaf,//, then there is a canonical isomorphism of BIZY with BIXY over Y, such
that the exceptional divisor in BIZY corresponds to the divisor E + n* D in BIXY,
with ? the exceptional divisor in BIXY.
B.6.9. If Ic y is a closed imbedding, and/: Y'-+ Y is a morphism, set
A" =/"' (X), g: X' -+ X the induced morphism. Then there is a closed
imbedding
Blx.Y'
BlxYxYY'
constructed as in B.6.1. If / is the induced morphism from BIXY' to BIXY, then
J~l(E) = E', where E and E' are the exceptional divisors.
B.7 Regular Imbeddings and l.c.i. Morphisms
437
In particular, if X c Y c Z are closed imbeddings, there is a-canonical
imbedding of 5/*y in BIXZ, with the exceptional divisor of Blxz restricting
to the exceptional divisor of BIXY. .
B.6.10. In case X c Y and Y a Z are regular imbeddings (cf. B 7)- let
Z = BIXZ, F the exceptional divisor in Z, g the projection from Z to Z Let
?= B/Xy Then ? c q~»G), F c q-\Y), and ? is the residual scheme to F in
Q'1(Y), i.e., the ideal sheaves of Y,F and q~*{Y) in Z are related by
,jr{Y)-jr{F)=jr{Q-'{Y)). ' ">; - ¦ ¦¦
In addition, the canonical imbedding of Y in Z is a regular imbedding^ with
normal bundle ..-,, ,
where n is the projection from Y to Y, and ? is the exceptional divisor on Y
To prove this, one may assume Z is affine with coordinate ring A, Y is defined
by a regular sequence tt,..., td in A, and X is defined by the regular sequence
t\,...,tn, n> d. If T\,...,Tn are homogeneous coordinates on P"-', then
(Lemma A.6.1) Z is the subscheme of ZxP" defined by the equation's
t1Ti-t!Ti = 0, all i<j, and Y is defined in yxP" by the equations
tt Tj—Tj Tj, d < i <j, where t, is the image of ?, in A/{t\,..., t<j). On the affine
open set U^ of Z where Tk =t= 0, the coordinate ring of Uk is ' -;:i
A[xu...,xk,...,xn]/{{ti-tkxi\i+k}),
and Q~'(Y) is defined in Uk by the ideal Ik generated by (?,,..., ^). If k S d,
h=('k), which is the ideal of F in Z. If k > d, Ik = (tk xt,..., tk xd) =
tk-{xx,...,Xd)\ since ?,t defines F on Uk, and (X|,...,xrf) defines Y, f is
residual to F in Q"'(^)- Since x,, ...,xd is a regular sequence in the ring of
Uk, k> d, the imbedding of Y in Z is regular. The asserted relation among
normal bundles follows from the relation among the ideal sheaves (cf.
Example 9.2.2).
B.7 Regular Imbeddings and l.c.i. Morphisms
Basic references for these concepts are: [EGA] IV. 16.9,17, 19.1, [SGA6JVIIL
and Grothendieck EI1.4. Here we consider only regular imbeddings -which
are closed imbeddings, and l.c.i. morphisms which admit factorizations into
closed imbeddings followed by smooth morphisms^
B.7.1. A closed imbedding /: X -+ Y of schemes is a regular imbedding of
codimension d if every point in X has an affine neighborhood U in Y, such
that if A is the coordinate ring of U, I the ideal of A defining X, then / is
generated by a regular sequence of length d. If J' is the ideal sheaf of X in Y, it
follows (Lemma A.6.1) that the conormal sheaf Jf/y2 is a locally free sheaf on
X of rank d. The normal bundle to X in Y, denoted Nx Y, is the vector bundle

438
Appendix B. Algebraic Geometry (Glossary)
on X whose sheaf of sections is dual to to SIS'1. The normal bundle Nx Y is
canonically isomorphic to the normal cone CXY. Indeed, by Lemma A.6.1, the
canonical map from Sym {SIS2) to © S"lf"+l is an isomorphism.
B.7.2. Let i: X -> Y be a closed imbedding, and let J be the ideal sheaf of X
in Y. If g : Y -> S is a morphism, there is an exact sequence of sheaves on X:
(i) SIS2-+i*Q[/s^Qx/s-+0.
If X is smooth over S, then i is a regular imbedding if and only if Y is smooth
over S in some neighborhood of X. ([EGA]IV.17.12.1). In this case the three
sheaves are locally free, and the above sequence is also exact on the left; this
corresponds to an exact sequence of vector bundles on X:
(ii)
0
Tx/S
i* T
r/S
B.7.3. If g : Y -> S is a (separated) smooth morphism of relative dimension
n, and i:S -* Y is a section, i.e. gi = ids, it follows that i is a (closed) regular
imbedding with normal bundle canonically isomorphic to i*Ty/s- More gen-
generally, if f:X-+ Y is any morphism, and y:X-> XxsY is the graph of/, i.e.
y = (\,f), then y is a regular imbedding of codimension n, with normal bundle
f*TY/s- (This follows since the base extension XxsY-*X of the smooth
morphism g is smooth.) In particular, the diagonal imbedding 5 : Y-* Yxs Y is
a regular imbedding with normal bundle TY/s-
B.7.4. If i: X -* Y and j :Y -+ Z are regular imbeddings, then the com-
composite k =j ° i is a regular imbedding, and there is an exact sequence of vector
bundles on X:
(iii) 0 -* NXY -* NXZ - i*NYZ -> 0
([EGA]IV.19.1.5).
If Y,a Z are regular imbeddings of codimension dit i= 1,..., r, and the
imbedding of X= f] y, in Z is regular of codimension ? 4> then
In particular, if Y -> 5 is smooth, the normal bundle to the r-fold diagonal
imbedding of Yin Yxs ... xs Yis the direct sum of r— 1 copies of 7Y/,y.
If iiA'-^y is a regular imbedding, and f:Y'-+Y is flat, then the
imbedding of X' =/"' (X) in y' is regular, with
NrY' = g*NxY,
where # is the induced morphism from X' to X (Example A.5.5).
B.7.5. If /: X -> y is a closed imbedding, g.Y-^Z smooth, such that g i is a
closed imbedding, then i is a regular imbedding if and only if g i is a regular
imbedding ([SGA6] VIII. 1.3). In this case there is an exact sequence
(iv) 0 -* /* TyiZ -+NxY-> NXZ -* 0
of vector bundles on X. (Apply B.7.3, B.7.4 to the imbeddings X^ XxzY^Y.)
B.8 Bundles on Imbeddable Schemes
439
B.7.6. A morphism f: X -* Y will be called a local complete intersection
(l.c.i.) morphism of codimension d if/ admits a factorization into a (closed)
regular imbedding of some codimension e, followed by a smooth morphism of
relative dimension d+ e. It follows from the preceding paragraph that if/= g i
is any factorization with i: X -> P a closed imbedding and g : P -* Y smooth of
relative dimension n, then i is a regular imbedding of codimension n — d.
lff=g i is such a factorization of the l.c.i. morphism/ the virtual tangent
bundle, denoted 7}, is defined to be the difference of the bundles i* TP/r and
NXP in the Grothendieck group of vector bundles on X:
Tf = \i*Tm]-[NxP]eK°X.
It follows from the exact sequences (iii) and (iv) that 7} is independent of the
choice of factorization (see the proof of Proposition 6.6(a), or [SGA6]VIII.2.5)
Note: In contrast to standard usage, we require a l.c.i. morphism to admit a
global factorization into a closed imbedding followed by a smooth morphism.
If X admits a closed imbedding/ into a scheme M smooth over Spec (K), then
any morphism f:X ~* Y admits such a factorization: X
i = (f,j), p the projection.
YxM^Y, where
B.8 Bundles on Imbeddable Schemes
B.8.1. If A' is a scheme which admits a closed imbedding in a non-singular
scheme, and &~ is a coherent sheaf on X, then there is a locally free sheaf E on
X and a surjection E -^3r. In case X is quasi-projective, some tensor product of
^"bya line bundle ^(n) is generated by its sections, and one may take E to be
a sum of copies of <^(— n), cf. Serre A) § 55. In general, one may assume X is
non-singular, in which case one may find such E which is a direct sum of
bundles 0 (/),), /), divisors on X, a result of S. Kleiman. The essential point is
that for an affine open covering ?/, of X, the complements X— U, are supports
of divisors. For details see Borelli A), or [SGA6JH.2.2.6.
B.8.2. If X<^> Y is a closed imbedding, and Y admits a closed imbedding in
a non-singular scheme, then there is a vector bundle ? on Y so that the projec-
projection from BIXY to Y factors into a closed imbedding in P(E) followed by a
projection from P(E) to Y. Indeed, if S is locally free and surjects onto the
ideal sheaf .f of X in Y, then Sym^ surjects onto ®J*, which imbeds
Proj (®J^) in Proj (Sym*?). One may take E to be the bundle whose sheaf of
sections is <Sy. In particular, BIXY also admits a closed imbedding into a non-
singular scheme. For if 7 c M, M non-singular, BIXY a BIXM, and by the
preceding assertion, BIXM imbeds in a projective bundle over M, which is
non-singular.
The homomorphism S'-^J'c^y corresponds to a section s of E whose zero
scheme Z (s) is X.

440
Appendix B. Algebraic Geometry (Glossary)
B.8.3. The fact that the canonical homomorphism from K°X to KOX is an
isomorphism for X non-singular, follows from several facts:
(i) For any coherent sheaf J*" on a non-singular variety X there is a locally
free sheaf Eo and a surjection Eo -+^F.8.1).
(ii) If Zis non-singular and n-dimensional, and
is an exact sequence of coherent sheaves on X, with Eo, ...,?„_, locally free,
then 4 is locally free. (This is a local assertion, and follows from the fact that
finitely generated modules over a regular local ring have finite free resolutions,
cf. Serre D)IV. th. 8.)
(iii) If 0 -+?'„'->... -* E'o' -* Jr" -> 0 is a resolution of T" by locally free
sheaves ?,', and 9~ -> T" is a surjection, there is a compatible resolution E. of
Sf, i.e., with a commutative diagram
4
4
4
E'[
4
0
> JT" -* 0
with the vertical maps surjective. (To achieve this, choose Ea to map onto the
kernel of the canonical map from E§ © ST to 3~". Having constructed the
diagram through Er, let fr be the kernel of the map from Er to Er_x, and fr' the
kernel of E" -* E"-\\ then choose Er+I to map onto the kernel of the canonical
map from E'r'+1 © $. to /?'.)
(iv) Let 3*~ -> 3~", E. -> ?" be as in (iii). Then the kernels E\ of the vertical
maps give a resolution of the kernel 9~' of S~ -* 9~'\ and
Z (-i)' [ea = Z (-1)' [??]+Z (-1)' [En
inK°X.
(v) A resolution E. of.9~ is said to dominate a resolution E" of J7" if they can
be related by a commutative diagram as in (iii), with &~" = Jr, y -* Jr" the
identity. Given any two resolutions of 3~, there is a third which dominates
them both. (The argument is similar to that in (iii), cf. Borel-Serre A) Lemma
11.)
From (i) and (iii) it follows that any coherent sheaf J*" has a finite resolu-
resolution E. by locally free sheaves. From (iv) and (v), the class ?( — 1 )'[?,-] in K°X
is independent of the resolution. From (iv), the resulting map [#"] -> ?( — 1)' [Et]
is well-defined on the Grothendieck group K0X. This determines a homomor-
homomorphism K0X -> K°X, inverse to the canonical homomorphism K°X -> K0X. For
details see Borel-Serre A) and Borelli A).
B.9 General Position
B.9.1. The following is a variation of a lemma of Serre.
Lemma. Let E be a vector bundle of rank r on
algebraically closed field, n the projection from E to
a scheme X over an
X. Let r be a finite
B.9 General Position
441
dimensional vector space of sections of E that generates E, and let Cl,...,C
be closed subsets of E, and Z|,..., Z, closed subsets of X. Then there is a non-
nonempty Zariski open subset F° of F such that for all s eF°, and all i,j, either
s'1 (C,-) is disjoint from Zj, or
dim(s
1 (Zy)) - r.
Proof. Since F generates E, the canonical morphism
cp:XxF-> E
is surjective and smooth of relative dimension v — r, where v = dim F. For
each irreducible component DiJk of CiC]n~1(Zj), (p'1(DlJk) is an irreducible
variety with
dim(p-l(Dijk) ^ dimiCiDn-^Zj)) +v-r.
Let p be the projection from Xx F to F. By the theorem on the dimension of
fibres of a morphism ([EGA]IV. 13, [H]II Ex. 3.22), there is a non-empty open
set FiJk c F such that, for all ^ e F^k, either p~x (s) is disjoint from (fl{Dijk),
or
dim(p-l(s)r\<p-l(DiJk))= dim (p-l(Dijk) - v .
The intersection of all the FiJk is the desired set F°. (See Example 12.1.11 for a
refinement.) D
B.9.2. The following is proved in Kleiman F), cf. [H]III,10.8.
Lemma. Suppose a connected algebraic group G acts transitively on a variety
X, over an algebraically closed field K. Let f:Y—> X, g : Z -* X be morphisms of
varieties Y, Z to X. For each point a in G, let Y" denote Y with the morphism
a ° f'from Y to X.
(a) There is a non-empty open set G° a G such that for all a in G°, Y" xxZ is
either empty or of pure dimension
dim (Y) + dim (Z) - dim (X).
(b) If Y and Z are non-singular, and char (K) = 0, then there is a non-empty
open set G° a G such that for all ainG°, Y" xxZ is non-singular.
For analogues of (b) in characteristic/), see Kleiman F) and Vainsencher C).

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Notation
Algebraic Geometry
Y^X,J-{Y) 426
A", IP", 0, oo 427
fiViX 426-427
R[X),R(X)* 7,427
dim(r), codim(F, X) 427
K(P),x(P) 427
F(/),F(/,,...,/m) 427
/=/,< = length 8,406
gf=g°f, id^l* 427
IxyjxJ 428
f-l(.Z),Xf\Z, fl ^ 428
^/y,^ 429
Tx/y,Tx 430
Z(s),/f(.s) 430,431
E®F,E®F, A'E 430
SiE = SymiE,Ev,f*E 430
|D| = Supp(D) 432
<?(?)) =^(D),^D 432
div(r) 432
C=Spec(S") 432-433,434
433
) 433
P(C®\) 433
/>(?) 434
GdE,Gd(Pn) 434,271
CJ.A'rl' 435
BIXY 435
^ 432
Pic(A-),Div(AO 14,30
KOX,K°X 17,280-281
Fl(A) 247
depth (X, Y) 251
Zm(JH 363
Intersection Theory
i(P,FG) 1
oxAv{r) 8
ZkX,Z*X 10
[div(r)] 10
Ra^JV, a~0 10
AkX,A^X 10
{a}* 10
/* HJ2
deg(a)Ja 13,144-145
[X] 15^
V(P) 15
c/ 17,372
i0 17,386
/* 18,32,98,113,131
ax,? 24
ord VD 29
[?»] 30, 32
|a| 33
D[V\,D-a 33
?>,
•?)„-a 38
...,Dn))x 39
c,(?)na 41
/¦ 43,98
s,(E)na 47
c{E),c,{E) 50,57
pr\<x,\p 53-54
x
c(F-E),ck(F~E) 57,243
si 65
s(X,Y),s{X,Y)k 73,155
exY,ePY 79
wn^ta) 8i
M^-r.Mly 86-87
X- V,XYV 94
(JV- V)s, (X]-...-Xr)z 95,154,201
i';f 98,113,134
i(Z,X- F Y) 120
^(«) = ^(ai,...,«,/) 121
i(Z,Dr...-Dd;V) 123
Notation
463
x -y,x yy,f*y n x 131
F- H^ 131
/i*jr,/i(JO 140
R 162
(/% 162
D(/),D(/) 166
(erj,,a, 176
lim(W,),lima, 196
;->0 (-0
\\mX,- V, 197
l->0
*«-,4r 199
A\X,A\X 211
degz.a, degz.F 211
Z(a),Z(s) 243,244
Z)t(o-),Dt(G) 243,254
D;o-),^D;o-) 243,249
c('>,...,cW) 243,419
= Ah Xi(c) 243,264
242,243
X(ao,...,ad),QhQ)i 253
|A|,X 263,264
Ax,AUfl 266
{A} = {A0,...,A.},^ 271
271
ch, td 56-57,282-283
282
293,354
aiX*- Y 305
/?°a,a', a»,a* 305-307
c = 4*' 320
282
,c(?.) 57,296
7),/!*(/) 320-321
Mc),g*c 322
324
[/] 327
chj(?.) 341-342
x,xx 287,349,353
HtX 371
Hy,uhp 371,372-373
372
A' 374
* 374
Nk{X),NS{X) 374,385
«.-=«^,y 378-379
c/r 378,380-381
dim^F 394,396
4fc(AVS) 394
A/,. ^, eA {q>, M) 407
^•(j),^t(j) 414
Xa(s) 415
^(^.....a./.A/) 416

Index
Abel-Jacobi map 14, 387, 390
Abelian variety 14, 60, 219
Adjunction formula 59,289,301
Affine bundle 22
Algebraic equivalence 185-186,373
Algebraic scheme 6, 426
Ample
line bundle 211
vector bundle 212
generically 217
n-ample 215
Analytic space 383-384
Arithmetic genus
modular law 292
uniqueness 292,293
birational invariance of 292, 294
and Todd class 288,354,361,362
Associativity
for intersection multiplicities 123, 129
for intersection products 132
Axioms for intersection products 97, 207
Basis theorem 268 - 279, 3 86
Bezout's theorem
classical version 101,144—152
for plane curves 14
refined 223-226
Bidegree 146
Bivariant rational equivalence class(es)
Gysin maps for 328
products of 322
pull-back of 322
push-forward of 322
Riemann-Roch for 365-366
for schemes 395
Blow-up 435—437, 439 (see also Monoidal
transform)
formula 114-117
Borsuk-Ulam theorem 239-240
Canonical class 67, 300, 302 (see also Chern
class)
divisor 60, 301, 432
of singular variety 77—79
Canonical decomposition of intersection
product 94-95,200-205,218-226
Cap product 131
refined 132-136,324-326
for Borel-Moore homology 371, 374-375
Cartesian product 24-25, 428
Carrier divisor(s) 29, 431-432
addition of 29
canonical section of 432
effective 19-20, 30, 432
intersection by 33-41
line bundle of 31-32,432
linear equivalence of 30
local equations for 29, 432
principal 30,432
representing a pseudo-divisor 31—32
support of 30-31,432
Weil divisor of 29-32
Castelnuovo-Severi inequality 311-312
Cellular decomposition 23, 25, 378
Characteristic
of a family of varieties 188 -194
homomorphism 199
section 199
variety 73
Chasles-Cayley-Brill-Hurwitz formula 310
Chern character 56-57,282-284
localized 340-355,363-366
in topology 367
Chern class(es)
blowing up 298-303
of dual bundle 54
of exterior power 55, 265
geometric construction of 252, 255—256
of line bundle 41-43,51
localized 255
of complex 347
top 244-246
MacPherson 376-377
Mather 79
of non-singular variety 60
of normal bundle 59—61
polynomials in 42, 53, 141
positive 216
ofP(E) 59
ofP" 59
of singular varieties 77—79, 376—377
of symmetric powers 57, 265
Index
465
of tensor products 54—56,265
total 50,60
uniqueness of 53
vanishing of 50,216-217,365,367
of vector bundles 50-63, 252, 255-256
of virtual bundles 57,296
Chern number 293-294
Chern polynomial 50
Chern roots 54
Chow ring 141 (see intersection ring)
Chow variety 16-17,186
Circle 192
Class
of hypersurfaces 84—85
of non-singular variety 84—85,253,
261-262,277-278
of plane curve 188
of surface 193
Codimension 427
Coefficient of a variety in a cycle 11
Cohen-Macaulay 81, 120, 13 7, 226, 251,
418-419
Cohomology
operations 377
for rational equivalence 143, 324- 326,
331,390
with supports 325
Coincidences, virtual number of 309
Commutativity
for Chern classes 41, 50
for divisors 35—38
for intersection multiplicities 123
for intersection products 106-107, 132
Compact supports for rational equivalence
184, 368, 378
Complete algebraic scheme 429
Complex
of vector bundles 431
perfect 365-366
Conductor 167,289
Cone(s) 70,432-434
construction, for moving lemma 206
exact sequence of 72—73
normal 71,73,435
projective 432—434
projective completion of 70
pull-back of 433
Segre class of 70-73
of sheaf 73
support of 433—434
zero section of 433
Conies, plane 63-64, 157-158, 182-183,
187-194,204,275
Conservation of number 180— 185,
193-194
Contribution of variety to an intersection
product 94-95,151,224-225
Correspondence(s) 308-318,384-385,
390-391
(a, b)- 309
birational 312
on curves 310-312
degenerate 308-309
degree (or codimension) of 308
fixed points of 315-318
on Grassmannians 317
indices (or degrees) of 309
irreducible 306
of morphism 306
principle 310
product of 305-308
push-forward and pull-back of 307
onP" 315-316
ring of 307-308
Severi's formula for 317
on surfaces 317
topological 384-385
transpose of 306
valence of 309-310
Zeuthen, generalized 310
Critical point
moderate 246
multiplicity of 124-125
Curve(s)
rational normal 84, 156
tangent to given varieties 187-194
Cycle
class 10
class homomorphism or map 371-378
and Gysin maps 382
and specialization 400
and topological intersections 378- 385
fundamental 15
k- 10
non-negative, positive 10, 180, 211
ofsubscheme 15
Deformation 198-200
generic 201-205
to normal cone or bundle 86-91
Degeneracy
class 254-263,329,330
positivityof 260-261
symmetric and skew-symmetric 216,
259-260
locus 243
existence of 215—216
Degree
of cycle or equivalence class 13
of cycle or variety on P" 42, 144, 149- 150
L- 211
of Schubert variety 274
Depth 251-252,255,256,418-419
Determinant 409-412

466
Index
Determinantal
class 249-253
formula 249-253
identities 419-424
locus 243,419
Differentials, relative 429-430
Dimension 427
pure 427
relative 429
for schemes over regular base 394, 396
Distinguished varieties 94-95, 120-121,
184,200-205, 218-226, 232
Double point
class 166
formula 165-171,289
scheme 166-168
set 166
Dual variety, degree of 62-63, 84, 144,
169-170
Duality
for flag manifolds 276-277
for Grassmann bundles 267, 269, 270, 271
Grothendieck 367-368
Poincar6 281,328,398,440
Serre 291,368
Dynamic intersections 128,195-209
Enumerative geometry 187-194,272-279
Envelope, Chow envelope 356
Equivalence
of variety for an intersection product
94-95, 203-204
of connected component 153—160
relations on cycles 374, 385-392
Euler characteristic
of fibres of map 245-246
of non-singular variety 60,136,362
Euler obstruction 78, 376
Exact sequence
for closed and open subschemes 21-22,
186
for monoidal transforms 115
Exceptional divisor 435—436
Excess
intersection of divisors 36
intersection formula 102-106, 113, 327,
382
normal bundle 102,113,225
Extension of ground field 101-102,240
Exterior products 24-25, 397
Families of cycles and cycle
classes 176-180,362
Fano varieties, schemes 80, 217, 275-276
Fibre
product 428
square 428
Fixed points of correspondence
perfect 315-317
virtual number of 315
Flag bundle 68,247-248
Flag manifold or variety 23, 68, 219, 270,
276-277,310
Flat 413 (see morphism, flat)
Function field 427
Functoriality
ofGysinmaps 108-112,113,134,330
of intersection products 108-112
on non-singular varieties 130—136, 141
of push-forward and pull-back 11,18
General position 440-441
Genus
of curve 60
Todd 60
Giambelli formula 262, 265, 267-268, 271
Graph construction
for complexes of vector bundles 340-346
for vector bundle homomorphisms 88,
347-348
Grassmann bundle 248-249, 266-270, 434
Grassmann variety 23,219,271-279
of lines in P3 272-273
Grothendieck group
of sheaves 17,281,285-286,354
of vector bundles 57, 280-281, 294-295
Gysin formulas
for flag bundles 247-248
for Grassmann bundles 248—249
for projective bundles 66
Gysin homomorphism or map
from bivariant classes 328
for divisors 43—45
for l.c.i. morphisms 113
for morphisms to non-singular varieties
131, 134,208
refined 97-102,112-114,131,134
for regular imbeddings 89-90, 98, 101
for zero section of bundle 65-67, 100
Herbrand quotients 407—411
Higher direct image 281
Higher K-theory 151,356,403-405
Hilbert-Samuel Polynomial 42, 81
Hodge theory 387-388
Homogeneous varieties 207, 219-221, 378,
441
Homological equivalence 374, 385-388
Homology theory
Borel-Moore 371,374-375
for rational equivalence 10, 324
for topological K-theory 367
Hyperplane at infinity 433
Index
467
Imbedding 426
obstructions to 60—61
Incidence correspondence or variety 189,
273, 275, 277, 295
Index theorem 289-290, 388-389 (see also
Signature)
Intermediatejacobian 80, 387, 390
Intersecting
with a pseudo-divisor 38
transversally 138, 206, 441
Intersection class (see also Intersection
product)
analytic 383-384
of cycle and pseudo-divisor 33, 38
infinitesimal 199
positivityof 218-223
refined 94
topological 378,381-385
Intersection multiplicity or number
for divisors 38-40, 123-124,232-233,
400
for divisors and curves 125—126
fractional 125,142-143
on non-singular varieties 137-139,
227-234
for plane curves 7-10, 14
on schemes over Dedekind rings 398
of Serre 401-403
on surfaces 40-41
uniqueness of 139,207
Intersection product
for curves on a surface 60, 76, 95—96,
198, 203
for divisors 111,208,224
on non-singular varieties 130—136
part supported on a set 95, 100-101,
201-202,222
onP" 144-151,202-203
on schemes over Dedekind rings
397-401
on singular surfaces 39—40,125,142
and Tor 401-403
Intersection ring
of algebraic scheme 324
of Cartesian product 141
of flag manifold 276-277
of Grassmann variety 270
of monoidal transform 142
of multiprojective space 146
of non-singular variety 140-144,
151-152
of projective bundle 141
ofP" 144-145
of quasi-projective scheme 143,324
of scheme over a Dedekind ring
397-399, 405
Inverse image scheme 428
Irreducible components of intersection 120,
148-149
Jacobian subscheme 84,168-169
Jacobian variety 14, 17, 79-80, 256-2 57,
309
Key formula 114
Kodaira-Spencer homomorphism 199
Koszul complex 121-122, 282, 414-416,
431
Lefschetz fixed point theorem 314
Length 8, 120-122, 137, 145, 363, 406-407,
411
Limit
cycle 196
intersection class, cycle 197
set 196
Line bundle 431
Linear system (s)
base of 83
on curves 43,262-263
and Segre classes 82-85
tangents to 61-62
Linkage (liaison) 159,165,173
Linking numbers 383
Littlewood-Richardson rule 265,267, 271,
423
Local complete intersection (l.c.i.)
112-114,439
Local ring of scheme along subvariety
426-427
Manin's identity principle 312—313
Milnor number 124, 245-246
Monoidal transform 114-117,142,
298-302, 435-437
Morphism 427-430
birational 428
dominant 427-428
finite 429
flat 18,429
of relative dimension n 429
local complete intersection (l.c.i.)
112-114,439
perfect 286,367
proper 11,429
separated 428-429
smooth 429-438
Motives 312-313
Moving lemma 26, 205-209
Multi pie point formula 160, 171 -174
Multiplicity
algebraic 79-82
geometric 15
of a module w.r.t. endomorphism
407-411,415-416

468
Index
one, criterion for 81, 126-127, 137-138,
207, 230
at a point 79 - 82, 227 - 234
along a subscheme 81
along a sub variety 79 - 82, 231
Multisecants 159-160
Neron-Severi group 385, 388, 400-401
Noether's formula 230
Non-singular
variety, scheme 130-152, 429-430
imbeddings into 438, 439
Normal
bundle 43,435,437-438
cone 73, 85-86, 435-436
projectivized 433-434,436
domain 419
variety 429
Normalization 9,429
Numerical equivalence 374, 376, 383
finite generation of 375—376
Order
function for domain 411 — 412
of zeros and poles 8-10
Orientation(s), orientation class(es)
326-331
compatibility of 330
for cones 336
for flat morphisms 326
for imbeddings of manifolds 372 — 373
for l.c.i. morphisms 326-327,382
for monoidal transforms 332-333
for morphisms to smooth schemes 329
for perfect morphisms 367
for regular imbeddings 326, 378-379
Osculation 43
Partition 236
conjugate 263-264
P-field 237-241
Picard
group 14,30,331
variety 385
Pieri
formula 264,266,271,422-423
theorem 315-316
Pinch point 168-170
Pliicker formula 43
Point 7, 427
rational 427
regular 427
Polar
classes of hypersurface 84-85
classes of non-singular variety 226-227,
261-262
locus 261-262
surface 170
variety 226-227
Porteous formula 254-263
excess 258
Positivity of intersection products 210-234
Principal
lines 315
parts 43, 144
tangents 315
Principle of continuity 180-185, 193-194,
207
Projection 47,428, 433
Projection formula
for bivariant classes 323
for Chern classes 41,51,53
for divisors 34, 39
for intersection multiplicities 123, 207
for intersection products 98, 132, 134,
135, 140
for sheaves 281
Projective bundle 434
cycle class of 61
rational equivalence on 64—65
Projective characters 252-253
Projective completion of cone or bundle
433
Proper component of intersection product
120, 137,138
Proper intersection 35, 119-126, 136,
137-139,205-209
Pseudo-divisor(s) 31 - 32
of Cartier divisor 31
line bundle of 31
pull-back of 32
section of 31
sums of 32
support of 31
Weil divisor class of 32
Pull-back
flat, of cycles or cycle classes 18-21, 113,
134
from non-singular varieties 131, 329
for pseudo-divisors 32
for regular imbeddings or l.ci. morphisms,
see Gysin maps
Push-forward, proper, of cycles and classes
11-14
Quadric surfaces 141,192-193
Quotient variety 20-21,23-24, 142-143,
313-314,331
Ramification 62, 82, 125, 164-165,
170-171,174,258
Rank 253,261-262,273
Rational equivalence 10, 15-17, 25-27, 31
for local rings 396
on schemes 394, 396
Index
469
Real
algebraic schemes 240-241
closed field 235-241
forms 240
Reduction to diagonal 130-131,151-152
Regular
imbedding 437-438
local ring 126-127,137,418
section 414,431
sequence 416-418
Residual intersection
class 162,163,221,231-232
formula for top Chern class 245
theorem 160-165,333-337
Residual scheme 160-161, 164-165,437
normal cone to 163—164
Residue 384,389
Resolution of sheaf 281-282,440
Resultant 8-9, 150-152,410-411,424
Riemann-Hurwitz formula 62
Riemann hypothesis for curves 312
Riemann-Kempf formula 79-80
Riemann-Roch 280-304,339-369
for abelian varieties 291, 292
bivariant 365—366
for Cartesian products 360
for curves 288-289, 360
without denominators 296-298,353
formula for complexes 363—365
Grothendieck- 286,354
and Grothendieck duality 367-368
Hirzebruch- 288-354
homomorphism 287, 349, 353, 357-359
uniqueness of 354, 360
for imbeddings 282-283, 287-288, 294,
364, 365
Lefschetz- 353
for l.c.i. morphisms 349, 353-354, 363
for projections 283, 284-285, 287,
291-292,295,304
for quasi-projective schemes 348-353
for schemes over regular bases 395
for singular varieties 339—369
and specialization 400
for surfaces 289,361
for threefolds 290-291
and topological K-theory 367
and Tor 364-365
Ruled
join 147-152
variety 143—144
Samuel multiplicity 79,81
Scheme(s)
algebraic 6,426
over a regular base 393-396
smooth 429
-theoretic intersection 428
Schubert
calculus 271-279
class 271
special 271,272
variety 269, 270, 271
in flag manifold 276-277
Schur polynomials (or S-functions) 216, 217,
243,263-266,419-424
skew 266,274
Scott, D. B., formula of 61
Section(s)
sheaf of 430
of vector bundles 430-431,440-441
zero scheme of 430
Segre class
birational invariance of 74—76
of cone 70-73
of sheaf 73
of subscheme, subvariety 67,73-79,
168-169
total 50
of vector bundle 47-49,63-64
Segre imbedding 61,142,146
Self-intersection
of divisors 67
formula 44,60,103,167
Serre's
lemma 440-441
intersection multiplicity 401-403, 405
Seven's formula 317
Signature 292-293,388-389
Singular (Jacobian) subscheme 84,
168-169
Singularity
effect on intersections 40, 125, 183 -184
ofmorphism 258,348
ordinary 168-169
Thom-Boardman 258
Smoothing cycles 297
Snapper polynomials 361
Solutions of equations, rationality and
existence of 235-241
Special divisors 79-80, 256-257
Special position, principle of 193
Specialization 33, 105-106, 176-180,
398-401
to normal cone 89—91
of Segre classes 180
Splitting
construction 51
principle 54—55
Stiefel-Whitney class 68, 377
Subscheme, subvariety 6-7,426
Support
of divisor 30,31
of cycle 33

470
Index
Symmetric product
of curves 27, 79-80, 180, 262-263
of projective schemes 16-17,386
Tangent bundle 59-61,219-220,430
relative 429-430,438-439
for projective and Grassrnann bundles
435
Tangent cone 227
Tangents to curves 187-194, 133-134,
273-274
Theta divisor 80
Thom-Porteous formula 254-263
Todd class
of algebraic scheme 354,360-361
ofcoverings 362-363
of curves 360
of fibres 362
of l.c.i. scheme 354
of non-singular variety 60, 288, 293
of surfaces 361
of vector bundle 56-57, 283
Todd genus 60
Tor 286
and intersection products 401 -403
and Riemann-Roch 364, 401-403
Triple points 168-170
Unirational variety 179-180
Universal bundle 434
Vanishing theorems of Serre, Kodaira,
Le Potier 291
Variety 6-7,426-427
Vector bundle(s)
Chern class of 51-61
direct sums of 430
dual of 430
exterior powers of 430
locally free sheaf of 430
pull-back of 430
rational equivalence on 64
Segre class of 47-50
tautological, universal 434
Veronese
imbedding 60,63,146
surface 60, 157-158, 187
Weil divisor
of a rational function 10,30
associated to a Cartier divisor 29-31
associated to a pseudo-divisor 32
Whitney
sum formula 51,57-59
umbrella 24
Young
diagram 263-264
tableau 264
expansion of 264—265
Zeuthen's rule 9-10
Zeuthen-Segre invariant 301
Zero-cycles 16-17,386-387,390
Zero-divisors of bilinear forms 238
Zero scheme
of section 243,430
of vector bundle homomorphism
243, 439
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