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Conference Board of the Mathematical Sciences REGIONAL CONFERENCE SERIES IN MA THEMA TICS supported by the National Science Foundation
Number 54
INTRODUCTION TO INTERSECTION THEORY IN ALGEBRAIC GEOMETRY
by
WILLIAM FULTON
Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island
Expository Lectures from the CBMS Regional Conference held at George Mason University June 27July 1,1983
1980 Mathematics Subject Classifications. Primary 14C17, 14C15, 14C40, 14M15, 14NlO,
13H55.
Library of Congress Cataloging in Publication Data Fulton, William, 1939lntroduction to intersection theory in algebraic geometry. (Regional conference series in mathematics, ISSN 01607642; no. 54) "Expository lectures from the CBMS regional conference held at George Mason University, June 27July I, 1983"T. p. verso. Bibliography: p. 1. Intersection theory. 2. Geometry, Algebraic. I. Conference Board of the Mathematical Sciences. II. Title. III. Series. QAl.R33 no. 54 [QA564] 510s [512'.33] 8325841 ISBN 08218.07048 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940. The owner consents to copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that a fee of SI.OO plus $.25 per page for each copy be paid directly to Copyright Clearance Center, Inc., 21 Congress Street, Salem, Massachusetts 01970. When paying this fee please use the code 01607642/84 to refer to this publication. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotion purposes, for creating new coUective works or for resale. Copyright © 1984 by the American Mathematical Society Printed in the United States of America
All rights reserved except those granted to the United States Government
Contents Preface I.
v
Intersections of hypersurfaces 1.1 Early history (Bezout, Poncelet) 1.2 Class of a curve (Plucker) 1.3 Degree of a dual surface (Salmon) 1.4 The problem of five conics 1.5 A dynamic formula (Severi, Lazarsfeld) 1.6 Algebraic multiplicity, resultants
1 2 2 4 5 6
Multiplicity and normal cones 2.1 Geometric multiplicity 2.2 Hilbert polynomials 2.3 A refinement of Bezout's theorem 2.4 Samuel's intersection multiplicity 2.5 Normal cones 2.6 Deformation to the normal cone 2.7 Intersection products: a preview
13 15 17
3.
Divisors and rational equivalence 3.1 Homology and cohomology 3.2 Divisors 3.3 Rational equivalence 3.4 Intersecting with divisors 3.5 Applications
19 19 21 22 24 26
4.
Chern classes and Segre classes 4.1 Chern classes of vector bundles 4.2 Segre classes of cones and subvarieties 4.3 Intersection formulas
29 29 32 34
2.
5. Gysin maps and intersection rings 5.1 Gysin homomorphisms 5.2 The intersection ring of a nonsingular variety 5.3 Grassmannians and flag varieties 5.4 Enumerating tangents iii
9 9 9 10 11
37 37 39 41 44
iv
CONTENTS
6.
Degeneracy loci 6.1 A degeneracy class 6.2 Schur polynomials 6.3 The determinantal formula 6.4 Symmetric and skewsymmetric loci
47 47 49 50 51
7.
Refinements 7.1 Dynamic intersections 7.2 Rationality of solutions 7.3 Residual intersections 7.4 Multiple point formulas
53 53 54 55 56
8.
Positivity 8.1 Positivity of intersection products 8.2 Positive polynomials and degeneracy loci 8.3 Intersection multiplicities
59 59 60 62
9.
RiemannRoch 9.1 The GrothendieckRiemannRoch theorem 9.2 The singular case
65 65 69
Miscellany 10.1 Topology 10.2 Local complete intersection morphisms 10.3 Contravariant and bivariant theories 10.4 Serre's intersection multiplicity References 10.
73 73 74 76 78 81
Preface These lectures are designed to provide a survey of modern intersection theory in algebraic geometry. This theory is the result of many mathematicians' work over many decades; the form espoused here was developed with R. MacPherson. In the first two chapters a few episodes are selected from the long history of intersection theory which illustrate some of the ideas which will be of most concern to us here. The basic construction of intersection products and Chern classes is described in the following two chapters. The remaining chapters contain a sampling of applications and refinements, including theorems of Verdier, Lazarsfeld, Kempf, Laksov, Gillet, and others. No attempt is made here to state theorems in their natural generality, to provide complete proofs, or to cite the literature carefully. We have tried to indicate the essential points of many of the arguments. Details may be found in [16]. I would like to thank R. Ephraim for organizing the conference, and C. Ferreira and the AMS staff for expert help with preparation of the manuscript.
1. Intersections of Hypersurlaces
1.1. Early history (Bezout, Poncelet). A most basic question in intersection theory is to describe the intersection of several algebraic hypersurfaces in nspace, i.e., the common solutions of several polynomials in n variables. The ancients certainly knew about the possible intersections of lines and conics in the plane, and they also knew that rational solutions of two quadric equations in three variables behaved like solutions of one cubic equation in two variables [61]. We do not know who first observed that two plane curves of degrees p and q should intersect in pq points. By 1680 Newton [48] had developed an elimination theory for two such equations. This produced a resultant, which was a polynomial in one variable of degree pq whose solutions gave an abscissa of the intersection points of the two curves. The corresponding construction and assertion for n equations in n variables were made in 1764 by Bezout [5, 6]. Bezout's treatment was entirely algebraic, although he briefly interpreted his result for n = 2 and n = 3: the number of intersections of two plane curves (or three surfaces in space) is at most the products of their degrees. By referring to the resultants, which are polynomials in one variable, one can also discuss the possibilities of nonreal solutions, asymptotic solutions, and multiple solutions. As geometry developed, the first two of these situations were subsumed by considering intersections of hypersurfaces HI"'" H" in complex projective space Pc. Now we assign an intersection multiplicity i{P)
=
i{P, HI ... H,,)
to a point P of the intersection n H,; if the H, do not meet transversally at P, this multiplicity will be greater than one. Although there was little early discussion of this multiplicity, the governing principle of continuity was well understood, at least since Poncelet [51]. If the Hi vary in families Hj(t), with H,(O) = H" and PI(t), ... , P,(t) are the points of the general intersection n H, (t) which approach P as t + 0, then
,
i{P, HI ... H,,)
=
E i(lj{t), HI{t)···
H,,{t)).
J=I
Varying the H, so that the H,(t) meet transversally, this determines the multiplicity i(P, HI ... H,,). In all the above discussion, it is assumed that the intersection of the hypersurfaces is a finite set, or at least that P is an isolated point of n H" 1
1.2. Class of a curve (Pliicker). An important early application of Bezout's theorem was for the calculation of the class of a plane curve C, i.e., the number of tangents to C through a given general point Q:
Equivalently, the class of C is the degree of the dual curve C v. If F(x, y. z) is the homogeneous polynomial defining C and Q = (a: b: c), then the polar curve CQ is defined by
FQ(x, y, z) = aFx + bF" + c~, where F, = aF(x, y, z)/ax, F", ~ are partial derivatives. This is defined so that a nonsingular point P of C is on CQ exactly when the tangent line to C at P (defined by XFxCP) + YFv(P) + Z~(P) = 0) passes through Q. One checks that C meets CQ transversally at P if P is not a flex on C, so class( C) = #C
nC
Q
= deg C deg CQ = n (n  I),
if n is the degree of C, and Cis nonsingular. If C has singular points, however, they are always on C n CQ' so they must contribute. For example, if P is an ordinary node (resp. cusp) and Q is general, then i(P, Co CQ)
=2
(resp. i(P, C· CQ)
= 3).
This gives the first PlUcker formula [SO)
n(n  1)
=
class(C)
if C has degree n, II ordinary nodes,
K
+ 211 + 3K,
ordinary cusps, and no other Singularities
1.3. Degree of a dual surface (Salmon). In 1847 Salmon [53) made a similar study of surfaces. If S c p3 is a surface, the degree of the dual (or" reciprocal") surface S v is the number of points PES such that the tangent plane to S at P
contains a given general line I. (This number is one of the projective characters ()f S, now called the second class of S.) For a point Q E p3, let SQ be the polar surface of S with respect to Q: if F(x, y, z, w) defines Sand Q = (a: b: c: d), then aFx + bFy + c~ + dFw defines SQ' Taking two points Q" Q2 on I, one sees as before that a nonsingulaI point P of S is on SQ, n SQ, if and only if the tangent plane to S at P contains [. Thus for S nonsingular of degree n, and Q" Q2 general, deg{SV)
= #S
n SQ, n SQ, = n{n  If
As before, all singular points of S are contained in S n SQ, n SQ,' If P is an isolated singular point of S, its contribution to the total n(n  1)2 is the intersection multiplicity i(P, S . SQ, . SQ,). For example, the contribution of an ordinary double point is two, so deg(S V) = n(n  V  2ll if S has II ordinary double points.
"
"
'I
~
A
1 . . /
"1//
If S is singular along a curve C, however, a new phenomenon occurs, a problem of excess intersection: how to compute the contribution of C to the total intersection n(n  1)2, so that n(n  1)2 diminished by this contribution, and by contributions of other singular points, yields deg(S V). Salmon initiates a study of the contribution of a curve C to the intersection of three surfaces in space when C is a component of their intersection. For example, if C is a line, he gives its contribution as m + n + p  2, where m, n, p are the degrees of the surfaces. Salmon justifies this by saying that the answer must be independent of the choice of surfaces of given degrees, and then he calculates it directly in the degenerate case when the first is the union of a plane containing C and a general surface of degree m  I. This surface meets the other two surfaces in (m  I)np points, m  I of which are on the line C. The plane meets the other two surfaces in curves of degrees n  I and p  I in addition to C; these curves meet in (n  I)(p  I) points. The total number of points of intersection outside Cis therefore
{m  I)np  (m  I) + (n  I)(p  I)
=
mnp  (m + n + p  2),
as asserted. In case C is a double line on the first surface, he calculates its contribution as m + 2n + 2p  4 by working out the case where this surface is the union of two surfaces containing C.
If C is a double line on a surface S of degree n, this analysis predicts 5n  8 as the contribution of C to the intersection of S with SQ, and SQ,' However, as Salmon points out, there are special points on C, called pinch points (or "cuspidal" points), where the two tangent planes to S coincide.
If C is the line x = y = 0, and S is the surface Uy2 + Vxy + Wy2 = 0, then these pinch points are the intersections of C with the surface V 2 = 4UW, so there are 2n  4 pinch points on S. Thus C, together with its pinch points, diminishes the degree of S v by (5n  8) + (2n  4) = 7n  12. For example, a cubic with a double line (e.g. y2 = zx 2 + Xl) has a dual surface of degree three. Salmon also considers more general curves. If C is a complete intersection of surfaces of degrees a and b, and C is a component of intersection of three Stt~ of degrees m, n, and p, then he finds that the contribution of C to the total number of mnp is ab(m + n + p  (a + b». Concluding this remarkable paper, he deduces that if such C is an ,fold curve on a surface S, then it diminishes the degree of the dual by
ab[(,  1)(3, + I)n  ,2(,  I)(a + b)  2,(,  I)]. 1.4. The problem of five conics. Problems of excess intersection arise frequently in enumerative problems. The famous problem of the number of plane conics tangent to five given conics in general position is a typical example of this. A plane conic is defined by a quadratic polynomial ax 2 + by2 + cxy + dx + ey + unique up to multiplication by a nonzero scalar, so the space of conics can be identified with p5. One checks that the condition to be tangent to a fixed nonsingular conic is described by a hypersurface of degree six in p5. The desired conics are then represented by the points in the intersection of five such H 5• There are not 6 5 = 7776 such conics, however, as hypersurfaces HI originally thought by Steiner and others. indeed, the Veronese surface V:; p2 of conics which are double lines is contained in n Hi' and one can show (cf. §4 below) that the contribution of V to the intersection is actually 4512, which leaves 3264, the actual number of (nonsingular) conics tangent to five given conics in general position.
t,
n ... n
Note that the conics tangent to a fixed line form a quadric hypersurface in p6. Given five general lines, the Veronese contributes 31 to the predicted intersection number 25 for the five quadrics. Since everyone knew that there is only one nonsingular conic tangent to five general lines (by duality, for example), it is curious that these false answers were proposed when the lines are replaced by curves of higher degree. In spite of the clear exposition of the importance of excess intersections in enumerative geometry by Salmon and Cayley, such considerations played little role in the great development of enumerative geometry at the hands of Chasles, de Jonquieres, Schubert, Halphen, Zeuthen, and others. For one thing, they avoided writing equations for varieties and, especially, for parameter spaces. In general, however, their work can be interpreted as calculating intersections on appropriate spaces so that the intersections become proper. Often these spaces are blowups of the naive spaces, which amounts to adding structure to degenerate figures. For example, a classical approach to the space of conics amounts to working on the space of complete conics, which is the blowup p5 of p5 along the Veronese; in this model a point in the exceptional divisor corresponds to a double line together with a pair of points on the line. The proper transforms of the hypersurfaces H, then meet properly on p5 outside the exceptional divisor, and once one knows an appropriate "intersection ring" for Ps one may calculate their intersection. The same approach works for quadrics of arbitrary dimension. The beautiful study of complete quadrics was initiated by Schubert, who found many enumerative formulas. The rigorous construction of these parameter spaces and their intersection rings has been carried out by Semple and Tyrell, with modern reexamination by Vainsencher, Laksov, and Lazarsfeld. Realizing the spaces as orbit spaces of suitable group actions, by Demazure and by De Concini and Procesi, has led to a clearer understanding of their structure. 1.5. A dynamic formula (Severi, Lazarsfeld). In general, if HI, ... , Hn are arbitrary hypersurfaces in pn, with d, = deg(H,), Severi [58] proposed to assign numbers i(Z) to certain distinguished subvarieties Z of the intersection locus HI. n ... n Hn, so that
Each irreducible component of n H, should be distinguished, and each isolated point should be assigned its intersection multiplicity. In general, as in Salmon's examples, there may also be imbedded distinguished varieties. Severi's dynamic procedure, corrected and completed by Lazarsfeld [40], can be summarized as follows. If F, is a homogeneous equation for Hi' consider deformations H,(t) of Hi defined by homogeneous polynomials F, + tG; + tZG; + .. '. For a given subvariety Z of n HI' let j( Z) be the number of points of n H, (t) which approach Z as t ? D, for a generic deformation; in fact,j( Z) of the points will approach Z for
6
WILLIAM FULTON
any deformation for which the first order parts (G I , ... , G,,) belong to a certain open set Uz of the space of ntuples of polynomials of degrees d l , •.. , d". For any point P set i(P) = )(P). Only finitely many points will have i(P) * O. For an irreducible curve C, set
i(C)=)(C)
L
i(P),
PEe
so i( C) is the number of points that generically approach C, but not any particular point on C. Inductively, i(2)
=
)(2)  Li(V),
the sum over all proper irreducible subvarieties Vof 2. Then D(2) = d l ... d", which achieves the desired decomposition. We will later see a static construction of this decomposition, which is also valid in contexts where such deformations are unavailable. It should be emphasized, however, that in spite of the existence of a rigorous general theory, and some explicit formulas, the actual computation of the contributions i(2) remains a difficult problem. For plane curves, following Segre [55], Lazarsfeld gives the following answer. If H, = D, + E, where DI and D2 meet properly, and P is a point in E, let Gi be generic as above, let A, be equations for D" and let F be the curve defined by AIG z  AzG I . Then
i(P)
=
i(P, E· F) + i(P, DI . Dz ).
For example, if HI = 2LI + L z, Hz = LI + 2L 2, with L I, L2 lines meeting at a point P, then the SegreLazarsfeld formula shows that
i(P)
=
i(LI)
=
i(L z ) = 3.
1.6. Algebraic multiplicity, resultants. For an isolated point P in the intersection of hypersurfaces HI,' .. , H" in pn, a modern static definition of the intersection multiplicity is
i(P, HI'" H,,)
=
dime (9P/(fI"" , [,,),
where (9 P is the local ring of pn at P, and [, is a local equation for Hi in (9 p. If P is the origin in en c P", (9p is the localization of C[XI , ... , X,,] at the maximal ideal (XI" .. , X,,). Or one may replace (9 p by its completion C[[ XI" .. , X,,]], or by the ring C{ XI" .. , Xn) of convergent power series. This algebraic construction of intersection multiplicity dates from Macaulay [42]. Let us verify the agreement of this definition with that obtained from elimination theory, at least for plane curves. Suppose the curves are defined by polynomials [(x, y) and g(x, y), and the two curves do not meet at infinity on the yaxis. Thus we may assume
[(x, y)
=
ao(x)yn + al(x)ynI + ... + a,,(x)
INTRODUCTION TO INTERSECTION THEORY
with ao(O)
*
O. Let A
=
7
C[x](X) be the local ring of the xaxis at the origin. Then
A[y]/(/) is an Aalgebra which is a free Amodule of rank n, and one may construct the resultant r
=
R(/, g) inA by
r = det( A[y]/(f):.! A[y]/(f»). (It is a formal exercise, left to the reader, to show that this agrees with the usual definition, as in [60].) We must show that the order of vanishing of r at x = 0 is equal to the sum of the intersection numbers of the two curves at aU points P on the yaxis:
Now A[yl/(/, g) is finite dimensional over C, so it is a direct sum of its localizations p / ( / , g), where P varies over the points on the yaxis on both curves. Therefore
e
Li(P) = dimeA[y l!(f, g). p
Since the order of vanishing of r at x proved is
=
dimeA[y]/(f, g)
0 is dime A /( r), the equation to be =
dimeA/(r).
This is a special case of an important algebraic fact: LEMMA. Let A be a onedimensionai Noetherian local domain, M a finitely generated free A module and : M ..... M an A homomorphism. Then
length A ( M/( M» = length A ( A/(det( »). The length of an Amodule N is d if there is a chain of sub modules N = No ::J N, :::> ••• :::> Nd = 0, where N;/N;+, is isomorphic to the residue field of A. In case A contains a sub field K which maps isomorphically to its residue field, then length A N = dim K N. When A is a discrete valuation ring, the lemma is an exercise in elementary divisors. For the general case see [16, A2.6].
2. Multiplicity and Normal Cones 2.1. Geometrie multiplicity. A subvariety X of C N is defined by a prime ideal I( X) in q XI'" .• XN1. The coordinate ring f( X) is the residue ring
f(X) = C[XI •..•• Xn]/I(X). A (closed) subscheme Z of X is determined by an ideal I = I(Z) of f(X). which is a subvariety if I(Z) is prime. In this case the local ring of X at Z is the localization of f(X) at I(Z). and is denoted 19 z .x . If Z is a subscheme of X. the irreducible components of Z are the subvarieties of X corresponding to the minimal prime ideals of f( X) which contain I( Z). If V is such a component. the geometric multiplicity of V in Z is defined to be the length of the Artinian ring 19 v. z = 19 v.x/I(Z)l9 v.x ·
The cycle of Z. denoted [Zl. is defined to be the formal sum r
[Z]
=
L m;[V;J,
where Vi •...• v,. are the irreducible components of Z. and mj is the geometric and Z is the schemetheoretic multiplicity of V, in Z. For example. if X = intersection of n hypersurfaces which meet properly. then
cn
[Z] = Li(P)[P],
the sum over the points P in Z. with i(P) the intersection number described in §1.6. For an arbitrary variety X. subschemes Z are defined by ideal sheaves g = g(Z). On any affine open U c X which meets Z. g is given by an ideal in the coordinate ring of U. which is prime if Z is a subvariety. The local ring of X along V. and the geometric multiplicity of a component V of Z can be defined using any such U. 2.2. Hilbert polynomials. A subscheme Z of pN is defined by a homogeneous ideal 1= I(Z) in qxo..... XN1. If qxo ..... XN1t denotes the homogeneous polynomials of degree t. such an ideal I is the direct sum of its intersections It with qxo •...• XN1 t . Two homogeneous ideals define the same subscheme when their homogeneous pieces are the same for all but finitely many t. The Hilbert polynomial of Z is the polynomial Pz (t) such that
Pz(t)
= dimdC[Xo.· ...
9
XNL!lt )
10
WILLIAM FULTON
for all sufficiently large t. Indeed, one shows (cf. [30, §1.7; or 57]) that the right side is a polynomial of degree equal to the dimension of Z, for t »0. If n = dim(Z), one may define the degree of Z, deg(Z), to be the coefficient of t"ln! in Pz(t), i.e.
(i)
+ lower terms.
pz(t) = deg(Z)t"ln!
Lm,[V,) is the cycle of Z, then
It also follows that if [Z)
=
(ii)
deg(Z) =
L
m,deg(V,).
dim(~)lI
If V is a subvariety of pN, and H is a hypersurface of pN not containing V, then
deg(V
(iii) It will geometric spaces. In played an
n H)
= mdeg(V).
later become clear that this definition of deg( V) agrees with the notion of counting intersections of V with complementary linear fact, we shall have no need for Hilbert polynomials, although they have important role in the modem algebraic development of multiplicity.
2.3. A refinement of Bezout's theorem. The elementary facts about degree in the preceding section, together with an important join construction, allow a ·simple proof of the following proposition. A stronger result will appear later when more intersection theory is available. PROPOSITION. Let VI' ... ' V, be subvarieties of irreducible components of VI n ... n v,. Then ,
L
pN,
and let ZI' ... ' Z, be the
s
deg(Z,) .;;;
;=1
TI
deg(~).
jI
By a simple induction, one may assume s = 2. Construct the ruled join pZN+ I as follows. Let X o, ... , X N ' yo, ... , YN be homogeneous coordinates on pZN+ I. Let pf (resp. Pi') be the linear subspace of p2N+ I defined by the vanishing of all Y, (resp. all Xi). Identifying p'N with pN, one has V, c p'N. Let J be the union of all lines from points of VI to points of Vz• Algebraically, the homogeneous coordinate ring of J is simply the tensor product of the homogeneous coordinate rings of VI and Vz• One verifies that PROOF.
J = J(VI' V2 ) in
(i)
deg( 1) = deg( VI) deg( Vz )·
Let L be the linear subspace of L = pN and
p2N+
I defined by Xi
=
1';, 0 .;;; i .;;; N. Then
(ii) Thus we are reduced to the case where one of the varieties being intersected is a linear subspace. Since a linear subspace is an intersection of hyperplanes, one is further reduced inductively to the case where one of the varieties, say VI' is a hyperplane. In this
11
INTRODUCTION TO INTERSECTION THEORY
case, either VI ~ V2 and the proposition holds with equality, or [VI n V2) = m,[Z,), where the Z, are the irreducible components of VI n V2, and by (ii) and (iii) of §2.2 (for any hypersurface VI not containing V2),
[~= I
Lm,deg(Zi)
=
deg(VI )deg(V2 ).
2.4. Samuel's intersection multiplicity. Suppose HI" .. , Hn are hypersurfaces in an ndimensional variety V, and P is an isolated point of nH,. Let A = 19 py be the local ring of V along P, and assume each Hi is defined by one element I; in A. Let I = (/1" .. ,/,,). Then A/I is finite dimensional over C, and if P is a nonsingular point of V, one may use dimcA/I to give a workable definition of the intersection multiplicity i(P, HI ... Hn) as in §1. The following is a standard example of the failure of this definition in general. EXAMPLE. Let V be the image of the mapping : C 2  C 4 defined by (5, t) = (54, s3 t, st 3, t 4 ), let P be the origin, and let HI and H2 be the hypersurfaces of V defined by the coordinates XI and X 4 respectively. By varying H1 and H 2 , the principle of continuity requires that the intersection mUltiplicity is 4. However, one calculates that the ideal of V is generated by X I X 4  X 2 X 3 ' X~X3  x~, X2X~  xj, and X~X4  X~XI' from which it follows that dimcA/(x l , x 4 ) = 5. Samuel [54) defines the multiplicity i(P) = i(P, HI ... Hn) to be the coefficient of t n /n! in the HilbertSamuel polynomial
(i)
pet)
= dimdA/I') = i(P)tn/n! + lower terms
for t » O. To see that dim(A/ I') is a polynomial of degree n in t, for t » 0, one may proceed as follows. Let A = A/I and consider the suIjection of graded rings 00
A[X" ... , xnl
(ii)
EB
1'/1 ,+ 1
10
which maps X, to the image of I; in 1/12. The kernel of this homomorphism is a homogeneous ideal which defines a subscheme P(C) of projective (n  I)space ~\I over A. (Those who feel uncomfortable with projective space over a ring such as A may realize P(C) in pnI X V, since A is a residue ring of A.) This scheme P(C) is the projective normal cone to nHi in V. We shall discuss normal cones in succeeding sections. Here we shall use the fact that P(C) has pure dimension n  I, so its Hilbert polynomial has the form (iii)
pp(C)(t)
= dimcI'/I'+1 = i(P)tnI/(n 
I)!
+ ...
for t » O. A simple calculation shows that this definition of i(P) is the same as that in (i). However, since P(C) cP..;=.!~ only component of P(C) is the underlying variety PCI of p,\'I and, therefore, (iv)
defines the multiplicity i( P) without reference to Hilbert functions. In addition, sinceP(C)C phI,and
it follows that
(v) We see also that equality holds in (v) if the morphism (ii) is an isomorphism. This is related to the important notion of a regular sequence. DEFINITION. A sequence of elements fl""'[d in the maximal ideal of a local ring A is a regular sequence if fl is a nonzerodivisor in A, and if, for i = 2, ... , d, the image of /, in AI([I""'/'_I) is a nonzerodivisor. (This is equivalent to asserting that the Koszul complex 0+ Ad(A d ) + AdI(A d ) + ••. + Ad + A defined by fl" .. , fd is exact,giving a resolution of A I I. In fact, the multiplicity i(P) may also be defined to be the alternating sum of the dimensions of the
homology groups of this complex, d. [57].) The dimension of a local ring A is the length n of a maximal chain of prime ideals Po ~ PI ~ ... ~ p. ~ A. If A is the local ring of a variety V along a subvariety W, the dimension of A is the codimension of W in V. The ring A is CohnMacaulay if its maximal ideal contains a regular sequence of dim(A) elements. For example if P is a nonsingular point of V, then (9py is CohenMacaulay. The following lemma contains the main facts from commutative algebra that we will need. For proofs, see [38 or 57]. LEMMA. (a) If A is CohenMacaulay, a sequence f l, . .. ,fd of elements in the maximal ideal of A is a regular sequence if and only if
dim(AI(fI"",[d))
dim(A)  d.
=
(b) Let fl'" . ,[d be a regular sequence in a local ring A, and let I = ([I'''',[d)'
Then the canonical homomorphism ac
AII[XI ,···, XdJ+
ffi
,=0
1'11'+1,
which takes X, to the image of/, in 1112, is an isomorphism. Moreover. the kernel of the canonical surjection 00
A[XI ,· ... XdJ+
is generated by the elements /; Xj 
~ X,.
ffi
,=0
I'
I .. i < j ~ d.
For example. with notation as at the beginning of this section. if (9 py is CohenMacaulay, it follows that i{P. HI'"
Hn)
=
dimcepyl(fl,· ... /,,).
i.e .. Samuel's sophisticated multiplicity agrees with the naive multiplicity of § 1.
INTRODUCTION TO INTERSECTION THEORY
13
2.5. Normal cones. If W is a subscheme of an affine variety V, defined by an ideal 1 in the coordinate ring A of V, the normal cone C = CwV to W in V is defined to be C
=
Spec(
EB
1'/1'+1).
The isomorphism of the coordinate ring of W with A/1 = 1°/1 I determines a morphism Pc: C + W, called the projection, and a closed imbedding sc: W + C, called the zero section, with Pc 0 Sc = id w' If Id" .. .Jd generate 1, the canonical surjection of A/1[XI , ..• , Xd ] onto tf) 1'/1'+ I determines a closed imbedding of Cin Wx Cd: Wx Cd
C
pr It" ?' zero
Pc '" '\ Sc
W If 11, ... .Jd is a regular sequence, it follows from the preceding lemma that C = W X Cd. In general, since C is defined by a homogeneous ideal, it is a subcone of W X Cd, i.e., C is invariant under multiplication by C* on the fibres Cd.
In spite of the marvelous brevity of this algebraic definition of normal cone, its geometry is not so simple. Considerable study, beginning with [32], has been devoted to the case where W is a nonsingular subvariety. For example, if W = P is a point, then CpV is the tangent cone to Vat P; if V is a hypersurface in Cd, and P is the origin, one may check that CpV is the hypersurface in Cd defined by the leading homogeneous term of an equation for V. However, as is evident from the preceding section, the normal cones of interest for intersection theory are usually defined by ideals which are not prime ideals, i.e. W is a subscheme of V, but not usually a subvariety. There has been extensive recent study of associated graded rings tf) 1'/1'+ I in commutative algebra; one hopes that useful criteria for identifying the irreducible components of C, with their multiplicities, may emerge. The projective normal cone P( C) = P( C wV) is defined by P(C)
Proj(
=
EB / /1 1
1 + 1 ).
In concrete terms, if generators for 1 are chosen as above, P(C) is the subscheme of W X p d  I defined by the same equations that define C in W X Cd. A closely related and equally important construction is that of the blowup of a variety V along a subscheme W. This is a variety V = Blw V, together with a proper morphism "IT: V + V, satisfying: (i) The inverse image scheme E = "ITI(W) is a Cartier divisor on V, called the exceptional divisor: at each Q E E, the ideal 1fJQ.vhas one generator. (ii) E is isomorphic to P( C wV), and the mapping from E to W induced by 'IT is the projection from P( C) to W: P(C)=E
.....
V=BlwV
.....
V
~
W
~"
(iii) The induced mapping from A quick definition of B\ w V is Blw
V
E to V  W is an isomorphism.
V= Proj ( ,0 ffi I')'
the mapping '!I determined by the isomorphism of A with 1°. If II'" . , Id generate I, BI w V is the subvariety of V X pJ I defined by the kernel of the canonical homomorphism from A[XI , ... , Xdl onto ffi I'. In case 11, ... ,fd is a regular sequence, BI w V is defined by the equations /, JS  J; X" i < j, by the lemma of §2.4. In general, BI w V is the closure of the graph of the morphism from V  W to pJ1 defined by (/1:'" :Id)' Note that, since A is a domain, ffi I' is also a domain, so Blw V is a variety. The identification of E = '!II(W) with P(C) follows from the canonical isomorphism
( EB I') ®A A/I = E9 I'/I'+ I. Over the subvariety of pd I where the coordinate X, is not zero, W is defined by the equation /" since J; = (JS/ X,) /,. One important consequence of this construction is that each irreducible component of E = P( C) has dimension d  I. Indeed, E is locally defined by one equation in the ddimensional variety V, and any such subscheme has pure codimension one. The above constructions globalize to the case of an arbitrary proper closed subscheme Wof an arbitrary variety V. If g is the ideal sheaf of W in V, they are written CwV= Spec( P(CwV)
=
EB
Proj(
Blw V = Proj(
1'/1'+1),
EB
1'/1'+1),
EB 1').
They may be constructed by covering V by affine neighborhoods, over which the preceding constructions apply, and gluing over the overlaps. In case the imbedding of W in V is a regular imbedding, i.e., local equations for the ideal of W in V form a regular sequence in local rings of V, then C wV is a vector bundle, called the normal bundle to W in V, and also denoted N wV, If V and Ware nonsingular, this agrees with the definition of NwVas the quotient of tangent bundles: 0 ..... T w ..... Tvlw ..... NwV ..... O.
When D is an effective Cartier divisor, on a variety X, NDX is the restriction to D of the associated line bundle (9 x( D) on X. If E = P( C) is the exceptional divisor on the blowup Vof a variety V along a subscheme W, then
N£V= (9v(E)I£=Bc (I) is also the dual line bundle to the canonical line bundle Be< 1) on P( C).
It is a useful exercise to examine a normal cone which is not a vector bundle. For example if W is the intersection of two curves which have common components in the plane V, then CwV will have irreducible components which lie over each irreducible component of W, and other varieties as well. If the curves· are written in the form DI + E, D2 + E, where DI and D2 meet properly, then CwV has components over each component of E and over each point in DI Dz , including those points which are in E. To see this, let d l , d 2 , e be polynomials in R = C[X, Y] defining D I , D2 , E, and set / = (die, d 2e), the ideal of W. One verifies that the kernel of the homomorphism A//[UI , U2 ]> fB In//n+ I, which takes U, to die mod /2, is generated by d 2U I  dp2. Therefore CwV is the subscheme of W X C 2 defined by dPI  dP2' from which one may read off the components of C. Note that these components lie over precisely the subvarieties of W which contribute to the intersection product by the SeveriSegreLazarsfeld prescription (§ 1.5).
n
2.6. Deformation to the normal cone. In light of the principle of continuity, a reason why one can expect to use norma} cones to compute intersection products is because there is a deformation from the given imbedding of a subscheme Wof V to the zero section imbedding of W in the normal cone C = CwV. The affine version of this is a closed imbedding
WxC i/' p
pc,"
C with M' = M;"Va variety of dimension one greater than dim(V), such that over t "" 0, the imbedding of W x {t} in M; is isomorphic to the given imbedding of W in V, while the imbedding of W X {O} in M; is isomorphic to the zero section imbedding of Win C wV. Suppose V is affine with coordinate ring A, and W is defined by the ideal / = (fl' ... , fd)· Let M be the closure of the graph of the morphism
(V  W) x C* by (P, t) inMby
>
>
pd
(UP): ... :fAP): t), in V X C X pd. Note that W X C is imbedded
Wx C
=
Wx C X {(O: ... :O: I)} C M.
Over t = 0, one sees that the fibre Mo of M contains the blowup V = Blw V, but this is disjoint from W X {O}. We shall see that the complement to V in Mo is the normal cone C = C wV, so that MO = M  V is the desired deformation space. An algebraic version of this deformation was studied by Gerstenhaber [24] using the graded ring B defined by
B
= ...
e I"r n e ... e /r I e
A
e
ATe···
e AT n e ... ,
rnT n, with /m = A for m .:;; 0, and T an indeterminate. One
i.e., B = fBoo may define M'U, be the affine variety whose coordinate ring is B. The projection
from MO to C corresponds to the canonical inclusion of qT) in B, and the imbedding of W X C in MO to the canonical surjection of B onto A/I[T). Since the canonical homomorphism from A[T) to B becomes an isomorphism after inverting T:
Br == A[Th, the imbedding W xC c MO  pI(O) is isomorphic to the trivial imbedding of W X C in V X C. Over T = 0, since oc
B/TB ==
ffi
1"//"+1,
nO
we see that M; = C wV, with W imbedded as the zero section. An equivalent construction of this deformation is to define M to be the blowup of V X C along W X {O). The normal cone to this imbedding is the cone C ED I
=
ffi "
Spec(
/"/1"+1 ®A/IA/I[Tl),
so the exceptional divisor is the projective completion P( C ED I) of C, where C = CwV. The blowup ii = Blw V is also contained in M as a divisor, and if p: M .... C is the projection to C, then the scheme Mo = p  1(0) is the sum of two Cartier divisors, P(C ED I) and ii, which meet in P(C). We have a commutative diagram: W
....i,
f V
WxC
.....'0
f
f
MI
....J,
M
Jo .....
Mo
.....
{O}
~ p
~
{I}
.....
C
W
p{ C ED I) + ii
~
v
This last construction of the deformation works equally well for arbitrary varieties V and arbitrary closed subschemes Wof V. The deformation to the normal cone often functions as an algebrogeometric analogue of the topologists' tubular neighborhood. Note, however, that even if V and Ware nonsingular, there is usually no neighborhood of W in V which is, even complex analytically, isomorphic to a neighborhood in the normal bundle. In the
singular case, the normal cone may be a bundle over Weven when no neighborhood of W in V is topologically a product of W with a disc. 2.7. Intersection products: a preview. Normal cones will be basic to our general construction of intersection products, even when the intersection is not proper. To intersect hypersurfaces (Cartier divisors) HI"'" Hd on a variety V, consider the setup: HI
n ... n
v
Hd
r
fa Vx ... x V
HI X··· X Hd
To intersect subvarieties VI"'" tion to the diagonal"):
V, of a nonsingular variety
X, consider (" reduc
vln···nv,
r
Xx ...
X
xX
Here adenotes the diagonal imbedding. Each of these is a special case of the situation: W
.....
V
X
.....
Y
r
r
f
Here V is an ndimensional subvariety of Y, and f: X > Y is a regular imbedding of codimension d, i.e. X is locally defined in Y by a regular sequence of d elements; and W is the intersection scheme X n V, the subscheme of Y defined locally by the equations for X and for V. Our goal is to construct and compute an intersection product X· V, which will be an equivalence class of cycles of dimension n  d on X (in fact on W). Since f is a regular imbedding, the normal cone to X in Y is a vector bundle of rank d on X, denoted N x Y. There is a canonical imbedding of C w V in N x Y: CwV
.....
NxY
!
!
W
X
In fact, if V and Yare affine with coordinate rings A and B, and Wand X are defined by ideals / and J, respectively, then JA = /, so there is a surjection
ffi
J'jJl+1
~
////1+1,
which corresponds to the imbedding of C wVin N xY. We saw in the previous section that C has pure dimension n = dim(V). By the procedure of §2.1, C determines a cycle [C] on NxY. The class X· V will be constructed by "intersecting the cycle [C] with the zero section of NxY". Explicitly, X· V will be represented by a cycle of the form I:nl[V;J, where V; are
WILLIAM FULTON
18
(n  d)dimensional subvarieties of X, and IC] is (rationally) equivalent to the pullback cycle Ln;(NxYl v,]. (The next few sections will explain these terms.) In fact if IC] = Lm,ICj], with C, the irreducible components of C, and Zj is the support of the cone C" then the intersection of Cj with the zero section is a welldefined cycle class a j on Zj' and
X· V = 'Em/a/.
We shall see that in the case of hypersurfaces considered in § 1.5, these Zj are the distinguished varieties found by Severi and Lazarsfeld, and the contribution i(Zj) is simply the degree of mja/. In case the intersection is proper, i.e. dim W = n  d, then X· V is a welldefined cycle on W. If WI' ... , w,. are the irreducible components of W, then
[Cl
= 'E m/[
NxYiW;],
so X· V = LmjlW;]. The coefficients mj agree with the multiplicities discussed in §2.4 in case n = d. In the opposite extreme, when V = X, so W = X, then C wV = X is the zero section of N x Y. The intersection of the zero section with itself will be the" top Chern class" of NxY, and we will have the selfintersection formula
X· X= cANxY) n [Xl.
3. Divisors and Rational Equivalence
3.1. Homology and cohomology. Before beginning to develop a theory of rational equivalence, Chern classes, intersection products, etc., let us look to topology for a model. A complex projective variety X has homology groups HqX and cohomology groups HP X. Homology is covariant, cohomology contravariant. There are cup
products
and cap products
which make H* X = €I) HP X a skewcommutative graded ring, and H. X = €I) Hq X an H* Xmodule. If f: X ..... Y is a mapping, f*: H*Y ..... H* X is a homomorphism of rings, and one has the important projection formula,
f.(f*a n b)
=
a nf.b,
for a E H*Y, b E H. X. Any kdimensional subvariety V of ~ determines a class denoted c1(V) in H2kX, This follows for example from the fact that V can be triangulated to be an oriented 2kcircuit. Iff: X ..... Yis a morphism, andf(V) = W, then
f.c1(V)
=
deg(V/W)c1(W),
where deg(V/W) is defined as follows. If dim(W) < dim(V), then deg(V/W) = O. If dim(W) = dim(V), there is an open W· C W such that V n I(W') ..... W· is a finite sheeted topological covering, and deg( V/W) is the number of sheets of this covering; algebraically, deg(V/W) is the degree of the function field R(V) of Vasa field extension of R(W). If X is nonsingular of dimension n, capping with c1(X) gives the Poincare duality isomorphism
r
nc1(X)
..
HPZ ..... H2n _ p X . 19
In the nonsingular case, H * X ;;;; H* X has an intersection product. If A and B are subvarieties of X of dimensions a and b, one may therefore define A . B in H2a+2b2n( X). This product may be refined by using relative groups:
cl(A) E H 2a (A);;;; H 2n  2a (x, X  A) and similarly cl(B) E H 2n  2b (x, X  B), so cl(A) U cl(B) E H4n2a2b(x, X  (A ;;;; H2a+2b2n(A
n
n
B))
B),
and thus A . B lives in H2a+2b2n(A n B). Note that if the intersection is proper, this last group is free on the classes of irreducible components of A n B, so this determines A . B as a cycle; in other words, this gives a topological construction of intersection multiplicities. Complex vector bundles E in X have Chern classes c,( E) in H 2 ,X, satisfying: (i) co(E) = I; c,(E) = 0 if i > rank(E). (ii) Iff: Y > X, c,U*E) = !*c,(E). (iii) If 0 > E' > E > E" > 0 is an exact sequence, then
c,(E) =
L
cj(E')· ck(E").
j+k=i
(iv) If L is an algebraic line bundle on X and s is a section of L with zeroscheme D, * X, then
c,(L) n cl(X)
=
cl([D,]),
where cl([D,D = Lmicl(D;) if[D,] = LmiDi is the cycle of Dr For a line bundle L E HI(X, (9n c,(L) E H2(X, Z) may be constructed as the coboundary of L from the exact sequence
For a vector bundle E of rank r, let pee) be the projective bundle of lines in E, with projection p: P( E) > X, and universal (tautological) exact sequence
o > L£ > p* E >
Q£
>
0
with L£ a line bundle. The line bundle (9£(1) is dfined to be the dual of L£. Let ~ = c l «(9(l». Following Grothendieck [28], one may then define the Chern classes of E by the identity ~r
(*)
+ p*cl(E)~r' + ... + p*dE) = O.
(Such an equation exists by the structure of H*P(E) as an H* Xalgebra.) The total Chern class c(E) is defined to be I + c,(E) + '" + cr(E). The total Segre class of E is the formal inverse:
so so(E) = I, sl(E) ( * ) shows that
see) = c(E)', =  cl(E), sz( E) = c l ( E)2 
site) n cl( X)
= p*(~r'+'
n
C2(
E), etc. A calculation using
cl(P(E))).
This gives an alternate construction of Chern classesor at least of their images in homology: define s;( E) n cl( X) by this last formula, and invert formally to obtain c(E) n cl(X). For complete (compact) varieties, singular homology is satisfactory. To extend to arbitrary varieties, BorelMoore homology, constructed from locally finite chains, is more appropriate. With this homology, every variety V has a fundamental homology class cl( V).
3.2. Divisors. A Wei! divisor on an ndimensional variety X is an (n  I)cycle on X, i.e. a finite formal combination En,[V;] of subvarieties of codimension I. A Cartier divisor on X is determined by local data consisting of a covering {u,} of X, and rational functions/, E R(U,)* = R(X)*, such that on overlaps U, n ~,/'I/j is a nowhere vanishing regular function, i.e., /,//j E f( U, n ~, e*). Local data {U,', f/} define the same Cartier divisor if/,/f; E f(u, n Uf, e*} for all i,j. If D is a Cartier divisor on X given by local data {u" /,}, and V is any subvariety of X, then the functions /, for U, n V * 0 are unique up to units in the local ring v. x' Such a rational function f is called a local equation for D at V. If V is of codimension one in X, and we write f = alb with a, bEe V.x' we may define the order of D at V by
e
ordv(D)
=
ordv(f)
=
1((9v.x/(a)) l(ev.x/(b)),
where I denotes the length; note that since (9 v. x has dimension one, (9 v.x/( a) and
ev.x/( b) have dimension zero, so finite length. It is not hard to verify that this is
independent of the choice of a and b. Each Cartier divisor D on X determines an associated Weil divisor, denoted [D], by
[D)
=
E ordv(D)[V).
the sum over the codimension one subvarieties V of X. The Cartier divisors on X form a group Div( X), the sum D + E of two Cartier divisors being defined by multiplying local equations for D and E. The mapping D ..... [D] defines a homomorphism Div(X) ..... Zn_I(X) from Div(X) to the group Zn_I(X) of Weil divisors on X. A Cartier divisor D on X determines a line bundle (9(D) = ex(D). If {u" /,} are local data for D, the transition functions for e(D) from the ~ neighborhood to the U, neighborhood are the units /,//;; thus a section of e(D) is given by a collection of regular functions s, on U, such that
s,
= (/,//;). sJ
on U, n ~. A Cartier divisor D is effective if it is defined by local equations /; which are regular; in this case s, = /, determines a canonical section of e( D). Equivalently an effective Cartier divisor is a subscheme of X whose ideal is locally defined by one equation; this subscheme is the zeroscheme of the canonical section of D).
e(
Any I E R( X)* defines a principal Cartier divisor div(f). Two Cartier divisors D and E determine isomorphic line bundles on X if and only if they differ by a principal divisor. It is not hard to show that any line bundle on a variety comes from some divisor, so Div( X)/Principal divisors;:;; Pic( X). Note that in topology a Cartier divisor D determines a cohomology class c l(t9(D» in H2X, while [D] determines a homology class cl[D] in H2,,2X. One can show that
c l (t9(D» n cl(X)
=
cl[D].
We will develop a rational equivalence theory with analogous properties. For this last formula to hold, note that it is necessary that the class of [div(f)] must be zero for any principal divisor div(f). 3.3. Rational equivalence. For any variety (or scheme) X over any field K, let Z" X be the group of kcycles Lnj[V,] on X, i.e. the free abelian group on the kdimensional subvarieties of X. Two kcycles are rationally equivalent if they differ by a sum of cycles of the form
L [div(/')], where /; E R(W,)*, with W, subvarieties of X of dimension k + I. (Strictly we speaking, [dive!;)] was defined in the preceding section to be a kcycle on freely use the same notation for the cycles they define on any larger variety.) The group of kcycles modulo rational equivalence on X is denoted Ak X, and we write
w,;
E9
A.X =
AkX = Z.X/ ,
where  denotes rational equivalence. Although the preceding definition is usually simplest to work with, it may be shown to be equivalent to the following more classical one. Two kcycles are rationally equivalent if they differ by a sum
Ln,([V,(O)]  [v,(oo)]) with n, integers, V, subvarieties of X x pi whose projections to pi are dominant, and V,(O) and V,(oo) are the schemetheoretic fibres of V, over 0 and 00, regarded as subschemes of X = X x {O) and X X {oo). Note that A"X = Z[X] = Z if X is an ndimensional variety. More generally, if X is a scheme of dimension n, then A.X is the free abelian group on the ndimensional irreducible components of X. If I: X > Y is a proper morphism, the formula
I.[V]
=
deg(V/I(V»(f(V)]
determines a homomorphism I .: ZkX theory, the following fact is basic:
>
ZkY' To have a covariant ("homology")
THEOREM. II f: X > Y is proper, and a and a' are rationally equivalent cycles on X, then I * a and I * a' are rationally equivalent cycles on Y.
Thus there is an induced homomorphism, the pushforward
f.: A.X ..... AkY' making A. a covariant functor for proper morphisms. For example, if X is a projective curve, and Y = Spec(K) is a point, the theorem asserts the familiar fact that, for a rational function r on X,
r. ordp(r)[R(P): K]
=
0,
i.e. r "has as many zeros as poles". Another important case is when X and Yare ndimensional varieties, andfis surjective: if r E R(X)·, then
f.[div(r)]
=
[div(N(r))].
Here N(r) E R(Y)* is the norm of r, the determinant of multiplication by r on the finite dimensional R(Y)space R(X). This formula is a consequence of the basic lemma in §1.6. The theorem, in fact, can be deduced from these two cases, cf. [16, § 1.4]. In particular, if X is complete, i.e. the projection p: X ..... Spec( K) is proper, the degree of a zero cycle is well defined on rational equivalence classes. We set
fa x
=
deg(a)
=
p.(a),
identifying Ao(Spec(K)) with Z. There is an important class of morphisms f: X ..... Y for which there is a contravariantpullbackf*: A.Y ..... Ak+nX, where n is the relative dimension off. For any kdimensional subvariety Vof Y, rl(V) will be a subscheme of X of pure dimension k + n, and we will define
f*[V)
=
[rl(V)].
r
(Note that I( V) denotes the inverse image scheme, defined by pulling back equations for V in Y; its cycle is defined as in §2.1.) This class of morphisms includes: (I) projections p: Y X T ..... Y, T an ndimensional variety; here p*[V] = [Vx T]; (2) projections p: E ..... Y (resp. P(E) ..... Y) from a bundle to its base; here p*[V] = [Elv] (resp. [P(Elv)]); (3) open imbeddingsj: U ..... Y, with n = 0, andj*[V] = [V n U]. If U is the complement of a closed subscheme X of Y, and i is the inclusion of X in Y, the sequences
are exact; (4) any dominant (nonconstant) morphism from an (n + I)dimensional variety to a nonsingular curve. A class of mappings including these for which this pullback is well defined on rational equivalence classes is the class of flat morphisms; f: X ..... Y is flat if each
24
WILLIAM FULTON I
local ring l'J v.x is flat as a module over l'J w.y, with W = f( V). This includes all smooth morphisms. For most applications here, the above examples suffice. The following proposition is needed to complete the construction of intersection product outlined in §2.7. PROPOSITION. Let E be a vector bundle of rank r on X, p E: E .... X the projection. Then the pullback homomorphisms
PE:: AkX .... Ak+rE
are all isomorphisms. If SE: X .... E is the zero section imbedding, and a is any kcycle or cycle class on E, define the intersection of a by the zero section, denoted sH a), to be the class in Ak_r(E) that pulls back to a:
p£{s£{a)) = a. Note that by the proposition, a is equivalent to a cycle of the form I:n;[Elv,], and clearly the intersection of such a cycle with the zero section should be I:n ,[ V;]. In particular, in the situation of §2.7, the intersection class X· V is a welldefined class in An_d(W), with W = X 11 V. Indeed, the normal cone C to Win V determines an ncycle [C] on the restriction N of NxY to W, and we may set X·V=s~[Cl·
As for the proof of the proposition, the surjectivity of PE: follows by a Noetherian induction argument, using the exact sequence of (3) above. The injectivity, and in fact a formula for the inverse sE:' uses Chern classes (§4). Another elementary operation on rational equivalence is the exterior product x
AkX ® A,Y .... Ak+'(X X y)
defined by [V]
x [W]
= [V
x
W].
3.4. Intersecting with divisors. If D is a Cartier divisor on X, and a a kcycle on X, we define an intersection class D . a E Ak_I(Z),
where Z is the intersection of the support of D (the union of varieties at which local equations are not units) and the support of a (the union of varieties appearing in a with nonzero coefficients). By linearity it suffices to define D . [V] if V is a subvariety of X. Let i be the inclusion of Yin X. There are two cases: (i) If V is not contained in the support of D, then by restricting local equations, D determines a Cartier divisor, denoted i* D, on V. In this case, set
D· [V]
=
[i*D],
the associated Weil divisor of i*D on V. In this case D . [V] is a welldefined cycle.
INTRODUCTION TO INTERSECTION THEORY
25
(ii) If V c Supp( D), then the line bundle 19 x( D) restricts to a line bundle i*l9 x( D) on V. Choose a Cartier divisor e on V whose line bundle is isomorphic to this line bundle: 19 v (C) == i*l9 x (D), and set
D·[V]=[e], the associated Weil divisor of C. Since e is well defined up to a principal divisor on V, [e] is well defined in Ak_,(V). In case D is an effective Cartier divisor on X, this class D . [V] agrees with the class D . V constructed in §2.7. In case (i) this is immediate, while in case (ii) it amounts to the fact that for a Cartier divisor e on a variety V, the cycle of the zero section [V] is rationally equivalent to the cycle [rr'( e)] in the line bundle L = 19 v( e), with rr: L ..... V the projection. When e is effective, corresponding to a section s of L, an explicit rational equivalence may be constructed as follows (ct. §2.6): let Z
= {(p,
(}\o: }..,))
E
Lx p'l}..osrr(p) = }..,p}.
Then Z(O) = rr'(e), and Z(oo) is the zero section. In general, because of the ambiguity in case (ii), D . (l is only defined up to rational equivalence. If the restriction of the line bundle 19 x( D) to D is trivial, however, D . (l can always be defined as a cycle. Namely, if V c Supp(D), set D . [V] = O. This applies when D is the fibre of a morphism from X to a nonsingular curve; the cycle D . (l is then called the specialization of (l. This intersection product satisfies the formal properties one would expect for a "cap product". For example: (I) If (l  (l', then D . (l = D . (l' in A.(Supp(D». (2) If D  D' is principal, then D . (l = D' . (l in A .(Supp( (l»). (3) (Projection formula) If f: Y ..... X is a proper surjective morphism of varieties, D a Cartier divisor on X, and (l a kcycle on Y, then f~(j*D·
(l) = D· f *(l
in Ak_,(Z), with Z = Supp(D) n f(Supp(l), and/,: r'(Z) ..... Z the morphism induced by f. There is a similar compatibility with flat pullbacks. From (I) and (2) it follows that the operation product D· (l determines products Pic( X) ® Ak X ..... A k,( X). For a line bundle L and cycle class (l we shall write c,(L) n (l for this product:
c,(l9 x (D)) n (l
=
D· (l.
This will be the basis for the study of Chern classes in the next chapter. If D is an effective divisor on X and f is the inclusion of D in X, it follows from (I) that [V] ..... D . [V] determines a "Gysin" homomorphism
/*: AkX ..... Ak_,D.
This is the key to showing that the general intersection product is well defined on rational equivalence classes. Note that this is a strong form of the principle of continuity: all such intersection operations, applied to rationally equivalent cycles, will give classes of the same degree. It also includes the statement that specialization respects rational equivalence. In fact properties (2) and (3) are straightforward to prove. Property (I) then follows from (2) and the following basic commutativity law, on which most of the subsequent theory depends. LEMMA.
Let D and E be Cartier divisors on an ndimensional variety X. Then
D·[E]=E·[D] in A,,_z(Supp(D)
n
Supp(E».
Consider the case where X is a surface, and 'IT: X > C 2 is a proper birational morphism which is an isomorphism except over (0,0), and Z = 'ITI«O, 0» is a curve. Let D and E be the inverse images of the two axes C X {O} and {O} X C. Then D = D' + D" and E = E' + E", where D' and E' map isomorphically to the two axes, while D" and E" are supported on Z. In this case D . [E] is the point where E' meets Z, and E· [D] is the point where D' meets Z. These two points may well be different, but one knows they are rationally equivalent, because Z is a connected curve, all of whose components are rational curves.
•m
~,
,eo,
Ij
0' ...__r
E'
Although one may fashion a proof along these lines (cf. [4]), there is now a simpler proof. Roughly speaking, one blows up X along various subschemes to reduce to the case where D and E are sums and differences of effective divisors D, and EJ , such that each intersection of D, with EJ is either proper (in which case the commutativity is easy) or D, = E} (when it is evident). See [16, §2.4] for details. 3.5. Applications. Let us apply the preceding results to a situation considered in the first two sections. If HI, ... , Hd are hypersurfaces (effective Cartier divisors) on an ndimensional variety X, we may define, for any kcycle a on X, a class HI ...
Z
=
n H, n
Hd · a
E
AkAZ),
Supp( a). Inductively, this class is defined to be H I ·(H2 ···Hd ·a).
The commutativity law says that this product is independent of the order of the Hi' If k = d, and Z is complete, this class has a welldefined degree, denoted
fH I •.• H d • (l. When (l = [X], we omit it from'the notation, and write simply HI ... Hd • Suppose a nonsingular point P on X, rational over the ground field (R(P) = K), is an isolated point of intersection of the intersection of n hypersurfaces HI,' .. , Hn , n = dim( X). Shrinking X, assume that H, meet only at P. Let 'IT: X ..... X be the blowup of X at P, E = pn I the exceptional divisor. Then
'IT'H; = m,E
+ G;,
where m, is the multiplicity of H, at P; the intersection of G; with E is the projective tangent cone P(CpH,). We will show that if these projective tangent cones do not meet, then
i(P) = i(P, HI ... Hn) = ml
...
mn'
Note first that, since local equations for H; at P form a regular sequence, the intersection product HI ... Hn is the cycle i(P)[P], with i(P) defined as in §1.6 or §2.4. Let T] be the projection from E to P. Then since n G; = 0,
O=T].(G I =
Gn } .
"·
T].(('IT·H I

mIE)'" ('IT'Hn  mnE».
Now, by the projection formula, T].( 'IT'H I ... 'IT'Hn) = HI ... Hn and
T] * ( 'IT'H'I ... 'IT'H'I. . E n k )
H ... H . T] (Enk). 'I lie * Since the intersection class pk = E ... E is in Ak(E), T].(pk) E Ak(P) = 0 if 0 < k < n. Thus the only terms that survive are =
0= HI ... Hn  (I)"m l
...
mnT].(En).
Now the restriction of fJx(E) to E = p n  I is the dual of the bundle fJ p ",(I), so E . E is represented by minus a hyperplane, and En is represented by (I)n I times a point. Therefore .
T].(P) = (_I)nI[p], and thus HI ... Hn = m l ... mn[P]. Let us also reconsider the case where three surfaces HI' H2 , H3 in a nonsingular threefold X contain a nonsingular curve C as a (schemetheoretic) component. Let 'IT: X ..... X be the blowup of X along C, and let E be the exceptional divisor, T]: E ..... C the projection. Let
'IT'H,
=
E
+ G,.
The hypersurfaces G; do not meet in E, and the intersection of the H; outside Cis represented by the class 'IT.( G I • G2 • G3 ). Expanding as above, and noting that again T].( E) = 0 for dimension reasons, we obtain
'IT.(G I
.
G2 • G3) = HI' H2 • H3
+ LH,· T].(E 2 )

T].(E 3).
The last two terms therefore determine the contribution of C to the intersection product. The problem is to compute T].( E;).
28
WILLIAM FULTON
We have seen that E = peN), where N is the normal bundle to C in X. With this identification, the restriction of x( E) to E (which is the normal bundle to E in X) is the dual of the bundle 0 N (I). Referring to §3.1, one expects the formulas
°
T].(E Z ) =
T].(E l )
=
T].( c l (0 N (I)) n [P(N)])
T].(c l (0 N (I))z n[p(N)1)
=
=

[CL
cl(N) n [C],
where cl(N) is the first Chern class of the bundle N. In the next section we will develop the necessary theory of Chern classes for rational equivalence. Combining these results, it follows that the contribution of C to the intersection product HI . Hz . H3 is
(~H,)· C  cl(N) n [C]. For example, if X = pl, and C is a complete intersection of surfaces of degrees a and b, N = 0(a) $ 0(b), deg(C) = ab, so the degree of cl(N) n [C] is (a + b)(ab), and the total contribution of C to the Bezout number is
ab (~ deg( Hj )

(a + b)),
as found by Salmon (§1.3). With the machinery of Chern classes, one can also compute the contribution when C is not a complete intersection. It is similarly possible to work out the contribution of the Veronese V in the intersection of five hypersurfaces H, representing conics tangent to five fixed general conics (§ 1.4). Blowing up p 5 along Vas above one has
'IT'Hj = 2E
+ G"
where G I , ... , Gs are hypersurfaces that do not meet in the exceptional divisor E. Knowing the Chern classes of the tangent bundles to V:;; pZ and p 5, one knows the Segre classes of the normal bundle N v Ps , and it is a pleasant exercise to verify that
fG
I ...
Gs = 3264.
This approach to excess intersection problems was developed primarily by B. Segre [56], with related work by Severi and Todd. All were searching for constructions which would yield invariants of varieties, generalizing the notion of genus for curves. For a subvariety V of a variety X, let X be the blowup of X along V, E the exceptional divisor, T] the projection from E to V. The classes T].(E') were called the covariants of the imbedding of V in X. We shall see that, up to sign, they are the inverse Chern classes of the normal bundle of V in X, at least when V and X are nonsingular. For example, Segre constructed the canonical classes of a nonsingular variety V by applying this construction to the diagonal imbedding of Vin X = V X V; the formal inverse of LT].(E') on V X V projects to the total Chern class of Tv on V.
4. Chern Classes and Segre Classes 4.1. Chern classes of vector bundles. Eventually one wants a contravariant "cohomology" theory A·X to go with the covariant theory A.X, and Chern classes of vector bundles on X should lie in A· X. Although such theories exist, at this time there is not yet a simple geometric construction of such a cohomology theory. Indeed, it would be extremely useful to have such a theory, perhaps analogous to Go~'s realization of ordinary cohomology via "geometric coqcles" [26]. .
()
At any rate, any such theory should have "c~~ts" A'X ® AkX .... Ak_,X, and Chern classes c,(£) E A'X. In particular, a bundle £ on X should determine homomorphisms     c,(E)()
~
Ak_iX,
>
by a .... c;( £) n a. In this section we construct such Chern class operations directly. They will satisfy properties expected from topo~ For a line bundle L on a variety (or scheme) X, to define cl(L) n a it suffices to define cl(L) n IV], for.~ Choose a Cartier divisor C on V such that the restrlciWn Llv of L to V is isomorphic to (9 v( C), and set
cl(L) n [V]
=
l£L
Note that if L = (9x(D), then cl(L) n a = D· a, the intersection product of §3.4. It follows from t e diSCUSSIOn of §3.4 that this operation respects rational equivalence classes, and satisfies the expected formal properties. For example, thereis~
. f.(cM·L) n ~ = cl(L) nf.a for f: Y .... X proper, L a line bundle on X, a a cycle c1as§ on Y. Similarly we have a commutativity property
cl(M) n (cl(L) n a)
=
cl(L) n (cl(M) n a)
for line bundles L, M on X, a EA. X. Thus any polynomial in first Chern classes of line bundles on X or on any variety that X maps tooperates on A. X. In addition there are the elementary formulas:
cl(L ® M) n a = cl(L) n a
+ cl(M) n
cl(CI)na= cl(L)na. 29
a,
Now if E is a vector bundle of rank r on X, define Segre class operatoriis..!E),
s,(E) n : AkX ..... Ak_,X, as follows. Let p: P(E) ..... X be the projective bundle of E, ('1£(1) the basic line bundle on P( E), and for a in A k X set
s;(E) n
IX = P.(cI(e.dDr I+, np·a).
One shows easily thats,(E) = 0 for i < 0, and thatso(E) = 1 (i.e. so(E) n a = a for all a). Basic properties such as projection formulas and commutativity of those classes follow readily from corresponding formulas for first Chern classes of line bundles. Now we define Chern class operators
c,(E) n _: AkX .... Ak_,X by formally inverting the Segre classes (cf. §3.1):
1 + cI(E) + c2(E) + ... i.e., set co(E) plicitly:
= I.
and cI(E)
=
=
(1 + sI(E) +s2(E) + ...
r
sI(E), c2(E) = sI(E)2  s2(E),
o
I,
. Ex
o
Properties such as the projection formula
f.( c,(j*E) n a)
=
c;(E) nf *( a)
and the commutativity property
c;(E) n (c/F) n a) = cJ(F) n (c,(E) n a) follow formally from the corresponding facts for Segre classes. Less obvious but also true are the vanishing property
c;(E)=O fori> rankE, and the Whitney sum formula
c;(E)
=
L:
cj(E')ck(E")
J+k;
for an exact sequence 0 .... E' .... E .... E" .... 0 of vector bundles. There are also formulas for Chern classes of ten~or, exterior or s~ts. Although we shall not carry out complete proofs of these statements here, a basic ingredient is ~ an equation among Chern classes of vector bundles in a given relation with each other is true if:
(i) the equation is valid when the bundles eaclr have lli!rations by subbundles such that the quotient bundles are line bundles. and (ii) the given rel;tio;is'~~llllback. This principle is a simple consequence of the fact that for any bundle E of rank
ronX. p.: AkX .... Ak+r_I(P(E)) Injective. which follows from the fact that so(E) = I. For on pee). p. E contains the universal line bundle L E• with quotient bundle QE: repeating the IS
process on Q E yields a composite f: Y .... X of projective bundles. so f* E is filtered. and f* injects A. X in A. Y. If E is filtered. with line bundle quotients L 1••••• L r • the vanishing and Whitney formulas reduce to showing that c,(E) is the ith elementary symmetric function of cl(L 1) ••••• c l ( Lr). For this. one first verifies directly that TIcl(L;) = 0 if E has a nowhere vanishing section; one then may apply this to the bundle p. E ® L[. which gives r
n (cl(p*L,) + c (l9 (I))) ,=1 l
E
=
O.
.~
from which the assertion follows easily. As usual we define total Segre and Chern classes by
s(E)= I +sl(E)+S2(E)+ ...• c(E) = I + cl(E) + c2(E) + .... These Chern classes may be used to prove the isomorphisms Ak_rX': AkE for a vector bundle E of rank r on a variety or scheme X (§3.2). One proves first the isomorphisms
rI which take ~Il, to ~CI(l9E(l»' n P*Il,. p the projection from pee) to X. The surjectivity of this mapping is proved by a Noetherian induction as in the affine bundle case; injectivity follows by applying operatorsp*(c l (l9 E (lW n _). using the identities that s;(E) = 0 if i < O. and so(E) = I. The projective completion peE ill I) contains E as an open subvariety. complementary to the hyperplane at infinity pee). From the above isomorphism and the exact sequence
AkP(E) .... AkP(E
ill
I) .... AkE .... 0
the injectivity of AkrX .... AkE follows easily. In addition. one derives a formula for the inverse isomorphism s!: AkE .... Ak_rX. Given a subvariety Vof E. let V be its closure in peE ill I). Then.
q.(cr(Q) n [f7]); I) to X. and Q is the universal rank r quotient
sH[V])
=
q is the projection from peE ill bundle of q*(E ill I) on peE ill I). (Note that Q has a canonical section which
32
WILLIAM FULTON
vanishes precisely on the zero section of X in E, mUltiplying by the top Chern class should correspond to intersecting with the zero section.) 4.2. Segre classes of cones and subvarieties. Let W be a subvariety of a variety V. If V and Ware nonsingular, or, generally, if the imbedding of W in V is a regular imbedding, one has a normal bundle NwV, and one may construct
invariants of the imbedding by using Chern and Segre classes
n [W]
c,(NwV)
and si(NwV) n [W].
In the general case, however, one has only a normal cone C = CwV. We shall see that, although one does not have a general Chern class formalism for cones, there is a useful notion of Segre class. We shall define a IOtal Segre class
s(W, V)
E
A.W
for any closed subscheme W of a variety V. If W = V, set s( W, V) = [V]. Otherwise, let V be the blowup of V along W, let E = P(C) be the exceptional divisor, and let T]: E ~ W be the projection. The ifold selfintersections E' = E ... E of the divisor E are well defined classes in Aki(E), k = dim(V) = dim(V), by the construction of §3.4. We set s(W,v) =
E (lr1T].(E'). i~1
At least in the nonsingular case, the images of these selfintersection Ei were basic for Segre's construction of invariants [56]. Identifying E with P(C), the restriction of !9ji(E) to E is the dual of the universal line bundle !9c (l) on P(C). It follows that Ei = (I)il cl «(9c(l»i1 n [P(C)], and hence
s(W, V)
=
E T].( c l (!9c (I)r n [P( C)]). ;;.0
Note that this last expression makes sense for any cone C on a scheme W; under the assumption that, for each irreducible component C of C, P( C) is not empty, we define the Segre class s( C) of the cone C by this formula:
For an arbitrary cone C, the cone C Ell I satisfies this assumption, and one may define s(C) to be s( C Ell I). Since the restriction of !9 ji( E) to E is also the normal bundle to E in V, another definition of the Segre class is s(W, V) =
T].(s(E. V»).
This is a special case of the following important formula.
INTRODUCTION TO INTERSECTION THEORY
33
PROPOSITION. Let '11: V' > V be a proper surjective mo" ,~ism of varieties, of degree d. Let W be a subscheme of V, W' = '1I i (W), and let T]: W' > W be the induced morphism. Then
T].(s(W', V')) = d s(W, V).
This is easily proved by blowing up to reduce to the case where Wand W' a.re Cartier divisors, in which case it follows from the formulaf .[W'] = d[W]. When d = I, the proposition expresses the birational in variance of Segre classes. When the imbedding of W' in V' is regular, e.g. if V' and W' are nonsingular, it gives a formula for s(W, V) in terms of Chern classes of the normal bundle of W' in V'. When all four varieties are nonsingular, it gives a remarkable relation among the Chern classes of the normal bundles; when these Chern classes are known, it can even be used to complete the degree d. If Z is an irreducible component of W, the coefficient of [Z] in the class s( W, V) is the multiplicity of V along Wat Z, and denoted (e wV) z. If A is the local ring of V along Z, and I the ideal of W, one may show that length{A/I') = (ewV)z(tn/n!) + lower terms for t » 0 and n = codim(W, V). In other words, this multiplicity agrees wi th Samuel's multiplicity for the primary ideal I in the local ring A. If Z = W, we write simply ewV. We shall see that other terms in the Segre classes also appear in intersection formulas. It is illuminating to apply these ideas to verify the RiemannKempf formula. Fixing a base point on a nonsingular curve C determines morphisms ud : C(d) > J from symmetric products to the Jacobian of C. If Wd is the image of C1d), 1 ~ d ~ g, and D E C(d) is a divisor, the RiemannKempf formula states that the multiplicity of Wd at the point ud(D) is (8~+r), where g is the genus of C, and r is the dimension of the linear series IDI of D. Indeed, one knows that IDI is the fibre Udi(ud(D», and one may calculate that
s(IDI, C(d))
=
(I + h)8 d + r n [IDI],
where h = c i «(9(1» on IDI = pro Since ud maps proposition applies, giving the multiplicity as
C(d)
birationally onto Wd ' the
Segre classes will appear frequently in these notes. In the study of holonomic "i)modules, important invariants are constructed by intersecting characteristic varieties in cotangent bundles with the zero section; as we shall see, all such intersections can be expressed in terms of Segre classes. It may be pointed out that, for a vector bundle E, some authors' si(E) correspond to our s;(E V) = (  l)is,(E). The necessity of enlarging their scope to include general cones dictates our convention.
4.3. Intersection fonnulas. Recall the situation of the basic construction of intersection products (§2.7):
W
.....J
q X
v ~ g
.....
Y
f
with j: X > Y a regular imbedding of codimension d, V an ndimensional variety, g a closed imbedding, W = X II V = gI(X). LetN = h*NxY, C = CwV the normal cone, which is a closed, ndimensional, subscheme of N. We have defined the intersection product
X· v=s~[cl EAn_d(W). Another description of X· V may be derived from the last formula in §4.1. Let Q be the universal rank d quotient bundle on P(N Ell I), and let q be the projection from P(N Ell I) to W. Then
X· V=q*(cd(Q)n [P(CEllI)]). We use the quotation {ah for the kdimensional component of a cycle or class a on a scheme. Using the Whitney and projection formulas, we have
X· V={q*(c(Q)n [P(CEllI)])}n_d =
{q*(c(q*(N Ell \)). c(0( _1))1 n [P(C Ell I)])} ,,d
= {
c(N) . q *( c( 0( I)) I n [P( C Ell I)])} ,,d'
This gives a basic
Intersectionjormula
X· V
=
{c(N) n s(W, V)},,_d'
Note first that if Z is an irreducible component of Wof the expected dimension n  d, then the coefficient of [Z] in X· V is just the multiplicity (ewV)z of V along Wat Z. In particular, in the case of proper intersection, i.e. dim W = n  d, we recover the formula
the sum over the irreducible components Z of W. Since A* W = EBA* W;, where the W, are the connected components of W, one has a corresponding decomposition for X· V:
X· V=L{c(N)ns(W;,V)}n_d' (Notation is abused by writing c(N) n s( W;, V) in place of c(N\W,) n s(W;, V).) If Y· is open in Y, and X·, V·, and W· are the intersections of Y· with X, V, and W, then the restriction homomorphism from A.W to A.W· takes X· V to X· . V·. By means of this localizing principle, which follows immediately from the construction, it suffices to consider the case when W is connected; similarly
INTRODUCTION m INTERSECTION THEORY
35
one may discard any closed subvarieties of dimension less than n  d, without loss of information. Consider the case when the imbedding of W in V is a regular imbedding of codimension d'. In this case the normal cone C is a subbundle of N. The quotient bundle E = N IC is called the excess bundle. Since s(W, V) is given by the i~verse Chern class of C, we deduce from the Whitney formula and the above intersection formula the
Excess intersection formula
X· V =
Cd d' (
E) n [W J.
In case d' = d, i.e. regular sequences locally defining X in Y remain regular sequences on V, we recover again the formula X· V = [W]. There is a simple but important refinement of these constructions and formulas. The morphism g: V ..... Y can be an arbitrary morphism; it need not be a closed imbedding. Defining W to be the inverse image scheme gI( X), h: W ..... X the induced morphism, one still has the normal cone C = CwV imbedded in N = h*NxY, and X· V E A*W can be constructed by intersecting [C] with the zero section in N. The preceding intersection formulas are equally valid in this generality. Combined with the birational invariance of Segre classes, this allows an important reduction procedure. To compute X· V, it suffices to find a proper birational 'IT: V' ..... V for which the class X· V' can be computed. For then the class X· V' pushes forward to given X· V. Indeed, if W' = 'IT1(W), and TJ: W' ..... W is the morphism induced by 'IT, then
X· V= TJ*(X, V'). This follows from the formula s(W, V) = TJ*(s(W', V'» and the intersection formula. For example, if W' is regularly imbedded inT' of codimension d', with excess normal bundle E = (hTJ)*NxYINwV', then X· V= TJ*(cd'd,(E) n [W']).
One may always reduce to this case, with d' = 1, by taking V' to be the blowup of V along W. Thus many difficult problems can be reduced to the case of divisors and Chern classes.
5. Gysin Maps and Intersection Rings
5.1. Gysin homomorphisms. Iff: d, we define Gysin homomorphisms
x ....
Y is a regular imbedding of codimension
/*: AkY .... Ak_dX by the formula /*([n,[V;]) = [n ,( X· V;), where X· V; E AkdX is the intersection product constructed in §§3.3 and 4.3. Verdier's proof that this formula respects rational equivalence [59] uses the deformation to the normal bundle to reduce to the known case where d = 1. It goes as follows. Let N = NxY, and let MO be the deformation space constructed in §2.6. Let i be the imbedding of N in MO (as a Cartier divisor). The complement of N in MO is identified with Y x C; let) be the inclusion of Y X C in MO. Consider the diagram: I.
Ak+,N
Ak+,Mo
j'l AkN
....j'
Ak+,(Y xC)
....
0
'" t pc'
..... 
AkY
Here j* is the Gysin homomorphism defined for divisors in §3.4. The row is exact (§3.3(3)), and j* 0 i * = 0 because the normal bundle to N in MO is trivial. Hence there is a specialization homomorphism 0 as indicated, with o( ct) = i*13 if )*13 = (pr)*ct. For a subvariety Vof Y, with W = V II Y, it follows that
o[V) = i*[M;"V] = [CwV), where C wV is the normal cone to Win V. One deduces that /* is the composite
which is evidently well defined on rational equivalence classes. There is a useful strengthening of these Gysin homomorphisms. If f: X .... Y is a regular imbedding of codimension d, and g: Y' .... Y is an arbitrary morphism, form the fibre square
X' d
x
..../' .... f 37
Y'
h Y
38
WILLIAM FULTON
i.e. X'
=X
X
y
Y'
= g I( X). We define refined Gysin homomorphisms t: AkY' ...... AkdX'
by the same formula t[V) = X· V. (This intersection product X· V was constructed in Ak_d(V n X') at the end of §4.3; as usual we use the same notation for its image in AkdX'.) Similar reasoning shows that t is well defined on rational equivalence classes. The main compatibilities of these Gysin homomorphisms are stated in the following theorems. THEOREM I. Consider a fibre square
f' ......
X'
Y'
g'! X f
with f a regular imbedding of codimension d. (a) If g is proper, and a
E
AkY', then
f*g.a
=
g~ta
inAk_dX.
(b) If g is flat of relative dimension n, and a E AkY, then
g'*f*a = tg*a
in Ak+n_dX'.
(c) If/' is also a regular imbedding of codimension d', set E = g'*NxY /Nx'Y', Then,for a E AkY',
ta = Cd_d·(E} n/,*a inAk_dX'. (d) If g is also a regular imbedding of codimension e, and a
g'f*a
=
tg*a
E
AkY, then
in Akd_.X'.
(e) If F is a vector bundle on Y', then for all a
E
AkY', and all i,
t( c, (F) n a) = ci(j'* F) n ta
in AkdiX'.
For example, if f is proper, then (a) and (c) yield the selfintersection formula: for a E AkX, f*f.a = cANxY} n a
inAk_dX.
Note that if d' = d in case (c), the assertion is thatta = /,*a. The proofs of (a)(e) follow quite easily from facts we have discussed before: (a) from the proposition in §4.2; (b) from an analogous formula for pullbacks of Segre classes by flat morphisms; (c) as in the excess intersection formula (§4.3); (d) is reduced, as in the discussion at the end of §4.3, to the case of divisors, which is the main lemma of §3.4; a similar reduction is used in (e).
INTRODUCTION TO INTERSECTION THEORY
39
THEOREM 2. Let f: X > Y and g: Y > Z be regular imbeddings of codimensions d and e. Then the composite gf is a regular imbedding of codimension d + e, and if a E AkZ, then
(gf)*a
=
f*(g*a) inAk_d_eX.
The equation (g{)* = f*g* also holds whenfis a regular imbedding and either (i) g and gf are flat, or (ii) gf is a regular imbedding and g is flat. For example, if p: E > X is a vector bundle of rank r, and s: X > E is a section, it follows from (ii) that so: AkE > Ak_,X is the inverse isomorphism to pO; in particular, s* is independent of the choice of s. The theorem and the variations stated after it are also valid for the refined Gysin morphisms.If h: Z' > Z is any morphism, and a E A.Z', then
(gf)'a
=
f'(g!a) inA.(X'),
with X' = X X z Z'. The theorem is straightforward when Z is a vector bundle over Y, and g is the zero section. The general case is reduced to this by a deformation to the normal bundle, cf.. [59]. We refer to [16, §§6, 17], for tne general statements and complete proofs. 5.2. The intersection ring of a nonsingular variety. If X is an ndimensional nonsingular variety (i.e., smooth over the base field), then the diagonal imbedding 8 of X in X X X is a regular imbedding of codimension n. Given a E AaX and 13 E AbX, a product a . 13 E AmX, m = a + b  n, is defined by
a·I3=8*(axl3). Thus the product on A. X is the composite
A.X ® AbX
>
Aa+b(X
X
X)
s' >
AmX.
Note that if a E AaV and 13 E AbW, with V, W closed subschemes of X, then tne product a . 13 has a natural welldefined refinement in Am(V n W), namely 8!( a X (3), with 8! the refined Gysin homomorphism constructed from the fibre square:
vn
W
! X
Vx W
! XxX
All the formulas of this section are valid for such refinements, but for simplicity we write them only in the absolute case. Define A P X to be An _p X. Then the product is a homomorphism
APX ® AqX .:. Ap+qX. Let 1 E AD X correspond to [X] E An X.
40
WILLIAM FULTON
If I: Y
>
X is a morphism of nonsingular varieties, then the graph morphism Y/ Y > Y X X
is a regular imbedding of codimension n the formula /*a
=
dim( X). Define /*: A P X
=
yj(a
>
A P Y by
x [Xl).
THEOREM. For X nonsingular, the above product makes A* X into an associative, commutative ring with unit 1. For a morphism f: Y > X 01 nonsingular varieties, the homomorphism /*: A* X > A*Y is a ring homomorphism. II also g: Z > Y, with Z nonsingular, then (fg)* = g*/*.
The theorem follows quite readily from the general properties of intersection products summarized in §S.l. For example, to prove the associativity of the product, consider the fibre square:
XxX
X 5~
UXI
XxX
xxxxx
!X5
Given cycles a, 13, y on X, the equality a . (13 . y) formula 8*(1
x 8)*(a x 13 x y) =
8*(8
x
=
(a . (3) . y is equivalent to the
I)*(a X 13
x y).
This follows either from Theorem I(d), (c), or from Theorem 2. We refer to [16, §8] for details and refinements. The formula for /* also makes sense when I: Y > X is any morphism, with X nonsingular. More generally, one may construct "cap products" AP X ® AqY .... Aq_pY
by defining /*a n 13, or 13 ., a, to be yj( 13 x a). This makes A. Y into a module over A* X, and one has the projection lormula I.(f*a n (3) '" I .( a) n 13, or
1.(13 ·,a}
=
1.(a)·13
in case I is proper. If Y is nonsingular, and a subvariety X of Y is regularly imbedded in Y by an inclusion i, then for any cycle a on Y, a· [Xl
=
i.i*(a)
inA*Y.
More generally, a· [X] = i'(a) E A.(X n Supp(a)). The commutativity property (Theorem I( d)) is used to prove this. Since the seminar of Chevalley [10], the intersection ring A* X has been known as the Chow ring of X. The construction in that seminar was for nonsingular quasiprojective varieties over algebraically closed fields, and was based on a
INTRODUCTION TO INTERSECTION THEORY
41
"moving lemma". Chow's work in turn was inspired by ideas and constructions of Severi, many of whose papers were devoted to intersection theory. B. Segre, Todd, Van der Waerden, Weil, and Samuel were among the others who studied rings of equivalence classes of cycles. One feature of the present approach, following [21), is the elimination of any need for a moving lemma. Our approach is closest to that advocated by B. Segre [56); related ideas have been proposed by many others, including Murre, Mumford, Jouanolou, King, Lascu, Scott, and Gillet. If V and Ware subvarieties of a nonsingular X, the refined intersection class [V). [W) = 1l![V X W) is in Am(V n W), m = dim V + dim W  dim X. In particular, any proper mdimensional component Z of V n W appears in [V) . [W) with a positive coefficient, the intersection multiplicity i(Z. V· W; X). Basic properties of this multiplicity, such as associativity, follows from the refined versions of the theorems in §5.1. If V and W meet transversally along a nonempty open subvariety of Z, it follows from our construction that i(Z. V· W; X) = I. The converse is also true. For this criterion of multiplicity one we refer to [34 and 16) for algebraic and geometric proofs. Although several of the abovementioned auth\lrs indicated that some of their constructions made sense on singular varieties, the attempt to bring singular varieties into the general picture was apparently diverted by the notion that it should be possible to intersect general cycles on a singular variety if rational coefficients are used. This is possible on normal surfaces and on quotients of nonsingular varieties by finite groups. For example, the intersection of two generating lines on a cone over a plane conic is then onehalf the vertex. But, as Zobel [62) points out, this is not possible in general. If X c p4 is the cone over a quadric surface Q, any two lines in Q are rationally equivalent in X, since they are rationally equivalent to generators of the cone. But the cone over a line in Q meets lines in one family of lines in Q transversally, but is disjoint from lines in the other family. It is interesting that this same cone is used in the example of Dutta, Hochster, and McLaughlin [14).
5.3. Grassmannians and flag varieties. In general the computation of the ring A* X, for a nonsingular projective variety X, is a very difficult problem. One has AD X = Z. and AI X = Pic( X),. but for p ;;. 2, there is little general knowledge of AP X. Mumford [45) showed that, for general surfaces, it is impossible to give A2 X any natural, finite dimensional, algebraic geometric structure. Collino [12) has calculated A* X for X a symmetric product of a curve, and Bloch and Murre [8) have done the same for certain Fano threefolds. Such calculations use all the special geometry of the varieties in question; there are very few general algorithms. There is an important class of homogeneous varieties, however, for which the groups AP X are finitely generated, and the rings A* X known, at least in principle.
42
WILLIAM FULTON
The grouptheoretic approach is probably most satisfactory (cf. [13. 31]). but we will give more classical descriptions. The Grassmannian G = Gd(pn) of dplanes in pn is a nonsingular variety of dimension (d + 1)( n  d). Fix a flag
¥ AI ¥ ... ¥ Ad C
Ao
of subspaces. with a,
=
P"
dim A,. and set
Q(Ao •...• Ad) = {L E Gjdim L
n A,;;, i. 0.;; i.;; d}.
Then Q(Ao •...• Ad) is a subvariety of G. called a Schubert variety. Its dimension is d
L (a
d
j 
j=O
i)
=
La,  d(d + 1)/2. jO
Its class in A*G depends only on the integers 0 .;; a o < ... < ad';; n. and is denoted (a o•...• ad)' A notation better suited to codimensions. and also used by Schubert. is to define. for n  d;;. }..o;;' ..• ;;'}..d;;' O. {}..o.···. }..d}
= (a o•· ... ad) =
[Q(a o•···• ad)]'
where a, = n  d + i  }..,. Then {}..o.·· .• }..d} is in AI)..IG. where I}..I = '[~o }..,. A third notation is 0)..0 •...•)...' In the usual Pli.icker imbedding of Gd(Pn) in pN. N = Ci!:)  I. the Schubert varieties are defined by linear equations. If eo, ...• en are points spanning pn. and A, is spanned by eo •...• ea' the Schubert variety Q = Q(Ao •...• Ad) has an open subvariety Q" consisting ~f those linear spaces L which can be spanned by vo •...• Vd with Vj in A,. but v, not in the span of eo •...• eo _I' The "reduced echelon" form of such a basis identifies Q" with the affine space of dimension '[(a,  i). The complement Q  Q" is a union of smaller Schubert varieties. An inductive argument. using the exact sequence of §3.3(3). then shows that the classes (a o•...• ad) generate A.G. We shall see that they form a free basis. As a first step toward understanding the ring structure on A*G. consider the intersection of classes (a o•...• ad) and (b o•...• bd ) of complementary dimension. The dual class to (a o•... , ad) is the class
(n  ad' n  ad_I ..... n  a o). One has the basic duality: if (b o•. ..• bd ) is dual to (a o •...• ad)' otherwise. To see this, one may represent (bo•...• bd) by Q(Bo •...• Bd ), where B j is spanned by the last points enb, •... ' en' The Schubert varieties are then seen to meet transversally in one point when the classes are dual; otherwise they are disjoint.
INTRODUCTION TO INTERSECTION THEORY
43
It follows that the Schubert classes form a free basis for A.G. Moreover, given any kcycle a on G, its expression in terms of this basis is described as follows. For each class (b o, . .. , bd ) of codimension k, set
Then a = [anUd .... .nuo(ao,···, ad)· One may use this principle to calculate general products. The reader is invited to work out A*G J (P3) this way. In this case Schubert used a special notation: I = (2,3), g = (1,3), gp = (0,3), g. = (1,2), g, = (0,2), and G = (0, I) form a basis, and
g2
=
gp
g; = g;
+ ge' =
G,
g. gp
=
g . ge
=
g"
gp. g. = 0, g. gs
=
G.
To see the geometry behind the first equation, note that g2 is represented by the variety of lines in space meeting two general lines. Moving the lines so that they meet, this variety degenerates to the union of the variety of lines through the point of intersection and the variety of lines in the plane of the two lines. Such arguments, standard in classical enumerative geometry, must be fortified with a verification of the multiplicities of intersection; for this one may intersect both sides with a dual basis. Other techniques will be discussed in the next chapter. With this one may calculate that g4 = 2G: there are two lines meeting four given lines in general position. If C is an irreducible curve of degree d in p3, and
then [Ve] = dg. To verify this, one checks that [Vel· g, = d, since there are d lines through a general point in a general plane which meet C. It follows that there are 2Ildeg(C/) lines meeting four curves C J , ••• , C4 in general position. One may similarly count the number of common chords to two space curves, and many other similar problems. From the fact that the projective linear group acts transitively on Gd(pn), one may deduce that, after putting varieties in general position via translations by this group, all intersections will be transversal, so that the naive geometric number agrees with the intersectiontheoretic multiplicity, at least in characteristic zero [37]. There is a similar description for a general flag manifold, dating from Ehresmann [15]. For ~ d J < d 2 < ... < d r < n, let F = F(d J, ••• , d r ; n) denote the flag manifold whose points are flags of subspaces
°
with dim L; = d;. Fix eo"'" en spanning pn as before. The Schubert varieties in F are described by an array with r rows
°
where each row is an increasing sequence of integers between and n, and each row is a subset of the next. The Schubert variety consists of all flags Lie ... c L, such that L, satisfies the Schubert condition prescribed by the ith row, with respect to the standard flag. The dimension of this variety is
L(a,  i) + L'(b;  i) + ... + L'(C,  i), where the primes denote that only tljose terms not counted in the preceding row are included. The classes of these cycles form a basis for A.(F), and the dual of such a class is obtained by replacing each row by the dual Schubert condition. 5.4. Enumerating tangents. Let I = F(O, d; n) be the incidence variety of points on dplanes in P n. Then A.(/) has a basis of classes of the form (a o , ... , ; k' ... , Here (a o,"" ad) is a Schubert condition for dplanes; if Ao c ... C Ad is a fixed flag, with dim A, = a" then (ao,' .. , ; k" .. , ad) is the class of the variety
ad)'
{ (P, L)
E
I\dim L n A; ~ i,
°~
i ~ d, and PEAk}'
whose dimension is ~(a;  i) + k. The dual class is
(n  ad'"'' n': a k , ... , n  ao). Let V be a subvariety of pn of codimension e ~ d + I. Let V' closure of
{(P,L)EI\PE Vreg,dim(L() TpV)~de+
C
I be the
I}.
Here V,eg is the nonsingular locus of V, and TpV is the tangent (n  e)plane to V at P. Then V' is a subvariety of I of codimension d + I, which measures the pointed dplanes that touch V. For many enumerative problems involving tangents, it suffices to compute the class [V'] in A d+ 1(/). If M is a linear subspace of codimension d  k + I, then the class of M' is one of the basic Schubert classes, which we denote fJ.k: fJ.k =
{I, ... ,
i ,0, ... , o}
=(ndI,nd, ... ,nd.tkI,nd+k+ I, ... ,nI,n). By calculating intersections with dual classes, one verifies that
[V']
= mde+lfJ.o
+
m d efJ.l
+ ... +
mofJ.de+I'
where m, is the ith class of V; namely m, is the degree of the closure of
{p E
V,eg!dim TpV
n
A ~ iI},
where A is a general (n  e  i  2)plane. For concreteness, consider the case where d = I, n = 2, and C = V is a plane curve. Then where n
[C'] = nv + mJ.L, = mo is the degree of C, m = m I the class of C (§ 1.2), and v = J.LI
= [{
{P, 1)l/is a fixed line}],
J.L
= [{
{P, l)IPis a fixed point}].
=
J.Lo
With this, one may calculate the number of curves in a given rparameter family of curves which are tangent to r given curves in general position. Let = {C')'ET be an rdimensional family of plane curves. Thl! characteristics J.L'V'' of the family are the numbers
e
J.L'V'' =
#{ tiC, passes through i general points and is tangent to r  i general lines}.
Given r curves C I, ... , C, in general position, let n; = deg(C,), m, Then the number of curves in the family tangent to C I, ... , C, is
= class(C,}.
This is evaluated by expanding formally, and substituting the characteristics for each J.L;v'' For example, if is the family of all plane conics, then J.L5 = 1, J.L4V = 2, and J.L 3V2 = 4, as one sees by the fact that the condition to be tangent to a line is a quadric in p5. For the others one has the Veronese as an excess component (d. §IA), but one may conclude by the duality of conics that J.L;v} = J.LJV'; so J.L 2V3 = 4, J.LV 4 = 2, v 5 = I. Thus if C I, ... , C5 are conics, one computes
e
{2J.L
+ 2v)5
=
3264
conics tangent to five given conics in general position. It should be pointed out that computation of characteristics can be very difficult. For the family of all curves of degree ~ 5, apparently no one has even guessed what the answers should be. On the other hand, the above tangency formula is easy to prove, including the generalization to arbitrary dimensions. Let (r) be the closure in I X ... X I X T (with r copies of l) of the set
e
{ (PI' II)
X ... X
(P" I,)
X
tl each P, is simple on C, and I; is the tangent line to C, at pJ
46
WILLIAM FULTON
Consider the projection
(r copies).
!:f{r)lx···XI
Using transversality, one sees that the desired number is the degree of the in tersection class
[8{r)] ,/([Cil x···
X
[C;])
constructed as in §5.2. Writing out the classes [Cf] = m,v. + n,v, the conclusion follows. Note that, by transversality, any lower dimensional subset of T may be discarded or added without changing these numbers; in particular, one may take T to be a projective variety. One may realize the equation [C'] = mv. + nv geometrically by deforming C to an nfold line 1 via projection from a general point Q:
The condition to be tangent to C deforms to n times the condition to be tangent to I, plus the sum of the conditions to pass through the points Pi where tangents from Q to C meet I. With this approach the intersection theory can be carried out on the original parameter space. The essential point is that, for generic such deformations, the contribution of the "Veronese" of multiple curves remains constant: no solutions enter or leave this locus of degenerate solutions at either end of the deformation. For details and other approaches see [16 and 18]. It may be pointed out that the basis for A .(1) used here and in other enumerative problems, is not the basis one obtains by realizing I as a projective bundle over Gd(pn), cf. §4.1. The notation for this basis follows Martinelli [43].
6. Degeneracy Loci
6.1. A degeneracy class. Let cr: E ..... F be a homomorphism of vector bundles of ranks e and f on an ndimensional variety X. For k ,..; mine e, f), set
Dk{cr)
=
{x
Xjrank{cr{x)),..;
E
k}.
This degeneracy locus has a natural structure as a closed subscheme of X, locally defined by the vanishing of (k + I)minors of a matrix representation of cr. One expects Dk ( cr) to be mdimensional, where
m=n{ek)(fk), but in general one can only state that each irreducible component of Dk ( cr) has dimension at least m. Our object is to construct a class Dk{cr)
E
A",(Dk{cr)),
to give a formula for the image of Dk ( cr) in A",X in terms of Chern classes of E and F, and to investigate when D k ( cr) is determined by the scheme Dk ( cr). To construct Dk(cr), let d = e  k, and let G = Gd(E) be the Grassmannian bundle of dplanes in E, with projection 'IT: G ..... X. On G one has a universal exact sequence 0 ..... S ..... Ec .... Q .... 0 with rank S = d, rank Q = k and Ec = 'IT. E. The composite S .... Ec : Fc determines a section, denoted s.' of the bundle S v ® Fc. The zero scheme Z(s.) of this section projects onto Dk ( cr); let 11: Z{s.) .... Dk{cr)
be the morphism induced by 'IT. If S v ® Fc ' one has a fibre square:
So
is the zero section imbedding of G in
Z{s.)
! G Since s. is a regular imbedding, we may construct the refined intersection class s:[G] E A",(Z(s.»; note that m = dim(G)  rank(S v ® Fe>. Set Dk{cr) =
11.(S:[G]) E A",(Dk{cr)). 47
Because D k ( 0) is constructed by a succession of our intersection operations, it is compatible with other such operations, e.g. by pullbacks by flat morphisms or regular imbeddings. In particular, D k (0) may also be constructed by pullback from a universal case. Let
H = Hom(E, F) = E v® F, a bundle over X. Inside H there is a subcone Dk consisting of mappings of rank ~ k. Locally Dk is a product of X and the variety of e x f matrices of rank ~ k; the latter variety is known [33] to be a reduced, irreducible CohenMacaulay variety of the expected dimension ef  (e  k)(f  k). Giving a morphism 0: E + F corresponds to giving a section t a of H. Then Dk ( 0) = t;I(Dk),and Dk(O) = t:[D k ]·
Note that the assertions about the dimension of Dk(o) follow from this statement. In addition, it follows that Dk(O) = [Dk(O)]
precisely when depth(Dk(o), X) = codim(Dk(o), X) = (e  k)(f  k); this means that for all x E Dk(o), the ideal of Dk(o) in \9x • x contains a regular sequence of length (e  k)( f  k). If X is CohenMacaulay, e.g. nonsingular, this is equivalent to Dk(o) having the expected codimension. Without this depth condition, even if Dk(o) has the right codimension, Dk(o) will be a cycle whose support is Dk(o) but whose coefficients are smaller than those in [Dk(o)]. It remains to compute the image of D k ( 0) in AmX. By the theory of §5, one has
s:[G]
= ctop(Sv®
FG ) n [G]
in AmG. The required class is then the image of this class in AmX. The answer, to be verified in the next section, is the GiambelliThornPorteous formula: Dk(O) = A1_/)(c(F E))
n [X].
Here 6,([>( c) denotes the determinant of the p by p matrix
and c(F  E) = c(F)jc(E) = c(F) . s(E). This formula yields a geometric construction for the Chern classes c,(F) of a bundle F of rank f. Let e = f  i + 1, and let E be the trivial bundle of rank e. Then 0: E + F is given by e sections Sl"'" s. of F, and D/_,( 0) by the locus where these sections become dependent. Then D/_,( 0) is a class in AII_,(D/_;( 0» which represents c;(F) n [X], since A(,I)(c(F  E» = c,(F). If F is generated by its sections, and s I' ... , S e are chosen generically, then D/_;(o) = [D/_,(0)1
= c;(F)
n [Xl.
49
INTRODUCTION TO INTERSECTION THEORY
6.2. Schur polynomials. For any formal series c = I + CI + Cz + ... with c, in any commutative ring, and any finite sequence )I. = ()I. I " ' " )I. d) of integers, define ~1.(c) to be det(c1.,+r,)' i.e. ~1., .....1.) c) = det
C1.,_1 C1. r d+ I
C1., c1. d
Note that adding a string of zeros to)l. does not change ~1.(c). Usually we will assume)l. is a partition, i.e. )1.1 ~ )l.z ~ '" ~)l.d ~ 0, so)l. partitions 1)1.1 = [)I. j' Then ~1.( c) is the Schur po(ynomial corresponding to )I.. If one represents )I. by a
Young diagram
I I I I I 1 I I
with )l.j boxes in the ith row, the conjugate partition is obtained by interchanging rows and columns. A basic formal identity is
(1) if Il is the conjugate partition to )I.. Another is
(2) where the sum is over all Il = (Ill"'"
fl.d+
I) with
fl.1 ~ )1.1 ~ fl.z ;;, ... ~ Ild ~
)l.d ~ Ild+
I ~ 0,
and 1fl.1 = 1)1.1 + m. More generally there is a LittlewoodRichardson rule for the coefficients N1. .•. p of an arbitrary product [41]:
~1.(c)· ~.(c)
(3)
=
LN1..".p~p(c).
Some useful formulas for top Chern classes can be expressed in terms of Schur polynomials:
(i) where f (ii) (iii)
= rank F,
and the subscript is repeated e
clop(SZE) ClOP (
A ZE)
= rank E times.
= 2e~e.e_I ... I(C(E)), =
~el.eZ ..... I(c(E)).
Here S2 E and A ZE are symmetric and exterior powers, and e = rank E [39]. We may use (i) to complete the proof of the GiambelliThomPorteous formula. It suffices to verify that, with the notation of §6.1, and any sequence )I. = ()l.z,· .. , )I. d)'
11.(~)..(C(FG  S))
n [G])
= ~~(c{F 
E)) n
[xl,
where V = (hi  k, ... , hd  k)., Note that A)..(c(FG  S)) = ~J.(c(FG  FG)· c(Q)). Expanding this determinant, one is reduced to showing that, for IX E A. X, ifil=···=id=k, otherwise. See [36, 3, or 16] for details. Schur polynomials have been used by Navarro Aznar [47] to define local invariants of a coherent sheaf '!J on a variety X at a point x. Let r be the generic rank of '!J, and h a partition with r ;;. hi;;' h 2 ;;. •••• Shrinking X if necessary, one can find a birational proper map 11: X  X such that the quotient of 11*'!Jby its torsion subsheaf is the sheaf of sections of a vector bundle E. Define
It follows from the birational invariance of Segre classes (§4.2) that this is independent of choices. When '!J is the sheaf of differentials, these classes were studies by Le and Teissier. MacPherson's local Euler obstruction is an alternating sum of some of these invariants.
6.3. The detenninantaI fonnula. There is a similar formula for a more general determinantal locus. Let 0: E  F be a vector bundle homomorphism as before, and let 1:' be a flag of subbundles of E:
oC Let v, = rank v" h, =
VI C ... C
v,. c
E.
f  v, + i, m = dim(X)  [h,. Set XI dim{Ker{ o{x)) n V, (x)) ;;.
n(.Y; 0) = { X E
i, I .;; i .;; r}.
A similar construction to that in §6.l constructs a class n(.Y; 0) in Am(n(.Y: a)), with analogous properties. The determinantal formula states that the image of n(.Y; 0) in Am( X) is the cap product of
det
c)..,{F VI)
c)..,+I{F v,)
c)..,_I{FV2 )
c)..,{FV2 )
c)..,+r_I{F VI)
with the fundamental class [X]. As in §6.1, the proof of Kempf and Laksov [36] carries over to arbitrary varieties, cf. [16, §14]. This applies to the Grassmannian X = G = Gd(P") = Gd+I(E), with E a vector space of dimension n + 1, and 0 the canonical projection from EG to the universal quotient bundle Q. A flag of subspaces Ao C ... CAd C P" corresponds to a flag Vo C ... C Vd C E of subspaces, with dim V, = dim A, + 1. The Schubert variety is the corresponding degeneracy locus
n{A o ,'''' Ad)
=
n(.Y; 0).
Set a,
= dim(A i ).
h,
=n
d
+ ia,. The deterlninantal formula then yields
Giambelli's formula:
{h o•· ... h d} = (a o•· ... ad)
=
~)..(c(Q)).
For example. the mth special Schubert class
am
=
{m}
= [{
L E GIL meets a given (n  d  m )plane)]
is equal to cm ( Q). for m = I •...• n  d. The formula (2) of §6.2 becomes Pieri' s formula {h o•· ... h d }' am = L{!lo ... ·• !ld}' the sum over n  d;;. !lo ;;. ho ;;. ... ;;'!ld;;' hd ;;. 0 with I:!l, = I:h, + m. Note that {!lo •...• !l,} = 0 if !lo > n  d or r > d. corresponding to the facts that ci(Q) = 0 for i> n  d and Si(QV) = ci(S) = 0 for i> d. Similarly the LittlewoodRichardson rule specializes to a general formula for mUltiplying Schubert classes. If X is a nonsingular subvariety of pn = P(E). there is a canonical vector bundle homomorphism
a: E ® (9x{i)
+
Nxpn
which is the composite of the quotient maps E ® 0(1) Tp"lx + Nxpn. If flagsA.'yare chosen as above,
U{.f; a) = {x
E
+
Tp " on pn. and
Xi dim TxX () A,;;. i.O ~ i ~ d}.
The degree of this locus is the projective character X(a o•...• ad)' The determinantal formula gives a formula for these extrinsic invariants in terms of the intrinsic Chern classes of Tx. and a hyperplane section. The classes and ranks of X (cf. § I ) are special cases. corresponding to partitions h = (I .. '..• I. 0•...• 0) and their conjugates h = (i. 0..... 0). 6.4. Symmetric and skewsymmetric loci. There are similar formulas for bundle maps a: E v + E which are symmetric (a v = a) or skewsymmetric (a v =  (J). Such correspond to sections ta of S2E or 1\ 2E . The locus Dk(a) is defined as in §6.1, but now its expected dimension m is. for k ~ e = rank( E). m
=
dim( X)  ( e 
~+
m
=
dim{ X)  ( e ;
k)
I)
(symmetric). (skewsymmetric.
keven).
There are classes denoted Dk(a) or Di'(a) in A",(Dk(a» in each of these cases. The analogous formulas are
Di(a)
= 2d~d,d_I,.. "I{C(E))
Di'(a)
= ~d_I".,I{C(E))
n [X],
n [xl.
These formulas also date from Giambelli; modern versions have been given by Barth. Tjurin. J6zefiakLascouxPragacz. HarrisTu. and Damon. A particularly
52
WILLIAM FULTON
simple treatment has recently been given by Pragacz. The calculations depend on calculating Gysin pushforwards for'TI: Gd ( E) + X as in §6.1. All such pushforwards are known "in theory", but it requires ingenuity to find useful general formulas. One such (35) is, for any}.. = (}..I .... ' }..d)' v = (Vi' .•• ' vk ), a E A. X, 'TI.{IlA(s(S)) ·1l.{s(Q))
n 'TI*a) = 1l.(s(E)) n a,
where Il = (}..I  k, }..2  k, ... , }..d  k, Vi'···' vk )· For applications, cf. HarrisTu (29), one needs generalizations to symmetric or skewsymmetric bundle maps a: E v + E ® L, for L a line bundle on X. Lascoux and Pragacz also makes these formulas explicit, as follows. Given partitions }.. = (}..I'.·.' }...) and Il = (Ill'···' Il.), say that Il .;; }.. if Il; .;; }..; for I .;; i .;; e, and define dA.=detl(>::=;)1 J
Set
t
.
I ~I.}"~
= (d, d  1, ... , 1) and II = (d  I, d  2, ... , 1). Then for d = e  k,
e = rank(E),
Df(a)
=
Zd)
E 2IAld,Alldc(E))CI(L)(Ji')Il\I, l\".
DkS(a)
=
2(J,')
E 21l\ld5l\1l~(c(E))cl(Lri')Il\I. l\,,5
Here i\ is the conjugate partition of }... When L = M*2, they follow from the preceding cases and the formal identity (39)
E dl\.Il~(c(E))cl(M)IAII.I . • "l\ The case for general L can be deduced from this case. 1l~(c(E ® M)) =
7. Refinements 7.1. Dynamic intersections. Consider our basic intersection theory setup W
.....
V
'+
Y
~
~
X
f
f
a regular imbedding of codimension d. V an ndimensional variety. W = X n v. The normal cone C = C wV is imbedded in the normal bundle N = NxY to Xin Y. Let with
Lm,[C,]
[Cj =
be the cycle of C. Each irreducible component C, of C is a subcone of N; let Z, = stY I( C,) be the support of C,. Let N, be the restriction of N to Z,. Then Cj is a subvariety of N,. and we may set a,
=
s~JCjj E And(Z,).
By construction. the class [m,a, represents the intersection product X· V in ;c W there is more information in the classes oc" with their multiplicities m,. than in the class [mja j on W. For any closed subset Z of W. set
An_d(W). Whenever some Z,
(X, V)z
=
L
mja,
E
And(Z).
Z,cZ
and call (X, V)z the part of X· V supported on Z. One way to refine this class further is to have a section s of the bundle N other than the zero section. Then s![C,] is a welldefined class on SI(C,) C Z,. which refines a, (i.e .• s![ C,] maps to a j by the inclusion of s  I (C,) in ZJ Suppose N is generated by a finite dimensional space f of sections. One can show that for any closed Z c W there is a nonempty open feZ) c f such that for all s E feZ). dim SI(C) = n  d, so s'[C] is a welldefined (n  d)cycle, and the part of s![C] contained in Z is precisely (X, V)z. Suppose the imbedding X > Y is deformed to a family~, > Y x T of imbeddings; we assume T is a nonsingular curve, 'X is flat over T, the imbedding of X in Y X T is regular. and the imbedding Xo > Y X {O} over 0 E T is the given imbedding. This deformation determines in a wellknown way a KodairaSpencer section of the normal bundle N. which we denote by sox. 53
If the generic intersection of XI with V is proper, one may define a limit intersection cycle lim l_ o XI . Vas follows. Consider the fibre square:
"l.ll" l
'X
VXT >
l YxT
The components of "1l\' that project dominantly to T have relative dimension n  d over T. The intersection cycle 'X.. (V x T) in A,,_d+ I ("1l\') therefore specializes to a welldefined cycle on the fibre over 0 (cf. §3.4); this cycle is denoted limr~o XI' V. One can show that lim l _ o XI . V is supported on s.:( I( C), and that this limit class refines s'[C). It follows that, if Z is given, then for any deformation 'X for which s~x belongs to f( Z), lim XI' V = so\[ C].
I~O
In particular, the part of the cycle liml~o XI . V supported on Z represents (X· V)z, for sufficiently general deformations, cf. [40, 16). For example, if X = HI X ... X Hd and Y = P" x ... x P" (d copies), with
Hr hypersurfaces in P", one may construct such deformations by varying equations for the H r , as in §l. In case d = n, V = P", it follows that the degree of (X· V)z is the number j( Z) constructed by the SeveriLazarsfeld method. Thus the (refined) static construction of intersection products (using normal cones) yields the same information as the dynamic construction (using deformations). 7.2. Rationality of solutions. In much of our geometric discussion, we have been tacitly assuming that the ground field is the complex numbers, or at least algebraically closed. No such assumptions are needed for the basic constructions, however. If one begins with cycles Ln,[V,) defined over a given ground field K, all our operations can be carried out with such cycles. The degree of a zerocycle LnrlP,) on X is Ln,[R(P,): K), where R(P) denotes the residue field of the local ring of X at P. For a zerocycle or class 0: on a complete variety X over K, we let fo: denote its degree. Suppose VI"'" v;. are subvarieties of a complete smooth variety X over K, with L codim( V" X) = dim X. Then our construction produces a cycle class VI ...
v;. E Ao( () v,)
whose degree is f (vd ... W)· For example, if K = R, and f(vd ... (V,) is odd, it follows that () V, must contain real points. Indeed, a zerocycle LnJPr ) on () V, which represents VI ... v;. cannot have all R(P,) = C. Note that this argument can be used on each component of () v,. For example, if certain points of proper intersections are known, their contributions can be subtracted; if an odd number remains, there are additional real points in () v,. Similarly one may subtract contributions from components of excess intersection.
55
INTRODUCTION TO INTERSECDON THEORY
A pleasant application of these ideas is to a simple algebraic treatment of the BorsukUlam problem [2]. Let S" be the sphere XJ + ... + = I in R"+ I. If gl"'" g" are odd polynomials in R[Xo, ... , X,,], then there is a point xES" such that all g,(x) = O. To prove this, for any odd g of degree d, set
X;
g* =
I (x5 + ... + x; )(dJl/2 gU),
where gU) is the homogeneous part of g of (odd) degree j. By the previous paragraph, have a common nontrivial solution (xo, ... , x,,). Multiplying by a positive scalar, one may assume this point is in S", in which case it is the required solution. It follows that for any n polynomials (or continuous functions, by approximation), there is a point xES" such that each takes the same value at antipodal points; one applies the preceding to the odd parts of the functions. For X = P" one may prove such results by using deformations, and the compactness of P"(R). The approach with refined intersections is simpler; it works for any X, and gives analogous results for any field all of whose finite extensions are a power of a fixed prime. The question of how many solutions of real equations can be real is still very much open, particularly for enumerative problems. For example, how many of the 3264 conics tangent to five general (real) conics can be real?
gr, ... , g:
7.3. Residual intersections. In our basic situation for constructing intersection products (§7.l), there may be a distinguished subscheme D of the intersection scheme W. A natural candidate for the contribution of D to the intersection product X· V is the classs
{c(NxY) ns(D,v)}m
E
AmD,
m = dimlY)  d, d = codim(X, Y). Our object is to construct a residual scheme R and a class, denoted R, in Am(R) so that one has a Residual intersection formlfla
X· V={c(NxY)ns(D,V)}m+R inAmW.
We have a diagram
D
a
W
>
j
>
h X
V
!/ >
Y
r
with i a regular imbedding, W = I(X). Assume first that the composite ja imbeds D as a Cartier divisor on V. The residual scheme R to D in W is th.e subscheme of V whose local equations are obtained by dividing local equations for W in V by a local equation for D in V; then W = D U R, with ideal sheaves on V related by 1(W)
=
1(D) ·1(R).
Set E = g*NxY ® j*fJ v(  D) = g*NxY ® (NDV) v. One verifies that the normal cone CRY is a subcone of the restriction ER of E to R. Then one may define the residual class R to be the intersection class of the ncycle [CRV] by the zero section of the bundle E R : R
= SE.[CRV] = {c{E) n s{R, V)}m.
With these definitions, the residual intersection formula is valid. To prove it, one blows up V along Z to reduce to the case where W = D + R as a divisor, in which case the excess intersection formula applies; see [16, §9] for details. For example, if the imbedding of R in Vis regular of codimenion d', then R
=
Cd_d,{E/NRV) n [R].
In particular, if d' = d, then R = [R]. For arbitrary D, one can blow up V along D to achieve the situation just studied. Let 'IT: V > V be this blowup, 'IT * D = D and 'IT  I (W) = W. Let R be the residual scheme to D in Wand R the residual class in AmR just constructed. If one sets R = 'IT ( R) and R = 'IT *(R), the desired residual intersection formula results. 7.4. Multiple point formulas. Laksov developed a version of this residual intersection formula to prove a double point formula for a morphism/: X > Yof nonsingular varieties of dimensions nand m. In this case (f X f)I(~y) contains ~ x; the residual scheme will be the locus D'(f) of double point pairs. If X is complete either projection from X X X to X maps D'(f) onto the double point locus D(f) in X. The projection of the residual intersection class R is the double point class, denoted D(f), in A 2n  m(D(f)). One deduces from the residual intersection formula the
Double point formula
D{f)
=
f*f *[X] (c(f*Ty)c{Tx))m_n n [X].
For example if X is a curve of genus g, Y = p2, and f maps X birationally onto a curve of degree n, one has the classical formula degD(f)
=
(n  l)(n  2)  2g.
By construction D(f) is constructed from the residual scheme R to the exceptional divisor P(Tx ) in the blowup of X X X along the diagonal ~ x' When this scheme R has the expected dimension 2n  m, then D(f) is the projection of the cycle [R]. Note that R may have components inside P(Tx ), as happens e,g. for plane curves with cusps. When m = n + I and f maps X birationally and finitely onto its image in Y, one expects D(f) to be the cycle determined by the conductor ideal, but this has only been proved for n = I. For triple point and higher multiple point formulas the situation is more complicated; however, when f is a proper immersion which is completely regular (i.e., for any distinct points x, E X with the same image y E Y, the images of the tangent spaces Tx, X are in general position in TvY)' the answer is quite simple. Let Y, be the set of points in Y which are the imag~s of k or more distinct points
INTRODUCTION TO INTERSECTION THEORY
57
r
of X, and let Xk = I(yk ), both with reduced scheme structure. Then one has an inductive formula of Herbert:
[Xkl
=
/*[YkIl
Cd
(/*Ty/Tx ) n [XkIl,
d = dim Y  dim X. Indeed, following Ronga, if X: denotes the set of unordered ktuples of distinct points of X with the same image on y, one has a fibre square
X;UX;_I t
X: t
X
Y
with X; mapping birationally onto JS. An application of the excess intersection formula (§4.3) then yields Herbert's formula. For more general mappings significant progress has been made, primarily by Kleiman, Le Barz, and Ran. Their results are most satisfactory when the multiple points occur only in a "curvilinear" way; they can be used to deduce enumerative formulas for secant lines to varieties in projective space. The excess intersection formulas can also be used to study fixed points of correspondences on a nonsingular complete variety X. If r is a variety, cycle, or equivalence class of cycles on X, with dim r = dim X, then the virtual number of fixed points is the intersection number Jr· .:l of r with the diagonal .:l in X X X. When X = pn, one recovers formulas of Pieri [49]. Conversely, Pieri's work may be seen as an important precursor of modem intersection theory.
8. Positivity 8.1. Positivity of intersection products. When cycles meet properly their intersection product will be an effective cycle, i.e. a sum Ln,[V;] with n, ;;. O. If tWQ cycles are equivalent to cycles which meet properly, their product is represented by (equivalent to) a positive (or zero) cycle. For general excess intersections, however, this is not possible: if E is the exceptional divisor of the blow up of a nonsingular surface at a point, then f E . E =  I. From our construction of intersection products via cones in normal bundles, it is natural to expect that suitable positivity of the normal bundle will guarantee the positivity of intersection products. Recall that a line bundle L on a variety X is ample if some positive power L OJ n is the pullback of (I) for a projective imbedding of X in projective space. If a is a kcycle on X, define
e
degL(a)= !c,(L)k na. x For a subvariety Vof X, let degL(V) = degdV]. Since hyperplanes can always be moved to meet subvarieties properly, deg L( a) > 0 whenever a is equivalent to a positive cycle, i.e. nonzero effective cycle. A vector bundle E on X is ample if the canonical line bundle (9 e(l) on P( E V) is an ample line bundle; note that eEv{I) is a quotient bundle of the pullback of E to P( E V). In general ampleness is preserved by direct sums, by tensor exterior and symmetric products, by passing to q)lotient bundles, and by pullbacks by finite morphisms. To investigate the positivity of intersection products it suffices to consider the intersection class of an irreducible cone C with the zero section in a vector bundle E on a variety X. Let a = sE[C] be this intersection class. We assume that dim(C);;. rank(E). Fix an ample line bundle L on X. THEOREM. (a) If E is generated by its sections, then a is represented by an effective cycle. (b) If E ® LV is generated by its sections, and Supp(C) = X, then degL(a);;;. degL(X). (c) If E is ample and generated by its sections, then a is represented by a positive cycle. (d) If E is ample, then deg L( a) > O.
59
Of these statements, (a) and (b) are quite easy to prove; they follow from the case where E is trivial, and one may induct on the rank. We refer to [16, §12] for the proofs of (a)(c). The most difficult is (d). For example, when C is the zero section, and dim X = rank(E) = n, then a = c,,(E) n [X]. The assertion that c,,(E) > 0 is a theorem of Bloch and Gieseker [7]. Their proof, valid in .:haraeteristic zero, usee resolution of singularities and the hard Lefschetz theorem. By using intersection homology [27] one may avoid resolution of singularities and extend the result to arbitrary characteristic, but we do not know a more elementary proof of the key assertion that c,,(E) ~ O. We do not know, in (d), if some positive multiple of a can be represented by a positive cycle. For the proof of (d), see [20]. The theorem applies to the intersection products X· V E Am(W) constructed from our basic construction, provided the pullback N of NxY to W has the required positivity. For example, if NxYis ample, and Vis a subvariety of Y with dim V ~ codim( X, y), such that [V] is equivalent to a positive cycle whose support meets X, then V must itself meet X. Indeed, deg L ( X· V) = deg L ( X . a) > O. If X is nonsingular and its tangent bundle Tx is generated by its sections, it follows from (a) that all intersections of effective cycles have effective representatives. Indeed, the normal bundle to the diagonal imbedding of X in X X ... X X (r copies) is the direct sum Tx 6l ... 6l Tx(r  I copies). If X = P" and L = 19(1), then Tx ® LV is generated by its sections, so (b) holds for all intersections on P". Let Vi'.·.' ~ be subvarieties of P", and let VI ... ~ be the intersection product constructed from the diagram
x
Xx ... x X
by the prescription of §7.1. If [C] = ~m;[C,] is the cycle of the normal cone to n ~ in VI X ... X ~, then we have a decomposition VI ... V, =
[m,a,
with a, a class on Z; = Supp( C;). By the theorem, each a, is represented by a positive cycle, and deg( a;) ~ deg( Z,) > O. In particular,
TI deg(V,) =
[m,deg(a;) ~ [m,deg(Z,).
Note that each irreducible component of n ~ appears as some Z" and each m, ~ I, so this refines the Bezout theorem of §2.3. 8.2. Positive polynomials and degeneracy loci. Let 0: E  F be a homomorphism of vector bundles of ranks e and f on an ndimensional variety X. For k ~ min(e, the expected dimension of the degeneracy locus Dk(o) was seen in §6.1 tobe m = n  (e  k)(f k).
n,
61
INTRODUCTION TO INTERSECTION THEORY PROPOSITION. Assume E v ® F is ample. and m :;. O. Then D, (0) any ample line bundle L on X.
"'" 0.
and [<Jr
deg L (~1_/1( c{ F  E))) > O. To prove this. let H = Hom(E. F) = E v ® F. and let D, c H be the cone of maps of rank .. k (§6.1). Then 0 corresponds to a section I. of H. and the degeneracy class Dk(o) is the intersection class I:[D.]. Since the normal bundle to t. is H. which is assumed 10 be ample. the theorem of §8.l yields deg L (D, » > O. The GiambelliThornPorteous formula for D k (0) completes the proof. A similar construction shows that if E is any ample vector of rank e on an ndimensional variety X. then for any partition A of n with e :;. AI :;. A z :;. ... ;:;.
«J
O.
To prove this one takes a vector space V of dimension n + e. and a flag of subspaces VI C Vz C ... C V with dim(V,) = e + iA,. Let H = Hom(Vx. E). and set 12).
= { E
Hldim Ker( 1. whose degree is positive by the theorem of §8.1. Any polynomial P(e l..... e.) E Q[c l ..... eel of weight n can be written uniquely in the form
the sum over partitions A with e :;. AI :;. ... :;. A" :;. O. and THEOREM
~A, =
n.
[20]. If a). :;. 0 for all A. and some a). > O. then
fxP{el(E) ..... ek(E)) > 0 for all ample vector bundles E of rank e on all ndimensional varieties X. Conversely. if some a). < O. there is an ample E for which fxP(c(E» < O. To see the last statement. let V be a Schubert variety representing the dual class to {AI •...• An} in G = G,,(p,,+e). Then
fvP{c(Q))
=
a). < 0
for Q the universal quotient bundle on G. Let L be a very ample line bundle on G. For any k > 0 there is a finite surjective morphism f: X + V such that f* L = M'H for a line bundle M on X. Then E = f*Q ® M is ample on X. and if F(t) is the polynomial
F(t)
=
fvP(e{Q
®
L''')).
then f xP( c(£)) = deg( XI V)F(\lk); lhis is negative for sufficiently large k, since F(O) = < O. A similar analysis shows the positivity of products of Schur polynomials in Chern classes in two or more ample bundles, and the existence of degeneracy loci DZ (0) or DZ"' (0) for symmetric or skewsymmetric bundle maps 0: £ v  £ ® L, when the expected dimension is nonnegative, and Sym2(£) ® L or /\ 2£ ® L is ample. Although we have been using rational equivalence, the natural equivalence for most of these questions is numerical equivalence. Two cycles a, a' on a complete variety X may be said to be numerically equivalent if f P n a = f P n a' for all polynomials P in Chern classes of vector bundles on X. When X is nonsingular, it follows from RiemannRoch that this is equivalent to requiring f~ . a = f~ . a' for all cycles ~ on X. If the expected dimension m of DI. (0) is at least I, and £ v ® F is ample, then DI. (0) must be connected [19]. The analogous assertion is open for symmetric and skewsymmetric bundle maps. These assertions would follow from the general conjecture that for any kdimensional subvariety V of an ample bundle of rank e, and any section s of V, SI(V) is connected, provided k > e. Note that 'the nonemptiness of s  I (V) follows from the theorem of §8.1, for any k ;;;,. e.
a,
8.3. Intersection multiplicities. Let Vi' ... ' v,. be subvarieties of an ndimensional nonsingular variety X meeting properly at a point P (assumed to be rational over the ground field). Let i(P) = i(P, VI '" V,; X) be the intersection multiplicity. By shrinking X, we may assume the Y, intersect only at P. Let 'IT: X  X be the blowup of X at P, £ = p,,I the exceptional divisor. Let V, c X be the blowup of Y, at p, i.e .. the proper transform. Note that V, n £ is the projective tangent cone P( CpY,), whose degree in £ = P"  I is the multiplicity epY, of V, at P (§4.2). Note also that the intersection product VI ... V, on X is a welldefined class in Ao(£). Then
For two curves on a surface the curves VI and V2 must intersect properly, and one may continue blowing up; this results in Noether's formula for i(P) as the sum of the products of the multiplicities at all infinitely near points. (One can prove ( *) in general using either deformation to the normal bundle to P in X, or the residual intersection formula.) In general, the V, need not intersect properly. Since intersections on X can be negative, it is not so obvious that Vi ... V, must be nonnegative. Using the theorem of §8.1, however, one can show that there is a decomposition
VI .. ·V="ma r I..J I
with m, > 0, a , a cycle on a subvariety Z, of
n V,
I
=
deg( a , ) ;;;,. deg Z; > O.
n P(CpY,),
INTRODUCTION TO INTERSECTION THEORY
63
The union of the Zi is n Vi' In particular, the "e;ror term" f VI V, is bounded below by the sum of the degrees of the irreducible components ()f np(Cp~) c E = pnI. When the V, are hypersurfaces the proof is an easy application of the theoreIIl, since the normal bundle to Vi in X is ample on Vi n E. For general ~ the proof is more complicated, since the normal bundle to the diagonal imbedding of X in X X ... X X is not ample near E. See [16, §12.4] for details.
9. RiemannRoc:h 9.1. The GrothendieckRiemannRoc:h theorem. If E is a vector bundle on a complete variety X, let X (E) denote its Euler characteristic:
X(E)
=
I( If dim H;(X, E).
Motivated by some ingenious calculations of Todd, Hirzebruch discovered the
HirzebruchRiemannRoch formula for expressing X(E) terms of Chern classes of E and of the tangent bundle of a nonsingular variety X: X(E)
=
fx ch(E)· td(Tx)·
Here ch and td denote the Chern character and Todd class respectively. The Chern character ch( E) of a vector bundle E of rank e on a variety X is the sum
ch(E)
= =
e + c, +
I
Hcf 
2c z ) +
Hd 
3c,c 2 + 3cJ + ...
Pk/k!,
where c, = c;( E), and Pk is the sum xf explicitly,
c, 2C2
+ ... + x; in Chern roots 0
of E;
0 0
C,
Pk = det
x,
c, kc.
c,
Then ch(E) = ch(E') + ch(E") if 0  E'  E  E"  0, and ch(E ® F) ch( E) . ch( F). Similarly the Todd class td(E) is defined by
td(E) = 1+
1C, + rr(d
+
cJ + ~c,cz + ...
e
=
f1x,/(I
exp(x;)).
They are related by the formal identity e
(I)
I(I)lch(J\'C)=ce(E)·td(E)'. 10
Note that td( E) = td( E') . td( E") if 0  E'  E  E"  O. 65
=
Both X and ch are additive on exact sequences of vector bundles. the former by the long exact cohomology sequence. Let K ° X denote the Grothendieck group of (algebraic) vector bundles on X; it is the free abelian group on isomorphism classes [E] of vector bundles on X, modulo relations
[E]
=
[E'] + [E"]
for any exact sequence 0 > E' > E > E" > 0 on X. If X is complete. then X determines a homomorphism from KO X to Z. For a nonsingular complete variety X. set K(X) = KOX. A(X) = A* X, and let
T: K(X)
>
A(X)Q
be the homomorphism given by T(E)
=
A(X) ® Q
ch(E)· td(Tx). So HRR reads:
=
x(E) =
f T(E). If I: X >
Y is a closed imbedding of nonsingular varieties. there is an induced homomorphism I • from K( X) to K( y), determined by m
I.[E]
=
L (I)'[£,]
,0
if 0 > £'11 > £'111 > ... > Fo > I. E > 0 is a resolution of the sheaf I. E vector bundles F,. Assume X and Yare complete. and consider the diagram:
x
f.
>
f.
~y
z r Q
One sees that to prove HRR on X it suffices to know HRR on Y (so the right square commutes). and the commutativity of the left square. When X is projective. one may take Y = Pll. Then K(PIl) is generated by [I:'(i)] for i = 0..... n, and the verification of HRR for these line bundles amounts to a formal identity. So the essential part of the proof of HRR. for X projective and nonsingular. is to verify the commutativity of the left square of the diagram. That is. for all cr E K(X).
(.)
ch(f.cr)·td(Ty)=I.(ch(cr)·td(Tx ))·
Let N be the normal bundle to X in Y. Since td takes sums to products. td(N)1 . td(f*Ty)
=
td(Tx).
By the projection formula. (.) is equivalent to
(** )
ch(f .cr)
=
1.(td(N)1 . ch(cr)).
Let us first verify ( •• ) on a simple example. v:here everything can be calculated explicitly: X is an arbitrary nonsingular variety, Y = P(N Ell I). where N is an arbitrary vector bundle on X, and I: X > Y is the zerosection imbedding of X in N, followed by the open imbedding of N in P(N Ell 1). Let p: Y > X be
INTRODUCTION TO INTERSECTION THEORY
67
the bundle projection, let Q be the universal quotiMt bundle on peN e I), aad let d = rank( N) = rank( Q). Let s be the section of Q determined by the projection of the trivial factor in p*( N e I) onto Q. The zero scheme of s is precisely X. It follows that for any pEA (Y),
(2)
1.(f*P}=P·I.[X]=cd(Q}·P.
The section s determines a Koszul complex
0+ 1\ dQv+ 1\ dIQv .... ....... 1\ IQ v .... l9 y
....
1.l9 x
....
0
which is a resolution of 1.19 x' It follows that for any locally free sheaf E on X,
I. E has a resolution 0+ I\dQ v®p*E .... ....... I\IQv®p*E .... p*E .... I.E+O. Therefore, ch(f .[E]) =
L: (If ch( 1\ iQ V) . ch( pOE}.
From (I), the right side is cd(Q) . td(Q)1 . ch( p. E). Using (2), and noting that f*Q = N and/*p*E = E, one has
(3) which proves ( •• ) in this case. This model also shows, via identity (I), where the Todd classes come from. To prove ( •• ) in general, consider the deformation to the normal bundle (§2.6). In order to have a projective parameter space, we deform from the given imbedding at 0 E pi to the normal bundle imbedding at 00 E pi; i.e., M is the blowup of Y x pi along X x {oo}:
X
....!
p(Nel}+
Y
k',.
.1.1
i" ~
Xx pi
=
{(0)
Moo
~
Ii' jex;
M
P +
pi
....
{O)
F
i X
i
io
Jo
Y f
= Mo
i
Here f: X + P( N e I) is the preceding model. Now let E be any vector bundle on X. We must show that equation ( •• ) holds. Let q: M + Y be the composite of the blowdown map from M to Y X pi and the projection to Y. Since qjo = id y, it will suffice to compute the image of ch(f .E) in A(M)Q' Let E = (pr)* E be the pullback of E to X X pi, and let
o ....
Gn
....
Gn 
I ...........
GI
....
Go
+
F.( E) .... 0
be a resolution of F .(t) on M. Since M is flat over pi, it follows thatJtG. is a resolution of I.E on Y and J!G. resolves I.E on Moo' In particular, since Y is
68
WILLIAM FULTON
Write ch( F.) in place of L:(  I)' ch( F,), for any complex F. of vector bundles. Using the projection formula, we have
= jo.(ch{jO'G.)) = ch{G.) ·jo'[X]. jo'[X] = k.[P(N Ell I)] + l.[y], since 0 = [div(p)] = [Mo] [Mx,]. jo.{ch{f .E))
Note that Therefore
ch(G.) ·jo.[X]
=
k.{ch(k*G.)) + 1.{ch(l*G.))
=
k.(ch(j.E)) + O.
But ch( j* E) was calculated for the model. So we have
jo.{ch{f.E))
=
k.(J.(td{N)I. ch(E)))
in A (M )Q' Applying q. to both sides yields the required formula (. *). since qkj = f. On an arbitrary variety X, let K.X denote the Grothendieck·group of coherent sheaves on X. Tensor product makes K·X into a commutative ring, and K.X into a module over K·X. Iff: Y lo X is a morphism, the pullback of vector bundles defines a ring homomorphismf*: K· X > K·Y. If f is proper, there is a pushforward
Here R'f •
conditions refer to Todd classes. When X is nonsingular, TX( a) is given by the formula ch( a) . td(Tx) of §9.1. For arbitrary X, and a E KoX, one may often calculate 1'x( a) by finding a proper morphism 1T: X' ..... X with X' nonsingular and a' E KoX' with 1T .a' = a. Then by (I), TX(a) = 1T.(ch(a')· td(Tx')). At least in characteristic zero sud} X', a' always exist, although X' may not be connected. Similarly using Chow's lemma, one may determine TX for arbitrary X from the construction of TX' for X' quasiprojective. What must be proved is that such constructions are independent of choices.
If X is a variety which admits a closed imbedding in a nonsingular variety M. then TX may be constructed as follows. For a coherent sheaf ,1' on X. let E. be a resolution of ~t on M. Note that ch( E.) = L(  1)' ch( E.) E A( M)Q restricts to zero in A(M  X)Q' Thus ch(E.) must be the image of some class in A.(X)Q' MacPherson's graph construction [4] produces such a localized Chern character ch~( E.) in A.( X)Q' Then
Txe'f) = td(
TMix) n ch~(E.).
The graph construction is an important generalization of the deformation to the normal cone. useful for constructing characteristic classes on their natural loci. For any scheme X. define the Todd class Td( X) 01 X by
Td(X) = TX(~X)
E
A.(X)Q'
If X is nonsingular. Td(X) = td(Tx) n [X]. Then GRR extends to arbitrary varieties in the following form. If f: X .... Y is a proper morphism. ~ E K· X. and there is an element I .( ~) E KY such that
I .(~
[~x])
®
=
I*(~) ® [('\.].
then by (I) and (2).
1.( ch( ~) n Td( X))
=
ch(f • ~) n Td( Y).
This includes the generalization of Grothendieck's theorem to nonprojective smooth varieties. In addition one has the HRR formula for a vector bundle E on an arbitrary complete variety X.
L:( l)'dim H'(X, E) =
fx ch(E) n Td(X).
Another corollary of the general RR theorem is that the induced homomorphism TX
® Q: K.( X)Q .... A.( X)Q
is an isomorphism. for any algebraic scheme X. When f: X .... Y is a regular imbedding there are pushforwards I .: K· X .... K· Y and pullbacksf*: K.Y .... K.X. Assume that Y can be imbedded in a nonsingular variety. Then any locally free sheaf E on X can be resolved on Y. and one sets
I.[E]
=
L:( I)'[F,]
if F. is a resolution of I • E. and f*[~]
=
L:( 1)I[X,(G.®y~)].
where G. is a resolution of I • ~ x. and X, (G. ® y ~) are the homology sheaves of the complex G.®t', ~. Then one has the RR formulas chi .[E] = 1.(td(N)1 . ch(E)), Txf*[~]
=
td(N)1 n f*Ty(~)
for a vector bundle E on X or a coherent sheaf ~ on Y; here N is the normal bundle to X in Y. Such formulas were first proved in Grothendieck's seminar
INTRODUCTION TO INTERSECTION THEORY
71
SGA 6 and by Verdier [59]. The second gives an adjunction formula relating Todd classes:
Td(X)
=
td(N)1 nf*Td(Y).
There are similar compatibilities with exterior products. In particular,
Td(XX Y)=Td(X)XTd(Y). Note that for any complete X,
L(I)'dimH'(X,0 x ) =
fx Td(X).
The above properties of Todd classes generalize classical facts about the arithmetic genus. For example, the constancy of arithmetic genus in flat families generalizes to the fact that Todd classes are compatible with specialization. For a recent interesting application of the singular RiemannRoch theorem in local algebra, see Morales [44].
10. Miscellany
io.1. Topology. For a space X, H'X denotes the ordinary (singular) cohomology of X, with integer coefficients. For a closed subspace Z of X, H'( X, X  Z) denotes relative singular cohomology. A useful homology theory for the study of possibly noncompact spaces is the homology with locally finite supports, or BorelMoore homology, which we denote by H, X. If X is imbedded in an oriented real nmanifold M, then H,X;: W'(M, M  X).
(I)
Taking M = R", this isomorphism may be used to define H;X. If Z is closed in X, and U = X  Z there is a long exact sequence
(2)
....... H,+IU .... H,Z .... H,X .... H,U .... ...
and there are cap products
(3)
H'( X, X  Z)
® ~
n
X .... HJ'Z,
Any complex kdimensional variety V has a fundamental class cl(V) which generates HuV ;: Z. For a complex variety X there is an induced homomorphism
(4)
cl x : AkX .... HuX
which takes Ln,[V,) to Ln, cl(V,). That cl x respects rational (or algebraic) equivalence is a special case of the proposition which follows. Any regular imbedding i: X .... Y of codimension d determines an orientation class u x.y E H 2d ( y, Y  X). If X and Yare nonsingular, u x.y is determined by the equality ux.y n cl(Y) = cl(X). If Y is a vector bundle over X, and i is the zero section, U x. y is the Thorn class of the bundle. For the general case see [4]. PROPOSITION. Let i: kdimensional variety,j:
X ....
Y be a regular imbedding of codimension d, V a 1( X). Then
v .... Ya morphism and W = r
f*(u x .y)ncl(V)=c1 w (X· V) in H u _2d (W). The proof can be achieved by reducing via familiar methods to the case of divisors, where it is straightforward [16, §19). It follows from this proposition that all our intersection constructions are compatible with those in topology; in particular, the (refined) intersection product on nonsingular varieties maps to the 73
topological product described in §3.1. For varieties meeting properly at a point in a nonsingular variety, it follows that the intersection multiplicity is given by the linking number of the intersections of the varieties with a small sphere about the point. One may define two cycles cr, cr' on a complex variety X to be algebraically equivalent if there are subvarieties V, of X X C, C a complete nonsingular curve, and t l , t2 E Cwith
(cf. §3.3). If X is nonsingular, one has a filtration of the codimension p cycles on X: Rat P X c AlgP X CHomP X c Num P X c ZP X conslstmg of cycles rationally, algebraically, homologically and numerically equivalent to zero. For p = I these groups and the factor groups are quite well understood: Algi X/Rat l X is the Picard variety, Algi X = Hom l X, Numl X/Algi X :;; H2( X)'ors is finite, and Zl X/Num l X is finitely generated and free. For p > I, ZP X/Hom P X is always finitely generated, so ZP X/Num P X is free abelian. We have mentioned that, for surfaces, Alg2 X/Rat 2 X can be "infinite dimensional". Griffiths showed that Alg2 X can differ from Hom2 X when X is a threefold, and Clemens [11) has improved this to show that Hom2 X/ Alg2 X need not be finitely generated. The principal tool for studying this problem is an AbelJacobi map Hom P X/Rat P X ... JPX
to the pth intermediate Jacobian of X. For p = 2, J. P. Murre has recently showed that the image of AlgP X/Rat P X is an abelian variety, universal for "regular" homomorphisms of AlgP X/Rat P X to abelian varieties.
10.2. Local complete intersection morphisms. Consider for simplicity the category of varieties which admit closed imbeddings into nonsingular varieties. Then any morphism/: X ... Y admits a factorization/ = pi
with i a closed imbedding and p smooth; if X c M, M nonsingular, one may take p = Y X M, P the projection. We call/a I.c.i. morphism if for some (and hence, in fact, for any) such factorization, i is a regular imbedding. If p has relative dimension nand i has codimension e, then d = e  n is independent of the factorization, and is the codimension of /. In addition, one has the virtual tangent bundle
INTRODUCTION TO INTERSECTION THEORY
75
One verifies that such notions are independent of factorization by comparing a factorization through P and through P' with the diagonal factorization:
If f: X ..... Y is a l.c.i. morphism of codimension d, j determines Gysin homomorphisms
f*: AkY ..... Ak_dX by f*Cl = i*( p*Cl). where p* is flat pullback (§3.3) and i* is the Gysin homomorphism for regular imbeddings (§5.1). Similarly there are refined Gysin homomorphisms /': AkY' ..... Ak_dX'
for any Y' ..... Y, with X' = X X y Y', by f*Cl = ;'( p'*( Cl». where p' is the induced (flat) morphism from P X y Y' to Y'. The RiemannRoch formulas of §9.2 generalize to l.c.i. morphismsj: X ..... Y:
=j.(td(1f)
ch(f.Cl) Tx(f*~)
=
·ch(Cl»),
td(1f) nf*Ty(~)
for Cl E K·X or ~ E K.Y. If i: X ..... Y is a regular imbedding of codimension d, and Y is the blowup of Y along X, then the morphism j: Y ..... Y is a l.c.i. morphism of codimension zer() (by the lemma in §2.4). Consider the fibre square:
~
X
Y ~f
d X
.....
Y
The exceptional divisor X is P(N), N = NxY' and the excess bundle E is the quotient bundle on P(N); E = g*N/NxY = g*N/fJ N ( I).
One can show that there are split exact sequences a
_
b_
0+ AkX ..... AkX Gl AkY ..... AkY ..... 0 with a( Cl) = (cd_,(E) n g*Cl.  i .Cl) and b(~, y) = j. ~ + j*y. Moreover thereis the following general formula for /*, involving Segre classes. For any kdimensional subvariety Vof Y, let ii be the blowup of Valong V n X. Then
f*[V]
=
[ii] + j.{c(E) n g*s(V n X, V)} k
in AkY. See [16, §6.7] for details.
10.3. Contravariant and bivariant theories. We have mentioned the problem of giving a geometric construction of a suitable contravariant ringvalued ("cohomology") theory A* to pair with the covariant (" homology") theory A * we have been studying. At present there are several definitions of such rings A* X, each with its uses as well as defects: (I) For quasiprojective varieties X, one may define [4, Appendix]
A· X
=
limA*Y,
the limit over all morphisms I: X ..... Y from X to nonsingular varieties Y, with A*Y as in §S.2. There are pullbacks /*: A* X ..... A* X' for any morphism f:
X' ..... X, cap products AP X ® AqX ..... Aq_pX,
with the usual projection formula, and vector bundles have Chern classes in A* X. For complex varieties there are homomorphisms cl:A*X ..... H 2 *X to cohomology. With this theory one also has the desirable properties
Pic( X);: A'X and
ch: KOXQ
':::'
A*XQ •
However there are few other functorial properties known. For example. one would like Gysin homomorphisms
I.:
APX ..... AP+dy
for a proper l.c.i. morphism I: X ..... Y of codimension d; it is nbt clear how to construct suchl * for this theory. even for smooth projections. Note that if X is nonsingular, A* X is the same as that constructed in §S.2. It follows that this theory A* is the linest possible contravariant theory agreeing with the given theory on nonsingular varieties. (2) For any X one may construct an operational theory A* X as follows [23, 16]: an element c of AP X is a collection of homomorphisms cX': AqX' ..... Aq_pX'
for all X' ..... X, and all q, compatible with all our other intersection operations. Precisely, one requires that if f: X" ..... X' is proper (resp. fiat, resp. a regular imbedding) with X' ..... X given, then for Cl E A * X" (resp. ~ EA. X')
cx·(f .Cl) = I *CX,,(Cl)
(resp. cx .. (f*~)
= /*cx,(~)),
The ring structure on this A* X is constructed by composing homomorphisms, This theory has pullbacks, cap products. Chern classes. and also Gysin homomorphisms I. for l.c.i. morphisms f. However, the map from Pic( X) to A I X need not be an isomorphism, and we do not know a homomorphism from these groups A* X to cohomology H* X, for complex varieties X. These operational groups are useful because of their formal properties, For any series of operations that finally end up with a class in a group A * X e,g, for any
enumerative problemthere is no loss at all in using them. Even less is known about computations of these A* X than in the classical case, however. One can at least show that A PX = 0 for p < 0 or p > dim X; our proof that A* X is commutative uses resolution of singularities, so is known only in characteristic zero. When X is nonsingular, this A* X also agrees with that in §5.1. Note that this A* is the coarsest theory with this property and with a theory of cap products compatible with intersection products. (3) Mumford [46] has used the image of the first of these theories in the second. (4) One may define APX to be HP(X, ~'p)' where ~"\'p is Quillen's sheaf of higher Kgroups [52]. When X is regular, Quillen proved Bloch's formula:
W( X, Jt.p)
=
AII_pX,
n = dime X). Gillet [25] has constructed Chern classes in these groups, cap products, and some Gysin homomorphisms. (5) Another possibility has been proposed by M. Levine, in order to extend results about vector bundles by Murthy and Swan, and Collino, to general singular varieties. It is most useful if the covariant and contravariant theories are part of a general bivariant theory [23]. This should assign to any morphism J: X ..... Y a graded abelian group
with products, for f: X ..... y, g: Y ..... AP( X~
z,
Y) ® Aq( y!. z) =. Ap+q( X~ z):
pushforwards, for J: X ..... Y proper, g: Y ..... Z.
and pullbacks, for J: X ..... y, h: Y' ..... Y.
h*:AP(X~
Y) . . . AP(XXyY'!: Y')
with /': X X y Y' ..... Y' the induced morphism. These three options should satisfy various compatibility axioms. Then one sets
id
id
id
whre pI. = Spec( K). The products for the composite X ..... X ..... X and X ..... X ...... pI. give "cup" and "cap" products. One point of such a theory is that orientations for morphisms J: X ..... Y should determine classes in A*( X
~
Y). For example. a flat morphism or a I.c.i.
morphism
f
of codimension d should determine a class [f] in A,/( X
I >
Y). And
such a class determines Gysin homomorphisms A~Y
/*:
>
A~ .dX,
f.:A~X>A~+'/Y.
/*cr.
=
[fl· cr..
f.cr.=f.(cr.·[J]).
When a class in A' X lives naturally on a locus Z eXas has been a frequent theme in these lectures. the class should really be a class in A*(Z > X). For closed imbeddings Z ..... X. A*( Z > X) should function as local cohomology groups A~X. In topology there is a natural bivariant theory H*( X
I >
Y). If one imbeds X in
R". one may define
w( X
~
Y) = w+"( Y
X
R". Y
X
R"  X).
At present we have only an operational bivariant theory for rational equivalence: a class in AP( X
I >
Y) is defined to be a collection of homomorphisms from
r Y') for all Y' > Y. all q. compatible as in (2) above. One ca~ show that A "( X > pt.) is isomorphic to Aq X. that there are orientation classes [f] for flat and I.c.i. morphisms. and that our constructions of degeneracy classes. residual intersection classes. local Chern classes. etc .. all belong to appropriate bivariant groups. When X > Y is a closed imbedding the local cohomology groups Hf{( Y. ~"\'p) used by Gillet look promising for a sharper bivariant theory. D. Grayson has pointed out. however. that for general f: X > Y. if one imbeds X in a nons in gular M. the groups Ht( Y x M. ~'.) are not independent of the imbedding. There is a satisfactory bivariant theory specializing to Ko and KO. which should agree with the ideal rational equivalence theory ® Q. The objects of Ko( X > Y) are complexes on X of finite Tor dimension over Y. On singular varieties the intersection homology theory of Goresky and MacPherson [27] have led to many new insights. One does not know an analogous theory lying between A* X and A. X. The place of algebraic cycles in their theory is not very well understood.
A" Y' to A,,_ p( X
X
10.4. Serre's intersection multiplicity. If two subvarieties V. W of a nonsingular variety X meet properly at a point P. Serre [57] showed that the intersection multiplicity i(P. V· W; X) is given by the formula i{P. V· W; X) =
I( 1)'length(Tor,A(A/I. A/f)).
where A is the local ring of X at p. and I and J are the ideals of V and W. A unique feature of this formulation is that. at least in its statement. it requires no reduction to the diagonal. This definition makes sense. in fact. for any regular local ring A. whether it contains a field or not. In this generality the positillity of this multiplicity remains an open question. Recently. Dutta. Hochster. and
INTRODUCTION TO INTERSECTION THEORY
79
MacLaughlin [14] have shown that the natural generalization of this conjecture to modules of finite projective dimension is false, even in the geometric case. In the process they produce some interesting resolutions of modules, which cannot be pullbacks of complexes of vector bundles from any nonsingular variety. For arbitrary varieties V, Won a nonsingular X, the virtual sheaf
Torx(V, W) =
L( 1)'[Tor,8
X
(l9 v,l9 w )]
is an element of K.( V n W). One can show that, if T is the RiemannRoch homomorphism (§9.2), then T(TorX(V, W))
=
V· W + lower terms
in A.( V n W)Q' even in the case of excess intersection. With Faltings' recent solution of the Mordell conjecture via solutions of conjectures of Tate and Shafarevich, one may anticipate a renewed interest in intersection theory on arithmetic varieties. For such applications it is important to bring in the infinite primes, as in [1].
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