Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties (NATO Asi Series)

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen Subseries: Mathematisc...

Author: Lothar Gottsche

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen Subseries: Mathematisches Institut der Universit~it und Max-Planck-Insitut ftir Mathematik, Bonn - vol. 19 Advisor: E Hirzebruch

1572

Lothar G6ttsche

Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Lothar GSttsche Max-Planck-Institut fiir Mathematik Gottfried-Claren-Str. 26 53225 Bonn, Germany

Mathematics Subject Classification (1991): 14C05, 14N10, 14D22

ISBN 3-540-57814-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57814-5 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10078819

46/3140-543210 - Printed on acid-free paper

Introduction Let X be a smooth projective variety over an algebraically closed field k. The easiest examples of zero-dimensional subschemes of X are the sets of n distinct points on X. These have of course length n, where the length of a zero-dimensional subscheme Z is dimkH~ Oz). On the other hand these points can also partially coincide and then the scheme structure becomes important. For instance subschemes of length 2 are either two distinct points or can be viewed as pairs (p, t), where p is a point of X and t is a tangent direction to X at p. The main theme of this book is the study of the Hilbert scheme X In] := Hilbn(X) of subschemes of length n of X; this is a projective scheme paraxnetrizing zero-dimensional subschemes of length n on X. For n = 1, 2 the Hilbert scheme X In] is easy to describe; X [1] is just X itself and X [2] can be obtained by blowing up X x X along the diagonal and taking the quotient by the obvious involution, induced by exchanging factors in X x X. We will often be interested in the case where X In] is smooth; this happens precisely if n < 3 or dim X < 2. If X is a curve, X In] coincides with the n th symmetric power of X, X(n); more generally, the natural set-theoretic m a p X ['t] --~ X (n) associating to each subscheme its support (with multiplicities) gives a natural desingularization of X (n) whenever X In] is smooth. The case dim X -- 2 is particularly important as this desingularization turns out to be crepant; that is, the canonical bundle on X In] is the pullback of the dualizing sheaf oi X (~) (in particular X (n) has Gorenstein singularities). In this case, X In] is an interesting 2n-dimensional smooth variety in its own right. For instance, Beanville [Beauville (1),(2),(3)] used the Hilbert scheme of a K3-surface to construct examples of higher-dimensional symplectic manifolds. One of the main aims of the book is to understand the cohomology and Chow rings of Hilbert schemes of zero-dimensional subschemes. In chapter 2 we compute Betti numbers of Hilbert schemes and related varieties in a rather general context using the Weil conjectures; in chapter 3 and 4 the attention is focussed on easier and more special cases, in which one can also understand the ring structure of Chow and cohomology rings and give some enumerative applications. In chapter 1 we recall some fundamental facts, that will be used in the rest of the book. First in section 1.1, we give the definition and the most important properties of X[n]; then in section 1.2 we explain the Well conjectures in the form in which we are later going to use them in order to compute Betti numbers of Hilbert schemes, and finally in section 1.3 we introduce the punctual Hilbert scheme, which parametrizes subsehemes concentrated in a point of a smooth variety. We hope that the non-expert reader will find in particular sections 1.1 and 1.2 useful as a quick reference. In chapter 2 we compute the Betti numbers of S In] for S a surface, and of

vi

Introduction

KAn-1 for A an abelian surface, using the Well conjectures. Here KAn-1 is a symplectic manifold, defined as the kernel of the map A [nl --* A given by composing the natural map A In] ~ A (n) with the sum A (n) --* A; it was introduced by Beauville [Beanville (1),(2),(3)1. We obtain quite simple power series expressions for the Betti numbers of all the S[n] in terms of the Betti numbers of S. Similar results hold for the KAn-1. The formulas specialize to particularly simple expressions for the Euler numbers of S[ n] and KAn-1. It is noteworthy that the Euler numbers can also be identified as the coefficients in the q-development of certain modular functions and coincide with the predictions of the orbifold Euler number formula about the Euler numbers of crepant resolutions of orbifolds conjectured by the physicists. The formulas for the Betti numbers of the S [~] and KAn-1 lead to the conjecture of similar formulas for the Hodge numbers. These have in the meantime been proven in a joint work with Wolfgang Soergel [Ghttsche-Soergel (1)]. One sees that also the signatures of S [nl and KAn-1 can be expressed in terms of the q-development of modular functions. The formulas for the Hodge numbers of S[ ~l have also recently been obtained independently by Cheah [Cheah (1)] using a different technique. Computing the Betti numbers of X[ nl can be viewed as a first step towards understanding the cohomology ring. A detailed knowledge of this ring or of the Chow ring of X[ nl would be very useful, for instance in classical problems in enumerative geometry or in computing Donaldson polynomials for the surface X. In section 2.5 various triangle varieties are introduced; by triangle variety we mean a variety parametrizing length 3 subschemes together with some additional structure. We then compute the Betti numbers of X[ 3] and of these triangle varieties for X smooth of arbitrary dimension, again by using the Well conjectures. The Well conjectures are a powerful tool whose use is not as widely spread as it could be; we hope that the applications given in chapter 2 will convince the reader that they are not only important theoretically, but also quite useful in many concrete cases. Chapters 3 and 4 are more classical in nature and approach then chapter 2. Chapter 3 uses Hilbert schemes of zero-dimensional subschemes to construct and study varieties of higher order data of subvarieties of smooth varieties. Varieties of higher order data are needed to give precise solutions to classical problems in enumerative algebraic geometry concerning contacts of families of subvarieties of projective space. The case that the subvarieties are curves has already been studied for a while in the literature [Roberts-Speiser (1),(2),(3)], [Collino (1)], [Colley-Kennedy (1)]. We will deal with subvarieties of arbitrary dimension and construct varieties of second and third order data. As a first application we compute formulas for the numbers of higher order contacts of a smooth projective variety with linear subvarieties in the ambient projective space. For a different and more general construction,

Introduction

vii

which is however also more difficult to treat, as well as for examples of the type of problem that can be dealt with, we also refer the reader to [Arrondo-Sols-Speiser

(I)]. The last chapter is the most elementary and classical of the book. We describe the Chow ring of the relative Hilbert scheme of three points of a p2 bundle. The main example one has in mind is the tautological p2-bundle over the Grassmannian of two-planes in pn. In this case it turns out hat our variety is a blow up of (p,,)[3]. This fact has been used in [Rossell5 (2)] to determine the Chow ring of (p3)[3]. The techniques we use are mostly elementary, for instance a study of the relative Hilbert scheme of finite length subschemes in a Pl-bundle; I do however hope that the reader will find them useful in applications. For a more detailed description of their contents the reader can consult the introductions of the chapters. The various chapters are reasonably independent from each other; chapters 2, 3 and 4 are independent of each other, chapter 2 uses all of chapter 1, chapter 3 uses only the sections 1.1 and 1.3 of chapter 1 and chapter 4 uses only section 1.1. To read this book the reader only needs to know the basics of algebraic geometry. For instance the knowledge of [Hartshorne (1)], is certainly enough, but also that of [Eisenbud-Harris (1)] suffices for reading most parts of the book. At some points a certain familiarity with the functor of points (like in the last chapter of [Eisenbud-narris (1)]) will be useful. Of course we expect the reader to accept some results without proof, like the existence of the Hilbert scheme and obviously the Weil conjectures. The book should therefore be of interest not only to experts but also to graduate students and researchers in algebraic geometry not familiar with Hilbert schemes of points.

viii

Introduction

Acknowledgements I want to thank Professor Andrew Sommese, who has made me interested in Hilbert schemes of points. While I was still studying for my Diplom he proposed the problem on Betti numbers of Hilbert schemes of points on a surface, with which my work in this field has begun. He also suggested that I might try to use the Weil conjectures. After my Diplom I studied a year with him at Notre Dame University and had many interesting conversations. During most of the time in which I worked on the results of this book I was at the Max-Planck-Institut fiir Mathematik in Bonn. I am very grateful to Professor Hirzebruch for his interest and helpful remarks. For instance he has made me interested in the orbifold Euler number formula. Of course I am also very grateful for having had the possibility of working in the inspiring atmosphere of the Max-Planck-Institut. I also want to thank Professor Iarrobino, who made me interested in the Hilbert function stratification of Hilbn(k[[x, y]]). Finally I am very thankful to Professor Ellingsrud, with whom I had several very inspiring conversations.

Contents

Introduction 1.

V

Fundamental

facts

1

1.1. T h e Hilbert scheme

1

1.2. T h e Weft conjectures

5

1.3. T h e p u n c t u a l Hilbert scheme . . . . . . . . . . . . . . . . . . . .

9

2.

Computation

o f the Betti n u m b e r s

of Hilbert schemes

.....

2.1. T h e local s t r u c t u r e of y[n] -~(n) . . . . . . . . . . . . . . . . . . . . . 2.2. A cell decomposition of P[2hI, Hilb~(R),

ZT, G T

. . . . . . . . . . .

2.3. C o m p u t a t i o n of the Betti n u m b e r s of S In] for a s m o o t h surface S 2.4. T h e Betti numbers of higher order K u m m e r varieties 2.5. T h e Betti n u m b e r s of varieties of triangles 3.

. . . . . . . . .

. . . . . . . . . . . . . .

The varieties o f s e c o n d a n d higher order d a t a . . . . . . . . . .

3.1. T h e varieties of second order d a t a

. . . . . . . . . . . . . . . . .

3.2. Varieties of higher o r d e r d a t a a n d applications

. . . . . . . . . . .

3.3. Semple bundles a n d the formula for contacts with lines 4.

....

. . . . . . .

12 14 19 29 40 60 81 82 101 128

The Chow r i n g o f r e l a t i v e H i l b e r t schemes of projective bundles

. . . . . . . . . . . . . . . . . . . . .

145

4.1. n-very arapleness, embeddings of the Hilbert scheme a n d the s t r u c t u r e of A I n ( P ( E ) )

. . . . . . . . . . . . . . . . . . . . . ~ 3

4.2. C o m p u t a t i o n of the Chow ring of Hilb (P2)

. . . . . . . . . . . .

146 154

4.3. T h e Chow ring of Hw~-flb3(P(E)/X) . . . . . . . . . . . . . . . . .

160

4.4. T h e Chow ring of H i l b 3 ( P ( E ) / X )

173

Bibliography

Index

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

184

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192

Index o f notations

. . . . . . . . . . . . . . . . . . . . . . . .

194

1. F u n d a m e n t a l

facts

In this work we want to s t u d y the Hilbert scheme X In] of subschemes of length n on a smooth variety. For this we have to review some concepts and results. In [Grothendieck (1)] the Hilbert scheme was defined a n d its existence proven. We repeat the definition in p a r a g r a p h 1.1 a n d list some results a b o u t X[ n]. X['q is related y["] ----* X(n). to the s y m m e t r i c power X (n) via the Hilbert-Chow m o r p h i s m wn :"'red We will use it to define a stratification of X [n]. In chapter 2 we want to c o m p u t e the Betti numbers of Hilbert schemes a n d varieties t h a t can be constructed from t h e m by counting their points over finite fields a n d applying the Well conjectures. Therefore we give a review of the Well conjectures in 1.2. T h e n we count the points of the s y m m e t r i c powers X ('0 of a variety X , because we will use this result in chapter 2. In 1.3 we s t u d y the p u n c t u a l Hilbert scheme H i l b " ( k [ [ X l , . . . , x 4 ] ] ) , p a r a m e t r i z i n g subschemes of length n of a s m o o t h d-dimensional variety concentrated in a fixed point. In p a r t i c u l a r we give the stratification of Iarrobino by the Hilbert function of ideals.

1.1. T h e Hilbert s c h e m e Let T be a locally noetherian scheme, X a quasiprojective scheme over T a n d s a very a m p l e invertible sheaf on X over T.

D e f i n i t i o n 1.1.1. [Grothendieck (1)] Let

7"liIb(X/T) be the contravariant functor

from the category Schln T of locally noetherian T-schemes to the category Ens of sets, which for locally noetherian T-schemes U, V and a m o r p h i s m r : V -----+U is given by f

7-lilb(X/T)(U) = I Z C X XTU closed subscheme, flat over U ) "Hilb(X/T)(r

: nilb(X/T)(U)

,7~ilb(X/T)(V); Z , ~ Z xu V.

Let U b e a locally noetherian T-scheme, Z C X XT U a subseheme, flat over U. Let p : Z ---* X , q : Z ~ U be the projections a n d u E U. We lJut Z~ = Hilbert p o l y n o m i a l of Z in u is

P.(z)(m) := x(Oz.(m)) =

x(o

o

q-a(u). T h e

p*bc")).

P,,(Z)(m) is a polynomial in m a n d independent of u E U, if U is connected. For 7"[ilbP(X/T) be the subfunctor of 7(ilb(X/T) defined

every p o l y n o m i a l P E Q[x] let by

TlilbP(X/T)(U) = (

Z CX •

U

closed subscheme

I

Z is flat ~ U and } P~(Z) = P for all u E U "

2

1. Fundamental facts

T h e o r e m 1.1.2 [Grothendieck (1)]. Let X be projective over T. Then for every polynomial P E Q[x] the functor 7-lilbP(X/T) is representable by a projective Tscheme HilbP(X/T). 7-lilb(X/T) is represented by

Hilb(X/T) : =

U HilbP(X/T)" PEQ[x]

For an open subscheme Y C X the functor 7"lilbP(Y/T) is represented by an open subscheme HilDP(Y/T) C HilDP(X/T).

D e f i n i t i o n 1.1.3. Hilb(X/T) is the Hilbert scheme of X over T. If T is spec(k) for a field k, we will write Hilb(X) instead of Hilb(X/T) and H i l b P ( x ) instead of If P is the constant polynomial P = n, then Hilbn(X/T) is the relative Hilbert scheme of subschemes of length n on X over T. If T is the spectrum of a field, we will write X In] for Hilbn(X) = Hilbn(X/spec(k)). X["] is the Hilbert scheme of subschemes of length n on X.

HilbP(X/T).

If U is a locally noetherian T-scheme, then Tlilbn(X/T)(U) is the set closed subschemes Z C X XT U

Z is flat of degree n over U}.

In particular we can identify the set X['q(k) of k-valued points of X In] with the set of closed zero-dimensional subschemes of length n of X which are defined over k. In the simplest case such a subscheme is just a set of n distinct points of X with the reduced induced structure. The length of a zero-dimensional subscheme Z C X is dim~H~ Oz). The fact that Hilbn(X/T) represents the funetor 7-lilbn(X/T) means that there is a universal subscheme

Zn(X/T) C X XT Hilbn(X/T), which is fiat of degree n over Hilbn(X/T) and fulfills the following universal property: for every locally noetherian T-scheme U and every subscheme Z C X XT U which is flat of degree n over U there is a unique morphism

f z : U -----* Hilbn(X/T) such that Z = ( l x XT f z ) - I ( Z . ( X / T ) ) . For T = spee(k) we will again write Z,,(X) instead of Zn(X/T).

1.1. The ttilbert scheme

3

R e m a r k 1.1.4. It is easy to see from the definitions that Zn(X/T) represents the functor Zn(X/T) from the category of locally noetherian schemes to the category of sets which is given by

Z,(X/T)(U) { (Z, a)

Z closed subschemes of X x T U, flat of degree n over U, a : U ----+ Z a section of the projection Z

Zn(X/T)(r

] / *U

: Z,(X/T)(U) ----+Z,(X/T)(V); ( z , ~),

(U, V locally noetherian schemes ff : V ~

, ( z • v v,

~0r

U).

For the rest of section 1.1 let X be a smooth projective variety over the field k. D e f i n i t i o n 1.1.5. Let G(n) be the symmetric group in n letters acting on X n by permuting the factors. The geometric quotient X (n) : = X"/G(n) exists and is called the n-fold symmetric power of X. Let ~ . : X n _ _ , X(") be the quotient map.

X (n) parametrizes effective zero-cycles of degree n on X, i.e. formal linear combinations ~ ni[xi] of points xi in X with coefficients ni E *W fulfilling ~ ni = n. X (~) has a natural stratification into locally closed subschemes: D e f i n i t i o n 1.1.6. Let u = ( n l , . . . , nr) be a partition of n. Let

i n l := {(Xl,...,Xn,)

Xl ~.X2 . . . . .

Xni} c X n'

be the diagonal and r

r

x : := I I

c II x"' = x"

i=1

i=1

Then we set

x~(") := + . ( x"~ ) and

:= x!")\ U Here # > u means that # is a coarser partition then u.

4

1. Fundamental facts

The geometric points of X (n) are

x(n)(-k)m(Zni[xi]Ex(n)(-k )

the points xi axe pairwise distinct }.

The X (~) form a stratification of X (n) into locally closed subschemes, i.e they axe locally closed subschemes, and every point of X (n) lies in a unique X (~). The relation between X [~] and X (n) is given by: T h e o r e m 1.1.7 [Mumford-Fogarty (1) 5.4]. There is a canonical morphism (the

Hilbert Chow morphism) y["]

COn : ~ L r e d

)

X(n),

which as a map of points is given by

z

Z xEX

~r y[n]. So the above stratification of X (n) induces a stratification . . . . red" D e f i n i t i o n 1.1.8. For every partition u of n let X In]

: : conl(x(n)).

Then the X[~n] form a stratification of y[n] into locally closed subschemes. .L red For u = ( n l , . . . , nr) the geometric points of X In] are just the unions of subschemes Z1 , . . . , Zr, where each Zi is a subscheme of length ni of X concentrated in a point xi and the xi are distinct.

1.2. The Weil conjectures We will use the Weil conjectures to compute the Betti numbers of Hilbert schemes. They have been used before to compute Betti numbers of algebraic varieties, at least since in [Harder-Narasimhan (1)] they were applied for moduli spaces of vector bundles on smooth curves. Let X be a projective scheme over a finite field F q , let J~'q be an algebraic closure of s and X := X x Fq ~'q" The geometric Frobenius

Fx : X - - + X is the morphism of X to itself which as a map of points is the identity and the map a ~-~ a q on the structure sheaf Ox. The geometric Frobenius of X over F q is Fq := Fx

x

l~q.

The action of Fq on the geometric points X ( F q ) is the inverse of the action of the Frobenius of Fq. As this is a topological generator of the Galois group Gal(F~, Fq), a point x E X ( F r is defined over F q , if and only if x = Fq(x). For a prime I which does not divide q let Hi(X, Q~) be the i th l-adic cohomology group of X and

bi(--Z) := dimq,(Hi(-x, Ql)),

p(Y, z) :=

b,(X)z

e(X) := b~(X) is independent of I. We will denote the action of Fq* on H ~ ( X , Q I ) by F~]Hr(~,Q~). The zeta-function of X over F q is the power series Zq(X't) := exp (n~>o 'X(Fq" )'tn / Here IMI denotes the number of elements in a finite set M. Let X be a smooth projective variety over the complex numbers C. Then X is already defined over a finitely generated extension ring R of 2~, i.e. there is a variety XR defined over R such that X n • n C = X. For every prime ideal p of R let Xp := X n • n R/p. There is a nonempty open subset U C spec(R) such that Xp is smooth for all p E U, and the l-adic Betti-numbers of Xp coincide with those of X for all primes l different from the characteristic of Alp (cf. [Kirwan (1) 15.], [Bialynicki-Birula, Sommese (1) 2.]. If m C R is a maximal ideal lying in U for which R / m is a finite field ~'q of characteristic p r l, we call Xm a good reduction of X modulo q.

6

1. Fundamental facts

Theorem

1.2.1. (Well conjectures [Deligne (1)], c]. [Milne (1)1 , [Mazur (1)1)

(1) z ~ ( x , t ) is a rational ]unction 2d

Zq(X, t) = ~ I Q~(X, t) (-1Y+' r~0

with Q~(X, t) = det(1 - tFr [Hr(~,q,))(2) Q~(X,t) e 2g[t]. (3) The eigenvalues ai,r of Fq*[Hr(~-,q,) have the absolute value tail[ = at~2 with

respect to any embedding into the complex numbers.

Zq(X, 1/qdt) = 4-qe(-x)/2t~(~) Zq(X, t).

(4)

(5) If X is a good reduction of a smooth projective variety Y over C, then we have

bi(Y) = bi(X) = deg(Qi(X, t)).

R e m a r k 1.2.2. Let F(t, sl,... ,sin) e Q[t, sl,... ,sin] be a polynomial. Let X and S be smooth projective varieties over F q such that

IX(Fq.)l

=

F(q", [S(Fq,)[,...,

IS(Fq-,-)t)

holds for all n E ~N'. Then we have

p(X, - z ) = F(z 2, p(-S, - z ) , . . . , p(-S, _zm)). If X and S are good reductions of smooth varieties Y and U over C, we have:

p ( V , - z ) = F ( z 2 , p ( U , - z ) , . . . ,p(V,-zm)).

P r o o f i Let a l , . . . , We put

Then we have

as

be pairwise distinct complex numbers and h i , . . . , hs E Q.

$

z ( ( a , , h,),) = I I ( 1 - a,) -h, i=1

1.2.

The Well conjectures

7

So we can read off the set of pairs {(al,hl),...(as,hs)} from the function Z((ai,hi)i). For each c 9 C let r(c) := 21ogq(Icl). By theorem 1.2.1 we have: for a smooth projective variety W over F q there are distinct complex numbers

(ii)~=l 9 C and integers (li)~=l 9 2g such that t

IW(Fq-)l = ~ li!~ i=1

for all n E / V . Furthermore we have r ( t i ) E ~_>0 and

(-1)%(w)= ~

l,

~(~)=k for all k E 2~_>o. Let ill,... ,it E C, for S. Then we have for all n E ZW:

ll,... ,lt

E 2~ be the corresponding numbers

t

IX(Fc)I

"

Let

~51,... , ~r

t

K"~l~n = F [kq n ,~...~ iPi ," "', E l i t i mn) 9 i=1

i=1

be the distinct complex numbers which appear as monomials in q and

the 7i in

(•

F q, "

lifli,...,

i=1

•

Ii!im

9

i=1

Then there are rational numbers h i , . . . , n~ such that

IX(Fqo)l = ~ n,e~ i=1

for all n E SV and

(-i)%(X)= ~

ni

r(~j )=k for all k E 2g>0. We see from the definitions that ~r(6~)=k z k in F(za,p(S,-z),... ,p(S,--zm)). [3

nj is the coefficient of

We finish by showing how to compute the number of points of the symmetric power X (n) for a variety X over F q . The geometric Frobenius F := Fq acts on X(n)('Fq) by

F(Eni[xi])

=Eni[F(xi)],

axtd X ( " ) ( F q ) is the set of effective zero-cycles of degree n on X which are invariant under the action of F.

8

1. Fundamental facts

D e f i n i t i o n 1 . 2 . 3 . A zero-cycle of the form r

E[Fi(x)]

with x 9 X(~b-'q. ) \ U Z(ZWq~ ) j[r

i=0

is called a primitive zero-cycle of degree r on X over Z~'q. The set of primitive zero-cycles of degree r on X over hrq will be denoted by Pr(X, ~'q).

Remark

1.2.4.

(1) Each element ( E X ('0 (~'q) has a unique representation as a linear combination of distinct primitive zero-cycles over F q with positive integer coefficients.

(2) IX(Fq.)l = y ] r . IP~(X, Fq)I tin (3) Zq(X,t) = ~

Ix(")(z~q)l~",

n>O

i.e. Zq(X, t) is the generating function for the numbers of effective zero-cycles of X over s r

P r o o f : (1) Let ( = ~ i = x ni[xi] E X(n)(lFq), where z l , . . . , xr are distinct elements of X(-~q). For all j let {j := En,kj[xi] 9 X(")(Fq). Then we have ( = y ] j (j, and it suffices to prove the result for the {j. So we can assume that ( is of the form ( = ~i~=l [xi] with pairwise distinct xi E X(-~q). As we have F({) = {, there is a p e r m u t a t i o n a of { 1 , . . . , r } with F(zi) = x~(i) for alli. Let M s , . . . , M s C { 1 , . . . , r } be the distinct orbits under the action of g. Then we set r/j := E [xi] iEMj for j = 1 , . . . s . Then ~ = ~j=ls r/j is the unique representation o f ~ as a sum of primitive zero-cycles. (2) follows immediately from the definitions. From (1) we have

Ix(")(Fq)lt" n>O

= I I ( 1 -T_>I

tr)-lP.(X,F,)l

= Zq(X, t). So (3) holds.

[]

1.3. T h e p u n c t u a l

Hilbert scheme

Let R := k [ [ x l , . . . , Xd]] be the field of formal power series in d variables over a field k. Let m = (Xl . . . . ,Xd) be the m a x i m a l ideal of R. D e f i n i t i o n 1.3.1. Let I C R be an ideal of colength n. The Hilbert function T ( I ) of I is the sequence T ( I ) = (ti(I))i>o of non-negative integers given by

ti = d i m k ( m l / ( I A m i + m i + l ) ) . If T = (ti)i>_o is a sequence of non-negative integers, of which only finitely m a n y do not vanish, we p u t ti < (d+i-1).

IT I =

~2 ti. The initial degree do of T is the smallest i such t h a t

Let Ri := m i / r n i+1 and Ii : = ( m I Cl [ ) / ( m i+1 (-I I). T h e n Ri is the space of forms of degree i in R and Ii the space of initial forms of I (i.e. the forms of m i n i m a l degree among elements of I ) of degree i, and we have:

ti(I) = d i m k ( R i / I i ) .

Let I C R be an ideal of colength n a n d T = (ti)i>_o the Hilbert function of I. Lemma

1.3.2.

(1) d i m ( m J / I N m / ) = E ti i>_j holds for all j > O. In particular we have IT] = n.

(2)

m".

P r o o f : Let Z : = R / I , a n d Zi the image of m i under the projection R ~ we have

Z. Then

N Zi = 0 .

i>0

As Z is finite dimensional, there exists an i0 with Zio = O. For such an i0 we have I D m i~ There is an isomorphism

Zj = m J / ( m j N I) ~- ~~vii=j ~ -~t/ of k-vector spaces, and R i / I i = 0 holds for i > i0. If we choose io to be minimal, then R i / I i 7s 0 holds for i < io. So we get (1). If t j = 0 for some j , then I D m j. T h u s (2) follows from Irl = n.

1. Fundamental fact~

10

In a similar way one can prove: Let X be a smooth projective variety over an algebraically closed field k. Let x E X be a point and Z C X a subscheme of length n with supp(Z) = x. Let Iz,, be the stalk of the ideal of Z at X. Then we have n

Iz,, D m x , ~. (Just replace R by Ox,~ in the proof above.)

R e m a r k 1.3.3. As every ideal of colength n in R contains m n, we can regard it as an ideal in R / m ~. Thus the Hilbert scheme H i l b n ( R / m n) also parametrizes the ideals of colength n in R. We also see that the reduced schemes ( H i l b ~ ( R / m k ) ) ~ d are naturally isomorphic for k _> n. We will therefore denote these schemes also by Hilbn(R)~d . Hilb~(R)~d is the closed subscheme with the reduced induced structure of the Grassmannian Grass(n, R / m ~) of n dimensional quotients of R / m ~ whose geometric points are the ideals of colength n of k [ [ x l , . . . , Xd]]/m ~.

Using the Hilbert function we get a stratification of Hilbn(R)red .

D e f i n i t i o n 1.3.4. Let T = (ti)i>_o be a sequence of non-negative integers with ITI = n. Let Z T C Hilbn(R)red be the locally closed subseheme (with the reduced induced structure) parametrizing ideals I C R with Hilbert function T. Let GT C ZT be the closed subscheme (with the reduced induced structure) parametrizing homogeneous ideals I C R with Hilbert function T. Let

PT : ZT

) GT

be the morphism which maps an ideal I to the associated homogeneous ideal (i.e. the ideal generated by the initial forms of elements of I). The embedding GT C ZT is a natural section of PT.

In the case d = 2 i.e. R = k[[x, y]] many results about these varieties have been obtained in [Iarrobino (2), (4)].

D e f i n i t i o n 1.3.5. The jumping index (ei)i>o of (ti)i>_o is given by ei = max(ti-1 ti, 0).

Theorem

1.3.6. [Iarrobino (4), prop. 1.6, thm. 2.11, thin. 2.12, thm. 3.13]

(1) ZT are GT non-empty if and only if to = 1 and ti <_ ti-1 for all i > do (here

again do is the initial degree of T).

1.3. The punctual Hilbert scheme

11

(2) GT and ZT are smooth, GT is projective of dimension

dim(GT) = ~ ( e i + 1)e~+1. (3) PT : ZT ~

GT is a locally trivial fibre bundle in the Zariski topology, whose fibre is an aj~ne space A n(T) of dimension n(T) = n - E (ei + 1)(ej+l + ej/2). j>_do

2. C o m p u t a t i o n

of the Betti numbers of Hilbert schemes

The second chapter is devoted to computing the Betti numbers of Hitbert schemes of points. The main tool we want to use are the Well conjectures. In section 2.1 we will study the structure of the closed subscheme X (-) ['] of X["] which parametrizes subschemes of length n on X concentrated in a variable point of X. We will show that (X(n))r,d is a locally trivial fibre bundle over X in the Zariski topology with fibre Hilb"(k[[xl,... xd]]). We will then also gtobalize the stratification of Hilbr'(k[[xl , ..., Xd]]) from section 1.3 to a stratification of X (~,). ["] Some of the strata parametrize higher order data of smooth m-dimensional subvarieties Y C X for m _< d. In chapter 3 we will study natural smooth compactifications of these strata. In section 2.2 we consider the punctual Hilbert schemes Hilbn(k[[x, y]]). We give a cell decomposition of the strata and so determine their Betti numbers. I have published most of the results of this section in a different form in [G6ttsche (3)]. They have afterwards been used in [Iarrobino-Yameogo (1)] to study the structure of the cohomology ring of the GT. We also recall the results of [Ellingsrud-Stromme (1),(2)] on a cell decomposition of Hilb"(k[[x, y]]) and p~n]. In section 2.3 we compute the Betti numbers of S ['~1 for an arbitrary smooth projective surface S using the Weil conjectures. This section gives a simplified version of my diplom paper [G6ttsche (1),(2)]. The auxiliary results that we prove here will be used several times in the rest of the chapter. We also formulate a conjecture for the Hodge numbers of the S In]. In a joint work with Wolfgang Soergel [G6ttsche-Soergel (1)] it has in the meantime been proved. Independently Cheah [Cheah (1)] has recently obtained a proof using a different method. One can see that the Euler numbers of the S In] can be expressed in terms of modular forms. By the conjecture on the Hodge numbers this is also true for the signatures. In section 2.4 we compute the Betti numbers of higher order K u m m e r varieties KA,,. These varieties have been defined in [Beauville (1)] as new examples of CalabiYau manifolds. While for a general surface S only the symmetric group G(n) in n letters acts on S n in a natural way by commuting the factors, there is also a natural action of G(n + 1) on An. KAn can be seen as a natural desingularisation of the quotient An/G(n + 1). To determine the Betti numbers we again use the Well conjectures. One can easily see from the formulas that the Euler numbers of the K A , can be expressed in terms of modular forms. It was shown in [Hirzebruch-HSfer (1)] that the formula for the Euler numbers of the S In] from section 2.3 coincides with the orbifold Euler number e(S", G(n)) of the action of G(n). We show that the Euler number of KA,, coincides with the orbifold Euler number e(A'*, G(n + 1)). As in section 2.3 we formulate a conjecture for the Hodge numbers. From this we also get an expression for the signatures of the KAn in terms of modular forms. In section 2.5 we study varieties of triangles.

As mentioned above X [3] is

smooth for an arbitrary smooth projective variety X.

So we can use the Weil

2. Betti numbers of Hilbert schemes

13

conjectures to compute its Betti numbers. We can view X [3] as a variety of unordered triangles on X. From X [3] we can construct several other varieties of triangles on ~ 3

X. The variety Hilb (X) of triangles on X with a marked side has been used in [Elencwajg-Le Barz (3)] in the case of Z = P2 to compute the Chow ring of p~3], and the variety H 3 ( X ) of complete triangles on X has been studied in detail in [Roberts-Speiser (1),(2),(3)], [Collino-Fulton (1)] for X = P2. For general X it has been constructed in [Le Barz (10)]. There is also a new functorial construction by Keel [Keel (1)]. We will construct two additional varieties of triangles. We show that they are smooth and study maps and relations among the triangle varieties. Then we use the Well conjectures to compute their Betti numbers.

14

2.1. T h e l o c a l s t r u c t u r e

o f X In] (,0

Let k be a (not necessarily algebraically closed) field and X a smooth quasiprojective variety of dimension d over k. In this section we s t u d y the structure of the s t r a t u m (X(n))~d which parametrizes subschemes of X which are concentrated in a (variable) point in X .

D e f i n i t i o n 2.1.1.

Let X be a smooth projective variety over a field k. Let A C

X x X be the diagonal a n d Z A / x x x

its ideal. Let A n C X x X be the closed

subscheme which is defined by Z~x/x xX" Let

pl,P2 : X • X

~X

be the projections and /51,/52 the restrictions to A n.

The (n - 1) th jet-bundle

Jn-1 ( X ) of X is the vector bundle associated to the locally free sheaf J,_,(X)

:= ( p 2 ) , ( O ~ o )

on X . More generally let Z ~ V A . be the ideal sheaf of A i in A n and J / _ a ( X ) be the vector bundle associated to Jn'_,(X) := for all i < n -

(p2),(ZA,/A.)

1.

We see t h a t the fibre

Jn_l(X)(x)

of J n - a ( X ) over a point x e X can be identi-

fied in a n a t u r a l way with Ox,~/mnx,x and similarly

Jn_l(X)(x)

with m xx, J m . x~, n

We have

Symi(T; ). H i l b n ( A n / X ) is a locally closed subscheme of Hilbn(X • X / X )

= Hilbn(X),

and there is a n a t u r a l m o r p h i s m r : Hilb~(An/x) ~

Lemma

2.1.2.

Hilb'~(An/X)r~d :

X.

YX (n))r~d ~ In] ~ as subschemes of X In] and 7r :

(n))~d ---* X is given by mapping a subscheme of length n which concentrated is (X['q in a point to this point.

2.1.

The local structure of X (n) ['q

15

P r o o f i Let k be an algebraic closure of k a n d X - := X x k k . Let Z C X be a subscheme of length n of X concentrated in a point, Iz its ideal in the local ring Ox,~ and m x , , the maximal ideal of Ox,~. Then we have Iz D m ~X , x (cf. 1.3.2). So we see that Hilb"(An/X)red and ~X ['q' t (n))r~d are closed subschemes of X [~] with the reduced induced structure, which have the same geometric points. Thus they are equal. The assertion on 7r follows directly from the definitions. D Let Grass(n, Jn-l(X)) be the Grassmannian bundle of n-dimensional quotients of J n - ~ ( X ) let and # : Grass(n, J n - l ( X ) ) ~ X be the projection.

L e m m a 2.1.3.

There is a closed embedding ~ : Hilb"(A"/X)r~a

, Grass(n, Jn-1 (X))

over X.

P r o o f : Let

Z,(A"/X) C A " x x H i l b " ( A " / X ) be the universal family (cf. 1.1.3) and let t52 : A " x x H i l b " ( A n / X ) ~

Hilbn(An/X)

be the projection. Then we have

(#2),(Oa.x,:Hilb.(a./x)) -- ~r*(Jn-l(X)). As Zn(An/X) is flat of degree n over Hilb~(A~/X), ([~2),(Oz.(A./x)) is a locally free quotient of rank n of 7r*(d~_a(X)). Thus it defines a morphism i: Hilbn(An/X)

,

Grass(n, dn-l(X)).

So we also get a morphsim 3: H i l b " ( A " / X ) r e d ~

Grass(n, Jn-l(X)).

Let T be the tautological subbundle of corank n of fr*(dn_l(X)). We abreviate Grass(n, Jn-l(X)) by Y. r is in a natural way an Oy-algebra. Let Q be the quotient of ~c*(ffn_l(X)) by the subalgebra generated by T. Q is a coherent sheaf on Y. For all x in Y let

q(x)

:=

dimk(G •

x Oy,./my,.)

2. Betti numbers of Hilbert schemes

16

be the rank of Q at x. From the definitions we see that q(x) <_ n holds for all x 9 Y. Let H C Y be the closed subscheme with the reduced induced structure, for whose points q(x) = n holds. T h e n we see

~(Silbn(A~/X)r~d) C H. Let ~" : H ~ 152 : h "

X be the restriction of the projection. Let A n := A n x x H

~ H be the projection. T h e n we have (/32),(Os

and

= # * ( J n - ~ ( X ) ) . As

/~2 is an homeomorphism, we can view Q as a quotient of OA- i.e. as the structure sheaf of a subseheme Z o f / ~ n , which is flat of degree n over H.

This defines a

morphism

j: H ~

Hilbn(An/X)~r

From the definitions it is clear that j is the inverse of i.

[]

For the rest of the section we want to assume in addition that there are an open cover (Ui)i of X and local parameters on each of the Ui defined over k. Let R := k[[x~,..., xd]], m := ( z ~ , . . . , xd) be the maximal ideal of R and let Hilb'~(R)~d be the Hilbert scheme parametrizing ideals of colength n in R / m ~ (cf. 1.3.3).

L e m m a 2.1.4. 7r : '~X (In] n ) )~r e d ------+ X i~ a locally trivial fibre bundle in the Zari~ki topology with fibre H i l b n ( R ) ~ d . Proof:

Let U C X be an open subset and Y l , . . . Yd local parameters on U. For

each g 9 k [ [ x l , . . . ,Xd]]/m" let

(P2),(PT(yd)-/5~(yd))) 9 r(Jn-~(U)).

Y : : g((f2),(~7(yl) -/5~(yl)),..., We see that the ~" are a basis of

Jn-l(U) in each fibre.

Thus there is an isomorphism

R / m n | Ou ~- J n - l ( U ) and so also an isomorphism e n : U x G r a s s ( n , R / m n) ~

Grass(n, J,~_, (U)).

We see that the image of U under en

x

Hilb"(n)r~d

is ~'-I(U) = ~ t< { t u [ n(n))red)" ],, :

an isomorphism r : U

x

H"J b n ( R ) r e d

C

U

x

G r a s s ( n , n / m n)

So the restriction of en to U x Hs9 n ( R ) ~ d is

] d. ~ ,Ub < (n)]re

[]

We can globalize the stratification of Hilbn(R)~,d to a stratification of

(Xtn], (n))red"

2.1. The local structure ~J ~ -~(~) Y[~]

D e f i n i t i o n 2.1.5. For

i = 1,...,

17

n -

~)i: J / _ l ( X ) ~

1

let

J~(X) ~- S y m i ( T * X )

be the canonical map. Let T = ( t o , . . . , t ~ - l ) be a sequence of non-negative integers. Let ~ : Grass(n, J,~-i (X)) , X be the projection as above. Let T be the tautological subbundle of ~r*(Jn-l(X)). For all i let Q, := ~*( J~_I(X))/(T n #*( Ji_l(X)) + ~ * ( J / + I ( x ) ) ) . Let WT C Grass(n, Jn-~ (X)) be the locally closed subscheme over which the rank of Qi is ti for all i. Let

zT(x) =

[~] ~-I(wT) c (x(~))~d

with the reduced induced structure. Let rrT : ZT(X) --~ X be the projection. Qilzr(x) is a quotient bundle of rank ti of ~@(Symi(T~()). Let Ti be the tautological subbundle on Grass(ti, Symi(T~). Let

7r1: H G r a s s ( t i , Symi(r~() ----+ X i be the projection and

V~(X) c H aTass(tl, Sym'(T~) i the closed subvariety over which

T~. ~;(T~) r T~+I holds for all i. Here T1- 7r~'(T~) denotes the image of 7'1 | ~r~(T~) by the natural vector bundle m o r p h i s m 7r~(Symi(T~) | T~) ~

Tr~(Symi+a(T~:)).

Let

pT(x):zv(x)

, cv(x)

be the morphism defined by the bundles Qi]ZT(X). Analogously to the proof of l e m m a 2.1.4 we can easily see:

Remark

2.1.6.

2. Betti numbers of Hilbert ~cherne~

18

(1)

Z T ( X ) and GT(X) are locally trivial fibre bundles over X with fibres ZT and GT respectively.

(2) W i t h respect to local trivialisations

ZT(U) ~ U X ZT,

GT(U) ~- U X GT over an open subset U C X we have pT(U) ~- 1u • PT.

Remark

2.1.7.

We can see from the definitions t h a t for all l _< d = dim(X) and

all s 9 JTV

G(1,I,(t+I ) ..... (t-I-;-,

Z(1,/,(,+l ~ , .-. ........('+;-1]](X)

))( X) =

Grass(l, Tfc ). '

.

is a locally trivial fibre bundle with fibre A r over

G(1,t,(,+l ) .....( , + : _ , ) ) ( X ) . Here r : - ( d - l ) ( ( t l s

) -l-

1).

P r o o f : By r e m a r k 2.1.6 we have to prove this only for

a(,,,,(,+,) .....(,+:_,)),Z(,,,,(,1,) .....(,+;_,)) c H i l b ( ' + ' ) ( R ) . The assertion forG(,,t,(,+,~,,, ..... (,+:_,)) is obvious. Now let Z 9 Z(,,t,(,+, ~ ,, ,

('+7')) and let Iz be the ideal of Z. Then there are y t + l , . . . , yd in R such that I z is given

by Iz = (Yt+l,... ,Ya) + m ~+1. T h e initial forms ui of the yi all have degree 1 a n d are linearly independent. We can assume t h a t x 1, 99 9 xl, ul+ 1, 9

u d are linearly

independent. We can modify the Yi to be of the form

Yi : Ui + f i ( X l , . . . , X l ) . The f i ( x l , . . . , xt) can be a r b i t r a r y polynomials in x l , . . . , xt of degrees < s, whose i n i t i a l forms hasve degree > 2. Thus the result follows.

Remark

2.1.8.

[]

Of p a r t i c u l a r i m p o r t a n c e is the s t r a t u m Z(1 ..... 1)(X) C X (~)['q It

is an open subvariety of X (n)['q It is however in general not dense in X (n) ["] if d -> 3 a n d if n is large. By the definitions it p a r a m e t r i z e s subschemes of X which are c o n c e n t r a t e d in a point z a n d lie on (the germ of) a s m o o t h curve t h r o u g h z. We will therefore also write X (n),c I'q instead of Z(1 ..... 1)(X). By r e m a r k 2.1.7 Y['q ~'(n),r is a locally trivial A ( d - O ( n - 2 ) - b u n d l e over P(Tx).

19

2.2. A cell d e c o m p o s t i o n o f p~n], Hilbn(R),

ZT '

GT

Let k be an algebraically closed field. In this section we review the methods of [Ellingsrud-Str0mme(1)] for the determination of a cell decomposition of p~n] and modify them in order get a cell decomposition and thus (for k = C) the homology of the strata Z T and GT of Hilbn(k[[x, y]]). Let R := k[[x, y]]. Let Hilbn(A 2, 0) be the closed subscheme with the induced reduced structure of (A2) In] parametrizing subschemes with support {0}. By lemma 2.1.4 we have Hilbn(A 2, 0) ~ Hilb~(R)r~d . In [Ellingsrud-Stromme (1)] the homology groups of p~n] A~,q and Hilbn(A 2, 0) are computed by constructing cell decompositions. We review some of the results and definitions on such cell decompositions. For a complex variety X let H . ( X ) be the Borel-Moore homology of Z with 2g coefficients. For each i let bi(X) = r k ( H i ( X ) ) be the i th Betti number and e ( X ) = ~-~(-1)%i(Z) the Euler number. Let A m ( X ) be the mth Chow group of X and cl : A . ( X ) , H . ( X ) the cycle map (cf. [Fulton (1), 19.1]). For X smooth projective of dimension d we put A m ( x ) = A d - m ( X ) . Definition 2.2.1. Let X be a scheme over a field k. A cell decomposition of X is a filtration X = X n D Xn--1 D ... D X o D X - 1 = 0

such that Xi \ Xi-1 is a disjoint union of schemes Ui,j isomorphic to affine spaces A n~,j for all i = 0 , . . . , n. We call the Ui,j the cells of the decomposition. P r o p o s i t i o n 2.2.2. [Fulton (1) Ex. 19.1.11] Let X be a scheme over C with a cell decomposition. Then (1) H2i+l(X) = 0 for all i. (2) H2i(X) is the free abelian group generated by the homology classes of the closures of the i-dimensional cells. (3) The cycle map el: A . ( X ) ~

g.(x)

is an isomorphism.

Ellingsrud and Str0mme have constructed the cell decomposition of p~n] using the following results of [Bialynicki-Sirula (1),(2)]. Let Z be a smooth projective variety over k with an action of the multiplicative group Gin. We will denote this action by ".'. Let x E X be a fixed point of this action. Let T+x,z C Tx,~ be the linear subspace on which all the weights of the induced action of (~,, are positive. T h e o r e m 2.2.3. [Bialynicki-Birula (1),(2)] Let X be a smooth projective variety over an algebraically closed field k with an action of Gm. Assume that the set of

20

2. The Betti numbers of Hilbert schemes

fixed points is the finite s e t {Xl,... , X m ) . For all i = 1 , . . . , m let Xi:={xeX

I l i m t . x = xi}. t~O

Then we have:

(1) X has a cell decomposition, whose cells are the Xi.

(2)

T x , , x , : T X,xi + "

For non-negative integers n k l we denote by p(n) the number of partitions of n and by p(n, l) the number of partitions of n into l parts. This n u m b e r coincides with the number of partitions of n - 1 into numbers smaller or equal to I. The main result of [Ellingsrud-Str0mme (1)] is: Theorem

2.2.4. [Ellingsrud-Str0mme (1)]

(1) For X = p~n], X = A~n] and X = Hilb'~(A 2, 0) the following holds: X has a cell decomposition. In particular i l k = C the cycle map cl : A , ( X ) ~H,(X) is an isomorphism, H 2 i + I ( X ) = O, and the H 2 i ( X ) are free abelean groups. If k

(2)

=

C the Betti numbers are

b21(P~hI) =

~

~

p(no,no - ko)p(nl)p(n2,k2 - n2),

noq-nl q-n2=n ko+k2=l-nl

(3)

b2,(n~ nl) = ;(~, l - ~), b2t(Hilbn(A 2, 0)) = p(n, n - l).

We will briefly review the ideas of the proof in [Ellingsrud-Strcmme (1)]. Let To, T1, T2 be a system of homogeneous coordinates on P2- Let G C Sl(3, k) be the maximal torus consisting of the diagonal matrices. Let A0, A1, A2 be characters of G such that all the g ~ G can be written as g = diag(~o(g),Al(g),A2(g)).

G acts on P2 by g 9 T / = Ai(g)Ti. The fixed points of this action are eo -- (1,o,o), P1 = (o, 1,o),

P2 =(o,o,1).

2.2. A cell decompostion of P~n], Hilbn(R), ZT, a T

21

The action of G on P2 induces an action of G on p~n], as G acts on the ideals in k[To, T1, T2]. Z E p~n] is a fixed point if and only if the corresponding homogeneous ideal I z C k[To,T1,T2] is generated by monomials. So the action on P~'q has only finitely m a n y fixed points. Let X be a smooth projective variety over k with an action of a torus H which has only finitely m a n y fixed points. A one-parameter subgroup 9 : Gm ~ H of H which does not lie in a finite set of given hyperplanes in the lattice of one-parameter groups of H will have the same fixed points as H. In future we call such a oneparameter group "general". Thus the induced action of a general one-parameter group q~ : G m ----* G has only finitely many fixed points on P~'q. Let 9 : Gm -----+ G be a general one-parameter group of the form @(t) = diag(tW~ with w0 < wl < w2 and w0 + wl + w2 = 0. Let F0 := {P0}, L C P2 the line T2 = 0, /;'1 := L \ P 0 e F2 := P 2 k L . Then ~5 induces the cell decomposition of P2 into F0, F1, F2. Ellingsrud and S t r c m m e apply theorem 2.2.3 to the induced Gm-action on P~]. We will modify their arguments in order to obtain a cell decomposition of the strata Z T of Hilbn(R). We denote by "." the action of Gm on p~nl induced by q~. As it has only finitely m a n y fixed points, it gives a cell decomposition of p~n]. Hilbn(R)~d = Hilbn(A 2, 0) C P~] is the subvariety parametrizing subschemes Z of colength n with support s u p p ( Z ) = {P0}. If Z e P~'q has support {P0}, then suppQin~( t . Z ) ) -- t--.olim(t . supp( Z ) ) = {P0}. If s u p p ( Z ) 7~ {P0}, then we have

upp(Z)) r (P0}.

z)) = So Z e Hilbn(A 2, 0) r

lim(t 9 Z) 9 Hilbn(A 2, 0).

So by theorem 2.2.3 Hilbn(A 2, 0) is a union of cells of the cell decomposition of P~'~] which belong to fixed points in Hilbn(A2,0). In particular Hilb'~(A2,0) has a cell decomposition. Using the identification := T I l T o ,

y :=T21To, R := \[Ix, y]] we have Hilbn(A 2, 0) = Hilbn(R)r~d . We identify the points of Hilbn(R)r~d with the ideals of colength n in R. The action of Gm on R and thus on Hilb'~(R) is given by t . x = twl-W~ t " y = tw~-W~

2. The Betti numbers of Hilbert schemes

22

Let I E Hilb"(R) be a fixed point. Then I is an ideal of colength n in R which is generated by monomials. Following Ellingsrud and Str~mme we put

aj

:=

rain{1 I xJY I E I}

for every non-negative integer j. Let r be the largest integer with ar > 0. Then (a0,. 99 at) is a partition of n, and y , 0 x y , l . . . , xr+l are a system of generators of I. So there is a bijection between the cells of Hilbn(R) and the partitions of n. In particular the Euler number of Hilbn(R) is p(n). Let T be the tangent space of Hilb"(A 2) in the point corresponding to I. Let F be a two-dimensionM torus acting on R by t. x = ~(t)x, t.v

= #(t)y

(here t e P and A, # are two linearly independent characters of r). We also denote by ,~ and # the corresponding elements in the representation ring of F. By [Grothendieck (1)] there is a F-equivariant isomorphism

T ~- gomn(I, R/I). Ellingsrud and Strcmme consider the corresponding representation of P on T. They get: L e m m a 2.2.5. In the representation ring of F there is the identity

aj -1

T =

( )~i--j--l #al--s--1 _]_ )~j--i #s--al ). E E O<_i<j<_rs=ai +t

We give a simple proof of this result: The lemma says that T has a basis of common eigenvectors to F with the eigenvalues as in the above formula. By

E.

2(aj-aj+l)=2 E

O
ai=2n=dim(T)

O
it is enough to give such linear independent eigenvectors. For f E R let [f] be the class in R/I. An R-homomorphism r : I ~ R / I is determined by its values on the xiy a~. They must however be compatible. It is easy to see that necessary and sufficient conditions for this are

r r

= [x]r = [Vo,-1-a,]r

2.2. A cell decompostion of p~n], Hilb~(R), ZT, GT

23

Let 0 < i < j < r and aj > s >_ aj+l. Let

[xJ+l-iy s+a'-a'] r

: I ----+ R / I ; xly ~ '

)

0

if l _< i, otherwise.

We can see immediately that the compatibility conditions are fulfilled and r a common eigenvector of F to the eigenvalue AJ - i # ~-a~. Let0<_~<~_r+landag-a~_

l+aT_<~
r ~ ~ : I - - ~ R / I ; x ' y ~' ,

5. We put

, {~x3+'-~Y a + ~ - ~ ]

' '

is

if/>_~, otherwise.

r is an eigenvector to the eigenvalue ~)-~#~-a~. The eigenvectors constructed this way are obviously linearly independent. The result follows by the substitution s:=~-a~

+a~

j:=~-I i:----~.

[]

We now formulate our result on the cell decompositions of Z T and G T in a form which has been influenced by [Iarrobino-Yameogo (1)]. In particular the formula for the Betti numbers of GT does not follow immediately from my original formulation. In [Iarrobino-Yameogo (1)] two combinatorical formulas are shown in order to derive this formula from my original one in [G6ttsche (4)]. Here we will give a direct proof. D e f i n i t i o n 2.2.6. Let (~ = (a0,... ,aT) be a partition of n. The graph of o~ is the set F ( a ) = {(i,/)E2g~_0 i ~ r , l < a i } . Picturally we can represent F(a) as a set of points, one point in position (i,j) for each ( i , j ) E F(a). The dual partition & = ( ~ l , . . - , ~ a 0 ) is the partition, whose graph is F(a) with the roles of rows and columns switched. The diagonal sequence is T ( a ) = ( t o ( a ) , . . . ,tl(a)), where

So it is the sequence of numbers of points on the diagonals of F(a). Let (u, v) E F(a). Then the hook difference h~,v(a) is

24

2. The Betti numbers of Hilbert schemes

I.e. hu,v(a) is the difference of the number of points in F(a) in the same column above (u, v) and the number of points in the same row to the left of (u, v). So we have h~,~(~) = (i~, + v - a~ - u.

For the partition c~ = (6, 3, 2) we get for instance the diagram

for F(c~) and & = (3,3,2, 1, 1, 1), T(c~) = (1,2,3,3, 1, 1). The hu,v(c~) are given by -1 --1 --3

0 0 --2

0 -2

-2

-1

O.

Theorem

2.2.7. Let T = (ti)i>_o be a sequence of non-negative integers with ITI = n. Then we have for X = GT and X = Z r : (1) X has a ceil decomposition.

I f k = C, then cl : A . ( X ) isomorphism and H . ( X ) is free.

~

H.(X)

is an

In case k = C we have for the Betti numbers:

(2)

b2,(Zr)=

(3)

b2~(Gr)=

a9

i{(u,v) eV(a)lhu,v(a)e{o,

{aEP(n)

T(a)=T;

1}}l=n_i

I{(u,v) E F ( a ) l h , ~ , ~ ( a ) = l } I = i } .

In particular the Euler numbers are =

=

9 P(n) I

: T}.

R e m a r k 2.2.8. In [Iarrobino (2),(4)] it has been shown that ZT and GT are none m p t y if and only if to = 1 and ti <<_ ti-1 for a l t i >_ d(T). If T = ( 1 , . . . , 1),

2.2. A cell decompostion of p~n], H i l b ~ ( R ) , ZT; G T

t h e n Z T is an A n - 2 - b u n d l e over GT = P1.

25

It is easy to see t h a t to(a) = 1 a n d

ti(a) <_ t i - l ( a ) for all i >_ d(T(a)) for each p a r t i t i o n a of n. If T = ( 1 , . . . , 1) the cell d e c o m p o s i t i o n of ZT of t h e o r e m 2.2.7 consists of one cell of d i m e n s i o n n - 2 a n d one of d i m e n s i o n n - 1 and t h a t of GT of one cell of d i m e n s i o n 0 a n d one cell of d i m e n s i o n 1 as e x p e c t e d .

As a b o v e let w o

(P:Gm-----4G; t ~ d m g ( t 9

,t W l ,t w 2 )

be a general o n e - p a r a m e t e r s u b g r o u p of G w i t h w0 < wl < w~ a n d w0 + W l + w 2 = 0. We also r e q u i r e t h e i n e q u a l i t y

n ( w l - wo) > (n -

l)(w2 - wo).

We consider the i n d u c e d G m - a c t i o n on H i l b ~ ( R ) . We k n o w a l r e a d y t h a t it gives a cell d e c o m p o s i t i o n of H i l b n ( R ) . Let T = (ti) be a s e q u e n c e of n o n - n e g a t i v e integers w i t h ITI = n.

Lemma

2.2.9.

( l ) ZT is it union of cells of the cell decomposition of H i l b ' ( R ) .

(2) PT : ZT

~ GT is equivariant with respect to the Gin-action.

(3) The Gin-action induces a cell decomposition of GT. Its cells are the intersec-

tions of the cells of ZT with GT. Proof:

Let I be an ideal in R w i t h Hilbert f u n c t i o n T. Let j E f g , s : = j + 1 - tj.

Let Ij be t h e space of initial f o r m s of degree j in I. We p u t J : = l i m t . I. t~O

For all i let Ji be t h e space of initial forms of degree i in J . Let T I = (t~)j_>0 be t h e H i l b e r t f u n c t i o n of J . C h o o s e f l , 99 9 f~ E I such t h a t t h e i r initial f o r m s g l , . 9 9 g~ are a basis of Ij. By r e p l a c i n g t h e fi by s u i t a b l e linear c o m b i n a t i o n s we can a s s u m e t h a t t h e gi are of t h e f o r m

gi = xl(i)Y j-l(i) ~- E

gi,mxmy j - m

rn>l(i) w i t h gi,m E k a n d t h a t l(1) > /(2) > . . . > l(s).

WO~Wl ~W 2

we

By t h e choice of t h e weights

get lim ~ ( t ) - (tt(i)(w~176 t~O

fi) = xl(i)y j-l(i).

2. The Betti numbers of Hilbert schemes

26

So the span of the xl(i)y j-l(i) is contained in Jj. So we have

t'i = i + 1 - dim(Ji) <_ tj. B y IT'I = n w e h a v e T = Z ' and thus (1).

(2) follows immediately from the definitions. GT is a smooth projective variety. I f I E Gr , then we have e 2 ( t ) . I E G:r for a l l t E Gin. S o G m acts on GT w i t h a finite number of fixed points and we can apply theorem 2.2.3. As the action on GT is the restriction of that o n ZT, (3) follows. E To determine the Betti numbers of Z T and GT we have to find out, which of the Gm-invariant ideals of R lie in ZT and what the dimensions of the corresponding cells of ZT and GT are. Let a = (a0 . . . . , at) be a partition of n and I the ideal of R generated by yaO, xy~l . . . , xr+l. L e m m a 2.2.10. For the HiIbert function T(I) of I we have T(I) = T(~). P r o o f i Let T(I) = (ti)i_>0. The monomials xiy I with i + l = j and l > ai form a basis of the space Ij of homogeneous polynomials of degree j in I. So we have:

tj--j+l=

{(i,/)E2g~_ 0 l i + l = j ,

l>_ai}

{(i,j) e r ( ~ ) l i + j = l}

= tj(.).

[]

Let again T be the tangent space of Hilbn(A ~) in the point corresponding to I.

The dimension of the subspace T + of T on which the weights of the action are positive is

L e m m a 2.2.11.

dim(T + )=n-

{(u,v) e V ( ~ ) ] h u , v ( ~ ) = 0

orh~,v(a)=l}

Proof." We apply lemma 2.2.5 to r -- G and AI

A2

Then we have for every character A~# b of G: (),a~b)(,~(t))

= to(,~,-~,o)+b(w,-wo).

.

2.2. A cell decompostion of P~'q, Hilb~(R), ZT, GT

27

By the choice of w0, wl, w2 the action of Gm has a positive weight on ~ # b , if and only if a + b > 0 or a + b = 0 and b > 0. Let i , j be integers satisfying

O < i < j < r , aj+l < s < aj. The weight of (,V-3-1 # ~ - ~ - 1 ) o ~ is positive, if and only if i + ai > j + s + 1, and the weight of ~ j - i ~ s - a i is positive, if and only if i + ai < j + s. From the definition we see that/z, is the smallest j satisfying s > a j, so/z, - 1 is the smallest j satisfying S > a j + l . So we have

E

dirnT + =

( aj - aj+ 1 -

{ s E 2~

O<_i<_j<_r

0 < j aJ+z + s - <--s
=

9

o

<

-

< z}l,

-

[]

Let To C T be the tangent space of GT in I. It is easy to see that the isomorphism T ~ H o m R ( I , R / I ) maps To to the space of degree-preserving homomorphisms in H o r n n ( I , R / I ) . In the representation ring of F the subspace To can be written as the sum of all terms in the representation of T with a + b = 0. Let To+ C To be the linear subspace on which the weights of the action are positive.

L e m m a 2.2.12.

d i m ( T + ) = {(u,v) C F(a) l hu,v = - 1 } , d i m ( T o / T + ) = {(u,v) E F(c~)[ h~,, = 1}.

Proof." Let i , j be integers satisfying

O < i < j < r , aj+l < s < a j . If i - j - 1 + ai - s - 1 = O, then the weight of ( A i - j - l # a ~ - 8 - , ) o ~ is positive. If j - i + s - ai = 0, then the weight of (M-i#s-a~)o~5 is negative. So we have

dim(T+)=

E

I{sE2~[aj+l

<-s
i-J+ai-s-2=O}l

O<_i<j<_r

= E o<_i
{sE2g

O<_s
i+ai--s--&,=l}

28

The Betti numbers of Hilbert schemes

2.

and

dim(T~

Z

I{sc2ZlaJ+, <-s
O
O<:i
By putting things together that theorem 2.2.7 1s proved.

R e m a r k 2.2.13. We can now easily determine the dimensions of they are both smooth. From lemma 2.2.12 we have:

GT and ZT, as

dim(GT(~)) = {(u,v) ~ F(c~) [h,,v(C~)l = 1} . Let T1 be the tangent space of ZT(~) in I. The isomorphim T ~ Hom•(I, R/I) maps T1 to the space of homomorphisms which preserve or increase the degree. So T1 can be written as the sum of the terms A,pb in the representation of T for which a + b > 0. In addition to the terms occuring in T + these are exactly the AJ-iy -a~ withj+s-ai-i=0. So we get:

dim( ZT ) = dim(T1) =dim(T+)+

Z {sE2~ O
ai+l<-s
=d/re(w+)+

Using theorem 1.3.8 we get for each partition c~ of n the combinatorical formulas:

{(?A,Y) ~ r(oL)

Ihu,v(O~)] -_

1} :

Z

(ci(T(o~)) -~

1)ei_l_l(T(o:,))

i>_do(T(c~))

{(u, v) E F(a)

h,,v(a) = 0} =

Z

ei(T(a))(ei(T(a)) + 1)/2.

i>_do(T(a))

Here (ei(T(a)))i>o is the jumping index and do(T(a)) the initial degree of T(a) (cf. definitions 1.3.5 and 1.3.1). In [Iarrobino-Yameogo (1)] these two formulas are proved eombinatorically.

29

2.3. C o m p u t a t i o n of the B e t t i n u m b e r s of 5:[~] for a s m o o t h surface 5: We want to use the Weil conjectures to compute the Betti numbers of S ["] for a smooth projective surface 5: over C. Let X be a smooth projective variety of dimension d over a field k. Let R = k [ [ x l , . . . , xd]]. We denote Vn := H i l b ~ ( R ) ~ a . We denote by

len(Z1) the length of a subscheme Z. For subschemes Z1, Z2 C X we

will write Z1 C Zz if Zl is a subscheme of Z2 (the same also if Z1 E Hilb ~ ( R ) ~ d and Z2 E Hilb~2(R)~d). For

(zl, Zl c (x •

(z2, we write

(x •

(xl,Z1) C (xz,Z2), if :cl = z2 and

Z 1 C Z2.

For a p a r t i t i o n u =

( n l , . . . , n~) we also write

I~,2~,...), where

ai is the n u m b e r of s u m m a n d s i in u. Let [a[ := E ai. i

Let

P(n) be the set of partitions of n.

We will assume for the following that there exist a finite open cover ( U i)i=l ~ of X and local p a r a m e t e r s on each of the Ui, defined over k.

Remark

2.3.1.

There is a sequence of bijections r

c o m m u t i n g with the action of the Galois group

Zl C Z2 "r

: Xl:l(k ) ~

( X • V,,)(k),

Gal(k, k) such t h a t

~len(Z1)(Zl) C ~)len(Z2)(Z2).

~[n] Proof: For i = 1 , . . . ,s let 7ri : r/U kt i)(n))~ea ----4 Ui be the restriction of the projection 7r : ttX["] ( n ) )~red -----+ X from l e m m a 2.1.2. By l e m m a 2.1.4 there are isomorphisms / U i)(n))red d n] ~ t : t~,~ ----4 Ui x Vn over

Ui for all n E 2V. Thus we have the required bijeetions r

:= aS~(k). For all

j=l,...,slet

w

:=vj\Uv i<j

['q (k~J there is a unique index For e a c h Z E X (n) put C n ( Z ) : =

r

--1 i(Z) such t h a t W C 7ri(z)(W,(z)). We

The result follows, as all the r

are bijective.

[]

2. Betti numbers of Hilbert schemes

30

D e f i n i t i o n 2.3.2. For any partition v = ( h i , . . . , n~) = (1 ~ , 2~%...) of n let

22:=(~,,,

•

.

..... , ) ) c xIo,l u,,) •

•

. x

u , ) = ~ H(xl:l)o, i

The symmetric group G(n) acts on X2 via its quotient

a(o~) := a ( , ~ ) x . . . • C(o,~,

)

by permuting the factors X (hi) [~i] with the same ni. Lemma

2.3.3.

There is a natural morphism r

:2n ~

X In], which induces a

bijection r

: X 2 ( k ) / a ( n ) --~ X~l(-~)

commuting with the action of GaI(k, k ). P r o o f : Let T be a noetherian k-scheme and let (Z~,..., Z~) C

r

z~)) := Zl

u...

u

22(T).

We put

z~.

This is obviously flat of degree n over T. ~ is compatible with base change, so it defines a morphism Cv : 2 ~ - - ~ X['q. The induced map r of geometric points maps )~2(k) to X[~n](k) and is invariant under the action of G(n). So we have a map

r

X2(k)/a(~) ~

X~J(~).

The image of Z e X[~n](k) is ~ 2 I ( Z ) = [Z1,...,Z~], where Z 1 , . . . , Z ~ are the connected components of Z and [] the class modulo G(n). []

D e f i n i t i o n 2.3.4. For an extension/~ of k we write o~

v(~) := U v,.(~). r=O

Let o be the point corresponding to the empty subscheme i.e. Vo(k) = V0(~) = {o}. For x E V~(Ic) we put len(x) := r. For a map f : X ( k ) ~ V(k) we put

Ion(f) :=

~

len(f(x)).

~x(~) Gal(k, k) acts on these maps by o-(f) := crofoo--1.

2.3. The Betti numbers of S ["1

31

We write fl C f2, if fl(x) C f2(x) for all x 9 x ( k ) . L e m m a 2.3.5. There exists a sequence of bijections

commuting with the action of Gal(-k/k) such that Z1 C Z2 ~

Olen(Z,)(Z1) C Olen(Z2)(Z2).

P r o o f : Let v = ( r t l , . . . ,rtr) be a partition of n, and let Z 9 X[n](k) with ~J~-l(z) = [Zl,...,Zr],

where len(Zi) = hi. We put

O,(Z) := f : X ( k ) ---4 V(k); X where P2 : X x V~ and lemma 2.3.3.

f p2(O,,(Zi)) o

if x = (Zi)red, if x ~ supp(Z),

Vn is the projection. The result follows from remark 2.3.1

D e f i n i t i o n 2.3.6. Now let k be a finite field F q , X a smooth projective variety over ~'q and F the geometric Frobenius of X over _gTq. Let

P(X,~'q) = U P~(X,Fq) r>0

be the set of primitive zero cycles of X over _~q (cf. 1.2.3). A map g : P(X, Fq) V(/Fq) will be called admissible, if g(~) 9 V(1Fq,.) for all ( 9 Pr(X, Fq). Let m (g) :=

~CP(X,Fq) and f

Tn(X, Fq) := / g : P(X,~'q) ---+ V(~q) Tnt(X,~gq), g2 e Tn2(X,J~q) 9 P(X, •q).

For gl

e

we

"1

g admissible with lea(g)= n~.

write gl C g2, if gl(() C g2(~) for all

2. Betti numbers of Hilbert schemes

32

L e m m a 2.3.7. There is a sequence of bijections rn : X [~] for all subschemea Z1, Z2 of X of finite length rten(zo(Z1) C "rlen(z2)(Z2) r

~ Tn(X, F q ) such that

Zl C Z2

and such that for all n C zW the following diagram commutes X["](Fq)

Z~

T~(X,_~q)

IIere g-~ is defined by § : T~(X, F ~ ) ----,

X(~)(~);

g~

~[n] and Wn : ~'red

le~(g(~)) ~,

~

X (n) is the IIilbert-Chow morphism.

P r o o f : Let (

Let

f'

'

Z

len(f(x))[x].

We have to find bijections ~ : N~(X, F q ) -----+T , ( X , Fq) satisfying

g-n(fl) C +n(f2) r

fl C f2,

such that the diagram

x(~)(Fqfl commutes. We choose a linear ordering < on the set X(•q). Then Cn(f) is in a unique way a linear combination 8

6~(f) = ~ a~r i=1

Let f e ;V~(X, Fq).

2.3. The Betti numbers of SD]

33

of distinct primitive zero cycles ~i E P r i ( X , ~ q ) with non-negative integer coefficients a,. For i = 1 , . . . , s tet xi E X ( ~ q ~ ) be the smallest element with respect to ri --1

j

_< satisfying ~i = ~ j = 0 [F (xi)]. Then we have

F ~' (f(xi)) = f ( F ~' (xi)) = f(xi), so f ( z i ) E V(Fq., ). We put r ~ ( f ) : P(X,~Cq) -----+V(I~'q);

f(xi)

I

[ o

~ = ~i for a suitable i, otherwise.

The inverse r~-1 is given as follows: let g C T~(X, Fq). For r E f g and ( P~(X, F q ) , let x(~) C X ( F q ) be the smallest element x C X ( F q ) with respect to r--1 _< with ~ = ~ j = 0 FJ(x) 9 Then we have

r[~(g) = f : X ( F q ) ---+ V ( F q ) ;

vJ(~(~)) ~-~ FJ(g(~)).

Lemma

[]

2.3.8.

n=O

r=l

n=O

P r o o f : For all (i,j) E tar x ZW we put

N(i,j) := {/: P~(X, Fq) ~

V(Fr

Z

len(f(~)) = j } .

Then by definition the number of elements of Tn(X, •q) is

Irn(X, Fq)l =

~ nl+2n~+3na+...=n

[ I N(~,~,) 9 s=l

On the other hand we have

Ivo(Fq)L,r~ i, (x F.), = Z N(r j),r' n=0

j=0

Now let S be a smooth projective surface over s and V~ := Hilbn(R).

Let k := •q, R := k[[x, y]]

34

2. Betti numbers of Hilbert schemes

L e m m a 2.3.9. For all 1 C W there is an mo C SV such thai we have for all multiples M of mo

ISt'q(rr

-

exp

( ~-~ tm L~(~qM~)I~ m 1 qMmtm] -

n=O

modulo

t 1.

rn=l

Proof." Let l E $V. There is an m0 E $V such that for all n G l the cell decomposition of V,Nq from theorem 2.2.4 is already defined over/Fq=0. Let M be a multiple of m0 and let Q qM. Because of the identity : =

II i=1

i)zit n

1 - z i - ' t i - 2-,

n=0 i=0

theorem 2.2.4 implies E

1

IVn(1FQ~)[ff~ =

n=0

"1 - Q~(i

1)tri

modulo t z.

i=1

By lemma 2.3.8 we have:

E

Is[n](FQ)[tn =

n=O

1 - Or(i-1)tri r=l

=exp

modulo t l

i=1

\ i=1

r=l

\i=1

m= 1

exp

h=l

~1--

r~lmr]Pr(S'-~Q)[)Qm('-l)'--m) O t )

For the rest of this section let S be a smooth projective surface over C. We can now compute the Poincar6 polynomial 2n

p(S M, z) = ~

dim(Hi(SM;Q))z i

i=0

of S ['q. Let again P ( n ) denote the set of partitions of n. T h e o r e m 2.3.10.

(1)

p ( S N , z) =

I I P( S (~'), z)z2("-I~D (1 a l , 2 ~ 2 , . . . ) E P ( n )

i=1

2.3. The Betti numbers of S['q

35

or equivalently: (2)

~ p( sI < _ z )t ~ = exp m=l

n~O

n=O

(1

=

z2mtm/

(1 + z2m-]tm)bl(S)(1 + z2m+ltm) bS(S)

fi (3) EP(S[~],z)t"

/32 ~ - -

-

-

z2rn~-27m~=z---2~mtm~b~=Z'~-~-m~2tm)b4(S)

rn=l

P r o o f i Let n 9 zW. Let S be a smooth projective surface over C and So a good reduction of S modulo q. Then (So) ['q is a good reduction of S In] modulo q. By replacing JT'q by a finite extension we can assume that for all h 9 zW I(S0)['q(~qh)l is the coefficient of t n in

exp

m

1 --

qhmtm

m--~l

Now (2) follows by remark 1.2.2. (3) follows from (2) by an easy computation and (1) follows from (2) and the formula of Macdonald [Macdonald (1)]

co dirnll(X) E p(x[n]' Z)tn = IX (1 "~-(--1)i+lzit)(--1)i+lbi(X)" n=O

[3

i=0

C o r o l l a r y 2.3.11. For the Euler numbera we have

(1)

~

~(sC~l)~.

n=O

(2) In particular,

= fI

(1 -

~)-~(~)

k=l

iI 4 S ) ----O, then 4SI<) = 0 for aU ~ 9 ~ .

For S a two-dimensional abelian variety (2) is already known (cf. [Beauville (1), p. 769]). R e m a r k 2.3.12. The Euler numbers of the Hilbert schemes can be expressend in terms of modular forms: let q := e 2'~i~ for r in the upper half plane

H:={zEC Im(z)>O}. Let A ( r ) be the cusp form of weight 12 for Sl2(2g) and r/(r) := A ( r ) '/2a the rkfunction. Then

ql/24 ~ e( S)

~ e(st-l)q- = \ 7(,)] rt~0

2. Betti numbers of Hilbert schemes

36

For a K3-surface we get in particular q

Z e(s[.l)q~ A(~) =

n~O

The Betti numbers bi(S In]) become stable for n > i:

Corollary 2.3.13. Let S be a smoo~h irreducible surface over C. Then

p(S[n],z) _ f i

((1 + z 2 m - 1)(1 _+ z_2m_+_1)) bl(..____s) modulo z n+l.

m=l

(1 --

z2m)b:(S)+l(1 --

Z2 m + 2 )

Proof." Let oo ((1 + z2m-ltm)(1 + z2m+ltm)) bl(S) G(z,t) := (1 - t) ml-I1 (1 -- ~----2t~)(1 - " Z ' ~ - m ~ --Z-2~'-+2~rn)

"

We have to show

P(S[~],z) - a ( z , 1 )

modulo z '~+'.

For a power series f C q[[z,t]] we denote the coefficient of zit j by ai,j(f). We see that ai,j(G(z,t)) = 0 holds for i > j. Let i _< n. By theorem 2.3.10(3) we have:

bi(S In]) =

ai,n

~fi-~t j a ( z , t ) j=0

/

= ~ a,,AC(z,t)) j=0 oo

= Z a, j(a(z,,)) j=O

= ai,o(G(z, 1)).

2.3. The Betti numbers of S ['q

37

T h e H o d g e n u m b e r s o f S [~] One would expect that similar formulas as for the Betti numbers of Hilbert schemes of points also hold for their Hodge numbers. For a smooth projective variety X over C let hP'q(x) := dirnHq(X, ftPx) be the (p,q)th Hodge number and let

h(x, x, y) := ~

P,q

hp,~(X)x%~

The xy-genus of X is given by xy(X) = h ( X , y , - 1 ) . By Hodge theory we have for

the signature ~ig~(X) = ~ ( X ) . Together with WoKgang Soergel I have computed the Hodge numbers of S [hI using intersection homology, perverse sheaves and mixed Hodge modules (cf. [GSttsche-Soergel (1)].) Independently Cheah [Cheah (1)] has recently proven this result by using a different technique, the so-called virtual Hodge polynomials. The result is:

T h e o r e m 2.3.14. oo

(1)

h(S In], z, y) =

Z

(xy) ~-I~l I I h(s(~'), x, y)

(l~l,2~2,...)Ep(n)

i=1

or equivalently

(2)

h(SM,-x,-y)<

= exp

n=0

-m=l

~

=(-;~y~

,

~

oo

(3) E h(S[n]'x'y)~n ~ ~Il-I (t -~-(--1)P+q+lxP+k--lyq+k--l~k)(-1)P+q+lhP'q(S) k=l

n~O

p,q

From this we get:

(4)

(5)

x_~(sM)t

sign(SE~ =

~ = exp

F_, (1"1,2~: ,...)6P(n)

x-~

(s)

(-1)"-I~l[I ~ig~(S("')) i=1

2. Betti numbers of Hilbert schemes

38

or equivalently (6)

E

sigTt(s[n])tn =

n=0

fI ( l

-- tkx~ (--1)ksign(S)/2 (1 -- t2k) -'(s)/:.

k=l \ 1 + t k J

(5) and (6) follow from (1) and (3) using sign(S) = XI(~) and e(S) = X - , ( S ) . Using these results we can also find formulas for the signatures of Hilbert schemes in terms of modular forms. Let again v be in the upper half plane and q = e 2~i~. Let e and 5 be the following functions:

n=l

din

dd

~=-~-anz= l

q~ dI

dd

(cf. [Hirzebruch-Berger-Jung (1)], [Zagier (2)].), and ~ are modular forms for r0(2) of weights 4 and 2 respectively. Both of them play an important role in the theory of elliptic genera. C o r o l l a r y 2.3.15.

~-~ sign(S[.])(_q). = q~(S)/24 ~(w) ~ig"(s) ~=o ~( 2r ) (~ig~(s)+~( s) ) /2

= (q)e(S)/24 For a K3 surface we get in particular

oo n=o

sign( s N ) ( - q ) . = A(T)2/3 qA(2T)I/3 z

q 512e(52

-

6)3/2

"

Proof." We set t := - q in 2.3.14(6). Then we get

~ign(st"J)(-q)" = I I \1 + qk ) n=0

(1 -

q~k)-e(s)/~

k=l ( 1 --

qk)sign(S)

= 11 (1- ~(~S))/2 k=l

= qe(S)/24

~/(v) ~ig"(s)

7i(2~-)(~ig.( s)+~( s) )/2

2.3. The Betti numbers of S In]

39

Using the formulas A(T) = 4096~(62 - ~)~ ~(~)16 _ 64(65 _ ') ,7(2,-)8

(cf. [I-Iirzebruch-Berger-Jung (1)]) we get r/(w) "ig'~(s) = (64(62 ~( 2T )( ,ig.( s)+~( s) ) /~

_

s162

40

2.4. T h e B e t t i

of higher order Kummer

numbers

D e f i n i t i o n 2.4.1.

varieties

Let S be a smooth projective variety over an algebraically

closed field. Let as above w,~ : S [~] ----+ S (') be the Hilbert-Chow morphism. Let A be the Albanese variety of S and a : S ----* A be the Albanese morphism. Let aN : S (n)

) A (~) be the morphism induced by a and let gn : A (n) ~

A be the

morphism which maps a zero-cycle ~-][xi] to its sum ~ xl in the group A. We put /k'Sn-1 = a)nl(anl(gnl(0))).

In the following two cases we want to compute the Betti n u m b e r s of t h e / x ' S n - l : (1) S = A is a two-dimensional abelian variety over C; then a = 1A : A - - ~ A, so we have K A n - 1 = g~l(0). In this case K A n - 1 has been defined in [Beauville (1),(2),(3)]. There it has also been shown that K A n - 1 is a smooth symplectic variety, and thus a new family of symplectic varieties was constructed. /x'A1 is the K u m m e r surface of A. So we can see the K A n - 1 as higher order K u m m e r varieties of A. This is the more i m p o r t a n t case. a

(2) a : S - - - ~ A is a geometrically ruled surface over an elliptic curve A.

Lemma

2.4.2. Let S = A be an abelian surface, or let a : S ----* A be a geometri-

cally ruled surface over an elliptic surface A over C or over 1Fq, where gcd(q, n) = 1. Then K S n - 1

is smooth.

P r o o f i For an abelian surface this has already been shown in [Beauville (1)]. We briefly repeat the argument: let (n) : A

A be the multiplication by n. T h e n we

have the cartesian diagram A • KS,-I

l

,

A [n]

~

n.

[]

(n)

A

1

This is true because the fibre product is {(b,Z) e A • A In] I g n ( w , ( Z ) ) = n . b}, and this is isomorphic to A • K S n - 1 via (b, Z) H (b, Z - b). Here Z - b denotes the image of Z under the isomorphism - b : A ----* A; X ~------+ x - b .

2.4. The Betti numbers of higher order Kummer varieties

41

As (n) is @tale, it follows that K A n - I is smooth. The case of a geometrically ruled surface can be treated by a modification of this argument. Analogously to the above we have AxK(A• , (AxP1)['q

[] (n)

A

l gn ~ ~

A.

So K ( A x P1)n-1 is smooth. Now let S-2-*A be a geometrically ruled suface. Let (Ui)i be an open cover of A such that a- 1(Ui) = Ui x P 1 for all i. We can assume that for every effective zero-cycle ~ of length n there is an i such that supp(~) C Ui x P1. Let K ~ - I := { Z E KSn-1

a(supp(Z)) C Ui}.

Then the K i _ l form an open cover of KSn-1 with .i 1 ~--(Ui x P1) [~] CI K ( A x P1)~-1. Kn_

[]

We will again use the Weil conjectures to determine the Betti numbers of the

KSn-1. To count the points we will use a result from representation theory, the Shintani-descent. Our reference for this is [Digne (1)].

D e f i n i t i o n 2.4.3. Let G be a group and {H / a cyclic group of automorphims of G. Let Gt,<(H I be the semidirect product. Let j be the set-theoretic map

j :G ----* G~<(H); g ~-~ (g, H). The H-classes of G are the sets j - l ( c ) , where c runs through the conjugacy classes of G. (G, H) has the Lang property, if the set of fixed points G g is finite and each g E G can be written as g = x - i l l ( x ) for an x E G.

Let L be a connected algebraic group over F q , let G = L ( F q ) and F the Frobenius over F q . Then (G, F ) has the Lang property by the theorem of Lang.

2.4.4. (c]. [Digne (1) Thin 1.4]). Let G be a group and H, H' two commuting automorphism8 of G such that both ( G , H ) and (G,H') have the Lang property. Then Theorem

(1) For all y E G we have y - i l l ( y ) e G H.

2. The Betti numbers of Hilbert schemes

42

(2) The map NH/H, : y - i l l ( y ) H yH'(y -1) definies a bijection from the set of

H-classes of G H' to the set of H'-cla~aes of G H. D e f i n i t i o n 2.4.5. Let S be a smooth projective surface over hTq and v =

(nl,...,n~)

=

(I~,2~,3~3...)

a partition of n. We write as above Ic~I := E c~i, and put Iv I := I~1 (obviously Iv[ = r). As above we denote the set of partitions of n by P(n). We put

i=1

-y. : s [ . ]

, s(")(Fq);

((~i),,v),

, ~ ] i . ~i i

and define

U s[~] vEP(n)

, s(n)(Fq)

by 7~ls[.l : = 7 . . By theorem 2.2.4(3) we can assume (maybe after extending F q ) that

IVt(Fq-.)l-- ~ qmr vEP(l) for all l < n and all m E PC. L e m m a 2.4.6. For all ~ E S(n)(~'q) we have I ~ 1 ( ~ ) 1 : I~;1(~)1 . Proof." Let

= ~ ni~i E s(n)(Fq), i=1

where the ~/ are distinct primitive cycles of degree di. Then we have Iw2~(~)[ = ~

[Vn,(FC, )l

i=1

: ~ i=1

Z qdi(ni-'#i') #iEp(ni)

For i = 1 , . . . , r let #i

i i = (ml,...,mlt~ q)

2.4. The Betti numbers of higher order Kummer varieties

43

be a p a r t i t i o n of hi, and let V = (nl,...

nl~l)

be the union of d / c o p i e s of each # / ( i . e . if #~ = (1 ~[ , 2 ~ , . . . ) , then v = (1 ~ , 2 ~ , . . . ) where ~j = }-~i dia}). Let

i--1 {llmi=j} Let r] be the sequence (rh,rl2,r?3,...). T h e n for all w E A ~-I"l the pair (r/,w) is an element of S[v] and =

In this way we can get all the elements of 7~-1(~). So we have ~nl(~)

~"~

E E "'" E qn-~dd.'l plCP(nl) p2EP(n2) p~EP(n~)

=

[]

For the next four lemmas let q be a prime power satisfying gcd(n, q) = 1 and a

let either S = A be an abelian surface over F q or let S - - ~ A be a geometrically ruled surface over an elliptic curve A over ~'q. In this case we assume t h a t there exist an open cover (Ui)i of A and isomorphisms a-l(Ui) ~- Ui • P1 over F q . In b o t h cases we assume that, for all l _< n, all t h e / - d i v i s i o n points of A are defined over F q . All these conditions can be o b t a i n e d by extending F q if necessary. Let F be the geometric Frobenius over F q . We put

try: A(Fq,) ~

A(Fq);

I

x

Z F/(x) i=0

for all l E SV.

Lemma

2.4.7. trl is onto and II~r/l(x)[ i,~ independent of x E A(Fq).

P r o o f : We have A ( F q ) F = (A(•q)),

and A ( ~ q ) F' = (A(Fq,)). Let x E A(Fq,).

Choose y C A ( F q ) satisfying x : F(y) - y (this is possible by the Lang property). T h e n we have

NF~ /F : y -- Fl(y) l--1 = E Fi(y - F(y)) i=0

= --trl(x

).

2. The Betti numbers of Hilbert schemes

44

As F t acts a s the identity on A ( F q ) , A(~'q) is the same as the set of Ft-classes on A(~;'q). Thus by theorem 2.2.4 trt is onto. For x E A(aCv~) and y 9 tr[-~(x) the m a p z ~ y + z gives a bijection between tr[ -1 (0) and tr~ l(x) []

Let hn = gn(.Fq)oan(.~i~q) : S(n)(.~q)

Lemma

} A(.Fq).

2.4.8. hn is onto and [hn~(X)l is independent of x E A(~'q).

P r o o f i For any partition t, = ( n ~ , . . . ,n~) = ( 1 ~ , 2 ~ , . . . ) of n let M(v) be the conjugacy class of the symmetric group G(n) whose elements consist of disjoint cycles of lengths nl . . . . , n~. Then we have liar/

n~

~1

H ic~'c~i! i=1

and ~-~veP(,~) IM(t,)l = X over F q we put

n!,

as G(n) is the union of the M(v). For a smooth variety

X(O, v) := h

X(G~

)'

j=l

x(0,n) ::

U

x(0,~),

vEP(n)

X ( n ) :=

U X(0, v) x M(v). vEP(n)

Let (I)x.0 : X(O,n) ----* X('0;

r ni --1 ( X I , . . . , X r ) V-'---+Z Z Fl(xi ) i=1

/=0

Cx : X ( n ) ---* X(");

((z~,..., ~), m) ~-, r r

A(n) ---* A(Fq);

((al,...,a~),m) ~

~ tr,,(a,). i=l

C l a i m (*). Ir

= n! for all ~ C X ( n ) ( F q ) .

x~)

2.4. The Betti numbers of higher order Kummer varieties

45

Proof of (*): Let ~ = ~ i ~ 1 mini E X ( n ) ( F q ) , where the {i are distinct primitive zero-cycles of lengths di. The points of ~xl0(~) can be obtained as follows: for any i let #i

i i = (ll,''',/].il)

i i = ("l~t'2c~2'''')

be a partition of mi, and put # : = ( ~ 1 , . . - ,

/2(~) = ( n l , .

# r ) . Let

, r t l v ( # ) l ) = (1 ~1 , 2~2,...)

be the union of the partitions di 9#' of dimi (i.e. flj = ~ i a}/d,)" Let

i=1

be a bijection satisfying no(i,u) = dil i and p(i,u) <_ p(i,v) for all i <_ r, u <_ v <_ [#i I. There are

ILgk! i-I,,j"}! such bijections. For all l E { 1 , . . . , ]v(#)]} we choose an xl E X ( F q - , ) satisfying

hi--1 Z rw(x ) : lj~i, ' w~O where p-l(1) = (i,j). There are di choices for xt. We see that

(~,,..., x,~(,),) e x(o, .(,)), and we have ( : ~ X , 0 ( X l , . . . , Xlv(#)l ) ~-- ~. All the elements of Oxl0(~), can be obtained this way, and all the possible choices lead to different elements. Obviously we have

f i kZ~ = Hr H (jdi)~'}" k=l

i=1 j

2. The Betti numbers of Hilbert ~chemes

46

Thus we get

pEP(ml)x...xP(mr)

(lr

9 IM(,-'(ff))l)

c~

k=l

#EP(ml)x...xP(rn~)

,~k, k~,

ioo----1

I I H oz;, i=l j=l

d ~1 -- n!.

~

~ 1

.

X i = l j----1

= ,fI 2 i---1 ,uiEP(mi)

i=1 j = l

/

'

0~!,]%

j=l

=n!. This shows (*). Let 5: S(n)

, A(n) be defined by being a(~'q.~ ): S(Fq.j ) ~

A(Fq-j )

on the factors S(Fq,j ) and the identity on the M(~,). The diagram

s(,~)

d~S

,

S(")(Fq)

an(Fq) A(n)

~A

,

A(n)(Fq)

A(Fq) commutes. By lemma 2.4.7. and by our assumptions before lemma 2.4.7 I ~ - l ( x ) l and la-l(~-l(x))l we independent of x E A(Fq). By (*) we have

Ih-'(~)l Thus the lemma follows.

[]

= la-a(~-'(x))l/n!.

2.4. The Betti number8 of higher order Kummer varietie~

47

For each l 9 zW let A ( F q ) t be the image of the multiplication (1) : A ( F q ) A(Fq).

Lemma

2.4.9. Let v = ( n l , . . . , n~) be a partition of a number m 9 SV. a. : A(Fq) ~ --* A(Fq)gcd(.); (Xl,.-.,Xr)

~-~nixi

~

i=1

i8 onto and I(r~-l(x)l is independent of x 9 A(Fq)gcd(~). P r o o f i Let x e A(~'q)gcd(,) and y C A(•q) with gcd(v)y = x. Let m ~ , . . . ,m~ 9 2~ satisfying

~-~ mini = gcd(u). i=1

Then we have

~.((m,~,

, m ~ ) ) = x,

and the m a p

is a bijection.

f , : ~-1(0) ~

~-l(x);

(Yl , . . . , Yr ) ~

(Yl -~- mly,...,yr-~-

torY)

[]

Observe that

bl(S) = 2dim(A) =

in case (1) (S is an abelian surface); in case (2) (A is an elliptic curve).

4 [ 2

L e m m a 2.4.10.

(1) [KS"-I(Fq)I-

IA(Fq) I

gcd(v)bl(S)qn-]"]

Z v----(lal,2a2,...)EP(n)

(2)

1

-LA(Fq)I

i=l

IS(~')(Fq)l

gcd(v)b~(S)z2(,_l,l) v=(l ~1,2 a2,...)EP(n)

\#i

=(1B1,2Z2 ,...) e P(cq) 3--1

j~j/~}!

48

2. The Betti numbers of HiIbert schemes

P r o o f i By l e m m a 2.4.6 we have

I K S ~ _ ~ ( F q ) I = 17~1(h~1(0))1 =

~

I%-l(h~l(0))l 9

vEP(n)

Let ~, = ( n l , . . .

,nl,i) = ( 1 ~ t , 2 ~ , . . . )

be a partition of n and let

# = (ml,...,mt):=

(la~,2~,...)

be defined by ~i = m i n ( 1 , a i ) for all i. Let f , : S[u] ---* A ( F q ) t ; ((~1,..., ~t), w) ~

(gam~ (aam~ (~1)),-.., ga~ t (aa~ t (~t))).

Then the diagram

1~~

r A(1Fq) t

~"

,

A(Fq)

commutes. By l e m m a 2.4.8 and l e m m a 2.4.9 a~ofu maps S[u] onto A(•q)gcd(v) = A(Fq)~cd(.), and If;l(~;l(x))l is is independent of x e A(Fq)gcd(,). As the multiplication with god(u) is an 6tale morphism of degree (gcd(u)) b~(s) of A to itself, we see

if;-1(~;-1(0))1_

IsMI

IA(Fq)gcd(~)l

(~=~Is(~,)(F.)I)q,~-I,l(gcd(u))b,(S) IA(Fq)I (1) follows by l e m m a 2.4.6, and (2) follows from this by remark 1.2.4(3) and an easy calculation.

Theorem

2.4.11.

(1) Let A be a two dimensional abelian variety over C. Then p(KA,~-I, z) - - (1 + z) 4

(gcd(.))4z 2("-I~l) II v(A(~ z) v=(l"l,2c'2,...)EP(n)

i=1

(2) Let S be a geometrically ruled surface over an elliptic curve over C. Then p ( K S , _ I , z)

-

(1

1 + z) 2

Z v=(1 ~1,2 ~2,,..)eP(n)

(g~(')) ~z~r

l]v(s(~), z) i=1

2.4. The Betti numbers of higher order Kummer varieties

49

(3) In both cases we can also write these formulas as

P(KS,-a, -z) = 1 (1

-

z)b,(s)

E

gcd(u)bl(S)z2(n-I~l)

v=(1 c'1,2 '~2 ,...) 6 P ( n )

i----1 \,u/=(1,o[,2

i~,...)6p(c~i)j=l

Proof." Let S be either a two dimensional abelian variety or a geometrically ruled surface over an elliptic curve over C. Let S be a good reduction of S modulo q, where gcd(q, n) = 1 such that the assumptions of lemma 2.4.7 hold. Then K S n - l i s a good reduction of KSn-1 modulo q. (3) now follows by lemma 2.4.10 and remark 1.2.2. (1) and (2) follow from this by the formula of Macdonald for p(S (n), z) (see the proof of theorem 2.3.10). [] In section 2.3 we have obtained power series formulas for the Betti numbers of the S ['q. We now also want to give power series for the KSn-1. They will however not be as nice as those for S ['q. We define a new multiplication Q) on the ring of power series 2g[[z, t, w]] by

znltmlw 1. @ zn2tm2w 12 :~_ znx+n=~gmt+m2WgCd(l~,12) and extension by distributivity.

Proposition 2.4.12. oo

p(USn-1)e r~=O

(1 + z)b,(s) w-d-s

-

k=l

1 -'}- w k

--1 -'}- (1 -- z 2 k - 2 t k ) ( 1

An equivalent formula is oo

T~,=O

_(

1

(wA)b' s)

(1- z)b,(S) \ dw )

-- z2ktk)b=(S)(1

--

z2~+2tk)]]l w = l

50

2. The Betti numbers of Hilbert schemes

P r o o f i It is easy to see t h a t the two formulas are equivalent. So we only have to show the following identity:

n=O

v = ( 1 ~1,2 ~ ,...) E P ( n )

C) ( =

i=1

( 1 + wk

,]

(l + z2k--ltk)bl(S)(1-bz2k+ltk)b3(S) - 1 + (1

-

z2k-2tk)(1

-- z2ktk)b2(S)(1

-- z2k+2t k)

k=l

This however follows i m m e d i a t e l y from the formula of Macdonald.

))

[]

We can now c o m p u t e the Betti numbers of the K S n - 1 for small n. We get the following tables:

Betti numbers b~(KA~) for higher order K u m m e r varieties: n

2

3

6

7

1 0 7 8 36 64 176 352 786 1528 2879 4496 7870 4496 2879 1528 786 352 176 64 36

1 0 7 8 36 64 176 352 809 1584 3327 6136 11298 16432 25524 16432 11298 6136 3327 1584 809

9

10

1 0 7 8 36 64 176 352 794 1592 3301 6416 12571 23456 43043 74040 118672 162808 198270 162808 118672

1 0 7 8 36 64 176 352 794 1592 3286 6424 12522 23680 44142 79920 140073 232368 354034 471712 538070

v

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 1 1 0 0 0 7 7 7 8 8 8 108 51 36 8 56 64 7 458 168 0 56 288 1 51 1046 8 288 7 168 0 64 1 36 8 7 0 1

7 8 36 64 191 344 915 312 748 312 915 344 191 64 36 8 7 0 1

1 0 7 8 36 64 176 352 794 1592 3278 6360 12202 21704 36440 51640 67049 51640 36440 21704 12202

2.4. The Betti numbers of higher order Kummer varieties

51

Betti numbers bv(KSn) for S a geometrically ruled surface over an elliptic curve:

n

1

2

3

4

1 0 6 2 6 0 1

1 0 3 4

1 0 3 4

1 0 3 4

6 8 6 4 3 0 1

13 14 45 32 45 14 13 4 3 0 1

10 16 30 48 90 72 90 48 30 16 10 4 3 0 1

10

I]

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

1 0 3 4 10 16 35 54 198 142 247 232 247 142 108 54 35 16 10 4 3 0

1 1 1 0 0 0 0 3 3 3 3 4 4 4 4 4 10 10 10 10 10 16 16 16 16 16 32 32 32 32 32 56 56 56 56 56 97 102 99 99 99 156 162 164 164 164 243 278 275 280 277 348 434 448 454 456 486 568 711 738 735 472 892 1C56 1146 1160 486 1206 1541 1763 1811 348 1232 2C48 2590 2764 243 1206 2557 3643 4089 156 892 2640 4704 5824 97 568 2557 5737 7903 56 434 2C48 5984 10O28 32 278 1541 5737 11788 16 162 1C56 4704 12288

Let al(n) be the sum of the positive integers dividing n. For

rEH:={a+biECb>O} let q := e 2'~i'. T h e n the eta function and the Eisenstein series E2 are given by 7](7") :----ql/24 l - I ( 1

_

qn)

n~--I

Be(r) : = 1 - 24 E

(Tl(n)q n.

n=l

We p u t O(r) : = q - ' / 2 4 ~ ( r ) . Corollary 2.4.13. (1) For an abelian surface A over C we have

e ( K A n - l ) ----n3•l(n).

2. The Betti numbers of Hilbert schemes

52

(2)

For a

geometrically ruled surface over an elliptic curve we have e(gSn_ 1) = 2ncrl (n).

(3) In both cases this can be expressed in terms of modular forms as

oo

Ec(I~-Sn_l)qn --

n=i

3 - bl(S)/2 ~ q d ~ bl(S)-I d 41Ti k dqJ log(iT(T))

b1(S)/2) \ dq]

= 1(3-

E2.

P r o o f i As p ( S , - z i) is divisible by (1 - z) b~(s), we see t h a t every s u m m a n d

(1 - z)b'(s) i=1 j=l J~ fl}!P(

-zJ)~i

in the sum of t h e o r e m 2.4.11(3) is divisible by (1 - z)b'(s)((~,J ~j)-l) Thus it does not contribute to the Euler number, except if u is (n~ ~ ) a n d #n~ is ( ~ - ) for some divisor nl of n. So we get from t h e o r e m 2.4.11(3):

nbl(S) nl p(S, - z n/n1) ~(~<s._,) : Z n, ln

---- E

nl[n

1

-

n

(1-z)b~(s)

I~=1

-bl(S) n l ( 3 - - bl(S)/2) it 1

n

= (3 -- bl(S)/2)nb~(S)-lcrl(n).

(~,,(s) n

\ nl /

[]

Table of the Euler numbers (A abelian surface, S geometrically ruled surface over an elliptic curve):

v n e(KAn) e(KSn)

1

2

3

4

5 27,:

24 108 448 750 2592 12 24 56 60 144

7 974:0

7680 112 240

10

18000 15972 234 360 264

We can again see easily t h a t the Betti n u m b e r s bi(KSn-1) become stable for

i<_n.

2.4. The Betti numbers of higher order Kummer varieties

53

Corollary 2.4.14.

(1f::z-~-z "~)2bl(S) Z2m'+1

p(KS._I,z) - ( i - 2 ) II

m=l

modulo z n.

P r o o f : For any p a r t i t i o n u of n satisfying [u[ > n/2 we see t h a t gcd(u) = 1. So we

have p ( S [nl, z)

p(KSn-1, z) = (1 + z) bds)

m o d u l o z ".

Thus we have by corollary 2.3.13

i

fi (i +

p ( K S n - a , z ) - (1 + z ) bl(s)

m~--I

z~m-1)bl(s)(1 (1-- z--~m~(-i

+

z2m+l) bi(S)

-

z 2m+2)

modulo zn

(1 + Z2m+l)2b1(S)

fi

= ~=~ (i - ~ w ~ -

T~+~)

oo = (I - z ~) I I (I + z~m+') ~b,(s) m=l (1 -- z2m) b2(s)+2 The result follows.

[]

In p a r t i c u l a r we have ba(KSn-1) = 0 for all n E ZW. In fact the K A n - 1 were proven to be simply connected in [Beauville (1)].

2. The Betti numbers of Hilbert schemes

54

The orbifold Euler number formula Let G be a finite group acting on a compact differentiable manifold X. Then there exists the well known formula for the Euler number of the quotient 1

g

~(x/a) = -~, ~ ~(x ), '

' gEG

where X g denotes the set of fixed points of g E G. If the quotient X / G is viewed as an orbifold, it still carries information on the action of G. In [Dixon-Harvey-VafaWitten (1),(2)] the orbifold Euler number is defined by

~(x, a) = ~1 ~

~(x. n x h)

gh=hg

(the sum is over all commuting pairs of elements in G). Now let X be an algebraic variety. We assume that the canonical divisor K x / a of X / G exists as a Cartier divisor. Furthermore we assume that there is a resolution ~G--Z--~X/G satisfying K x~ / a = ~r* K x / c . Then it has been conjectured that A

e(x,a) = e(x/a). This formula we will call the orbifold Euler number formula. In the case that the group G is abelian this conjecture has been proved in [Roan (1)] under certain additional hypotheses. In [Hirzebruch-Hbfer (1)] some examples of this formula are studied. First they give a reformulation:

~(x, a) = ~ , e(xg/c(g)). [g] Here C(g) is the centralizer of g and [g] runs through the conjugacy classes of G. Hirzebruch and Hbfer consider in particular the action of the symmetric group G(n) on the n th power S n of a smooth projective surface S by permuting the factors. The quotient is the symmetric power S (~), and w,, : S ['q ~ S ('0 is a canonical resolution of S ('). The canonical divisor K s , is invariant under the G(n) action. Thus it gives a canionical Cartier divisor Ks(,) on S ( ' ) , and it is easy to show that

~ * ( K s ( . ) ) = Kst.~. So the assumptions of the conjecture are fulfilled, and in fact Hirzebruch and Hbfer use my formulas (corollary 2.3.11) to prove that

~(s~-]) = e(s -, o(~)).

2.4. The Betti numbers of higher order K u m m e r varieties

55

Another case in which they check the formula is that of the K u m m e r surface KA1 of an abelian surface as a resolution of the quotient of A by G(2) = 2g/2 acting by x ~ - x . We will now generalize this result to the higher order K u m m e r varieties K A n - 1 . Let A be an abelian surface. Let

A'~:={(Xl,...,x~)EA

n

Exi=O}

cA~

with the reduced induced structure. Then A~ is isomorphic to A '~-1. The G(n) action by permutation of the factors of A n maps A~ to itself. So we can restrict it to A~ and the quotient is A~n). Let w := COn[Ix'AN_,. Then w : K A n - 1 ~ A~ '0 is a canonical desingularisation of A~n). The canonical divisor of A~n) is trivial, and by [Beauville (1)] KA,~-I is a symplectic variety; in particular we also have K K A . _ , = O. So the conjecture says that e ( K A n - 1 ) = e(A n-a, G(n)) should hold. For a permutation a of {1,... ,n} let

be the partition of n which consists of the lengths of the cycles of a. It determines the conjugacy class of a. The fixed point set is given by ( A n ) a = { ( X l , . .. ,Xn) 9 d n or

x~,, . . . .

xv, for all cycles ( , 1 , . . . , ui) of a }

~ I I A~i (~) = i=1 The centralizer C(a) acts by permuting the cycles of a of the same lengths. So we get

- 1-I i

= HA~'(~)/G(c~i(a)). For

h = (hi, h2,...) e 1-[ i the fixed point set ((An)~') h consists of the ( z l , . . . , xn) 9 A n satisfying zl = xj for all i , j for which the following holds: either i and j occur in the same cycle of a, or they occur in two different cycles of the same length l, and these are permuted by hr. So we get that ((An)~) h = (An) ~ for some 7 9 G(n) and

((An)~') h = (An) (1 ...... ) --"~A,

56

2. The Betti numbers of Hilbert schemes

if and only if

p(~) = ((~/~)~), and ha is a cycle of length a in G ( a ) for a positive integer a dividing n. Remark

2 . 4 . 1 5 . Let ~r E G(n). Then we have

{n 4 p(~) = e((A~)~) =

0

(n);

otherwise.

P r o o f : Let B be an abelian variety and h : B

~ B an a u t o m o r p h i s m of B. T h e n

every connected component of B h is either an isolated point or a t r a n s l a t i o n of an abelian subvariety of positive dimension of B. In p a r t i c u l a r e(B h) is the n u m b e r of isolated points in B h. For a cycle a of length n we have

=

xEA

nx=

,

and this has Euler n u m b e r n 4. Let a E G(n) with p(a) = ( n l , . . . , n r ) ,

r >2.

Then we get

(A~)a~ {(Xl,...,Xr) Znixi =0}. Let ( z ~ , . . . ,x~) C (A~) ~. For every y e A the point (Xl + n2y, x2 - n l y , x 3 , . . . , x r ) lies in the same connected component of (A~) a as ( x a , . . . ,x~). By the above we have e((A~) ~) = 0.

T h e o r e m 2.4.16.

[]

e(An-l, C(n)) = n 3 o ' l ( r ~ )

e(I(An-1).

=

Proofi e(A~, C ( n ) ) = Z e ( ( A ~ ) ~ / C ( a ) ) M

Ili ~(~)! ( ( ( 0 ) ) ) n4

Mn = n30"l ( n )

[2]

2.4. The Betti numbers of higher order Kummer varieties

57

Conjectures on the Hodge numbers of the KSn-1 Similar to the results of theorem 2.3.14 we can formulate conjectures on the Hodge numbers of the KSn-1.

Conjecture 2.4.17. h(KSn-l,x,y) 1 ((1 + x)(1 + y))b,(S)/2

~_,

(gcd(~'))b'(S)(xy) "-H

v=(l~t ,2~2 ,...)EP(n)

91] h(S(~ i

or equivalently h(KSn-a, - x , - y ) 1

(1 x)~,(s)/:(1 -

90(

u)b,(s)/z

-

g~d(v)~l(s)(x~,) " - H v=(l':' 1,2c'2 ,...)EP(n)

,

/~'=(I p , ,2~2,...)ep(c~i)

J ~/%.

J

]

In the case of the K A , _ a the conjecture has been verified in [GSttsche-Soergel

(1)]. R e m a r k 2.4.18. From the proven part of conjecture 2.4.17 we get for the )/y-genus and the signature: (1)

X-y(KA,-1) = n E

nla(1 + Y " " + yn/nl--1)2yn--n/n,

(2)

sign(gA,_l) = (-1)~-ln

E

d3"

din , n / d odd

We can again express the signatures of the KAn-1 in terms of modular forms (notations as in 2.3.15).

sign(KAn)(-q) n = ~q.

(3) rt~O

Proof." As in 2.4.12 only the terms with u = (n~/n'), #,~In, = (n/nx) give a contribution to the xu-genus. So we get

x-y(gA._l) = ~ ~ ( 1 - x"/"')(1 - ~"/",) ~--1 -,I-

(1

-

x)(1

y)

2. The Betti number~ o] Hilbert scheme~

58

(1) follows by easy c o m p u t a t i o n and (2) by p u t t i n g y = - 1 . (3) is obvious from the definition of e. [] By applying the same a r g u m e n t to the case of a geometrically ruled surface over an elliptic curve we get that sign(KSn-1) = O. This was however clear from the beginning as the dimension of K S~ - I is not divisible by 4. It seems r e m a r k a b l e t h a t in all cases the signatures and the Euler numbers can be expressed in terms of the coefficients of the q-development of m o d u l a r forms. For the first few of the X - y ( K A ~ - I ) we get: X - y ( K A 1 ) = 2 + 20y + 2y 2,

X-v(KA2) = 3 + 6y + 90y 2 + 6y 3 + 3y 4, X-y(KA3) = 4 + 8y + 44y 2 + 336y 3 + 44y 4 + 8y ~ + 4y 6, X-y(KA4) = 5 + 10y + 15y 2 + 20y 3 + 650y 4 + 20y 5 + 15y 6 + 10y 7 + 5y s, X-y(KA~) = 6 + 12y + 18y 2 + 72y 3 + 288y 4 + 1800y ~ + 288y 6 + 72y 7 + 18y s, + 12y 9 + 6y 1~ Let b+ be the n u m b e r of positive eigenvalues of the intersection form on the middle cohomology and b_ the n u m b e r of negative ones. T h e n we get the following table:

b2n(KAn) sign(KA,) b+(KAn) b-(KAn) 1 2 3 4 5 6 7 8 9 10

22 108 458 1046 3748 7870 25524 67O49 198270 538070

-16 84 -256 630 -1320 2408 -4096 6813 -10080 146521

3 96 101 838 1214 5139 10714 36931 94095 276361

19 12 357 208 2534 2731 14810 30118 104175 251709

We can also determine the Chern numbers of KA2: C4 = O, C~C2 :

O, ClC3 :

O,

c4 = 108, c~ = 756, This is true because cl = --KtcA._I = 0 and

s i gn( K A , _l ) = l (7p2(KA2) - p2(KA2)) = 84,

2.4. The Betti numbers of higher order Kummer varieties

p~(KA~) = ( ~ - 2~)(IZA~) = - 2 ~ ( K A ~ ) ,

59

60

2.5. T h e B e t t i n u m b e r s of varieties of triangles Let X be a smooth projective variety of dimension d over a field k. For d _> 3 and n > 4 the Hilbert scheme X['q is singular. However X [3] is smooth for all d E ~W. In this section we want to compute the Betti numbers of X [3]. X [3] can be viewed as a variety of unordered triangles on X. We also consider a number of other varieties of triangles on X, some of which have not yet appeared in the literature. As far as this is not yet known, we show that all these varieties are smooth. We study the relations between these varieties and compute their Betti numbers using the Weil conjectures.

Definition 2.5.1. [Elencwajg-Le Barz (5)] Let Hil'---b'~(X) C X [~-1] x X In] be the reduced subvariety defined by Hil--'bn(X) = { ( Z . _ I , Zn) C X [~-1] • X[~]

~ 3

Zn-1 C Z . }.

~ 3

Here we will be interested in Hilb (X). Let i : Hilb (X) ~ X [2] • X [3] be the embedding. If one interprets X[ 3] as a variety of unordered triangles on X, then ~ 3

Hilb ( X ) parametrizes triangles Z3 with a marked side Z2. In the case k = C it was ~ 3

~ 3

shown in [Elencwajg-Le Barz (5)] that Hilb (X) is smooth. Hilb (X) represents the contravaria~t functor from the category of Schln k locally noetherian k-schemes to the category Ens of sets ~ 3

7-lilb (X) : Schln k T, (r

---~ T2),

Ens; ,

, ((z~,z~)

~

xt l(T) • XE I(T)

c

r

• r

((ix

•

So for a smooth variety X over C and a reduction X0 of X modulo q the variety ~ 3

~ 3

Hilb (X0) is a reduction of Hilb (X) modulo q. Let P2 : Hii-b3(X) ----* X [31 be the projection. For any partition • of 3 ( i . e . . = (1, 1, 1), J v = (2, 1), J ~ = (3) ) we put ~ 3

Hilb~(Z) := p~-l(x[~3l). In [Elencwajg-Le Sarz (5)] a residual point of a pair (Z2, Z3) E ~ b 3 ( X )

Definition 2.5.2. [Elencwajg-Le Barz (5)] Let (Zn-1, Zn) ~ X [~-1] • X [~].

is defined.

2.5. The Betti numbers of varieties of triangles

61

Let I n - 1 be the ideal of Zn_ 1 in Oz.. Then the residual point r e s ( Z n - 1 , Zn) E X is the point whose ideal in Oz. is the annihilator Ann(I,,-1, Oz.) of It,-1 in Oz..

Elencwajg and Le Barz show t h a t the m a p (Zn-a, Zn) ~-* res(Z,_a, Z,) gives ~ n

a m o r p h i s m res : Hilb ( X ) ~ a r b i t r a r y field.

X , if the ground field is C. We show this for an

L e m m a 2.5.3. The map (Z,~-a,Zn) ~ res(Zn-l,Zn) defines a morphism res : ~ n Hilb ( X ) ~ X. ~ n

P r o o f i Let T be an integral noetherian scheme and (Zn-1, Zn) E 7-filb (X)(T). Let I be the ideal sheaf of Z,,-1 in Oz.. Then for all t C T the dimension of the annihilator Ann(It, Oz.,t)is 1, so Ann(I, Ozn) defines a subscheme res( Zn-1, Zn) C Zn, which is flat of degree 1 over T, i.e a T-valued point of X . So res is given by a m o r p h i s m of functors.

Remark

2.5.4. We can also describe the residual point as follows: for ( Z n - a , Zn) E

~ 3

Hilb ( X ) the zero-cycle wn(Zn) - w n - a ( Z , - a ) is Ix] for some point x e Z and ~ n

r e s ( Z , _ l , Zn) = x. If we consider Hilb ( X ) as a variety of triangles with a m a r k e d side, then res m a p s such a triangle to the vertex opposite to the m a r k e d side.

Via ~ 3

il : = res x i : Hilb ( X ) ~

X x X [2] x X [a]

~ 3

we will in future consider Hilb ( X ) as a subvariety of X x X [21 x X[3]:

~ 3

This means we consider Hilb ( X ) as a variety of triangles with a side a n d the opposing vertex marked. Let ~ 3

t51 : H i l b ( X ) ~ 3

t52 : H i l b ( X )

, X, ~ X [21,

~ 3

Pa : Hilb ( X ) ~ 3

Pl,2 : I-Iilb ( X ) ~ ~ 3

/~1,3 : Hilb ( X ) ~

, X [a], X x X [21, X x X [a]

2. The Betti number8 of Hilbert schemes

62

be the projections. From the definitions we can see that the support of the image ~ 3

of/51,3 coincides with the support of the universal subscheme Z3(X). As Hilb (X) is reduced, this defines a morphism ~ 3

/~1,3 : Hilb (X) This morphism is birational,

, Z3(X).

as its restriction gives an isomorphism from

~ 3

~ 3

(Hilb (X))(1,1,1) to a dense open subset of Z3(X). So/51,3 : nilb (X) ----* Z3(X) is a canonical resolution of Z3(X). We can consider Z3(X) as the variety of triangles with a marked vertex. Then i51,3 is given by forgetting the marked side. ~ 3

Pl,2 : Hilb

is birational,

as

it

gives

an

(X)

, X • X [2]

isomorphism

of the

dense

open

subvariety

~ 3

(Hilb (X))(1,1,1) onto ints image. Let Z2(X) C X • X [2] be the universal subscheme. As a set Z2(X) is given by

z (x)

=

x • xc l

x c z}.

One can also verify easily that it carries the reduced induced structure and that it can be described as X x X blown up allong the diagonal. Let w : X • X [21 ~

Z3(X)

be the rational map which is defined on the open dense subvariety (X • X [2]) \ Z2 (X) by w((x, Z)) := (x, x t2 Z). Then obviously the diagram N 3

Hilb (X)

l

lbl,2

w

X x X [2]

4.

z (x)

~ 3

commutes.

So Pl,3 : Hilb (X) ~

Z3(X) is a natural resolution of the indeter~ 3

minacy of w. We will see later that Hilb (X) is the blow up of X • X [2] along

2.5. The Betti numbers of varieties of triangle~

T h e varieties o f

complete triangles

63

on X.

Semple [Semple (1)] has constructed a variety of complete triangles on P2This variety has been studied and its Chow ring was determined in [Roberts (1)], [Roberts-Speiser (1),(2),(3),(4)], [Collino-Fulton (1)]. (The Chow ring coincides with the cohomology ring in case k = C). Le Barz has generalized this construction in [Le Barz (10)] to general projective varieties and shown that the resulting varieties of complete triangles are smooth. Keel [Keel (1)] also gave a functorial construction of these varieties. Let X be a smooth projective variety of dimension d over a field k. We want to define other varieties of complete triangles. Because of this we call the variety defined by Le Barz the variety of complete ordered triangles on X. We also want to show that our varieties of complete triangles are smooth by using results fl'om [Le naxz (10)]. Definition 2.5.5. [Le Barz (10)] Let X be a smooth projective variety over a field k. The variety H3(X) of complete ordered triangles on X is the closed subvariety of X 3 x (X[2]) 3 • X [3] defined by Xi,Xj C Zl; Zi C Z; ~(x)

:

(Xl,X2,z3,Z1,Z2,Z3, Z) ~ ( x ~ • (xI~l)~ • xI~l)

x, : r ~ ( ~ , , Z j ) = r ~ ( Z t , Z )

for all permutations (i,j,l) of (1, 2, 3)

In [Le Barz (10)] /t3(X) is shown to be smooth for X a smooth variety over C. H3(X) represents the obvious functor 7~3(X) : Schln k > Ens:

(x1,~2,z3,zl,z2,z3,z) 7~3(X)(T) =

E (Z 3 x (X[2]) 3 X X[aI)(T)

x . xj c z~; z~ c z ; ~t = r ~ ( x , , Z j ) = r ~ ( Z t , Z ) for all permutations (i,j,l) of (1, 2,3)

(see also [Collino-Fulton (1) rem. (5)]). So if X is a smooth projective variety over C and X0 is a good reduction of X modulo q, then H3(X0) is a reduction of H3(X) modulo q. Let j : 9~(x) , x ~ • (x~21) ~ • xE31 be the embedding. Let :~1 : ~q3(X) - - ~ X 3, p~ : 9 ~ ( x )

- - , (x[~]) ~,

b~ : fi-z(x) __~ xC~l, be the projections. From the stratification of X [3] we get one of H3(X). Let u be a partition of 3. Then we put

2. The Betti numbers of Hilbert schemes

64

We can view the xi as the vertices of the triangle Z and Zi as the side opposite to xi. Thus s parametrizes the complete ordered triangles on X (i.e. together with a triangle we are given all its vertices and all its sides together with an ordering). The projection t51 : H 3 ( X ) -----+X 3 is birational. D e f i n i t i o n 2.5.6. [Le Barz (10)] For a pair (i,j) satisfying 1 < i < j < 3 let

Z~i,j := {(ZI,N2,Z3 ) E (X[2]) 3 Zi : Zj} c be the diagonal between the i th and jth factors. Let

be the small diagonal in X 3, and ~2 the small diagonal in (X[2I) 3. Then we put

E~,j(X) :=/5;~(/x~,~), D~(X) := (/~1 x/52)-1((~1 X (~2).

In [Le Barz (10)] these varieties are shown to be smooth for X a smooth variety over C. The Ei,j(X) are irreducible divisors in H3(X). D~(X) is the variety of second order data on X , which we want to study in more detail in chapter 3. For x 6 X let m x , . be the maximal ideal in the local ring Ox,x and

q. : m x , .

~ mx,~:/m2x,.

'the natural projection. We can describe the subscheme Z(1,2)(X) C X Is] (cf. section 2.1) as the closed reduced subvariety given by

Z 6 X [3]

supp(Z) = x for an x E X, and there is a ] 2-codimensional linear subspace V C rnx,x/m2x,, such that / " the ideal Iz of Z in Ox,. is of the form Iz = q[l(V)

Obviously Z(1,2)(X) is isomorphic to the Grassmannian bundle Graas(2, T } ) of two-dimensional quotients of the cotangent bundle of X. We put E

:=/5;~(z(i,2)(x)) [

I

supp(Z) -= x; Z1, Z2, Z3 C Z

j "

2.5. The Betti numbers of varieties of triangle8

65

Let Z E Z(1,2)(X), x : - supp(Z). Then the ideal Iz of Z in Ox,~ is of the form Iz = q~-l(V) for a suitable 2-eodimensional linear subspace V of m x , x / m ~ , z. Let

qz : mx,~: ---+ m x , ~ / I z be the natural projection. The ideals Iz2 of subschemes Z2 of length 2 of Z are given exactly by the qz I ( W ) for the one-dimensionM linear subspaces W of m x , ~ / I z . Let 7r: Z(1,2)(X) = Grass(2, T~) ~

X

be the projection. Then the subschemes Z2 C Z of length 2 are given by the onedimensional linear subspaces of the fibre of the tautological subbundle T1 of 7r*(T~) over the point V. Thus we get

Remark

2.5.7. E ~ P(T1) •

. . . . (2,T;r P(T1) XCrass(2,T;r P(T1).

Let X be a smooth projective variety over C. Pl,2 : H~b3(X) ----+ X x Z [2] is the blow up along Z2(X). Proposition

2.5.8.

Then

Proof: ~3 /51,2 : Hilb (X) , X x X[ 2] is an isomorphism over (X x Z [2]) \ Z2(X). Let F :=/5~,1(Z2(X)). Then F can be described as the set:

F : {(X, Z l , g )

E X x X [2] x X [31 x 1 C Z1, Z1 C Z, r e s ( Z l , Z ) ~ - x 1}.

Let Pl,4,7 : / ~ 3 ( X ) (Xl,X2,x3,Zl,Z2,Z3,Z),

, X x X [21 x X [31 "' ( x l , Z l , Z )

~3 be the projection. We see immediately that the image of this morphism is Hilb (X) so we get a morphism

:~,,,~ : ~ ( x )

--~ n~i-b3(X).

Let

(xl, z2, z3, Z1, Z2, Z3, Z) E E1,2(X). Then we have Z1 = Z2 and thus Xl = x~. So we get Xl C Z1. We see that P1,4,7(E1,2(X)) C F.

So we get a m o r p h i s m q : E1,2(X) , F. Let ( x l , Z 1 , Z ) 6 F. We put x2 := x~, If supp(Z) consists of two points, we see that Xl # x3 and

x3 := res(xl,Z1).

Z = Z3 tJ x3 for a unique subscheme Z3 of length 2 with support xl. If supp(Z) is a

2. The Betti numbers of Hilbert schemes

66

point but Z does not lie in ZO,2)(X), then Z has a unique subscheme Z3 of length 2. In both cases we get

q--l(xl, Z1, Z) = {(Xl, g2, x3, Zl, Z1, Z3, Z)}, If Z lies in Z(1,2)(X), then it is given by a two-dimensional quotient W of the cotangent space T~;(xa) of X at xl, and the subschemes Za of Z are given by the one-dimensional quotients V of W. So we get

q-l(xl,Zl,g) = {(.TI,X2,x3,ZI,ZI,Z3,Z ) 23 C Z} "~ P l . Putting things together we see that q is onto and a bijection over the open set F\paa(Z(1,2)(X)). As Ea,z(X) is an irreducible divisor of s F is an irreducible ~3 divisor on Hilb (X). Let e : X x-'X[~] ---+ X • X [21 be the blow up of X • X [2] along Zz(X). Let Z be the ideal of Z2(X) in X • X [2]. From p ~ ( Z 2 ( X ) ) = F we get that plaZ. --1 OH_~b3(x ) is the invertible sheaf corresponding to F. By the universal property of the blow up (cf. e.g. [Hartshorne (2), II. prop.7.14]) there is a morphism

N3 g : Hilb (X) -----+X x-X[2] such that the diagram

~3 Hilb (X)

~

,

X x'--X[2]

X x X[ z] commutes, g is a birational mo..rphism. By [Hartshorne (1) II Thm. 7.17] g is the blow up of a subscheme of X • X[2]. g is an isomorphism outside F, F is irr__._~educible, and the image g(F) is the exceptional divisor of the blow up g : X • X[ 2] X • X [21. Thus g is an isomorphism and the result follows. D In a joint work with Barbara Fantechi [Fantechi-G6ttsche (1)] we use proposition 2.5.8 to compute the ring structure cohomology ring H*(X [3],Q) of the Hilbert scheme of three points on a smooth projective variety X of arbitrary dimension in terms of the cohomology ring of X. We also compute the cohomology ring of ~3 Hilb (X).

2.5. The Betti numbers of varieties of triangles

67

Proposition 2.5.8 also follows from [Kleiman (3)] t h m 2.8. I have learned that Ellingsrud [Ellingsrud (1)] has proven independently the following: if S is a smooth surface, the blow up of S x S In} along the universal family

zo(s)

=

s • stol

x

z}

is a smooth variety m a p p i n g surjectively to S[ n+l] (proposition 2.5.8 is essentially the case n = 2 of this). One can see easily that E1,2(X) is obtained from F by blowing up along

p31(zr D e f i n i t i o n 2.5.9. For all n E zW let ~ x , n : X n -----+ X (~),

excel,, : ( X [21)" ---+ (X [21)(") be the quotient morphisms. Then let )~[31 C X (3) • (X [2])(3) • X[3] be the image of H s ( X ) under ~X,3 • ~X[21,3 X ~X[Sl : X 3 • (X[2]) 3 • X [31 ---+ X (3) • (x[2l) (3) • X [3] with the reduced induced structure. Let ZCl : H 3 ( X ) ~ this morphism t o / I 3 ( X ) C X s x (X[2]) 3 x X [3].

.~[3] be the restriction of

The symmetric group G(3) acts on X 3 • (X [2])3 • x[3l by permuting the factors in X 3 and (X[2]) 3 simultaniously. 7rl : H 3 ( X ) ~ 2 [3] is the quotient m o r p h i s m with respect to the induced action on ~r3(X). We can consider )~[3] as a variety of complete unordered triangles on X, as together with a triangle Z E X [3] we axe given all its vertices [xl] + Ix2] + [x3] and all the sides [Z1] + [Z2] + [Z3] (however without an ordering). The projection P3 : X (3) • ( X ( 2 ) ) (3) x X [3] ~

X [31

induces a birational morphism p : )~[31

~ X[3]

(p is an isomorphism over the open dense subset Y ([3] ) 9 We can again give a 1,1,1) stratification of )~[3] by putting

L31 := p-l(x 31)

2. The Betti numbers of Hilbert schemes

68

for all partitions v of 3. We put

F_~:~. p-l(z(1,2)(X))

f(a[4,[z,]+[z,]+[z~l,z)

/

x ~ x; ZI,Z2,Z3 ~ X[2]; a E Z
~

Then we have /) = rq(E). The action of G(3) on -~3(X) maps E to itself. The induced operation of G(3) on E is by permuting Z1, Z2, Za a n d / ) is the quotient . So we get from remark 2.5.7:

R e m a r k 2.5.10. ----(P(T~) •

P(T,) •

P(T1))/G(3)

= P(Syma(T1)).

P r o p o s i t i o n 2.5.11. Let X be a smooth projective variety over C. Then

(1)

2{ai

is smooth.

(2) p : 213J ~

x{31 i~ the

bto~ up aZong Z(~,2)(X).

P r o o f i It is clear that p is an isomorphism over the open dense subset (X[a])(x,xj). g[3] i.e. Let Z = (Z2 U x) E "'(2,1), Z2 E r[21 "'(2), x E X, y := supp(Z2) 5~ x. Then we have

p-'tz)= {(2[yl+[4,2t(xuyll+[zcz)}.

Now let ( .g[a} \

Z ~ \==(3) ~-Z(1,2)(x)) = Z(1,1,1)(X) and x := supp(Z). Then the ideal of Z in O x , , is given by

Iz = (*~,x2,...,xd) for suitable local parameters Xl, x 2 , . . . , xa. The subscheme Z2 given by

is the only subscheme of length 2 in Z, and we have

p-l(z = {(3L 1,3Ez21, z)}.

2.5. The Betti numbers of varieties of triangles

69

As X [3] is smooth, p is an i s e m o r p h i s m over X [31 \ ZO,2)(X ) by Zariski's m a i n theorem [Hartshorne (2), V. 5.2]. Now we show (1). As p is an i s o m o r p h i s m over X [3] \ Z(1,2)(X), it is enough to prove the smoothness at the points o f / ~ = r l ( E ) . Le Barz has given analytic local coordinates a r o u n d any point e E E a n d so proved the smoothness of J~a(X). To simplify notations we will assume t h a t the dimension of X is 3. The a r g u m e n t for general dimension d is completely analogous, only more difficult to write down. Now let E = (o,o,o, Zl,Z2,Z3,Z

) c: E

and g : = 7r1(E). We choose local coordinates x, y, z on X centered at o. By choosing x, y, z suitably we can assume t h a t

Iz := (x 2, xy, y~, z) is the ideal of Z and t h a t

Iz, := (z ~, y, z) is the ideal of Z1. W'e have to distinguish 3 cases: (a) Z1 = Z2 = Z3. Then Le Barz constructs the chart ( r l , 81, tl, (31, C2, C3, V, p, IT)

a r o u n d e as follows: let

r := (ol, o~, o3, Zl, Z~', Z;, Z') be a point of .~3(X) near e. T h e ideal Iz, of Z ' can be written as:

Iz, = (x~ + u x + v y + w , x y + u ~ x + v t y §

+u"x+v'yWw",z+px+ay+O)

for suitable u, v, w, u ~, v ~, u", v", w", p, a, O. Let

(rl,81,tl),

(r2,8~,t~), (r3,83,t3)

be the coordinates of the points Ol, o2, o3. T h e ideal Iz~ of Z~ can be written

Iz~ = (x 2 + aix + b i , - y + cix + di, - z + eix -I- fi) for suitable ai, bi,ci,di, ei,fi. constants can be c o m p u t e d

Now Le Barz shows t h a t from r l , s ~ , t l , c l , c 2 , c a , v , p , a ,

all other and that

( r l , Sl, t l , el, c2, c3, v, p, (7) is a local chart of H 3 ( X ) a r o u n d e. Because of the

s y m m e t r y we can replace r l , s l , t l by r2,s2,t2 or r3,s3,t3 and so also by r : = r l -l-r2 + r 3 , 8 : = 81 -I-82-t-83, t : = t l -t-t2-l-t3.

2. The Betti numbers of Hilbert schemes

70

So we get the local chart (r,s,t, c l , c 2 , c 3 , v , p , a ) around e. With respect to this chart the action of 7" C G(3) on ~r3(Z) is given by ~ ( r ) = r, ~(~) = 8, ~(t) = t,

r(ci) = c~(i), T ( v ) = ~, ~ ( p ) = p, ~ ( o ) =

~.

So we see that (r,s,t, cl + c2 + c3,clc2 + cac3 + c2c3,cac2c3,v,p,a) are local coordinates of )~[3] around ~. (b) Za = Z2 # Z3. In this case we can choose the local coordinates x, y, z in such

a way that the ideal Iz3 of Z3 is given by

Iz, = (x 2, y - x, z). So the ideal Iz~ of Z~ is of the form

Iz,s = ( x 2 + a x + b , - y + ( 7 + l ) x + d , - z + e x + f ) . By [Le Barz 10] (r, s, t, Cl, c2,7, v, p, a) are local coordinates around e. The stabilizer of the operation around e is G(3)e --= {1, (1,2)}. We can choose the coordinate neighbourhood so small that we have ~(Y) n U # 0

*=* ~ G ( 3 ) e .

r, s, t, 7, v, p, a are fixed by the action of G(3)r and we have (1,2)(C1)

= C2, (1, 2)(C2) = C 1,

So (r, 8,t, cl + C2,ClC2,7, v,p,o') form a local chart of )~[3] at ~. (c) The Zi are pairwise distict. We can assume that the ideals Iz2, Iz~ of Z2, Z3 are of the form

Iz2 = (z 2, - y + x, z), Iz~ = (x2,z + y,z).

Then the ideals Iz~, Iz,3 of Z~, Z~ can be written in the form

Iz~ = (z 2 + a2x + b 2 , - y + (7 + 1)x + d 2 , - z + e2x + f2), Iz,a = (x 2 + a3x + b 3 , - y + (7' - 1)x + d 3 , - z + e3x + f3)Le Barz shows that (r, 8, t, cl, 7, 7', v, p, a) form a local chart of H 3 ( X ) around e. The stabilizer of the action of G(3) at e is G(3), = {1}. Again we can choose the coordinate neighbourhood around e to be so small that we have

r(U) n V # O r

r=l.

2.5.

The Betti numbers of varieties of triangles

71

Then (r,s,t, cl,7,7',v,p,a) is also a local chart of )~[31 at ~. Putting things together we have proved (1). We already know from remark 2.5.10 that /) := p-I(Z(~,2)(X)) is a locally trivial Pa-bundle over Z(~,2)(X) = Grass(2, T}). In p a r t i c u l a r / ) is an irreducible divisor on )~[3]. So we can complete the proof of (2) in the same way as that of proposition 2.5.8. [] Keel [Keel (1)] has proved by a different method that the symmetric group G(3) acts on H a ( X ) and that the quotient is the blowup of X [a] along Z(1,2)(X). Let

~3

Hilb (X) C X x X (2) x X [21 x (X[2]) (2) x X [a] be the scheme-theoretic image of .~a(X) under 1x x ~I'x,2 x lxE~l x and let :r2: H 3 ( X ) , Hil"'b3(X)

~xE2~,2 x 1xE~I

be the restriction of this morphism. 2g/22g acts on X a x (X [21)a .x X [a] by permuting the last two factors in X a and (X [~])a simultaniously. This action restricts to an action on Ha(.X). factorizes into

Let 7r2 : H a ( X )

, Hil""ba(X) be the quotient morphism. ~rl

~3(x )

~1

,

~E31

~3

Hilb (X). ~3

We can view Hilb (X) as the variety of complete triangles on X with a marked vertex (or equivalently with a marked side). The projection

Pl,3,5 : X

x X (2) x X [21 x (X [2])(2) x X [31

~ X x X [2] x X [3]

restricts to a birational morphism ~3

~1,3,~: Hilb ~3

Let 10a : Hilb (X)

(X)

~3

, Hilb (X).

, X [31 be the projection. We put:

B(X) := ~;1(Z(I,2)(X))

~3

C Hilb (X)

with the reduced induced structure. B(X) is a P l - b u n d l e over Z(1,2)(X) ---Grass(2, T~c). In fact we can see in the same way as above that B(X) = P(T1) holds, where T1 is the tautological bundle on Grass(2, T} ). We put /~ :=

!~I,~(B(X)) f(x,2[x],Zl,[Z2] t-[Z3],Z)

/

x ~ X;

Zi,Z2,Z 3 ~ 2[2]; Z ~ / ( 1 2 ) ( 2 ) ; "~ Z1, Z2, Z3 C Z' f

s u p p ( Z ) = x;

72

2. The Betti numbers of Hilbert schemes

Then we h a v e / ) = ~r2(E). The action of 2g'/2Zf on hr3(X) restricts to an action on E by permuting Z2, Z3, and the quotient i s / ) . So we get from remark 2.5.7: Remark 2.5.12.

/) ~ (P(T1) x c . . . . (2,T7r P(T1) • = P(T1) x c

....

P(Sym2(T1)),

(2,T})

and the restriction 151,a,s : / ) ~

. . . . (2,T~r P ( T 1 ) ) / ( 2 Z / 2 , ~ )

B ( X ) is the projection onto the first factor.

P r o p o s i t i o n 2.5.13. Let X be a smooth projective variety over C. Then

~3

(1) Hilb ( Z ) is smooth. ~ 3

~ 3

(2) Pl,a,s : Hilb (X) - - ~ Hilb ( X ) is the blow up along B ( X ) . Proof:

~ 3

/51,3,5 is obviously an isomorphism over Hilb ( X ) ( 1 j j ) .

Let (x, Z 2 , Z ) E

~ 3

Hilb (X)(2,1). Then there are two cases: (c0 Z2 = x U y for a point y :fix and Z = 1472U y for a subscheme W2 of length 2 with supp(W2) = x. Then we have

Pl,3,5((z, Z 2 , Z ) )

=

x,[x]4-[y],xUy,[xUy]4-[W2],W2

Uy

9

(fl) supp(Z2) : y # x. Then we have

Z ,Zl)

= {

z ,2[x u

u

}

Now let ~ 3

(x, Z2, Z) C Hilb (X)(3) \ B ( X ) . Then Z2 is the only subscheme of length 2 contained in Z. So we have ~--1 Pl,a,5((x, Z 2 , Z ) ) .=

{( x,2[x],Z2,2[Z2],Z) } .

~ 3

~ 3

As Hilb (X) is smooth, this shows that/51,a,s is an isomorphism over Hilb ( X ) \ B ( X ) . We now show (1). As above we only have to show the smoothness of ~ 3

Hilb (X) in points of/~. We again use the local charts of Le Barz around a point e = (o, o, o, Z1, Z2, Z3, Z) C E. Let g := 7r2(e). We use the same notations as in proposition 2.5.11. There are four cases: (a) Z1 = Z2 = Z3. We see anologously to the proof of proposition 2.5.11 that ~ 3

(r, s, t, cl, c2 + c3, c2c3, v, p, a) form a local chart of Hilb (X) around ~.

2.5. The Betti numbers of varieties of triangle~

73

(b) Z1 # Z2 = Z3. We switch the role of Z1 and Z3 in the case (b) in the proof of proposition 2.5.11. So we see i m m e d i a t e l y that (r, s, t, ^{, c2 + c3, c2c3, v, p, a) form a local chart a r o u n d ~. (c) Z1 = Z2 ~ Z3. We switch the role of Z1 and Z2 in (b) in 2.5.11. This way we see that (r,s,t, cl,7, c3,v,p,a) form a local chart at ~. (d) Z1, Z2, Za are pairwise distinct. Then (r, s, t, cl, 7, 7', v, p, a ) form a local chart near ~. We have proved (1). By r e m a r k 2.5.12 E - - ~ B ( X ) is a locally trivial P2~ 3

bundle. In p a r t i c u l a r / ~ is an irreducible divisor on Hilb (X). Now (2) follows in the same way as in the proof of 2.5.8 and 2.5.11(2).

[]

If we put our results together, we get the following d i a g r a m for the triangle varieties of a smooth projective variety X. x a

XxX

,

(2)

P~

,

Z3(X)

,

p~,~

~(x)

~3 (X) Hilb

l X (a)

, p .....

~3 (Z) Hilb

l ~

~,

X[3]

~

P

Here the horizontal arrows are birational morphisms.

TM

~[a]

2. The Betti numbers of HiIbert schemes

74

Computation

of the Betti numbers

of some of these varieties

To compute the Betti numbers of some of these varieties we will again use the Weil conjectures. So we have to count their points over finite fields. F i r s t we look at the local situation. Let k be a field and R = k[[xl,..., Xd]]. As above H i l b " ( R ) parametrizes the ideals of colength n in R. D e f i n i t i o n 2 . 5 . 1 4 . For all l E zW let W~ C (Hilb2(R)) z x Hilba(R) be the reduced closed subscheme defined by

w; = ~(Ii,.,.,It,.])E(HilbZ(R)) (

' xHilba(R)

I,,...,ItD

J/. )

Now let k be a finite field F o. 2 . 5 . 1 5 . There is a finite field extension FQ of ~'O such that for all finite extensions ~'q of ~'Q:

Lemma

(1)

- qd) (1

(1 - qd-1)(1

Iw~(F~)I= il--q--~-q~ "-+q)'+

qd-1 1 -- qd

1-q

In particular

(2)

i w O ( F q ) l = (1

-

qd)(1

iT-

Iw~(Fq)l

l

-

qd+l)

'

- (1 - qd)2

5-?V' I w ~ ( z % ) l = (1 - ( ) ( 1

+ 2q + q2 _ 3 (

- (+1)

(1 - q)2

P r o o f i We have the stratification

Hilba(R) =

Z(1,1,1 ) [-J Z(1,2).

Over the algebraic closure F q , the stratum

Z(1,1,1 ) is a fibre bundle over P d - 1

with fibre A d-1. We choose the extension ~b-'Q in such a way t h a t the fibre bundle structure and a trivializing open cover are already defined over F Q . Now let ~'q be a finite extension o f / F Q . Let m = ( z l , . . . , Zd) be the m a x i m a l ideal in R. An ideal d I E Z(1,2)(~Tq) corresponds to a 2-codimensional linear subspace of ( m / m 2) = Lb~q. So we have Z(1,2)(Fq) ~ Grass(2, F~). A n ideal I E Z(1,1,1)(Fq) is contained in a unique ideal I ~ = I + m 2 of colength 2 in R. Let I E Z(1,2)(Fq). Let f : m

~ m / I be the canonical projection. T h e n the

2.5. The Betti numbers of varieties of triangles

75

ideals of colength 2 in R containing I are those of the form f - 1 (V) for V e P ( m / I ) . So we get iWg(Fq) I = [Z(1,2)(X)(Fq)I( 1 4- q)t 4- [Z(1,1,1)(X)(Fq)[. (1) follows. (2) follows from (1) by an easy computation.

.~itPq be

From now on let /F'Q be as in 2.5.15 and let Let X be a smooth projective variety over F q .

G a finite extension of F Q .

D e f i n i t i o n 2.5.16. We write V~ instead of H i l b n ( R ) ( F q ) and put

T 1 l= X ( F q ) ,

T2 := { M c X(IFq) [MI = 2} U P2(X, Fq) U (X(Fq) x V2),

IMI = 3} u (X(Fq)

Ta := { M C X ( F q )

[--J{ {Xl,(x2,b)}

x P2(X,~'q)) u Pa(X, F q )

Xl r x2 ~ X ( ~ q ) , b

~ g2 } [.J (X(~i'q) x g3) 9

Recall the notations from 2.3.6. We identify a m a p f :

P(X, Fq)

----+ V(.~gq) with

the set

{(~,/') E P ( X , ~ q )

X ( V ( F q ) \ Vo(Fq))

f(~) = I }

and the set M x VI(Fq) with M. In this way T2 is identified with T2(X,_Fq) and T3 with T3(X, Fq) (see definition 2.3.6). Via these identifications the relation C carries over to T1, T2, T3. So by 2.3.7 there are bijections r

= 1x(Fq) : X ( F r

' T1,

r

: xI2l(Fq) ~

T2,

r

: X[3I(Fq) ----* T3,

respecting C.

Lemma

2.5.17.

d-1 Ixt3](G)l = Ix(3)(Fq) I + q k ~ l- X (Fq)l q

(1)

2

4- q2 (1 -- qd-1)(1 -- qd) Y( ~ ,,

( ~ _~ q--~ -- q q

(2)

~3

IHilb (X)(Fe)I = I(X

x

X(2))(Fq)I

, . t,.l~ q ) [,

_ d-1

4-

2qllq~qlX(Fq)[ 2

76

2. The Betti numbers of Hilbert schemes

q_ q2 (1 -- qd-1)2

t~r3(x)(Eq)l = l X 3 ( F q ) l + 3 q = l ~ q

(3)

(1 -

+ q

IX(Eq)l u

qd-J)(1 + 3q _ 3qd _ qd+l (1 -

q)2

) [x(zv~)l.

Proof: Immediately from the definitions we get

~3

(~2 • e3)(Hilb (X)(1Fq))= pairwise distinct

{x],x2}, {Xl,X2,Xa})

v (z(~q) • P~(X,~))

U{({ZI,Z2},{(xl,b),x2})

X, # X2 e X(.~q) be V2}

and ((1X(E,)) 3 x Cza • r

{

=

(Xl,X2,x3, {x],x2}, {x2,xa}, {xa,x,}, {Xl,X2,X3})

U{(xl

,Xl,X2,{Xl,X2},{Xl,X2},(Xl,b),{(xl,b),x2})

Xl'X2'xaeX(~q)} pairwise distinct

X(Fq), t xl ~ x2bee V2

}

I

U{(x 2, Xl , Xl,(Xl.,b),{Xl,X2},{xl , x2},{(Xl,b),x2}) o

(.,x,x,(x,~),(.,<),(x,~),(x,c))

x I ~X2ebew2X(.~q),}

b~ >~, ~ >c, < > e

"

~3

We sum the numbers of elements of Ta, (r • e3)(Hilb (X)(~q) and (13(Eq) x r x ea)(/ta(X)(/Fq)) respectively. Then we use remark 1.2.4 and lemma 2.5.15 to get

IX[3](~q)[ = (]X(f q)l) -~-IP2(X, ff2q)l[X(.~q)I -}-[P3(X,-~q)[ 1 qd + ~ _ q X(Eq)I(IX(~'q) I - 1) -

(1 -

+

q~)(1 - q~+~)ix(E~)l,

(1 - q)(1 - q~)

2.5. The Betti numbers of varietiea of triangles

~3 IHilb

(X)(.~?q)l

= 3

(IX(fq)l)

+

~- I P 2 ( X ,

JFq)IIXOFq) I

1 a- d

2-;~:IX(.Fq)I(IX(JFa) I 1 _ ( / -

I) +

(1-qd)2 ~ 7~

1-- d

I~r~(x)(F~)l = 6 ( I X ( f q ) l ) +

+

77

3~qqlX(Fq)](lx(.~'q)l

-

IX(~)l,

1)

(1 - qa)(1 + 2q + q2 _ 3qd _ qd+l) (1

-

IX(Fq)l.

q)2

By remark 1.2.4 we have

IX(2)(Fq)l =

+

IP~(X, Fq)IIX(F~)I + IPz(x, Fa)l,

+

IP~(X~)l.

So we get

qd

1) Ix(F~)I z 1-r - _~qd)(1 q--~-i-~-qd+l) q~l--q/lx(F~)l, 1

-

IX[3](Fq) I _- IX(~)(Fq) I + k -~ _-q -

+((i

-

-

--3

(21-q d

IHilb ( X ) ( F q ) I = IX(2)(Fq)IIX(Fq)I + k

1~ q

2

)

[X(Fq)I 2

((1:qd) 2 21--qd~ + \ (1 - q)2 + 1 1 - q / IX(Eq)l,

1Lr3(X)(Fq)l -- lx(l~q){3 '~ (~3 1 1 _- qd q _ 3 ) IX(Fq)l 2 + ( ( 1 - qd)(1 + 2q + q2 _ 3qd _ qd+l) 31 - qd~ + 2 - 1_-77 / [x(F~)I, (1 - q)2 and the result follows by an easy calculation.

Theorem

(1)

[]

2.5.18. Let X be a smooth projective variety over C. Then we have: p ( X [a] , z) = p ( X (3), z) + z 21 - z 2d-2

1 - z 2 p(X,z) 2

+

__ z 2 d - 2 ~ l ~ __ 2d

-

)(-

)

)p(X,z),

1 X , - z 2 ) p ( X , z ) + ~1 p ( X , - z a ) p(X E~1,- z ) = ~ p ( X , - z ) ~ + ~p(

78

2. The B e t t i numbers of Hilbert schemes

1 + z2 -

+z 4

(2)

z 2d-2

-

-

1 - z2

p(X,

-z)

2

(1 - z Z d - 2 ) ( 1 - z 2~)

p(X,-z),

p ( Z [31, z) = p ( X (3), z) + z 2 1 - z 2d-2 1 - z 2 p(X,z) 2 + z2 (1 - Y - 2 ) ( 1

+ z2 -

z 2~ -

z2d+2)p(X

' z),

(1 - z 2 ) 2 p ( ) ~ [ a ] , _ z ) = -1~ p ( X , - z ) 3 + ~p( 1 X ,-z2)p(X,

z) + l p ( x , - z a )

1 -- Z 2d-2

+ z2 -

-

1 - z2

- z) 2

p(X,

+ z2 (1 - z ~ d - 2 ) ( 1 + z 2 - z 2d - z 2d+2) (1 -

p(X,-z),

z2) ~

3

(3)

Z 2d-2

p ( H i l b ( X ) , z) = p ( X , z) • p ( X (2), z) + 2z 21

-1 -

z2

p(X,z)

2

(1 - z 2 d - ~ ) 2

p(Hilb (X),-z)

= ~ (p(X,-z)

3 + p(X,-z2)p(X,-z))

+ 2z 2 1 - z 2d-2

(1 - z2d-2) 2

1 - z2 P(X,z)

+z4

3

(4)

.p(X,-z),

z 2d-2

p ( H i l b ( Z ) , z) = p ( X , z) x p ( X (2), z) + 2z 2 1 1 - z 2 p(X,z) 2 + z2 (1 - z2d-2)(1 + 2z z -- 2z 2d -- z 2 d + 2 ) p ( X ' z), ( 1 - z2) 2 p(Hilb (X),-z)

=

p(X,-z)

3 + p(X,-z2)p(X,-z))

+ 2z 2 1 -- z 2d-2 1- z 2 p(X,-z)

2

+ z2 (1 - z2d-2)(1 + 2z 2 -- 2z 2d (1 - z 2 ) 2 (5)

~2d+2~ -

~

Jp(X,

-z),

p ( H 3 ( X ) , z) = p ( X , z) 3 + 322 1 1- -z 2d-2 z 2 p(X,z) 2 + z2 (1 - z2d-2)(1 + 322 - 3z 2d - z 2 d + 2 ) p ( X ' z).

(I Proofi

-z~) 2

X is defined over a finitely g e n e r a t e d ring e x t e n s i o n T of 2~, i.e. t h e r e

is an X T over s p e c ( T ) satisfying Z T

X T

C

=-

X.

~ 3

Let Y = X T

XT

(T/m)

be a

~ 3

good r e d u c t i o n of X m o d u l o q. T h e n y[3], Hilb (Y) a n d / ~ 3 ( y ) X[3], Hilb ( Z ) and H 3 ( X )

are also r e d u c t i o n s m o d u l o q, a n d we can choose the m a x i m a l ideal

m E s p e c ( T ) in such a way t h a t t h e y are all g o o d r e d u c t i o n s (see the r e m a r k s

2.5. The Betti numbers of varieties of triangles

79

before theorem 1.2.1). Choose m in such a way that furthermore l e m m a 2.~15 holds. Then (1), (3) and (5) follow immediately from l e m m a 2.5.17, remark 1.2.2 and Macdonald's formula. Z(~,2)(X) is a Grass(2, d)-bundle over X. So we have

p(z(1,~)(x))

(1~1 z2d-2)( 1 _ z 2d)

--~)-5-z~

=

p(X,z)

By proposition 2.5.11 w e get

p(2t31z)=p(xE~Jz)+

(1 -- z2d-2)(1 -- z2d)l 2

(~:~-~/:z~

~ +

Z4

+

z6)p(X,z).

So (2) follows from (1) by an easy computation. B ( X ) is a P l - b u n d l e over Z(1,2)(X). So we have by proposition 2.5.13

A ~ ( X ) , z ) = p(Hilb ~ p(Hilb (X),z) +

(I

- ~ ) (1l __ - _z4 z ~)) ~ z,~ + z4)p(X,z) + z2)(1(1: _z ~ z2)(

(4) follows again by an easy computation.

[]

For a smooth projective surface S over C these formulas can be written as follows: p ( S [3] , Z) : p ( S (3), Z) -Jr-z2p(S, Z) 2 -4- z4p(S, Z) ~3 p(Hilb (S) = p(S • S (2), z) + 2z2p(S, z) 2 + z4p(S, z)

p(~[31,z)

= p(S (~), z) + z~p(S, z) ~ + (z ~ + 2z 4 + z~)p(S, ~)

~3

p(Hilb (S), z) = p(S x S (2), z) + 2z2p(S, z) 2 + (z 2 + 3z 4 -{- z6)p(S, z)

p(.ff~(s), ~) = p(S, ~)~ + 3z~p(S, z) ~ + (z ~ + 4z ~ + z~)p(S, z) Now we consider the case of projective space Pd. The Chow groups A i ( P [3]) and A i ( H 3 ( p d ) ) have already been determined in [Rossell6-Xambo (2)].

Proposition

2.5.19. p[3], --3 (Pal), H~b3(pd) and _~3(pd) all have a cell d ~[3], d Hilb decomposition. In particular for Y one of these varieties H2i+I(Y, 2Z) = O; the groups Ai(Y) = H2i(Y, 2~) are free, and their ranks can be computed by theorem 2.5.18. Proof." Let To,... ,Td be homogeneous coordinates o n P d . For i = 0 , . . . n let Pi be the point for w h i c h T i = l a n d T j = O f o r i # j . Let r C S l ( d + l , C ) be the maximal torus of diagonal matrices and let A0,..., Ad be the linearly independent characters o f t for which any g E I" is of the form g = diag(Ao(g),..., Ad(g)). Then F acts on Pd by g. Ti := Ai(g)Ti. T h e fixed points are p 0 , . - - , P d . We have an induced

80

2. The Betti numbers of HiIbert scheme~

action of P on p~n] for all n, as F acts on the homogeneous ideals in To,... ,Td. A subscheme Z E p~n] is a fixed point of this action, if and only if its ideal is generated by monomials in To,..., Td. So the action of F has only finitely many fixed points on P[dhI. The same is true for a general one-parameter subgroup of F. We fix a onep[2] parameter subgroup 9 of F which has only finitely many fixed points on Pd, ~d and P[d3]. The induced action of 4) on P(d3) x (P[d2])(a) x p~3] and P d x P[d21 X p[a] and on the quotients Pd • p~2) x P[d2] • (pd[2])(2) • Pd[3] and (pd)3 • (Pd[2])3 X p!3] restricts to an action on the subvarieties ~3] , Hilb - - 3 (Pd), Hilb 3 (Pd) and Ha(Pd). As the action on Pd, P[d2] and P[d31 has only finitely many fixed points, it has only finitely many on Pd x P[d2] • p~3] and (Pd) 3 x (P[d2])3 x p~3]. The fixed points on the quotients P d X p~2) x P[d2] x (p~2])(2) x p[a] and (Pd) 3 X (P[d2])a x P[d3] are the images of the fixed points on (Pd) 3 X (p~2])a • p~a] under the quotient map. So there are also only finitely many. In particular the action of 9 has only finitely many fixed points o n ~ 3 ] , Hilb ~ 3 (Pal), Hilb ~ 3 (Pd) and Br3(pd). As these are smooth, they have a cell decomposition. [] ~

3. T h e varieties o f s e c o n d a n d higher o r d e r d a t a The second part of this work (chapters 3 and 4) is devoted to the computation of the cohomology and Chow rings of Hilbert schemes. In chapter 3 we define varieties of second and higher order data on a smooth variety X and study them. In section 3.1 we consider the varieties D~(X) of second order data of m-dimensional subvarieties of X. We define D~(X) as a subvariety of a product of Hilbert schemes of zero-dimensional subschemes of X. Then we show that D~(X) can be described as a Grassmanian bundle over the Grassmannian bundle of m-dimensional subspaces of the cotangent bundle of X. D~(X) is a natural desingularisation of X (3)" [3] Using the description as a bundle of Grassmanians we compute the ring structure of the cohomology ring of D~(X). Then we descibe in what sense D~(X) parametrizes the second order data of m-dimensional subvarieties of X and the relation to second order contacts of such subvarieties. In section 3.2 we consider the varieties of higher order data D~(X). Their definition is a generalisation of that of D~(X). We show that only the varieties of third order data of curves and hypersurfaces are well-behaved, i.e. they are locally trivial bundles over the corresponding varieties of second order data with fibre a projective space. In particular D3(X) is a natural desingularisation of ~c[4] "~(4)' Then we compute the Chow ring of these varieties. As an enumerative application of the results of chapter 3 we determine formulas for the numbers of second and third order contacts of a smooth projective variety X C PN with linear subspaces of PN. In section 3.3 we introduce the Semple bundle varieties Fn(X), which parametrize higher order data of curves on X in a slightly different sense. We use them to show a general formula for the number of higher order contacts of a smooth projective variety X C PN with lines in PN. Arrondo, Sols and Speiser [Arrondo-Sols-Speiser (1)] have independently constructed new contact varieties for m-dimensional subvarieties of a given variety X, for which they also give a number of applications. Their approach is different from the one of sections 3.1 and 3.2 and is in fact a generalization of the Semple bundle construction. This approach is more general then mine, as it gives varieties of arbitrary order. It has however the disadvantage of not taking the commutativity of higher order derivatives into account, and thus, except in the case m = 1, the actual data varieties are given as subvarieties (by requiring "symmetry") of considerably bigger varieties. The precise description of these subvarieties appears to be not a very easy task, and as far as I know has been carried out only in the case of second order data of surfaces in P3.

82

3.1. T h e varieties o f s e c o n d o r d e r data. Let X be a s m o o t h projective variety of dimension d over an algebraically closed field k. In this section we want to define a variety

D2m(X) of second order d a t a of

m - d i m e n s i o n a l subvarieties of X for any non-negative integer m < d. A general

D2m(X) will correspond to the second order d a t u m of the germ of a smooth m - d i m e n s i o n a l subvariety Y C X in a point x C X , i.e. to the quotient Oy,~/m3x,~ point of

of Ox,~. Assume for the m o m e n t t h a t the ground field is C and x 6 Y C X , X is a smooth complex d-manifold and we have local coordinates

zl,..., Zd at x. Then Y

is given by equations

f i ( Z l , . . . , Zd)~-O Then the second order d a t u m

i= 1,..., d-m.

Oy,,/m3x,, is

C[zl, . . . ,

Zd]/((fl,..., fd-m) + m3),

and giving the second order d a t u m is equivalent to giving the derivatives

0f~

Ozj(X), i = l , . . . , d - m , 02 fi , , O~jOzt~X), i = 1 , . . . , d - m ,

j= l,...,d j,l = 1,...,d

N o t a t i o n . In chapter 3 and 4 we will often use the G r a s s m a n n i a n bundle associated to a vector bundle. So we fix some notations for these. Let S be a scheme and E a vector bundle of rank r on X . For any m < r let

Grass(m, E) denote the G r a s s m a n n i a n bundle of m - d i m e n s i o n a l quotients of E. Let 7rm,E : Grass(m, E) -----* S be the projection, Qm,E the universal quotient bundle of ~rm,E(E) a n d T~-m,E the tautological subbundle. T h e n the projectivization of E is P ( E ) = Grass(r-1, E) and Op(E)(1) = (T1,E)*. We also p u t 15(E) : = Grass(l, E). We write Grass(re, r) for the G r a s s m a n n variety of m-dimensional quotients of C ~. Let Qm,~ and T~-m,~ be the universal quotient bundle a n d the tautological s u b b u n d l e on Grass(m, r ).

Notation.

For subschemes

respectively in

Z1, Z2 of a scheme S with ideal sheaves :Z'zl, Iz2

Os, let Z1 9 Z2 denote the subscheme Z of S whose ideal sheaf I z is

given by 2"z : = Zzl " I z 2 . As above we will write Z1 C Z2; to mean t h a t Z1 is a subscheme of Z2. In this case we will write

Zzl/z2 for the ideal of Z1 in Z2.

3.1. The varieties of second order data

Definition 3.1.1.

83

Let 7)2re(X) be the contravariant functor from the category of

noetherian k-schemes to the category of sets which for noetherian k-schemes S, T and a m o r p h i s m r : S ~

T is given by:

Zo,Z1,Z2 C X x T closed subschemes fiat of degrees 1, m + 1, (%+2) over T Z o C Z a c Z 2 , Z1 c Z 0 " Z 0 , Z2 C Z0 9 Z1

"D2(X)(T)= { (Z~ Vi(X)(r

:

VI(X)(T)

(Zo,Zl,Z2) ~

,

] ,

~i(x)(s)

(go XTS,Z1 XTS, Z2 XTS).

L e m m a 3.1.2. :D2(X) is representable by a closed subscheme D 2 ( X ) C X x Zlm+11 • X[C+2)]. P r o o f i Let

Z l ( X ) := A c X • X

Zm+l(X) C X z(,o:~)(x) c

x

• X [m+l] •

x('~t ~)

be the universal subschemes. To shorten notations we write

w := x • xEm+'I • X [(mt~)] For i = 1,2,3 let Pi be the projection of W to the i th factor. Let 2-0, 2-1, 2"2 be the ideals of

Wo := ( i x • p~)-~(Zl(X)), Wa : = (1x • p2)-l(Zm+a(X)),

w2 := ( i x • p3)-l(z(m+~)(x)) respectively in Ox • w. Let U0, U1,0"2 C W be the subschemes defined by 2-0 + 2"1, 2-1 + 2"2 + 2"~ u n d 2"2 + 2-0 92"a respectively. T h e n we have obviously Ui C Wi for i = 0,1,2. As X is a closed subvariety of a projective space P N , Wo,W1,W2, [To, U1, U2 are in a n a t u r a l way subschemes of P N • W . The Wi are flat of degree (i+m) o v e r W f o r i = O , 1,2. W e p u t ~" := Ouo @ Ou, G Ou2. For may m o r p h i s m g : T

, W of a noetherian scheme to W we p u t

~

:=
84

3.

The varieties of second and higher order data

on P N x T. Let rrT : P N • T ~ T be the projection. By [Mumford (1) Lecture 8] there is a closed subscheme D2m(X)C W such that the following holds: (~rT).orgis locally free of rank

r1:=1+m+1+ ( m ; 2) over T if and only if g factors through D2m(X). (D~(X) is closed and not only localty closed as each Ui is a subscheme of the corresponding l/Vi, and so (TrT).(.~g) can at most have rank rl in points of T.) By the relations U0 C W0, U1 C W1, U2 C W2, (Yrr).(Yg) is locally free of rank rl if and only if (1p N

xg)-l(ui)=(1pN •

i =0,1,2.

Now we can easily see from the definitions that

D~(X)

represents the functor

[]

D e f i n i t i o n 3.1.3. Let

D2m(X) C

X x X [m+l] x X [('~:~)]

be the subscheme representing the functor Z)~(X) by lemma 3.1.2. As a set it is given by

{

(X, Zl,Z2) E X x X [m+l] x X [ ( m ~ 2 ) ]

X"C Z1 C Z2' } Z 1 Cx.x, Z2 C :E' Z 1

Later we will see that D~(X) is reduced and even smooth. D2m(X)is called the variety of second order data of m-dimensional subvarieties of X. Analogously we define D~(X) as the closed subscheme of X x X [m+l] that represents the functor given by

~I(X)(T)

:=

(No, Z1)

flatZ0,Z1 C X x T closed subschemes } of degree 1, m + 1 respectively over T . Z0 c Z 1 c Z 0 " Z 0

D~(X) is the variety of first order data of m-dimensional subschemes of X. As a set it is obviously given by

{(x , Z ) E X • We will also see that

D~(X) is

[m+l]

xCZCx.x}.

smooth.

For a surface S the variety D~(S) is considered in the literature (using a slightly different definition). It is called the variety of second order data on S and denoted

3.1. The varieties of second order data

85

by D(S). D2(p2) was studied extensively in [Roberts-Speiser (1),(2),(3),(4)] and [Roberts (1)] to find enumerative formulas for second order contacts of families of curves in P2. For a surface S the variety D~(S) has been studied in [Collino (1)], and there its eohomology ring was determined. In [Le Barz (10)] D21(X) has been defined for a general smooth projective variety X over C as a subvariety of ~r3(X) (see section 2.5). We now give another definition / ) ~ ( X ) of D2m(X), which will enable us to compute the Chow ring of this variety. We then have to show that D~(X) and /~2m(X) are isomorphic.

Let again JR(X) be the rtth jet-bundle of X. Let ~'1 : Grass(m,T}) -----+ X be the projection and T1 := Ta-m,T}, Q1 : = Qrn,T;;. We also write /~lm(X ) for arass(m,T~). Let jl : #~(T}) , #~(JI(X)) be the canonical inclusion and D e f i n i t i o n 3.1.4.

0

----,

T1

it

---.

~,

,

~I(T})

~

Q1

~

0

the canonical exact sequence. We define the vector bundle ~)1 on ] ~ I ( X ) by the following commutative diagram with exact rows and columns. 0

0

)

T1

il '

~ , (T/~) , rrl

)

T1

i '

7]'• ( J 1 ( X ) )

0

~

0

~

Q1

~

0

q~ ----+

O1

----+

0

~

0

Ob~(x )

O~(x )

I

I

0

0

Let s2(q,) : Sym2(#~(T~)) ---+ Sym2(Q1) be the morphism induced by the quotient morphism q, : ~'~(T~) ---+ Q1, and let j2 : #~(Sym2(T~)) , ~~(J2(X)) be the canonical map. We define the vector bundle T1 on Grass(m, T~) by the following diagram

3. The varietie~ of second and higher order data

86

with exact rows and columns in which the right lower square is cartesian 0

0

T 0

'

(~)1

T 0

--~

01

T

~ (Sym: (T~())

J~,

To~

~(J2(X))

--~

l ~1 0

--~

Sym2 (5; (T~())

~2

-----~ 0

T1

~(Jl(X))

[]

l'

--~

T1

T

T

T

0

0

0

~

0

---*

0

and W~(X) and (T1 9 T~() by the following diagram with exact rows and columns in which the left lower square is cartesian

0

0

0

0

0

l

l

(T1.T~)

( T1. Tj, )

1

l

~t(Sym2(T])) l.~(ql) Sym2(QI)

~ --~

T1

[]

IP

~

~

0

l --~

W2m(X) ~

T1

, 0

T1

, 0

l

1

1

0

0

0

Obviously W2m(X) is a vector bundle of rank r=

rn + 1) 2 +d-rn

over Grass(m, T~). (TI" T~c) is also the image of the subbundle T1 | h~'(T~) under the natural vector bundle m o r p h i s m s2 : # t ( T ~ | T~() , ~-~(Sym2(T~)). We can see easily that (T1 9 T~c) is a vector bundle, and from the diagram we get Sym2(Q1) -- Sym2(T~ )/(T1 9T~ ).

3.1. The varieties of second order data

87

s 2(q,): Sym 2(T~) - - ~ Sym2(Q1)is t h e quotient map. Definition 3.1.5. We put

b~(x)

::

ar~((~t'), W~(X)).

Let ~2 := D 2 ( X ) --~ Grass(m,T~c) be the projection. Let T2 := Td_m,w~(x). Let o

--~

T2

i2

---~

~;(Ws

-~

Q~

--,

o

be the natural exact sequence. We define the vector bundle T2 o n / 9 2 ( X ) by the following diagram with exact rows and columns in which the upper right square is cartesian.

0

~

0

0

0

T

T

T

T2

i2 ,

~r~(W~(X))

--~

0

--~

T

T

~;(T~. T~)

~ ( T ~ . T~)

T

T

0

0

q: ,

Q2

--,

o

,

Q2

--,

0

l ,

0

The vector bundle Q2 is now defined by the following diagram : 0

0

0

l

1

l 0

1~;(;1)

0

~;2

~ (Q1)

_, N ) Ir2(Q1

1

1

0

0

--*

0

---*

0

3. The varieties of second and higher order data

88

From these diagrams we can read off the exact sequences 0

---*

Sym2(Qi)

W~(X)

~

02

T1

----+

and

on D I ( X )

and -----+

2 ((~1)

'

0

on D ~ ( X ) .

D e f i n i t i o n 3.1.6. For any n E iN let as above Z~(X) C X x X In] be the universal subscheme with the projections:

z.(x) Jp.

"N qn

X

X In] .

Let

~1 : D ~ ( X ) ~

XI~+il,

r 2 : D ~ ( X ) ---, X [(~2+2)] be the projections. We put

(ox)~

: : q(q,.+~).(Oz,~+~(x)) = q(q,.+~).v;~+~(ox), * )) ,(Oz
rank ( ' 2+2) on D ~ ( X ) . Let A C X x X be the

sheaf. For all n E iW let A n be the subscheme of X x X defined by ( Z a ) n (which has support A --~ X). Let sl,s2 : X x X ~

y,(x) : (~2),(Ox• Let r : T ~

X be the projections. T h e n we have

)n+')

: (~2),(o~o+~).

X be a morphism from a noetherian scheme. We define Ar C X • T

by

A

---*

T [] Ar ~

XxX

T l~x~ x xT.

3.1. The varieties of second order data

89

Then the projection PT : A S ~ T is an isomorphism. Analogously we define for all n C SV the subscheme A~ C X x T by A '~

T

,

[]

3.1.7.

l lxxq~

, XxT

A2

Theorem

X xX

There exist isomorphisms ~)1 : D I ( X ) ~

r

G r a s s ( m , T ~ ),

D~(X) ~

D2m(X),

for which the diagram D~(X)

D~(X)

,

Grass(m,T~()

X commutes such that a;(O,) = ( o x L ,

Proof." With the notations of definition 3.1.6 we can rewrite the functors

Z~L(X)as:

DI(X)(T) := { (r Z,)

~L(X)(T)

:={(r Zl, Z2)

D~(X),

r ,X } Z1 C X x T closed subscheme flat of degree m + 1 over T with ' A s C Z1 C A~

$ : T ----~ } Z1, Z2 C X x T closed subschemes flat over(m+2Tof degrees m + 1 and respectively with A o C Z1 C A~; Z1 C Z2 C z2kr 9 Z 1

"

3. The varietie~ of ~econd and higher order data

90

Let Zl C X x D~(X) be the universal family of subschemes fiat of degree m + 1 over DI(X). Then we have A,,~ C Z1 C A 27r1 9 Let ql : Art ) Dim(X) be the projection. (ql),(2"/N~/z~ ) is a locally free quotient of (ql),(ZA~/A~) ----~r~'(2r~() of rank m. This defines a morphism

Dtm(X)

r

>Grass(m,T~ )

over X, We get the inverse as follows: for the variety A~, C X x Grass(m,T~() the projection Pl to Grass(m, T}) is an isomorphism, and we have

+l(J.(X))

(p,).COA_.+,).

=

The quotient (~1 Of fr~(Jl(X)) = (p,),(O/,~) defines a subscheme Zl C A2,~, satisfying A ~ C Z1. The pair (~'1, Z1) defines the required m o r p h i s m

~i : Grass(m, T~) ~ over X. We see that r To construct r

r

DI(x)

is the inverse of (~lwe proceed in a similar way.

Let Z1, Z2 C X x D2m(X) be the universal subschemes of degrees m + 1, (,,+2) over D~(X). Via r we identify Grass(m, T~:) with Dim(X). By definition we have

A+,,.-,+o.,.,.2C Z1 Zl

C

Z2

C

A.lorr

2 9

Z1

C A ~,ri-i o .ri-2 , C

A 37pl o 1T2 .

Let ql be the projection of A . . . . ~ to D2m(X). (qa)*(Zz, Iz~ ) is a locally free quotient of

(ql ).CSz;IA.,..o.:,.Z~ ) = 7r~CW2(X) ) of rank (m2+1). This defines a morphism r

DZm(X) ----+D~m(X) over DI(X).

Let Z1 := ~-1 (W1) where W1 is the universal subscheme over D ~ (X) of degree m + 1. Aeioe, C X x / 9 ~ ( X ) is via the projection to the second factor isomorphic t o / ) ~ ( X ) . We have #~(~-~'(J,~(X)))

=

(p2)*(OA;+),~).

T2 is a subbundle of ~ ( T 1 ) and #;(T1) is a subbundle of (p2).(OA]10+2). By the definitions g'~(T1 9 T~) is a subbundle of T2. Let I2 C OA~10+2 be the O/,~1o+ 2submodule with (P2).(I2) = T2. As T2 is a subbundle of ~ ( T a ) , we have /2 C

ZZ, IA~to, ,. As ~'~(T1 9T~:) is a subbundle of :F2, we have

3.1. The varieties of second order data

91

So we have in particular

Oa~_~1 o~-2 912 C 12. 3 So 12 is an ideal in O ~ o , 2 , and thus defines a subscheme Z2 C A~o~ 2. By I2 C Iz~/zX~o, ~ we have Z1 C Z2 and by 9 2"a.

_ ,a~

C

12

we get Z2 C A~o~ 2 9 Z1. The triple (#lo#2,Z1,Z2) defines the morphism ~2 : b~(X) , D~(X) over r satisfying r

*

2

Obviously we have ~b2 ---- (~2-1 9

= (#~r~(J2(X))/T2 = Q2. U]

In future we want to identify /91(X) with D~(X) a n d / 9 ~ ( X ) with D~(X) via r and 42. Remark

3.1.8.

(1) The closure of Z 0,m,(.,2+~)) (X) in X [(,,:+2)] is Z(1,m,(,~+~)) (X) = { Z C X[ (m+2)] therewitharexXcEz1X'cZ1zEcZO,m)(X)x 9Z1 } " (2) The projection r2 : D2m(X)

~

Z(1,rn,(m:,))(X) is a natural resolution of

9

(3) r2 : D~(X)

, --(a) y[S] is a natural resolution of y[a] "'(a)- It is the blow up along

z(1,2)(x). Proof:

D~(X) is closed and irreducible. By the

image of the projection r2: D~(X) ~

definitions

Z(1,ra,(m+,)) (X) is the

X [(m~)]. As D2m(X) and Z0,m,(,,+l)) (X)

are smooth, we can easily see that r2 is an isomorphism over the open subset Z(1,m,(m+l))(X ) of-Z(1,m,(m+l))(X ). r21(Z(1,m,(m+l)))(X ) is dense in D2~(X), as D2m(X) is irreducible. So Z(,,m,(m2+,))(X) is the closure of Z(1,m,(m+,) ) (X) in X[(~+2)]. As D2m(X) is smooth, it is a resolution of Z0,m,(,~2+~))(X). It is easy to see that "'(3)Y[a]is the closure of Z(1,1,1)(X). r2 : D21(X) . ' v[a] "'(a) is an isomorphism over Z(1A0)(X ). Z(la)(X) has codimension 2 in y[3] "'(3), as X[3] (a) = Z(1,1,1)(X

) IJ

Z(1,2)(X),

92

3.

The varieties of second and higher order data

and Z(1,1,1)(X) is an A d - l - b u n d l e over P ( T } ) and Z ( 1 , 2 ) ( X )

=

Grass(2, T~). We

have the exact sequence

o

~

O~ ~ ~

w?(x)

~

r~

--~

o.

Let

D~(X)oo := P(T~) C O(W~2(X)) = D~(X). We see that

D{(X)~ is an irreducible divisor in D~(X) and (D12(X)oo) =

r;l(z(1,2)(X)).

(3) follows with the same argument as in the end of the proof of proposition 2.5.8. []

For the rest of section 3.1 let di, el, fi,gi be variables of weight i. Each class b ~ Ai(X) will also be given weigth i. Let E be a vector b u n d l e of rank r over X. T h e n it is well known (cf. e.g. [Fulton(I) ex. 14.6.6]) that we have for the Chow ring

A*(arass(m, E)) =

A*(X)[dl,..., d . . . . e l , . . . , em] djei-j = ci(E),

(1 < i < r

,

where we have formally put do = 1,e0 = 1 and dj = 0, e~ = 0 for j > r - m, l > m respectively. One can summarize these relations to (1 + dl + . . . +

dr-m)(1 +

el + . . .

"~ ern) =

c(E).

One has to note that the relation holds for every weight. We have

e(Tr-m,.) = (1 + dl + . . . + dr-m), c(0m,E) = (1 + el + . . . + ~m). In the case of a projective b u n d l e P ( E ) we get in particular A * ( P ( E ) ) --

A*(X)[P]

where P = cl(Op(E)(1)).

For the Chern classes of a symmetric power of a vector bundle we have the well-known relation:

3.1. The varieties of second order data

93

R e m a r k 3.1.9. Let E be a vector bundle of rank r over X with total Chern class c(E) = 1 + el + . . . er. Let c(E) = (1 + Yl)... (1 + y~) be a formal splitting of c(E). Then we have c(Symm(E)) =

( l + y i , +...+Yim).

H il <...
If E has rank 2, we have c(Sym2(E)) = (1 + 2e, + 4e2)(1 + e,) = 1 + 3el + (2e~ + 4 e 2 ) H- 4ele2 c(Syma(E)) = (1 + 3el + 9e2)(1 + 3el + 2e~ + e2)

= 1 + 6e~ + (lle~ + 10e:) + (6el + 30e,e:) + lSe~e~ + 94 c(Sym4(E)) = (1 + 2el)(1 + 4e1 + 16e2)(1 + 4e1 + 3el2 + 4e2)

= 1 + 10el + (35e~ + 20~2/+ (50e~ + 120e,e~) -t- (24el4 + 20Sere2 + 64e 2) + 96e31e2 + 128e, e~ c(SymS(E)) = (1 +

5e I

-~- 25e2)(1 +

~- 4e~ + 9el)(1 +

5e I

5el

-~ 6e~ + e2)

= 1 + 15el + (85e~ + 35e2) + (225e~ + 350ele2)

+ (274el~ + llSae% + 2594) + (274e~ + 1540e~e2 + 1295ele~) + 600e14e2 + 1450e~e~ + 225e~. If E has rank 3, we get c(Sym2(E)) = (1 -F 2el -F 4e2 -{- 8e3)(1 -F 2el + el2 -t- ele2

--

e3)

= 1 + 4e, + (5~ + 5e~) + (2e? + n e l e : + 7e3)

+ (6e~e: + 44 + 14ele3) + ( S e ~ + 4e14 + ~e:e~) + (8ele2e3 -- 8e2).

Definition 3.1.10. Let Yl,... yr be variables and f l , . . - , fr the elementary symmetric polynomials in the Yi. Let

cm(f',',f~)

:=

1-I

(l+y~,+...y~,o)

il <_...<_im

viewed as a polynomial in the fi. Each fi has weight i. Let c m ( f l , . . . , part of weight i in c m ( f l , . . . , f,.).

f~)

be the

From the above we see that for a vector bundle of rank r over X with Chern classes e l , . . . , er the formula c(Symm(E)) =

crn(e,,..., e,.)

3. The varieties of second and higher order data

94

In future we don't want to distinguish between classes a E A*(X) and zr;(a) e A*(Dlm(X) and also not between b e A*(DI (X)) and Try(b) e A*(D~(X)).

holds.

Proposition 3.1.11. A*(D2m(X)) A*(X) [dl , . . . , d d - m , e l , . . . , e m , f l , . . . f d - m , g l , . . . , g ( ~ l ) ]

d

I

(1 +dl + . . . + rid-m)(1 + el + . . . + era) = ~ ( - 1 ) ' c i ( X ) , (1 q- fl -4-.

-4- fd-m)(1 Jr-gl q-.. q- g(~+~))

= (1+dl +... +dd-m)C2(el,...,em) where

c(T,) = (1 + d, + . . . + d~_m), c(Q1) = (1 + el ~-...-~- era), c(T2) = (1 + fl + - . . + fd-m), c(Q2) = (1 q- gl + . . . q- g(.~+l)).

If X is a smooth projective variety over C, the same result holds, if we replace the Chow ring A*(.) by the cohomology ring H*(., 2g) everywhere. Proof."

By the above D ~ ( X )

is isomorphic to the Grassmannian bundle

Grass(("+'), W2m(X ) ) over Grass(m, T~ ). The exact sequence 0

-~

Sym2(Q1)

---*

W~(X)

~

gives C(Wm(X)) = c(Sym2Q1)c(T1). The result follows.

T1

~

0

[]

Two cases axe somewhat simpler: (1) the variety D~_I(X ) of second order data of hypersurfaces on X. (2) the variety D~(X) of second order data of curves on X. C o r o l l a r y 3.1.12. For a variable P we write qi(P) := ~-~(-1)Jcj(X)P i-j, j
Then we have "i ~, d 1, , , ~ _--

(d

0 < i < d - 1.

A*(X)[P, Q]

E(-1)ipd-ici(X), i=O

(~)

~ _,

(Q - PI ~_, O (~) C?(q~(P),..., q~-~(PI) i=0

)

3.1.

The varieties of second order data

where P = cl(Op(T~)(1)), q

=

95

Cl(Ot(w~_,(X))(1)).

D2d_I(X) is the projective bundle P ( W ~ _ I ( X ) ) over P(T)~), and we have ci(Qd-l,T~,) = qi(P). Thus the result follows immediately from proposition 3.1.11. Proof: [3

C o r o l l a r y 3.1.13.

H*(X)[P, Q]

A*(D~(X)) =

)

i=0

\ i=0

1

where P = e,(Oe(Tx)(1)), Q = c,(Op(wp(x).)(1)). Proof." This follows immediately from proposition 3.1.11.

[]

If X is a smooth projective variety over C, then corollaries 3.1.12 and 3.1.13 also hold, if we replace the Chow ring by the cohomology ring. We will write the above formulas explicitely for X of dimension smaller or equal to four. (1) Let X = S be a smooth surface. Then we have

A*(S)[P, Q] A*(D~(S))= (P2+cl(S)P+c2(S),

Q2 + (c,(S) where P =

-

)

P)Q + 2c2(S)

Cl(OP(Ts)(1)), q = Cl(Op(w~(s).)(1)).

(2) Let X be a smooth variety of dimension 3. Then we have

A*(D~(X)) =

A*(X)[P,Q] p3 + c,(X)p2 + c2(X)P + c3(X), Q3 _~ ( c l ( X )

where P = Cl(op(~x)(1)),

A*(D2(X) =

_

Q =

"~

p)Q2 + (c2(X) - c~(X)P - p2)Q + 2c3(X) ) '

c~(Op(w~(x).)(1)).

A*(X)[P,Q] pa _ c~(X)P: + c2(X)P - c3(X), (Q

P)(Q

+p

-

+ 4(P 2 -

cI(X))(Q 2 + 2(P - Cl(X))Q ca(X)P + c2(X))

)

3. The varieties of second and higher order data

96

where P = q(Op(T~)(1)), Q = Cl(Op(w;(x))(1)). (3) Let X be a smooth variety of dimension 4. Then we have

A*(D~(X)) =

A*(X)[P, Q]

P4-k-cl(X)pa+c2(X)P2+ca(x)p+c4(X)' 0 4 At_ ( e l ( X )

p)Qa + (c2(X) - c l ( X ) P

_

)

- -

p2)o~

,

q- (ca(X) - c2(X)P - ci(X)P 2 - p3) q_ 2c4(X) where P

=

Cl((.QP(Tx)(1)),

Q = ca(Op(w~(x).)(1)).

A.(D2(X) ) = A*(X)[pl, P2,7"1,r2] (R1, R2, R3, R4 ), where R1 :=p~ - 2plpz + p~cl(X) - p2cl(X) + pac2(X) + ca(X),

R2 :=P~p2 - p 2 + plp2ci(X) + p2e2(X) - c 4 ( X ) , R3 :=r~ - ar~r~ + ~i + r ~ ( - 2 p l

+ e l ( X ) ) + ~1~:(4;, - 2 c 1 ( X ) )

+ r~(-2plcl(X) + 3p2 + c2(X)) + r2(-3p2 + 2plCl(X) -- c2(X)) -}- rl(PlP2 -- 2plC2(X) -~- 3p2el(X) -}- ca(X)) -

2p22+ 2p2c2(X) - 2pica(X) + 2c4(X),

R4 :=r31r2 - 2rlr 2 + r~r2(-2px + cl(X)) + r~(2pa - cl(X)) + r~r2(3p2 - 2p, e~(X) + c2(X)) + r2(pap2 - 2plc2(X) + 3p2q(X) + ca(X)) + 4pac4(X), with

c(T~) : (1 + p, + p2), c(T2) :

(1 + rl + ~2).

A*(D2(X)) A*(X)[P,Q] p 4 _ C l ( X ) p 3 ~_ c 2 ( X ) p 2 _ c 3 ( X ) p ~_ e 4 ( X ) , (Q _ p)(Q3 _ 2(P -

q(X))Q 2 +

4(P 2 - cl(x)P

+

c2(X))Q

+ 8(P 3 - e1(X)P 2 + c2(X)P - ca(X)) 9 (Qa + 2 ( P - c~(X))Q 2 + (p2 _ 2 q ( X ) P + cl(X)2)Q -- c l ( X ) P 2 -}- c21(X)P -- c l ( X ) c 2 ( X ) where P =

ca(Op(T~)(1)), Q = Cl(Op(w~(x))(1)).

-- c 3 ( X ) )

3.1. The varieties of second order data

97

D2m(X) as t h e v a r i e t y o f s e c o n d o r d e r d a t a o f m - d i m e n s i o n a l s u b v a r i e t i e s of X. We want to see in what respect D2m(X) parametrizes the second order data of m-dimensional subvarieties of X. First we will more generally consider the Ith order data of germs of smooth subvarieties. D e f i n i t i o n 3.1.14. Let Y be the germ of a smooth subvariety of dimension m at x 9 X. Let Iv C O x , , be the ideal of Y in Ox,,. The Ith order d a t u m of Y at x is the subscheme DI,,(Y) := spec(Ox,,/(Iy + m xl+l , , ) ).

R e m a r k 3.1.15. The l th order data of germs of smooth subvarieties of X are the points of

z(1,.,,(mt, ) ..... <./_l))(x)

c x[<,*')]

(see 2.1.5, 2.1.6, 2.1.7). P r o o f i For Y C X a smooth subvariety defined in a neighbourhood of x E X we have Dt,,(Y) 9 Z(1,,~,(m+l ) ..... (my_l))(X). Now let Z 9 Z(1 ....... (m+z~_,))(X) and supp(Z) = x 9 X. Let Iz be the ideal of Z in Ox,,. Then there are local parameters ( X l , . . . , Xd) near x such that ~l+l

I Z = ( X m + l , . . . , X d ) -~ "'X,x"

Let Y C X be the smooth subvariety defined in a neighbourhood of x by the ideal Iy := (Xm+l,..., Xd). Then we have DI,,(Y) = O x , z / I z . [] Because of remark 3.1.15 we write

#m(X)o := Z(1 ....... (m+/_l))(X). We see that D~(X)o = y[n+l] ~(n+l),c (see remark 2.1.8). So D~(X)o parametrizes Ith order data of smooth m-dimensional subvarieties of X. It is easy to see that

Dlm(X)o -= Grass(m,7~(). For 1 _> 2 and d _> 2 however D ~ ( X ) is not compact. Remark

3.1.16. Let

Pl : Dlm(X)o DI,,(Y),

, Dm l-1 (X)o , Dt-I,,(Y)

3. The varieties of second and higher order data

98

Then DIm(X)o is via Pt a locally trivial fibre bundle o v e r D~-l(X)0 with fibre A(d-m)(m+t-1). This is only a reformulation of remark 2.1.7. Now a variety of Ith order data should be a natural smooth compactification of D~(X)o. This is for instance the case for D~(X), as this is given in a canonical way as a subscheme of a product of Hilbert schemes, it is smooth, compact and contains D2m(X)o as a dense open subvariety. There is a morphism r

D~m(X) ~

Dim(X) = Crass(m,7~ ),

extending P2- The fibres of r are obtained by compactifying the fibres of p2 to the Grassmannian Grass(("+'), (,~+1)+ (d - m)). Now we want to compute the class of the complement D~(X)oo := D~(X) \ D~(X)o. It parametrizes in a suitable sense the second order data of singular m-dimensional subvarieties of X. We will use a tool that will play a major role in the enumerative applications of higher order data in section 3.2, the Porteous formula. We will not quote the result in full generality but in the formulation in which we are going to use it. D e f i n i t i o n 3.1.17. Let X be a smooth variety and E and F vector bundles on X of ranks e and f respectively. Let c(E), c(F) E A*(X) be their total Chern classes. We write

c(F

-

E):= c(F)/c(E)

and ci(F - E) for the part of c(F - E) lying in AJ(X). The total Segre class s(E) of E is given by 4 E ) := c ( - E ) = 1/4E), and the jth Segre class s / ( E ) of E is the part of s(E) in AJ(X). Let a : E -----* F be a morphism of vector bundles on X. For all x r X let a(x) be the corresponding m a p on the fibres. Let :Dk(cr) C X be the subscheme

with its natural scheme structure, i.e. with respect to local trivialisations of E and F it is defined by the vanishing of minors of the matrix representing a. We call ~ k ( a ) the k th degeneracy locus of a. Let [~Pk(a)] r A*(X) be the class of ~Pk(a). We call [T)k(a)] the k th degeneracy cycle of a.

Theorem

3.1.18[Fulton (1) Thm. 14.4].

3.1. The varieties of second order data

99

(1) Each irreducible component of :Dk(rf) has codimension at most r := ( e - k ) ( f k) i n X .

(2) If the codimension of T)k(a) in X is r, then we have: [:/)k(a)] ----dct((cf-k+i-j(F - E))l<_i,j<_e-k).

We consider the m o r p h i s m r

7r~(Sym2(Q1)) ---+ Q2

of vector bundles on D ~ ( X ) which is defined by the diagram 0

1 T2

1 0

,

~r~(Sym2(Q1))

~

W~(X)

,

T1

~

0

Q2

l 0. Then D ~ ( X ) ~ is the degeneracy cycle l?(mr162

:= {v e D ~ ( X )

r

not onto }.

The intersection of each fibre 7r~-l(v) with D2m(X)~ is a divisor in rr~-l(v). So we get by the Porteous formula: [ D ~ ( X ) ~ ] -- c1(Q2) - ~r~(cl(Sym2(Q1)))

= c1(Q2) - (rn + l)Tr~(cl(Q1)).

R e m a r k 3.1.19. Let Y C X be a smooth locally closed subvariety of dimension m0 >_ m. Then D2m(y) is in a natural way a locally closed subvariety of D 2 ( X ) . If Y has dimension rn, then D2m(Y) is isomorphic to Y in a natural way.

Definition 3.1.20. Let Y1,Y2 C X be smooth locally closed subschemes of dimensions dl, d2 > rn and x C X. We say Y1, Y2 have Ith order contact along an

3. The varieties of second and higher order data

100

m-dimensional subvariety at x, if x C Y1 A Y2, and there is a germ of a smooth subvariety Z C X at x satisfying Dl,x(Z) C Y~ and Do:(Z ) C Y2. We say Y1, II2 have m-dimensional l th order contact, if there is an x C X such that they have Ith order contact along an m-dimensional subvariety at x. If m is the m i n i m u m rain(d1, d2), we say in this case that Y1 and Y2 have l th order contact

(at ~). From the definitions we get immediately:

R e m a r k 3 . 1 . 2 1 . Y1 and Y2 have m-dimensional Ith order contact at x, if and only if D~(Y1)o, and D~(Y2)o intersect as subvarieties of Dtm(X)o in points lying over

xEX. In case dl = m <_ d2, Yt and Y2 have second order contact at x if and only if D2m(Y1) a n d D ~ ( Y 2 ) i n t e r s e c t as subvarieties of D2~(X) in points lying over x E X. (In this case the intersection point automatically lies in D2~(Y1)o N D2m(Y2)o, as Y1 is smooth of dimension m, and D~(X)o A D2m(Y2) = D~(Y2)0.)

101

3.2. Varieties of higher order data and applications We now want to t r y to generalize the definition of the varieties of second order d a t a to a definition of varieties of higher order data. We will however only have p a r t i a l success. This means t h a t we give a general definition of the variety D ~ ( X ) of n t h order d a t a of m - d i m e n s i o n a l subvarieties of a smooth variety X, which however does not behave very well in general. The varieties of t h i r d order d a t a of curves and hypersurfaces on a smooth variety X t u r n out to be projective bundles over the corresponding varieties of second order data. However the varieties of t h i r d order d a t a of subvarieties which have b o t h dimension and codimension greater or equal to two are not locally trivial fibre bundles over the corresponding varieties of second order data. Also, even if X is a surface,

D~(X)

is not a locally trivial fibre bundle

over D~(X). At the end of this section we give some enumerative applications of our results. As a straightforward generalisation of the definition of the varieties of second order d a t a of m-dimensionM subvarieties of a smooth variety X we get the following definition:

D e f i n i t i o n 3 . 2 . 1 . Let X be a smooth projective variety of dimension d over a field k. Let n,m E 2~>_owith 1 < m < d. Let ~),~(X) be the contravariant functor from the category of noetherian k-schemes to the category of sets which for noetherian k-schemes S, T and a m o r p h i s m r : S ~

T is given by:

Zi C X x T closed subscheme flat of degree (re+i) over T (i = O , . . . , n ) I)~(X)(T) : { (Zo,..., Z,~)

Zo C Z1 C . , . C Z n

and Zi " Zj D Zi+j+l for all i,j with i + j < n -

p~,(x)(r

:

~(X)(T) (Zo,...,Zn)

,

, ,

1

}

~(X)(S) (Zo•

Here we use again the notations we have introduced in definition 3.1.1. In the same way as in l e m m a 3.1.2 we can show t h a t : / ) n ( X ) is represented by a closed subscheme D ~ , ( X ) c X x XEm+,1 x . . . x X

We call

D~(X)

D~(X)

is as a subset o f X [m+l] x . . .

D~m(X) :=

the variety of

n th

order d a t a of m - d i m e n s i o n a l subvarieties of X. x X [ ( '~"+n)] given by:

(z0 ..... zn) c x • xEm+,l •

[(into)].

• x

[(~:")]

Z0 C Zl C Z2 C . . . C Zn / and Zi - Z j D Z i + j + l / for all i,j with i + 3 _< n - 1

3. The varieties of second and higher order data

102

Obviously we have D ~

= X. For i = 0 , . . . , n let r i : D~(X) ~

X [(~'+')]

be the projection. We also consider the projection

~r. : D~(X)

, Dnm-I(x) 9

It is not clear in which cases D,~(X) is reduced, irreduble or smooth. In cases in which it is reducible a b e t t e r c a n d i d a t for the variety of higher order d a t a is the closure the image of D~(X)o under the obvious embedding. A C X x X be the diagonal.

Let

T h e n Hilb('+m)(Ai+l/X) is a closed subscheme

of Hilb('+m)((X x X ) / X ) = X[('+m)] for all i. We can see i m m e d i a t e l y t h a t the projection ri: D~(X) ~

X [ ( ' + ' ) ] factors through Hilb('+~)(Ai+I/X), as we see

from the definitions t h a t Hilb('+'~)(Ai+~/X) represents the functor

TI

, {(Zo,zd

Z0, Zi C X • T closed subschemes ) fiat of degrees 1, (re+i) over T /

Zi C Zg"}-1 We now want to show t h a t D~(X) and D3d_a(X) are again G r a s s m a n n i a n bundles corresponding to vector bundles over D~(X) and D ~ _ I ( X ) respectively. Before doing this we want to show t h a t these two cases are the only ones in which we can expect such a result (exept for the trivial case m = d).

R e m a r k 3.2.2. (1) Let 2 < m _< d - 2. T h e n ~r3 : DO(X ) ~

D~(X) is not a locally trivial fibre

bundle 9 (2) Let S be a s m o o t h surface. T h e n 7r4 : D4(S)

, D~(S) is not a locally trivial

fibre bundle. Proof.* (1) Let x E X . Let xa . . . . Xd be local p a r a m e t e r s near x a n d let m x , ~ : = (xl . . . . , x a ) be the m a x i m a l ideal at x.

Let Z1, Z2 be the subschemes of X with s u p p o r t x

defined by the ideals

I1 = (xm+l,.. 9 ,xd) T m Xp~g~ 2 I2----(Xm+l,.

9 9

,Xd)Tm 3

X~x

3.2. Varieties of higher order data and applications

103

in Ox,,. Then we have (x, Z1,Z2) E D ~ ( X ) . The fibre rc31((x,Z,,Z2)) consists exactly of the subschemes Z3 C X with support x whose ideal/3 in Ox,, is of the form I3 = ( V ) -[- (Xm.t- 1 Xd)" m x , . + m 4 for some (d - m)-dimensional linear subspace V of

(Xm+l ...Xd> -t- <XiXjXl [ i,j,l <_m). (Here we denote by ( f l , . . . , f~) the span as a vector space in contrast to ( f l , . . . , h ) , which denotes the ideal generated by the fi.) So we have

~ ; ' ( ( x , Zl, z2)) "~ a r a ~ ( ( " ? ) , d - m + ('22)). Let Z~ C X be the subscheme with support x defined by the ideal 1; : = ( X m - 4 - 3 , . . 9 ~ Xd,X2,XlX2)-~-(XiXj[i

>m)+m

3X,x

in Ox,,. Then (z, Z1, Z ~ ) i s a point of D ~ ( X ) . The fibre rr31((x, Z1, Z~)) consists of exactly those subschemes Z~ with support x whose ideal I~ in Ox,, is of the form

/~ =(w) + ( z ~ z j l i >_ m + 3) + ( x ~ x i l i , j > m + 1) + ( x ~ z i , X l ~

i < d)

+ ( ~ z ~ x , l i > m + 1) + m~x,. for an (m+2)-codimensional linear subspace W of

V := 2[_<Xm+lXi,Xm.l_2Xi li < m> + (xixjxz[1 < i < j < l < m ; j > 2>. By dim(U) = d - m + (,,,+2) + 1 we have

% ' ( ( z , Z , , Z ' ~ ) ) ~- Grass((~+2),d - m + 1 + (~+~)). (2) Now let S be a smooth surface, s E S and x, y local parameters near s. Let Z1, Z2, Z3 be the subschemes of S with support s defined by I1 :-- (x, y2),

I2 := (~,y~), 13 := (x,y4). Then we have (s, Z1, Z2, Za) e D3(X). Thus r~-l((s, Z1, Z2, Za)) consists of the subschemes Z4 with support s whose ideal /4 in Os, s is of the form /4 = w + (x 2 , xy, yh) for a one-dimensional linear subspace w C (z, y4 ). So we have

7r41 ((s, Zl, Z2, Z3)) ~-~ P1.

3. The varieties of second and higher order data

104

Let Z~, Z~ be the subschemes of S with support s defined by I ; := (x2,xy,y2), /~ := (x2,xy, y3). Then (s, Z1, Z~, Z~) is a point of D~(S).

zr4~((s, Z1, Z~, Z~)) consists of the sub-

schemes Z~ with s u p p o r t s whose ideal is of the form

I'~ = (t) + ( x ~ , ~ > x y ~ , ~ ~) for a two-dimensional linear subspace t C (x 2, xy, y3>. So we have

7F41((W, Zl,Z2, Z3)) ~ P2.

[]

D e f i n i t i o n 3.2.3. Let X be a smooth projective variety of dimension d over a field k. Let m be a positive integer with m < d. We will again use the notations from the definitions 3.1.3, 3.1.4 and 3.1.5. Let ~2 := #loft2. We define the s u b b u n d l e T2 of ~ ( J a ( X ) ) by the d i a g r a m

0

0

'

0

T

T

Q2

Q2

T

T

~ o

l

0

---+

~(Syma(T~))

~ ~-;(da(X))

--~

0

--~

~(Syma(T~))

~

--~

T2

~(J:(X))

T2

T

T

T

0

0

0

Let again A C X x X be the diagonal and Zt, C O x x x

be the projections. For all non-negative integers i _< j let

J j ( X ) : = (S2).((Za)i/(zA )J+I). Then J~(X) is locally free, and we have the exact sequence

,

J~(X)

--*

Jj(X)

--+

0

,

0

its ideal sheaf. Let

sl,s2 : X x X ----* X

0

)

Ji-~(X)

,

O.

39

Varieties of higher order data and applications

105

We see J~ = Jj(X) and Jj(X) = SymJ(T~). Let il <_ i2 < j2 _< jl be positive integers9 The multiplication in Ox • gives a morphism

9 : (z~)~I/(zA)Jl | (ZA)'~/(ZA) j2 ---* (ZA)~I+~=/(Z,,)i'+J~ of sheaves on X • X. So it gives a morphism of locally free sheaves 9 : 4:(x

) | 4:(x

Ji~+j2 ~+,2 (X).

) __~

For locally free subsheaves F C J;:(X), G C J]:(X) we denote by F . G the image of F | G under ".". This is a coherent subsheaf of Jit+i2{x]il+j2 ~ j. By definitions 3.1.4 and 3.1.5 we have

W2m(X) = T1/(T1. # ; ( ~ ) ) , and T2 is a subbundle of #~(W2~(X)). T2 is the preimage of

T2 C 7r~(W2m(X)) C -~(J2(X))/(~;(T1)" V~(T~)) under the natural morphism p : W~(Ja(X))

, ~(J2(X))/(~(TA).~(~I~))

9-~*( J ~ 1 (x))).

= ~(J3(X))/(~i(rl)

Here for coherent sheaves F, G we write F C G to mean that F is a subsheaf of G. So ~r~(T1).-~(J~(X)) is a subbundle of T2, and we have

~;(Jl(X))) = T2.

T-2/(~';(T1)"

From T2 C ~r~(T1) C ~'~(J~(X)) we get r~2

-* 1 ~ . 9-7r2(J2(X)) C ~;(T1) V;(J~(X)),

~ ~* ~ 1 +~(T1)" +~(T1) C rr2 (T1) 9--* u2(J2(X)) C T2 C --* u2(Ja1 (X)).

a W~m(X) a We define the coherent sheaves Uam,V~, on b 2 ~ ( X ) by

u~

:= (~(~)

73

:= (#;(rl)"

w~(x)

:=

* 1 9 -71"2(,]2 ( X ) ) ) / ( T r ~*2 ( T i ) "

~'; (5~1)),

7c2(J~(X)))/(Tr2(T~)" 9~(T1) + T2 .~(J~(X))), ( T ~ ) / ( ~ ( ~ ) . #~(~) + (:~). ~ ( j l ( x ) ) ) .

Then we obviously have the exact sequence o

Lemma

3.2.4.

,

Let m =

v~

~

1 or m = d-

ws

1.

~

T~

---,

O.

3. The varieties of second and higher order data

106

m+2 (1) U3m is locally free of rank r e ( d - m) -[- ( 3 )" (2) Vain i~ locally free of rank (m+2] 3 /"

(3) W~m(X) is locally free of rank e - m + (m+~9" Proof:

By t h e e x a c t sequence

o

,v~

,

w~(x)

,

T~

it is e n o u g h to show these results for [/am a n d V~. fibrewise.

Let v 6 D ~ ( X )

,

0

It is e n o u g h to check t h e m

be a p o i n t lying over x 6 X .

Let x l , . . . , X d

be local

p a r a m e t e r s n e a r x. For i = 1 . . . . , d we d e n o t e by ~'i the class of xi in O x , ~ / m ~ : , ~ . We can a s s u m e t h a t t h e fibre ~;(Tl(v)) is of the f o r m (Xm+l,... ,Xd)/m2x,x. T h e n we h a v e for t h e fibres:

(~1

. . . ~1( x ) ) ( v ) = <~22j[i > m ) + ('2i~jxt[i,j,l <_ d), 9.~r2~r2J

(e~(T~) e~(T1))(~)

: <~,~ li,j > .~> + <~{~j~, li > - J .

Let A0 : : (~iYcj I i < m , j > rn} + (Yzi~j2z [i,j,l < m). T h e n t h e r e s t r i c t i o n of the natural projection ~ : (~TI

~J~(x))(~)

, V~(v)

to A0 is an i s o m o r p h i s m , and (1) follows.

T2(v) is an ( m - d ) - d i m e n s i o n a l linear s u b s p a c e of ~(w~,(z))(v) Let p : fr~(W~(X))(v) ~

-*. . . 9. 7r2T}))(v . = (7c2T,/(rc2T, ).

fr~(T1)(v) be t h e p r o j e c t i o n . As we h a v e a s s u m e d t h a t

rn = 1 or m = d - 1 holds, we h a v e either p(T2(v)) = ~r~(T1)(v), or p(T2(v)) has c o d i m e n s i o n 1 in ~r~(T~)(v). (~~(T1)(v) is o n e - d i m e n s i o n a l in case m = d - 1, a n d

T2(v) has c o d i m e n s i o n 1 in ~r~(W~(X))(v) in case m = 1.) (a) p is onto. T h e n we h a v e

f~(v) = (ym+~,... , w h e r e x l , 9 9 Xm,

Y m + l ,

.

-

9 ,

y d ) / m x , ~3,

Yd are local p a r a m e t e r s n e a r x. So we can a s s u m e

t h a t xi = yi for i = m + 1 , . . . , d .

T h e n we h a v e

(~. ~;(J~(x)))(v) : <},~ li > ~)+ <~,~j~ li > ~). Let A1 : = (~2i2.j24[i,j, 1 <_ m). T h e n t h e r e s t r i c t i o n of t h e n a t u r a l p r o j e c t i o n

ql : (~r~(Tx). ~ ( J ~ ( X ) ) ) ( v )

, V3m(V)

3.2. Varieties of higher order data and application~

107

to A, is an isomorphism, and (2) follows. (b) p(T2(v)) has codimension 1. By changing the local coordinates if neccessary we can assume T2(v) = ((Xm+2,...,Xd, f ) + ( X m + l X j [ j < _ ' m + l ) ) / m x ,

x3

for an f C (xixj l i,j <- - rn) \ m 3X ~ : c " Let f denote the class of f in Ox,x/m4x,~. Then we have (T2" ~;J~(X) )(v) = (.~ixj ]j >_m + 2) + (x,ix, j~,l [i > m) + ( f 2t ]l <_ m). Let A2 := (2~m+l~i [i <_ m) + (~iY:j~tli,j, l < m). Then the restriction of the natural projection q2 : ( # ~ ( T , ) ' ~(J~(X)))(v)

--~ V2(v )

to A~ is a surjection with kernel (f2~ [i _< rn), and (2) follows.

[]

D e f i n i t i o n 3.2.5. We put

/)la(X) := P(W~(X)) = P((Wla(X))*), /~_,(X)

:= P ( W ~ _ , ( X ) ) = I~((W~_,(X))*).

For m = 1 or m = d - 1 let ha : ba~(X) ~ b2m(X) be the projection and ~a := hi oh2og'a. Let Ta := Td-m,wam(X ) be the tautological subbundle and

0 -----+ T3 ~ia

h;(W3m(X))

_2~

Q3

---,

0

the canonical exact sequence. Let It" := h ; ( T 1 ) ' h ; ( T , ) + T~. V;(ZJ (X)) be the kernel of the natural vector bundle morphism ~ : T2 - - ~ W~(X). We define the vector bundle T3 o n / ~ ( X ) by the diagram

0

,

T3

T 0

,

0

---,

T

0

T

1

i3

,

0

,

0

[3 ~

i3

h;(K)

0

)

?r~(T2)

~

Qa

T -~;(K)

T

T

0

0

T ,

0

108

3.

and the vector bundle

03 o n / ) a ( X ) 0

0

----+

T3

The varieties of second and higher order data

by

0

- -is~

0

-#~(T2)

4~ '

Q3

--~

0

(~3

;'

0

773(Q2)

'

0

l 0

~

eft3

"--')"

~:~(J3 ( X ) )

~

0

0

In particular we get the exact sequence 0

~

03

)3 '

#3

03 '

~ .z(QJ) --

We now generalize the definition of the bundles

D e f i n i t i o n 3.2.6.

For any I C /V let again

Zt(X)

----* 0.

(Ox)~

and

(Ox)2~

from 3.1.3:

C X x X[q be the universal

subscheme with the projections

z~(x) /p,

\q,

X

X[O.

For any vector bundle E of rank r on X we put /~t := This is a vector bundle of rank

rl

(qt).(p~(E)).

on X [z]. For all n E ~W and all m _< d we put

(E)~ := rn(E(,,+,)), where r,~ :

D~(X) ~

Z[(~+n)] is the projection from D~(X). We call it the contact

bundle of rank r . (m+,~) on

E,X

and m.

3.2.1. ( E ) ~ is a vector b u n d l e corresponding to

3.2. Varietie, of higher order data and applications

Theorem

109

3.2.7. Let m = 1 or m = d - 1. Then there is an isomorphiam <

:=

D~m(X) --~ b~m(x),

for which the diagram

commute, such that r

Dam(X)

r

,

Dam(X)

Dam(X)

r

,

s

=

(Ox)am.

Proof." We use the' notations from definition 3.1.6. T h e n we can write 1)am(X) as r :T

~ X m o r p h i s m over k Z1, Z2, Z3 C X x T

1)~(X)(T)={

(r Z1, a2, Z3)

closed subschemes flat of degrees rn + 1, (m+2), (m+3) over T with Aq5 C Zl C Z2 C Z3, Z1 C A~, Z 2 C A r 9 Z1,

Za c A r

ZacZ~-Z1.

/ .

Let

Z1, Z2, Z3 C X x Dam(X) (m+3"~ be the universal families of degrees m + 1, t(re+z) 2 J, ~ a J over Dam(X). Via r we identify/gzm(X ) with Dam(X) and #1 and #2 with rrl and 7r2 respectively. We put 71" ; ~ 7rloTr2o7r3~ # : = #1o#2o#3 = 71-1o7r2o#3,

The subvariety A,~ C X x Dam(X) is via the projection p to the second factor isomorphic to Dam(X), and we have

p.(Q:+,)

:

~*(&(x)).

For a subscheme Z C X x D a ( x ) let Zz be the ideal of Z in X x Dam(X). By definition we have Z2 C Z a C A ~ . Z 2 c A 4, Z3 C Zl 9 ZI C /N4.

So p,(Zz2/za) is a locally free quotient of rank (m+2) of 3

p,(Zz~/(za..z~ + Zz,.z, )) = ~ ( w ~ ( x ) ) . This defines a m o r p h i s m Ca : D 3 ( X ) ~

D 3 ( X ) over D 2 ( X ) .

3. The varieties of second and higher order data

110

Let Za := gal(W~), Z2 := ~-31(W2) for the universal subschemes W1 and W2 over D2m(X) of degrees m + 1 and (m+2) respectively. The subvariety Ae C X x/gam(X ) is via the projection i5 to the second factor isomorphic t o / ? a ( X ) , and we have

~ , ( o ~ ; + , ) = ~*(J,(x)). T3 is an O b ~ ( x ) - s u b m o d u l e of #*(J4(X)). Let Ia C Ozx~ be the Ozx~-submodule with/5.(I3) = Ta. By the inclusions

rta(rr2(T1))" ~-*( . : (*T ,-) )

c ra,

(see definition 3.2.5) we have

:r/,~lA~ 9:rZ~lA, C Ia, $2 Z, la~ C Ia,

So we have in particular OA~ - Ia = ira. So Ia is an ideal in Oz~ and defines a subscheme Za C A 4 satisfying

By the inclusions

:I~,,l/, ~ "TUZ~IAI c I3, I~,l,, ~ c h, Ia c ~Z=la~ we have

Z2 c Za c A~. Z2, Z3 C Z1 " Z1,

Z2 C Za. (~, Zl, Z2, Z3) defines a morphism ~b3 : Dam(X) - - + D~(X) over D2m(X) satisfying r = Qa. It is easy to see that r = r [] In future we want to identify

Da(X) with

/?la(X) and also Da l(X) with

bLI(x). As D~(X) is smooth, we see first that the projection ra : D~(X) ~ X [4] factors through "'(4)" y[4] As also X (4),c [4] = Z(1,1,1,1)(X) is smooth, ra is an isomorphism over Z(1,1,1,1)(X ). The preimage D~(X)o parametrizes third order data of

3.2. Varieties of higher order data and applications

111

(germs of) s m o o t h curves on X . Here the n th order d a t u m of a s m o o t h subvariety Y C X in a point x C Y is the quotient O Y,x/~ m x,~ ~+1 of Ox,x.

In a similar way

one can t r e a t D ~ _ I ( X ) : the preimage r ~ l ( z ( 1 d-1 (~ ( ~ + ~ ( Z ) )

is an open dense

\ '

'k2/'\ 3 ]l

subset Dad_l(X)o in D ~ _ I ( X ) , and the restriction r3[D~_~(X)o is an isomorphism.

r31 (D~_l(x)0 )

p a r a m e t r i z e s third order d a t a of (germs of) s m o o t h hypersurfaces

of X .

Remark

3 . 2 . 8 . Let Y C X be a s m o o t h closed subvariety, Then for all n E ZW the

Hilbert scheme y b ] is a closed subscheme of X b ] . So for all n, m E ZW with m _< d

D ~ ( Y ) is a closed subscheme of D ~ ( X ) . F r o m the definitions of the vector bundles ( O x ) ~ and ( O y ) n we see that (Ox)nlD,~(y) = ( O y ) n. So/~2 ( y ) C / 9 ~ ( X ) a n d /9~(Y) C / ) l a ( X ) are closed subvarieties with

Q,i(x)[y = Q.i(Y), Q i ( X ) ] v = Q i ( Y ) , Here we write Q i ( X ) , Q i ( X ) for the classes Qi, Qi

(i = 1,2,3). on

Di(X)

and similar for Y.

In case m = d = d i m ( X ) we see i m m e d i a t e l y t h a t D ~ ( X ) is isomorphic to X via its projection. T h e universal families are Zi = A i+1 C X x X for i = 0 , . . . , n. So we have ( O x ) ~ = Zn(X), n

( O X ) m / ( O x ) ~n+l

Sym"(T~).

=

Now we can compute the Chow rings of D ~ ( X ) and D 3 _ I ( X ) . For this we first

have to determine the Chern classes of

WI (X) and wL (x).

Lemma 3.2.9. (1) In case m = 1 we have V~ ~ ~r~(Q1) | Q2, and so there is an exact sequence

0

---*

~r~(Q1)|

--*

W3t(X)

,

T2

--+

on (2) In case m = d -

0 o

1 there are exact sequences

---+

T2|

--,

0

on 5 L I ( X

).

----*

Vda_l

,

V#_ 1

---+

U~_ 1

~

---4"

W~_I (X)

~

Yd3_l

ff~(T1)| ----4

r 2

)

0

0

3. The varieties of second and higher order data

112

Proof:

(~) Let ~ : ~ ; ( ~ ) ~ ~ ( : J ( X ) ) --~ V? be the ~atural homomorphism. We see i m m e d i a t e l y t h a t w is onto and

~ ; ( J I ( X ) ) @ T2 -[- ~ 2 ( r l ) @ 7r2 (T1) lies in the kernel of w. As all the sheaves we are considering are locally free and have the right rank, we have

and obviously this is also the kernel of the n a t u r a l m a p w : 9i(:F1)| ff~(j1 ( X ) ) - - ~ #i(Qi) | Q:. (2) We a l r e a d y know the lower sequence.

The middle sequence comes from the

diagram 0

0

l

0

0

~

~;(T1).~(J~(X))

~

1 ~'~ ( r l ) "~'~ ('~1)

~

1 ~r~ (Sym2 (T1))

0

~

~ (Syma(r~))

~

~r~ ( ' ~ 1 ) - ~ ( J1 ( X ) )

~

~-~ ( r l ) . ~

0

~

~r~ ( S y m a ( Q 1 ) )

--

U~_ 1

~

0

(T~)

(~r~(T1).~(T~))/Sym~(fr~(T~))

0

0

if we use (T1 9#~(T~i))/Sym2(T1) ~= T1 | Q1. Let w2 : U~_ 1

) V)_ 1 be the n a t u r a l h o m o m o r p h i s m .

We have to show

~. ker(w2) ex~ = T2 | % (QI). We consider the exact sequence

~2+~(jl(X))

~

uL~ - ~

v)_~

O,

where w0 is the obvious map. We see t h a t (~';(T1) 9 -%* ( T ~*) ) Q -~*( J ~ 1 ( z ) ) + ~ e ~ ' ~ ( r l ) c

]r162

and

(T~|

(X)))/((~;(T~). ,~(T~)) 0 ~ ( j I ( x ) ) + ~ 0 ~ ; ( ~ ) ) ~- ~ | ~(T:,.)/((~;(T~). ~(T~)) 0 ~;(T~) + ~ 0 ~(T~)) ~- T2 | ~;(O,).

So there is a surjection of vector bundles

3.2. Varieties of higher order data and applications

113

Because the bundles have the same rank, it is an isomorphism. So (2) follows.

[]

Again for i = 1,2, 3 we don't want to distinguish notationally between a0 in A * ( D i - ' ( X ) ) and ~r~'(ao). We formulate our results (proposition 3.2.10 and proposition 3.2.11) only for the Chow rings, but it is clear that they also hold if we replace the Chow rings by the eohomology rings everywhere. P r o p o s i t i o n 3.2.10. Let X be a smooth projective variety of dimension d. Then

A*(D~(X)) =

A*(X)[P, Q, R] d

Z

Pd-ici(X),

i=0

•

ci(X) - 2 c i - l ( X ) P - E cj(X) j=O

i=0

pQi-l-j Here P = C l ( O P ( T x ) ( 1 ) )

,

Q

=

i--2--j

'_ '_ x

, R = cl(Op(wa(x).)(1)).

Cl(Op(w~(x).)(1))

P r o o f : This follows immediately from lemma 3.2.9(1).

[]

P r o p o s i t i o n 3.2.11. Let X be a smooth projective variety of dimension d. As an abbreviation we write qi(P) := ~j<_i(-1)Jcj(X)P i-j, 0 < i < d - 1. Then we

have with the notations of definition 3.1.10 and corollary 3.1.12 9

a

A (Dd_a(X))=

A*(X)[P, Q, R] d

Z(-1)ip"-ici(X), i=0

(Q - P)

qd-l(P)), i=O

(R-e)

E /=0

n=O

\ i + j < d - 1 qi(P) --

j /--Tt

3. The varictie~ of second and higher order data

114

tlere (.),, de~ote~ the part of degrer ,~ (P,Q,R have each degree 1 and ci(X) ha~ d~gree i).

We have P

=

~(Op(r~)(1)),

Q = c~(Op(wL,(x))(1)),

n

cl(Op(w2_,(x))(1)). P r o o f i By the exact sequences from l e m m a 3.2.9(2) we get

e ( W 3 _ I ( X ) ) = c ( S y m 3(Tc:(Q~))) -, c ( #*2(Q1)Q~r~(T1))c(T2)/c(T2 | -* and for a vector bundle E of rank r and a line bundle A we have

e(E | A) = ~

( r - ji ) c i ( E ) c l ( A ) J .

[]

i+j=r

We will rewrite these formulas explicitely for d < 3. If X is a surface, then

A*(X)[P, Q, R] p2 + Cl(X)P + c2(X),

A*(D~(X)) =

Q2 + (c~(X) - P)Q + 2c2(X),

] ,

R 2 + (c~(X) - 2 P ) R + c~(X) - 2c~(X)P - P Q

]

If d = d i m ( X ) = 3, then

A*(X)[P, Q, R] A * ( D s ( X ) ) = liPS + c l ( X ) p 2 + c2(X)P + cs(X), Qa + (Cl(X) _ p)Q2 + (e2(X) - c~(X)P - p2)Q + 2ca(X),

L

R 3 + ( c l ( X ) - 2 P ) R 2 + (c2(X) - 2Cl(X)P - P Q ) R

+ ca(X) - 2c2(X)P - c l ( X ) P Q - p(Q2 _ pQ).

9and

A*(Ds(X)) =

A*(X)[P, Q, R]

(Sl, $2, $3) where Sx : : P ~ - e ~ ( x ) P ~ + e : ( X ) P -- c~(X), $2 : = ( Q -

P)(Q + P

+ 4(P 2 -

-

cl(X))(Q 2 +

2(P -

c~(X))Q

cl(X)P + c2(X)),

$3 :=R 5 + ( 4 P + Q - 6 c l ( X ) ) R 4 + (12P 2 + 2PQ - 23Pe1(X) + Q2 _ 3Qcl(X) + 11e 2 + 10c2)R 3

=

3.2. Varieties of higher order data and applications

+ (4p2Q

_

35p2cl(X)

_

pQ2

_

115

2PQcl(X) + P(41c,(X): + 23c~(X))

q_ Q3 + 2Q2cl(X) + Q(_4Cl(X)2 + 8c2(X)) - 6 c l ( X ) 3 30cl(X)c2(X) + llc3(X))R 2

-

+ (_p2Q2 + 9p2Qci(X) + p2(24c1(X)2 + 15c2(X)) - 3PQ 3 + 14PQ2cl(X) + pQ(-21c~(X) 2 - 25c2(X)) + P(-24c~(X) 3 - 57cl(X)c2(X) + 6c3(X)) -I- 6Q3cl(X) q- Q2(-18cl(X)2 q- 9c2(X)) + Q(12c~(X) 3 + 3cl(X)c2(X) - 13c3(X)))n 2 1 p 2 Q 3 + 99p2Q2cl(X) + P2Q(-66c1(X)2 - 19c2(X)) -

-

"4-P2(-168Cl(X)c2(X) -4- 56c3(X)) -4- 75pQ3 c1(X) + PQ2(-137ci(X)2 - 126c2(X)) + PQ(66cl ( x ) 3 + 99cl(X)c2(X) + 6c3(X)) -4- Q3(-68ci(X)2 + 5c2(X)) -4- Q2(36cl(X)3 + 111c1(X)c2(X) - 54c3(X)).

Let E be a vector bundle of rank r on a s m o o t h projective variety X .

Now

we want to s t u d y the vector bundles (E),~ from definition 3.2.6. For this p u r p o s e we first consider the bundles/~t on the Hilbert scheme X Ill. We can associate in a n a t u r a l way to each section s of E a section ~) of fist and thus also a section (s)"m of

(E)7.: Definition 3.2.12. vector space H~

For any point Z E XM the fibre /~t(Z) of fist over Z is the

E |

Oz). Let

evz : H ~

~H ~

|

be the evaluation morphism. For any section s E H ~

Oz) E ) we define a section ~t of

fist by ~ t ( z ) :=

~vz(s)

and p u t ( s ) ~ := r*(s~(,~+,)). This defines the evaluation m o r p h i s m

eVE: H ~

|

O o ~ ( x )

~

(E)~n.

R e m a r k 3 . 2 . 1 3 . Let s be a section of E and Y C X its zero locus. F r o m definition 3.2.12 we see i m m e d i a t e l y t h a t Y['q C X In] is exactly the zero locus of ~l, and thus

D ~ ( Y ) C D,~(X) is the zero locus of ( s ) ~ . To begin with this is only true settheoretically, i.e. without considering the possible non-reduced structure. If rn = 1 a n d n < 3 or n < 2 one can however show by c o m p u t a t i o n s in local coordinates on

D ~ ( X ) t h a t the s m o o t h subvariety D~,(Y) is the zero locus of (s),~ in D,~(X), if

3. The varieties of second and higher order data

116

Y is smooth of codimension r. In particular we see in this case for the top Chern classes c,.(E) = [Y] C A~(X), c (m+.)((E),~) = [ n ~ ( Y ) ] E A"(m'+~")(D~(X)).

The vector bundles ( E ) ~ can be related to the simplest case E = Ox: let A C X x X be the diagonal, 2-zx its ideal and A n+l C X x X the subseheme defined by 2"n+l. Let Hilbt(A'~+l/X) C Hilbt(X x X / X ) = X[q be the relative Hilbert scheme of subschemes of length n of A n+l over X. Let ~r : HilbZ(An+l/X) ~ X and r0 : D,~(X) , X be the projections. L e m m a 3.2.14. Let E be a vector bundle on E. (1) /~t]Hilb,(An+Ux ) = 7:*(E) | ((Ox)tlHilb,(zxn+~/x)).

(Ox)~.

(2) For all ~ c ~V, .~ _< d we have ( E ) ~ = r~(E) |

P r o o f : (1) For all l E ZVVwe put Zt(X)(,~) := H i l b t ( A n + i / X ) XxvJ ZI(X). Let

zz(x)(.) /q

~p

X

Hilbl(An+l/X)

be the projections. Then we have the commutative diagram

Hilbt(An+l/X)

'P

Zt(X)(n)

X

X,

and by the projection formula we get Ez[Hilb,(A.+l/x) = p,(q*(E))

= p.(p*Qr*(E))) = 7r*(E) | (Ox)t]Hilb~(A.+l/x). So we get (1). The projection

r. : D~(X)

~ X [(r.+.)]

factors through Hilb("+-:~)(A"+I/X) (see the remarks after definition 3.2.1). So (2) follows from (1). []

3.2. Varieties of higher order data and applications

117

Now we specialize to the case X = PN and to the hyperplane bundle H = 0(1). Proposition 3.2.15.

(I)

Let

H :-- CI(OPN(1)) ,

P := cl(Op(rr, N)(1)), O :=

q(oP(we(i,N,)(1)),

R := cl(Op(we(pN).)(1)).

Then we have in A*(D~(PN)) c((H)~) = (1 + (3H + P + Q) + (3H 2 + 2H(P + Q) + PQ) + (H a + H2(P + Q) + HPQ)

and in A*(Da(pN)) c((H) a) = 1 + (4H + P + Q + R) + (6H 2 + 3H(P + Q + R) + PQ + PR + QR) + (4H a + 3H2(P + Q + R) + 2H(PQ + P R + QR) + PQR) + (H 4 + Ha(p + Q + R) + H2(pQ + PR + QR) + HPQR). (2) Let dl,...,dm

be the Chern classes of the universal quotient bundle on Dim(X) = Grass(m, T~N ) and f l , . . . , f(m+~) the Chern classes of the universal quotient bundle on D 2 ( X ) = Grass(~m+~ W m2\ fX ]~) 9 Then we have \\ 2 )~

i+j<_(%+:)

k+~=i

(3) Ifm = N - l , let in addition h~,..., h(N+l) be the Chern classes of the universal

quotient bundle on D3_I(X) = Grass((N+l),

W~_I(X)).

Then we have c((H)aN_l) =

i+J
k+t+s=i

Here in (2) and (3) we yor.~aUy set dk = 0 for k > .~, f, = 0 for l > (re+l) and hs = 0 ior s > (N+I)

3. The varieties of second and higher order data

118

P r o o f : This follows from l e m m a 3.2.14 and the exact sequences 0

,

Q1

---*

(Ox)L

~

Ox----~

0

'

Q2

--~

(Ox)~

--*

(Ox)k

0

'

Qa

~

(Ox),3,, -----+ (Ox)2,.

0 0

--~

0

[]

Now we want to c o m p u t e the class [D2(C)] C A3N-3(D~(PN)) for a s m o o t h curve C C P N .

Proposition 3.2.16. [D12(C)]

=

deg(C)HN-1pN-1Q N-1 + ((N + 1)deg(C) + 2g(C) - 2)(HNpN-2Q N-1 + H N p N - 1 Q N-2)

Proof: We have H . [D~(C)] = deg(C). By remark 3.2.8 we also have P . [D~(C)] = 2g(C) - 2, Q . [D~(C)] = 4g(C) - 4. On the other h a n d we can use the relations to c o m p u t e the intersection table:

HNpN-1QN-2 HNpN-2QN-1 H

1

P Q

HN-1pN-1QN-1

1

This proves the result.

[]

1

-N-1

1

-2N - 2

3.2. Varieties of higher order data and applications

119

Enumerative applications for contacts of projective varieties with linear subvarieties of P N Now we want to a p p l y our considerations to o b t a i n formulas for the numbers of higher order contacts of a smooth projective variety X C P N of dimension d with linear subvarieties of P N of dimension m. We have to distiguish two cases: m _> d and m < d. We will see t h a t the first case is the simpler one, as in this case we have X = D n ( x ) , and so the c o m p u t a t i o n s can be carried out directly in the Chow ring of X . In case m < d we have to consider the more complicated Chow rings of

D ~ ( X ) and D~(X). We again want to use the Porteous formula. Let H = 0 p N ( 1 ) be the hyperplane bundle on P N . We will denote by the same letter its restriction to X a n d its first Chern class.

Contacts with linear subvarieties of higher dimension Let X C P N be a s m o o t h m - d i m e n s i o n a l subvariety. We can in a n a t u r a l way identify D,~(X) with X for all n E W , and with this identification we get

(H)~ = H | ((gx)~ = H | Jn(X). On X = D ~ ( X ) we consider the evaluation m o r p h i s m

eVm : H ~

OpN(1)) | O x

) (H)~.

This is the composition of the restriction r: H~

Ov~(1))

~H ~

H)

with the evaluation m o r p h i s m

evil : H ~

H) | O x

, (g)~

from definition 3.2.12. Over every point x E X the kernel of the induced m a p evm(x) : H ~

OpN(1)) ~

H~

H | (Ox,,/m~x+a,))

on the fibres consists of the sections s e H ~ P(ker(s)) C P(H~

OvN (1)) for which the h y p e r p l a n e

O p N (1))*) has n th order contact with X . A linear subvari-

ety V of P N of dimension m l > m has n th order contact in x, if and only if each h y p e r p l a n e of P N containing V has nthorder contact with X at x. So the locus where X has n th order contact with a n / - c o d i m e n s i o n a l linear subvariety of P N is the degeneracy locus

~)N+l--l(eVm) = {X E X

r]g(evm(X)) ~ N -}- 1 - / } .

120

3. The varieties of second and higher order data

So we get by Porteous formula (see. theorem 3.1.18): Proposition 3.2.17. Let X C P N be a smooth closed subvariety of dimension m. The locus where X has n th order contact with l-codimensional linear subvarieties of P N has at most codimension

\ \

7~ /

in X . If its codimension is r, then its class is

det((C(m+n)_N_l_t_l+i_j(,Jn(X)@g))l
~Ar(-u

9

In particular we have:

(1) The class of the locus, where X has n th order contact with a hyperplane in P N is

E

f('+") - i) "j~

E

i4_j=(ra+nn)_ N

(-1)i ~

il+...+in=i

[ I ci'(SymI(TX))Hj" /=1

(2) Let C C P , be a smooth curve9 The number of n th order contacts of C with hyperplanes in P~ is 2

(2g(C)-2)+(n+l)deg(C).

(3) Let S be a smooth surface in P N If N = { n + l ~ (n - 1) th order contacts with hyperpIanes in PN is 9

~

2

]

_

1, then the class of the

n--1 k=l

If N = (n+l) __2, then the number of (n - 1) th order contacts with hyperplanes in P N is

n--1

E E (~- 2k)2c2(s) ra=l O_
4-(~ \

( 2 ) (n -- i)2 +

k=l

E

ij(n--i)(n--j))cl(')

l
i=2

2

t

2

3.2. Varieties of higher order data and applications

121

If N = (n+a~ then the number of ( n - 1 ) th order contacts with 2-codimensional linear aubapaces in P N is

Z

ij(n -- i ) ( n

-- j ) -- ~

i=2

l<_i<_j<_n--1

n-1

- Z

Z

(rl --

i) 2 cl(S) 2

- 2k)2c2(S)

m:l 0_
_ Zkk=(ln _ k )

n +2l

/

) +1

(4) Let X be a smooth threefold in P9. with hyperpIanes is

/ cl(S)H +

)

(n+ 1)2+ 1 H 2.

The class of second order contacts of X

- 5 c 1 ( X ) + 10H.

Let X be a smooth threefold in Ps. with hyperplanes is

The class of second order contacts of X

9cl(X) 2 --[-6c2(X) - 45c1(X)H + 45H 2.

Let X be a smooth threefold in P7. The number of second order contacts of X with hyperplanes is - 7 C l ( X ) a - 20Cl(X)c2(X) - 8c3(X) + 72Cl(X)2H

+ 48c2(X)H - 180Cl(X)H 2 + 120H a. Let X be a smooth threefold in Plo- The class of second order contacts of X with 2-codimensional linear subvarieties is 16c1(X) 2 -- 6c2(X) -- 55c1(X)H Jr- 55H 2.

The number of second order contacts of X with 3-codimensional linear aubvarieties is - 4 2 c 1 ( X ) 3 + 40cl(X)c2(X) - 8ca(X) + 192cl(X)2H -

72c2(X)H - 330cl(X)H 2 + 220H a.

Obviously (1)-(4) only hold in the case that the locus where the contact occurs has the right codimension in X . P r o o f : By (H),~ = J , ( X ) | H the total Chern class satisfies

E

Z

i + j < ( ~ + ~) il+...+i~=i

(-1)' ("+ )-i iic,,(Sym,(Tx))g,. /=1

3. The varieties of second and higher order data

122

From this we immediately get (1). (2) follows by an easy computation. (3) By (1) and remark 3.1.9 the coefficients of cl(X) and cl(X) 2 in c((H)~ -1) are the coefficients of xl and x 2 in n--1

I-[(1 -- (n

-

#)Xl)

k

k=l

respectively, and the coefficient of c2(X) is the number TZ--1

E E (m_2k2

m=2 0
The rest follows by an easy computation. (4) follows from (1) and remark 3.1.9 by an easy computation. []

Contacts with linear subvarieties of lower dimension Let X C P N be a smooth projective variety of dimension d. Now we want to treat the second order contacts of X with linear subvarieties of P N of dimensions m < d and also the third order contacts of X with lines. We first study the case of second order contacts. On D2~(X) we consider the evaluation morphism

evm : H ~

OPN(1 ) | ODL(X) ~

(H)2~ 9

This is the composition of the restriction

r: H~

~

H~

with the evaluation morphism

evil: H~

H) | ODL(X) ---* (H)2m.

Over each point w = (x, Z1, Z2) C D2~(X) the kernel of the induced map evm(w) : H ~

0 p N ( 1 ) ) - - ~ H~

| Oz~)

on the fibres consists of the sections s E H ~ 0 p N (1)) for which the hyperplane P(ker(s)) C P ( H ~ 0pN(1))* ) contains Z2 as a subscheme. A linear subvariety V of P N of dimension rn contains Z2 as a subscheme, if and only if each hyperplane containing V also contains Z2. So the locus

{w = (x, Z1,Z2) E D2m(X) Z2 lies on an m-plasle ~

3.2.

Varieties of higher order data and applications

123

is exactly the degeneracy locus

~)rn+l (eVm ) =

D

_< m + 1}.

(x)

Let r0 : D2m(X) , X be the projection. From the above we get for the image of the degeneracy locus {

ro('Dm+l(eVm)) =

x EX

there is an m-plane } having second order contact with X in x "

So (ro).(Dm+l(ev,,)) E A*(X) is the class of the locus where X has second order contact with m-planes counted with multiplicities. Let W be an irreducible component of ro(59m+l(evm)). The multiplicity of W in (ro).(Dm+x (evm)) is the degree of r0 IDm+l(~vm) over W (or zero if this degree is infinite), i.e. the number of m-planes having second order contact in a general point of W counted with multiplicities. So we call (r0).(Dm+l (evm)) the class of second order contacts of X with m-planes in PNWe can also determine this class in a dual way: let ev*: ((H)i)* ~

(H~

|

OD~(X))*

be the dual morphism of eVm. For w = (x, Z1, Z2) E D~(X) the subscheme Z2 lies on an m-plane if and only if ev*(w) has at most rank m + 1. So the set

{w=(x,Z,,Z2) ED2m(X) Z2 lies on an m-plane } is the degeneracy locus :Drn+l(eV~n). So we get: 3.2.18. Let X be a smooth projective variety of dimension d in PN. If the locus where X has second order contact with m-planes has codimension at least Proposition

then its class is

_ J_t

2 )

In particular the class of second order contacts of X with lines is

(ro).(SN-l(((H)~)*)) E AN-2d+I(x),

3. The varieties of second and higher order data

124

if this locus has codimension N - 2d + 1. In a similar way we can argue for third order contacts with lines. Let X C P N be a smooth projective variety. On D~(X) we consider the evaluation morphism Ov^,(1)) | O,9~(x)

ev : H ~

' (H) a.

Let

ev*: ((H)~)* ---+ ( H ~

COPN(1)))* | OD~(X )

be the dual morphism. For w = (x, Z1, Z~, Za) ~ D 2 ( X ) the subscheme Za lies on a line 1 C P N , if and only if ev*(w) has rank 2. So the locus of third order contacts of X with lines in PN is the degeneracy locus "l)2(ev*). Let r0 : D } ( X ) ----+ X be the projection. Then we get as above: P r o p o s i t i o n 3.2.19. Let X C PN be a smooth variety of dimension d. If the codi-

mension of the locus, where X has third order contact with lines, has codimenaion 2N - 3d + 1, then its class is (ro).(SN_l(((H)am).)2

-

-

3 ) 9)SN-2(((H),~) a 9)) E A2N-ad+2(X). SN(((H)m

As we know the Chow rings of D ~ ( X ) and D a ( x ) , and the Chern classes of (H)2m and (H) a can be expressed in terms of the generators of these eohomology rings, we can in principle compute the classes of second order contacts with m-planes and the classes of third order contacts with lines. Note however that the Chow ring of D2m(X) is quite complicated for m ~ 2. For the explicit computation we will therefore restrict ourselves to the case of contacts with lines. We compute these classes for small N with the help of a computer. The total Segre class of ((H)~)* is s(((H)12) *) := (1 - H ) - I ( 1 - ( P -b H ) ) - I ( 1 - (Q q- H)) -1, and the total Segre class of ((H)~)* is s(((H)~)*) := (1 - H ) - 1 ( t - ( P + H ) ) - ' ( 1 - (Q + H ) ) - I (1 - (R + H)) -1 So we get the following formula: The class of second order contacts of a smooth surface X C P4 with lines is 2 ( - 3 c 1 ( X ) + 5H). The number of second order contacts of a smooth surface X C P5 with lines is 2(7c1(X) 2 - 5 c 2 ( X ) -

18cl ( X ) H q- 15deg(X)).

3.2. Varieties of higher order data and applications

125

This formula has been obtained in [Le Barz (4),(9)] using a different method. The class of second order contacts of a smooth threefold X C P6 with lines is 4 ( - 3 c 1 ( X ) + 7H). The class of second order contacts of a smooth threefold X C P r with lines is 4(7c1(X) 2 - 5c2(X) - 2 4 c l ( X ) H + 28H2). The n u m b e r of second order contacts of a smooth threefold X C P8 with lines is

1 2 ( - - 5 c 1 ( X ) 3 -}- 8 c I ( X ) c 2 ( X

) - 3ca(X) + 21c,(X)2 H - 1 5 c 2 ( X ) H

- 3 6 q ( X ) H 2 - 28deg(X)).

The class of second order contacts of a smooth fourfold X C P9 with lines is 8(7c1(X) 2 - 5c2(X) - 3 0 c l ( X ) H + 45H~). The class of second order contacts of a smooth fourfold X C P10 with lines is 8 ( - 1 5 c 1 ( X ) a + 2 4 q ( X ) c 2 ( X ) - 9ca(X) + 77c1(X)2H - 55c~(X)H -

1 6 5 q ( X ) H 2 + 165H3).

The n u m b e r of second order contacts of a smooth fourfold X C P11 with lines is 8(31c1(X) 4 - 7 9 c l ( X ) 2 c 2 ( X ) + 21c2(X) 2 + 4 4 q ( X ) c a ( X ) - 17c4(X) - 1 8 0 c , ( X ) 3 H + 2 8 8 c , ( X ) c 2 ( X ) H - 108ca(X)H + 4 6 2 C l ( X ) 2 H 2 - 3 3 0 c 2 ( X ) H 2 - 6 6 0 c 1 ( X ) H 3 + 495deg(X)).

The class of third order contacts of a smooth threefold X C P s with lines is 85c1(X) 2 - 49c2(X) - 3 3 0 q ( X ) H + 411H 2. The class of third order contacts of a smooth fourfold X C P7 with lines is - 5 7 5 c 1 ( X ) a + 7 9 0 c l ( X ) c 2 ( X ) - 251c3(X) + 3 4 0 0 q ( X ) 2 H -

1 9 6 0 c 2 ( X ) H - 8 2 2 8 c l ( X ) H 2 + 8680H 3.

In section 3.3 we will develop a new method of determining a formula for higher order contacts of a smooth variety X C PN with lines in PN. At the end we will obtain a general formula which contains the ones above as special cases. We briefly want to consider the contacts of a projective variety with more general families of subvarieties of PN.

3. The varieties of second and higher order data

126

D e f i n i t i o n 3 . 2 . 2 0 . Let T be a smooth projective variety a n d Y ~

T a smooth

m o r p h i s m of relative dimension m. Here we asume Y to be quasiprojective over T. We p u t Dlm(Y/T):= Grass(m,f~y/T). We define the vector bundle W~(Y/T) on

s

in an analogous way to W2m(X), replacing the bundles by their relative

versions relative to T. Then we put 5 ~ ( Y / T ) : =

arass((W1),W~(Y/T)).

It is obvious from the definitions, t h a t b o t h

D~m(Y/T)a n d b~(Y/T)

are iso-

morphic to Y. D e f i n i t i o n 3 . 2 . 2 1 . Let T be a s m o o t h variety and YT C P N • T a flat family of m - d i m e n s i o n a l subvarieties of P N , i.e. we have the projections

YT /P,

\P~

PN

T

P2 is flat, and for all t ff T the fibre Yt = P2-1(t) has pure dimension m. In a d d i t i o n we assume t h a t YT is irreducible, and there is a dense open subset Yr,o C YT such that the restriction Yr, o ~T is a smooth morphism. T h e n I)~(YT,o/T) is a locally closed subvariety of / 9 ~ ( ( P N x T)/T) = / ) ~ ( P N )

x T = D ~ ( P N ) x T,

if w e again identify D ~ ( P N ) a n d /~2m(PN) via r

Let

]D2m(yT) b e

the closure of

D2,,(YT,o/T) in D ~ ( P N ) x T and [/)~(YT)] its class in A*(D~(PN) x T). Let p: D~(PN) x T ~

D2m(PN)

be the projection. Let X C P N be a smooth projective variety of dimension d > m. Let i : D~(X) ~ projection. We p u t

D~(PN)

be the e m b e d d i n g and r0 : D~(X) ----* X be the

K(X, YT) : = (ro).(i*(p.([D~(VT)]) ~ A*(X).

Remark 3.2.22.

K(X, YT)

is a c a n d i d a t e for the class of the locus where X and elements of the family YT have second order contact.

Proposition 3 . 2 . 2 3 . Let n,d ff iN. Let YT C PN • be a family of re-dimensional projective varieties satisfying the conditions of definition 3.2.21 with dim(T) = t.

3.2. Varieties of higher order data andapplications

127

Let e = (N - d)(m+z2 ) - t + ( d - m), and assume 0 < e < d. For all partitions ((~) = (1~1,2~2,...) of numbers s < e there are integers no such that we have for all smooth projective varieties X C P N of dimension d: :

o,

s=0 c,EP(s)

P r o o f : Let f : = ( m+2 2 ) ( N - m) - t. We will show more generally t h a t for every class W E A I ( D 2 ( p N ) ) there are integers n~ for all p a r t i t i o n s a of n u m b e r s s < e such t h a t the above formula holds for ( r 0 ) . ( i * ( W ) ) . As A * ( D ~ ( P N ) ) i s generated by H and the Chern classes of the universal quotient bundles Q1 a n d Q2, it is enough to show the result for the monomials M in H a n d the Chern classes of Q~ and Q2.

Using our conventions we can write i*(H) = H, i*(Q1) = Q1 and

i*(Q2) = Q2. Let M = MoMIM2, where M0 is a m o n o m i a l in H and the Chern classes of X , M1 a monomia] in the Chern classes of Q1 a n d M2 a m o n o m i a l in the Chern classes of Q2. We assume t h a t M1 e Adl(D2m(X)), M2 e Ad2(D2m(X)). If dl = r e ( d - m ) and d2 = ( m + l ) ( d - r n ) , then we have ( r 0 ) . ( M ) = aMo for a suitable integer a depending only on the monomials M1 and M2 and not on X . (Let ql . . . . , qm and r l , . . . , r(,~+l) be the Chern classes of the universal quotient bundle on Grass(re, d) a n d on Grass((~+'),("~+~) + d - m) respectively. T h e n a is the p r o d u c t of the intersection n u m b e r s M1 ( q l , . . . , qm) and M ~ ( r l , . . . , r(~+~)) on these G r a s s m a n n i a n s . ) If d2 < ( m + l ) ( d - m) or d2 = ( m+] 2 )(d - m ) a n d dl < r e ( d - m ) , rn+l then we have ( r o ) . ( M ) = 0. If d2 > ( 2 )(d - m), then we use the relations of propositions 3.1.11 to express M as a linear combination with 2g-coefl:icients of monomials N = NoN~N2, where N2 E A~2(D~(X)) with e2 < d2.

If dx =

( m-i-1 2 ) ( d - m) and dl > m(d - m), then we use the relations of p r o p o s i t i o n 3.1.11 to express M as a linear combination with 2g-coefficients of m o n o m i a l s N = NoN1 M2, where N1 C A ~ I ( D 2 ( X ) ) with el < dl. So the result follows by induction.

[]

128

3.3. S e m p l e b u n d l e s a n d t h e f o r m u l a for c o n t a c t s w i t h lines In this section we introduce the Semple bundle varieties Fn(X) of a smooth variety X . They p a r a m e t r i z e in a slightly different sense t h a n D~(X) the n th order d a t a of curves on X . Like D~(X) and D31(X) they are smooth compactifications of y"~(n+l),c [ ~ + l ] by a tower of Pal_l-bundles over X (d = dim(X)). R e m e m b e r t h a t (n+l),~ p a r a m e t r i z e s the rtth order d a t a of germs of smooth curves on X. We will X[n+l] use the F~(X) to o b t a i n a general formula for the higher order contacts of a smooth variety X C P N with lines in P N as a linear combination of monomials in the h y p e r p l a n e section H and the Chern classes of X. We finish by considering more generally higher order contacts of X with a family of curves. For simplicity we will assume during the whole of section 3.3 that the ground field is C.

D e f i n i t i o n 3 . 3 . 1 . Let X be a smooth variety of dimension d. We define inductively varieties F~(X) and vector bundles Gn(X) on F.(X). Let f0(x)

:=

x, Go(X) : = Tx.

Assume inductively t hat F0 ( X ) , . . , F~_ 1(X) and G0 ( X ) , . . . , G n - 1( X ) are already defined. Assume furthermore t h a t G n - I ( X ) is a s u b b u n d l e of the tangent bundle

TF,_~(x) of rank d. T h e n we put Fn(X) :----P ( G n - I ( X ) ) . Let f~,x : P(Gn-I(X)) ~ be the projection.

Yn-l(X)

Let 8n := Op(G,~_KX))(--1 ) be the tautological s u b b u n d l e of

f*,x(Gn_l(X)).

Let TF.(X)/F._dX) = (f~F.(X)/F~_,(X))* be the relative tangent bundle. We define the subbundle Gn(X) of TF.(x) by the d i a g r a m

0

~ TFo(X)/F._I(X)

'

TF,dx)

0

'

)

Gn(X)

T TF.(X)/F,~_KX)

df.,x,. f~,x(TF~_~(X)) .

[]

,

0

)

O.

lJ '

an

* Sn ~ ) fn,x(TF._,(x)) is the composition of the n a t u r a l inclusions s~' ,f*,x(G~_I(X)) and f*,x(G~_l(X))r ~f*,x(TF._,(x)). We call G~(X) the n th Scruple bundle and F~(X) the rt th Semple bundle variety of X .

Here j

Let the divisor D~+I C F n + I ( X ) be defined by

Dn+l = P(TF.(X)/F._I(X)) C P ( G ~ ( X ) ) = F~+I(X).

3.3. Semple-bundles and the formula for contacts with lines

129

For 0 < i < n - 1 let

gi,x :----fi+l,X . . . . . fn,x : F,~(X) ~

Fi(X)

0 If this does not lead to confusion, we will not write the index X and gn,X :~ gn,X" of the maps fn, g/. We put n

Fn(X)o

::

Fn(X) \ (U(g~)-l(D'))

9

i=2

The Semple bundle varieties were first introduced in [Semple (1)]. In [Collino (1)], [Colley-Kennedy (1),(2)] they are considered for arbitrary smooth surfaces. The construction of Fn(X) for an arbitrary smooth projective variety X is an obvious generalisation. For our purposes it appears to be slightly more practical to use the tangent bundles instead of the cotangent bundles in the construction. We can easily determine an inductive formula for the Chow rings of the Fn(X). Proposition 3.3.2.

A*(F~_,(X)[.~])

A*(F,~(X)) =

( i : < c'(Gn-l(X))pd-i)" ~/r

we have P~ = c l ( s ; ) = c l ( O p ( G o _ , ( x ) ) ( 1 ) ) ,

and the Chern clas~es ci(a,~(X))

are computed inductively by the formula

c(Gn(X)) = (1-Pn) c(ao(X))

:

Z ( d ; i ) f*(ei(Gn-l(X)))PJn i+j<_d--,

4X).

P r o o f i This follows immediately from the exact sequence o

~

Trn(x)/F,,_~(x) ~

Gn(X)

8n ~

0

and the Euler sequence

0 ~

OF,(X)

~

f*(Gn-l(X))@s;

TF~(X)/F._I(X)

~

O.

[] Let Y C X be a smooth closed subvariety of codimension r. Let Ny/x be the normal bundle of Y in X. We now want to show that Fn(Y) is a closed subvariety

3. The varieties of second and higher order

130

of F,~(X), and want to describe its class in the Chow ring A*(F,~(X)). We suppress g~,x* and (g~,x)i * in the notation. L e m m a 3.3.3. Fn(Y) is a closed subvariety of

f:,~x(F,~-~(Y)) C g:,~x(Y) C F,,(X), and its cla~s

[F.(Y)]

~ A ( g . , x ( Y ) ) i~

[F,(Y)] = c~(Ny/x | s~)c~(Xy/x | s; | s~).., c~(Xy/x | s~ |

| s*).

P r o o f i We assume by induction that F,~(Y) is a closed subvariety of F , ( X ) . On Fn(Y) we have the diagram

0 --~

TF.(X)/F._~(X)]F.(y)

---+ a,~(X)lF=(y )

T 0

~

, s,~ --~

0

, s,~

0

T

TF,~(y)/F,~_,(y

)

)

G,,(Y)

(*) .,

So F,~+I(Y)

= P(Gn(Y)) is a closed subvariety of f ~ I , x ( F ~ ( Y ) ) = P(Gn(X)IF,(y)). To determine the class [Fn+I(Y)] ~ A*(g~_~I,x(Y)) we have by induction only to determine the class of F,+a(Y) in A*(f~.~I,x(F,~(Y)) ). For this we consider the canonical injection

~r : s,,+a~ , f~,+l,x(G,,(X)lF,(y))

on f:)-l,x(F,,(Y)). The subvariety F.+l(V) C f[~-I,x(F.(Y)) is the locus where ~r factors through the subbundle f*+a,x(Gn(Y)) of f,~+l,x(Gn(X)lF.(y)), i.e. the vanishing locus of the composition

s,+l

'~',

f*+l,x(G,(X)]F,~(y))

~ f*+I,x(G,(X)IF.(y)/Gn(Y)).

As Fn+I(Y) has eodimension r in f~_,,x(Fn(Y)), its class in Ar(f~_~I,x(F,,(Y))) is the Chern class

c~(s~+l | f*+I,x(G,dX)M.(v)/G,(Y))), and by the diagram (*) we have:

G n(X)IF~(y)/Gn(Y ) "~ (TF.(X)/F._t(X)lF.(y))/TF.(Y)/F,,_,(y)" It is well known that the relative tangent bundle of a projectivized vector bundle E of rank r is

Tp(E)/y = Op(E)(1) Q Qr-I,E.

3.3. Scruple,bundles and the formula for contacts with lines

131

So we have

a.(X)l~o~r)/c.(Y) ~_ ** | ( ( f * , y ( G . - , ( X ) l v . _ ~ ( r ) ) / s . ) / ( f * , r ( G . - a ( Y ) ) / s . ) ) ~-- s* | f . , y ( G . - I ( X ) I F . _ I ( y ) / G . - I ( Y ) ) . So we get by induction

1 @ (g.,x) (~,) | g . , x ( T x l y / T r )

*

*

G . ( X ) I F . ( y ) / G . ( Y ) ~- s. | ,',., * = 8n |

|

~- **.|174

1 (gn,x)

*

*

*

*

*

(81) |

|

[]

In the case of a s m o o t h curve C C X we want to describe the e m b e d d i n g

Fn(C) C F , ( X ) a little more precisely. 3 . 3 . 4 . Let C C X be a smooth curve. As G~(C) has r a n k 1 over F~(C), the projection fn,c : Fn(C) ----* Fn-a(C) is an i s o m o r p h i s m and so also g~,c : F~(C) ----* C is an isomorphism. The e m b e d d i n g Remark

jn,c : Fn-I(C)

f.-,~c ,Fn(C) ~ ,IQ~x(F._I(C))

is defined by the sub line bundle TF._,(C) of Gn-l(X) C TFn_I(X ). Let iv : C be the e m b e d d i n g of C into X and in,c the e m b e d d i n g

g-~" .,c

9

,.,c: c

,F.(C).

~X

,g:,~(C).

Then we obviously have

i~,c = j . , c . . . . . jl,coic.

Remember

that

Xl,~,c

C

X["I

parametrizes

subschemes of the form

s p e c ( O c / m ~ # ) for s m o o t h locally closed curves C C X a n d x E C. L e m m a 3.3.5. The map

sVec(Oc/m"~+J)~-* i . M x ) defines an open embedding y[n+l] i~ : ~(~+1),r

with image F.(X)o (see definition 3.3.1).

--~ F.(X)

3. The varieties of second and higher order

132

P r o o f i We have to show that this map is well defined (i.e. does not depend on the choice of the smooth curve C) and defines an isomorphism. For this we introduce local coordinates on ~(n+l),c y-[n+1] Let (Xl,.. ,xd) be y [ ~ + l l and F,~(X)o. Let Z E ~'(n+l),o" local coordinates on U C X such that

I z := (Xl + l , x 2 , . . . , x d ) g[n+l]

is the ideal of Z. The subsehemes Z ' E ~'(~+1),c near Z are defined by ideals

j=O

So al,0 and the ai,j ( i = 2 , . . . , d, j = 0,.

'

/ i=2,...,d/

, n) are local coordinates of ~(~+l),~Y[n+l]near

Z. We want to suppress the pullback in the notation. We write x ~ := xi. V C / I - I ( u ) be the open subset on which d x ~

dx~

1

xi

Let

# 0 holds. T h e n

:=

dx01~,

is regular for i = 2 , . . . , d. x ~ and the dx~ (i = 2 , . . . , d) form a basis of the relative differentials f~F1(X)/XIV. Let by induction x ~ and x{, (i = 2 , . . . , d ,

j = 0,...,n)

be local coordi-

nates on ( g l ) - a ( y ) A Fn(X)o such that the dx'~ (i = 2 , . . . , d ) form a basis of T h e n we have

~F,~(X)/F._I(X)[(g~)-I(V)oFn(X)o.

dx~

7s 0 o n ( g n1 + l ) --1 (V) N

Fn+l (X)o, and the functions d X n/ I~.+~ x~ +1 . _ dxOl~,+~ are regular on ( g 1 + 1 ) - l ( V ) N Fn+a(X)0. x ~ and the x Ji ( i = 2 , . . . , d; j = 0 , . . . , n + 1) are local coordinates on ( g 1n + a ) - l ( V ) A F , + l ( X ) 0 .

The dx'~ +1 (i = 2 , . . . , d )

form a basis of ~-~Fn+~(X)/F.(X)[(g~+I)-I(V)c.IFn+I(X)o. These coordinates have been introduced in the case of a surface in [Colley-Kennedy (1)]. Now let C C U be a smooth locally closed curve such that Xl is a local parameter on C. From the definitions we get that in our coordinates the m a p in,c is given by

So the m a p

spec( O c /m~x+l. ) ~

i.,a(x)

3.3. Scruple-bundles and the formula for contacts with lines

133

can be described in our local coordinates by

( al,o, ( ai,j )i=2 ..... d;j=0 ...... ) ~-+ (el,o, ( bi,j )i=2 ..... d;j=0 ...... ) where

hi'J= (~--~)J (~=oai'kXk) . . . . o

= jIai,j + E

k>j

k! k-j (k - j)! a''kal'~ "

So it is well-defined and an isomorphism on ( g ~ ) - l ( V ) . As the inverse i,~,c(x) H spec(Oc/m~x +1) is well-defined and does not depend on the local coordinates, i,~ is an isomorphism onto its image. We see that we can cover all of F,~(X)o by changing the local coordinates. As is is an isomorphism in all coordinate charts, its image is the whole of F , ( X ) o . [] So we see that F~(X) is a smooth compactification of X (n+l),c" In+l] Now we want to compute the number of n th order contacts of a smooth variety X C P N with lines in PN.

Definition 3.3.6. Let Aln+I(PN) nA-1 C p~+a] be the closed subvariety ~+1

{

AIn+I(PN ) :=

p~+al Z C

Z is subscheme of a line, } and the suppport of Z is one point

with the reduced induced structure.

) is a subvanety " In+l] Obviously AIn+I(PN ,~+1 of (PN)(n+l),c" Now we want to describe AIn+I(PN). ~+1 By definition it parametrizes subschemes of the form spec(Ol/m~x+,~) for lines l C P N and points x E l. Let A ( N ) C P N • G(1, N) be the incidence variety

A ( N ) : = {(x,I) E P N •

xEl}.

L e m m a 3.3.7. Let n > 1. The application

spec ( Ol / m ~_ ,+1 _),

, (x,l) c P n • G ( 1 , N )

gives an isomorphism n+l e~ : AIn+I(PN )~

A ( N ) C P N x G(1,N).

3. The varieties of second and higher order

134

P r o o f : Let X l , . . . , x N be the s t a n d a r d coordinates on A N C P N . Let Z E nq-1 Aln+a(PN ). We can assume t h a t Z C A N, a n d that the ideal of Z is of the n+l form Iz : = ( x ] + l , x 2 , . . . ,XN). A subscheme Z' E AIn+I(PN ) near Z has an ideal of the form IZ,

:=

((Xl -- al,0) n+l, x2 -- a2,1Xl

-- a2,o,...,

XN

-- aN,oXl

-- aN,o),

and al,0 a n d the ai,o, ai,a, (i = 2 , . . . , N) are local coordinates on A l nn++l I ( P N ) near Z. Let l be the line defined by ( x z , . . . ,XN). A line near I is given by X2 - - a 2 , 1 X l

- - a2,0~ 9 9 9 ~ X N

-- aN,0),

-- aN,oXl

and the ai,o,ai,1, (i = 2,... , N ) are local coordinates on G ( 1 , N ) near I. So the application n+l

e n : Aln+I(PN ) ---* P N • G ( 1 , N ) is given in our local coordinates by (al,o~a2,o,.

. . ,aN,o~a2,1

...,aN,l)

~

((al,0,a2,0,...,aN,0),(a2,0,...,aN,0,a2,1...,aN,1)),

a n d this defines an isomorphism with the subvariety

A(N) C PN • G(1,N)

Remark

3.3.8.

[]

Let X C P N be a smooth subvariety of codimension r.

From

n+l

the definitions we can see t h a t the intersection points of X (n+l),c ['~+1] and A / n + l ( P n ) -[,+a] in (PN)(n+l), c correspond exactly to the n th order contacts of X with lines in P N . More precisely we have: the image ~- n l At v( n[+nl )+, q (x,l) E X • G(1,N)

r (']

n+l AI,.,+I(PN)) is

1 has n *h order contact with X at x } .

Now we want to describe the incidence variety A ( N ) C P N • G ( 1 , N ) more precisely. We have the projections

A(N) ~/ Pl

PN

~ P2

G(1,N)

3.3.

Semple-bundles and the formula for contacts with lines

135

R e m a r k 3.3.9. Let OPN(--1 ) be the tautological line bundle on P N and T2 :---T2,N+I the tautological subbundleon G(1, N) = Gr(2, N + I ) . Let Q1 := QN,N+I :-@N+I or,, /OpN(-1). Then we can see easily that Pl and P2 can be described as the natural projections P(Qa) =

A(N)

= P(T2) P2

G(1, N)

PN (see also [Fulton (1)1 Ex 14.7.12). We put /~ := p~(OpN (1)), t5 := Op(Q~)(1),

H := c1(/~), p :_-- c1(/5 ).

Then we can see easily t h a t / t = Op(T2)(1), and/5* is the universal quotient bundle

/5*

=

* * ' T 2)/ "/t* 9 QI,T~ = P2(T2)/Op(T~)(--1) =P2(

We have

p~(c(Q1)) = 1 + H + H 2 +... + H N, and so 2~[H, P]

A*(A(N))= ( N HN+I'

(.)

)

E Hi pN-i

9

i=0 GN+I /T 2)~ be the universal quotient bundle on G(1, N). Let Q2 : = Q N - 1 , N + I := ~( 0 GO,N)/ is the pullback of a Schubert cycle

p~(ck(Q2))

(x, l) E A(N) /~* = Op(T2)(--1) a n d / 5 . =

I intersects a fixed } linear subspace "

(k + 1)-codimensional Q1,T~ imply

p~(Q2) =

p~(Q1)/.P* and

so

k

P2( * C k(Q2)) = ~ - ~ H J P k-j. j=0

The relative tangent bundle is * 1 )) = ~I (~ /5*. TA(N)/G(1,N ) = Op(T2)(1 ) ~ (P2(T2)/Op(T2)(--

136

The varieties of second and higher order

3.

Now we want to describe the restriction

inl.N+lz n

: ~N+l/n

For this we give embeddings an :

A(N) --~ F,(PN with Z"n IAN+~(pN Nq-1 )

D e f i n i t i o n 3 . 3 . 1 0 . We want to define a , : A(N)

~--- O/nO~ n .

, F n ( P N ) for n _> 1 inductively.

We have TP N =

OPN(1)

@ QI.

So there is a n a t u r a l isomorphism a l : A ( N ) = P ( Q 1 ) ------4P(ZpN ) with

a~(sl) = a~(Op(TPN)(--1))

=

TA(N)/G(I,N

).

W e put AI := FI(PN). A1 is mapped to G(I,N) by line bundle of

commutes. So

p2oa~ I. TA,/G(I,N) is a sub

TF~(PN)' and the d i a g r a m

TA1/G(],N )

'

)

TFI(pN)

~1

,

,

I ; ( T I , N)

T&/G(I,N) is a sub line bundle of G I ( P N ) C TFI(PN)"

We assume by induction t h a t a,~ :

A(N)

, F n ( P N ) is an embedding. Let

An C F n ( P N ) be its image. Am is m a p p e d to G ( 1 , N ) by p2oa~ 1. We also assume that

TA~/G(1,N) is a sub line bundle of Gn(PN)[Ao. Let fln+l :

An

-1 fn+l(An) C Fn+I(PN)

Gn(PN)IAn. Let An+l C F n + I ( P N ) be the image of fln+l Then TAn+I/CO,N) is a sub line bundle of TFn.I-I(PN)]A.+t, and the d i a g r a m

be the e m b e d d i n g defined by the sub line bundle TAn/G(1,N ) of

TAN+I/G(1,N )

)

TFu+I(pN)

1 ~N+~

~dfN+l '

f;!+lCrPN)

commutes. So TAN+~/a(1,N ) is a sub line bundle of

GN+J(PN)[An+, C TF.+I(pN)]An+ 1.

3.3. Semple-bundIes and the formula for contacts with lines

137

We p u t OLn+1 : = ~n+100~n- This is a closed embedding. We get inductively for all n:

a*(sn) = TA(N)/G(1,N ) = !ft @ P*.

Lemma

3.3.11.

i.l~+

....

~,N+Ik-C~N)

= a~0e~.

P r o o f : We only have to show t h a t inltw+11 = (N+I)

holds for every line.

C*.oenl/[N+~] (N+I)

Here ~(n-lfin+l]1) i8 the closed subvariety of l["+1] p a r a m e t r i z i n g

subsehemes of length n + 1 which are concentrated in a point of I. The projection ][N+I] P : "(g+l) ~ l m a p p i n g such a subseheme to its s u p p o r t is an isomorphism and 6 :~- enlltN+l ] op--1 is the m a p (N+l)

So we have to show t h a t a . o e = in3 holds for the e m b e d d i n g i~ 3 : I~-----+g21pN(I) C

F,(X). By definition 3.3.10 a l : A ( N ) ---. P ( T p N ) is defined by the sub line bundle

TA(N)/G(1,N) C

T p N. So the sub line bundle

~10~: l ---, f v l ( l )

Tl C Tp NII defines the e m b e d d i n g c FI(PN).

By r e m a r k 3.3.4 this also defines il,i. By induction we assume t h a t anoe = in,l. In p a r t i c u l a r we have

(a~oe)(l) = r~(1) C g~,~,N(l). The e m b e d d i n g ~ + 1

: A,, TA./a(1,N) C G~(PN)IA., i.e.

~ f~a(A~)

/~-+llF,~O) : r,(1) ~

is given by the sub line bundle

f211(F,(1))

is given by TF~(0 C G n ( P N ) I Y . ( 0 . By r e m a r k 3.3.4 this also defines the e m b e d d i n g j n + l , l : Fn(1) ----+ fj~l(Fn(l)). So we have/3n+lIF~(0 = j~+lj, and thus by r e m a r k 3.3.4 an+lOe = Jn+l,loin,l = in+l,l.

[]

Now we can show a general formula for the numbers of higher order contacts of X with lines in PN,

3. The varieties of second and higher order

138

D e f i n i t i o n 3.3.12.

A l nn ++l l ( P g ) ~

Let X C P N be a smooth projective subvariety.

Let p :

P N be the projection. We put Aln+l,X := p - I ( X ) .

Let Px : Aln+l,X

, X be the restriction of p. Let kn,x : Aln+l,x ~

g~,~N(X)

~[n+l] be the restriction of the embedding in : (PNJ(n+l),c ~ F n ( P N ) to Aln+l,X. Let [Fn(X)] be the class of F n ( X ) in A*(g:,IpN(X)). The class of n th order contacts of X with lines P N is defined as

K n ( X ) := (px).(k*,x([Fn(X)])) E A*(X). The class of n th order contacts of X with lines in P N which intersect a general linear subvariety of dimension l + 1 is

Kn,t(X) := (px).(k*,x([Fn(X)]) " e*(p~(ct(Q2)))) c A*(X). For a closed subvariety X C P N we put A x := p11(X) c A ( N ) . Let qj : A x ----* X and q2 : A x ~ G ( 1 , N ) be the projections. Let --1

an,x : A x --~ g,~,pN(X) be the restriction of a n : A ( N ) --~ F~(PN).

R e m a r k 3.3.13. By lemma 3.3.11 we get

K n ( Z ) = (ql ).(~*~,x([Fn(Z)])), g n 3 ( X ) = (ql)*(a*~,x([Fn(X)])" q~(cl(Q2))).

R e m a r k 3.3.14. Let h C P N be a general linear subspace of codimension l + 1 and W ( h ) C G(1, N) the set of lines intersecting h. By remark 3.3.8, lemma 3.3.11, definition 3.3.12 and remark 3.3.13 we have:

a~,lx(Fn(X)) = { ( x , I ) C A x

l hasnth order contact } with X at x

{

ql(a'~,~(Fn(X))) =

there is a l i n e / with which

x EX

X has n *h order contact at x

}

3.3. Semple-bundtes and the formula for contacts with Iine~

and

139

ql(,~lx(F.(X)) n q;~ (W(h))) =

there is a line l } intersecting h and having n th order . contact with X at x

x 6 X

Let W be an irreducible component of q l ( c r ~ a x ( F n ( X ) ) ~ q ~ l ( W ( h ) ) ) . The multiplicity of W in (qx).(a~,lx(Fn(X)) N q f l ( W ( h ) ) ) is the degree of qll~:,~x(F,(X))nq[l(w(h))) over W (or 0 if this degree is infinite), i.e. the number of lines intersecting h having n th order contact with X in a general point x C W counted with multiplicity. In particular we have: let Y C X be a closed subvariety of dimension d where d = l + nr - N + 1 so that there are only finitely m a n y n th order contacts of X with lines intersecting h in points of Y. Then the n u m b e r of these contacts counted with multiplicities is the intersection number K n , t ( X ) " [Y].

T h e o r e m 3.3.15. Let n be a positive integer. Let X C P N be a smooth projective variety of codimension r, let NX/pN be the normal bundle of X in P N and H the class of a hyperplane section. Let O < l < N and d := l + n r - N + l. We assume O < d < N - r. Then we have:

K.j(X)=E

E

())

(_1) ~ N + k - l

.

8

s=max(O,k-l)

k=O

n

E Ae(X).

In particular we have in the case l = O:

,

Kn(X)=E(-1)k(N+k

k

)

k=O

( l~j~-~

E il+...+in=d-k

j=l

)

/_/k II%(Nx/p,,) . j=l

Let Y C X be a closed subvariety of dimension d and [Y] E A N - r - d ( X ) its class. Let h C P N be a general (l + 1)-codimensional linear subvariety. If there are only finitely many n th order contact8 of X with lines intersecting h in points of Y, then the number of these contacts counted with multiplicities is

,(k

E

k=O

E

s=max(O,k--I)

(>) (

(_1) 8 N

+k-I

S

.

)

3. The varieties of second and higher order

140

If in particular l = 0 and d = N - r = d i m ( X ) , and so 2 N - 1 = (n + 1)r, then the number of n th order contacts of X with lines in P N counted with multiplicities is N-r k=O

il+...+i~=N-r-k

We first show the following lemma:

L e m m a 3.3.16.

(I)

(ql),(pN-,+k)

=

0,

k
1,

k = 0,"

-H, O,

k = 1; otherwise.

l

(2) t=0 k

s=max(O,k-l)

Proof:

(1) By remark 3.3.9(*) we get

A*(X)[P] A * ( A x ) = (i=~o H i p N _ i )

"

(**)

The result is clear for k < 0 and k > N. By (**) and the projection formula we get ( q l ) , ( P N) = ( q l ) , ( - g P

N-l) = -g.

Now let N > k > 2, and assume the result holds for k - 1. Then we get by (**) and the projection formula N

(ql)*(pN-l+k) = - E

g~(ql)*(PN-l+k-8)"

By induction and the above this is - H k + H k = O.

3.3.

Semple-bundle~ and the formula for contacts with lines

141

(2) By (1) we have

(

(ql). (P - H) N-1+k-' E HtP'-t t=O

)

= (ql)* ( P - H) N-l+k-I E

Hk-=pl--k+=

s=k-I

=Hk

E

(-1)'

N-l+k-I s

s=max(O,k--l)

Hk

(-1)=

+ s-I

s

s=max(O,k-l)

Proof of theorem

3 . 3 . 1 5 : We only have to show the formula for

KnA(X). By

lemma 3.3.3 and definition 3.3.10 we have

~ , x ( [ r ~ ( x ) ] ) = ~,.(q;( N x / l , = ) e ~ L x ( s ~ )) . .. .

9c,-(qr(Nx/p= ) @ a~,x(s~) |

| a~,,x(s*,)))

j=1 =

n(•

j=l

3

ql ( c i ( N x / P N ) ) ( P

-

H ) r-z

)

i=1

So we get by the projection formula and remark 3.3.9:

K=,~(x)

= (ql).(a*,x([Fn(X)]) 9q~(cdQ2))) =

(i -/

*

'

=Z k=0

3

j=l E

(ql)*

il+...+in=d-k

ql ( C i ( J Y x / P N ) ) ( P

-

i=1

(

EHtPt-t(P-H)

n'-d+k

H) r-'

)n

t=0

))

jr-i'%(NX/PN)"

j=l

By the definitions we have n r - d + k = N - 1 + k - I. The result now follows by lemma

3.3.16.

[]

So we have found formulas for the contacts of X C P N with lines in P N as linear combinations of monomials in H and the Chern classes formula

c(Nx/PN ) = (1 + H)N+I/c(Tx)

ci(Nx/pN ). Using the

3. The varieties of second and higher order

142

we can replace the Chern classes of Nx/pN by those of X if we want. T h e result will however be more complicated this way. It is easy to check t h a t the formulas after proposition 3.2.19 can be o b t a i n e d as special cases. Now we want to show that more generally the class in A*(X) of the locus where a smooth projective variety X C P N has rtth order contact with a given family of curves is p r o p e r l y interpreted a linear combination of monomials in H and the Chern classes of X . This linear combination will depend on the familyCT. We will not treat here the much more difficult question how to determine this linear combination for a given family CT. The argument is similar to t h a t at the end of section 3.2. First we will generalize the Semple bundles to a relative situation.

Definition

3.3.17.

Let T be an irreducible algebraic variety.

Let X - - ~ T

be a s m o o t h m o r p h i s m of relative dimension d. We will inductively define varieties Fn(X/T) and vector bundles Gn(X/T) on Fn(X/T). Let Fo(X/T) :=

X, Go(X/T) : = Tx/T. By induction assume that Fo(X/T),... , F , _ I ( X / T ) are already defined. Assume t h a t G n - 1( X / T ) is a subbundle of rank d of TF._,(X)/T. Then we put

Fn(X/T) : = P ( a n - 1 (X/T)). Let

f,,X/T : P(Gn-I(X/T)) -----+Fn-l(X/T) be the projection.

Let sn be the tautological subbundle of f*,X/T(G,~_I(X/T)).

Now we define the s u b b u n d l e Gn(X) of TF,(X)/T by the d i a g r a m 0

~

TFn(X/T)/Fn_I(X/T ) ~

0--"* TF,,(X/T)/F,~_I(X/T ) ~

TFn(X/T)/T

Gn(X/T)

---+ f * , x / T T F n _ I ( X / T ) / T

~

0

--*

~

O.

s,,

If Y C X is a (locally) closed subvariety such t h a t the restriction to Y of the projection X

) T is a s m o o t h m o r p h i s m of relative dimension m, then we see

in a similar way as in the proof of l e m m a 3.3.3 t h a t Fn(Y/T) is a (locally) closed subvariety of F,~(X/T). D e f i n i t i o n 3 . 3 . 1 8 . Let T be a smooth projective variety of dimension m - 1 and

CT C P N x T a flat family of curves, i.e. we have the projections CT

PN

T,

3.3. Semple-bundles and the formula for contacts with lines

143

p2 is flat and for all t E T the fibre C, = p~-I (t) is a curve. We assume in a d d i t i o n t h a t there is a dense open subset CT, O C CT such t h a t the restriction CT, O ~

T

is a smooth morphism. T h e n Fn(CT, o / T ) is a locally closed subvariety of F ~ ( ( P N x T ) / T ) = F n ( P N ) x T. Let f'n(CT) be the closure of Fn(CT,o/T) in F , ( P N ) x T a n d [Fn(CT)] its class in A(N-1)(n+I)(Fn(PN) X T). Let p: F~(PN) x T ~

F~(PN)

be the projection. We p u t

K ~ ( C T ) : = p.([Fn(CT)]) E A r ( F ~ ( P N ) ) , where r : = N + ( N -

1)n-re.

Let X C P N be a s m o o t h projective variety of

dimension d. Let i,~,x : F , , ( X ) ~

F n ( P N ) be the embedding. We p u t

K . ( X , CT) := (g,~,x).(i*~,x(K.(CT)) E A ~ ( X ) , where e = N + ( N - d)n - m.

R e m a r k 3 . 3 . 1 9 . K n ( X , CT) is a c a n d i d a t e for the class of the locus where X and curves in the family CT have n th order contact. We have for example

g , , x ( p ( F n ( C T , o / T ) M ( F ~ ( X ) • T))) =

there is a t E T such t h a t x is a s m o o t h point of } Ct and Ct has n th order contact with X in x "

x EX

Assume in p a r t i c u l a r e = d, i.e. m = (n + 1)(N - d), and assume the subset

Fn(CT,o/T) N ( F n ( X ) • T) C F n ( P N ) • T to be finite a n d to coincide with fi',~(CT) f) (F,~(X) • T). T h e n the n u m b e r of n th order contacts of X with curves in the family CT counted with multiplicities is

K.(X, CT).

Proposition

3.3.20.

Let n , d E ZW. Let

CT

be a family of curves satisfying the

conditions of definition 3.3.18 with d i m ( C T ) = m. Assume e = N + ( N - d)n - m and 0 < e < d. For all partitions

(~) = (1~, 2 ~ , . . . )

3. The varieties of second and higher order

144

of numbers s <_ e there are integers n~ such that for all closed subvarietiea X C P N of dimension d

Kn(X, CT) : ~

~

rlc~Se-Scl(X) cq ...ce(X) c~e.

s = o c~E P ( s )

Proof:

W e show m o r e generally t h a t for any W G Ae+n(d-1)(Fn(PN)) a n d for all

p a r t i t i o n s a of s < e t h e r e are integers n~, satisfying

(g,,,,:i,(i:,.,.(w)) =

Z s=0

As A * ( F n - I ( P N ) )

c~CP(s)

is g e n e r a t e d by H, P1 : = C l ( S l ) , . . . , P , , - 1

: = c1(~,~_1), it is

e n o u g h to p r o v e the result for m o n o m i a l s in H, P1, 9- 9 P,,. We will now suppress

i*,x in t h e n o t a t i o n a n d write g~ i n s t e a d of g~,x. Let M = MoP~ 1 ...P~" be a Here M0 is a m o n o m i a l in H, c l ( X ) , . . . ,cd(X). If li = d - 1 for all i = 1 , . . . , n, t h e n we h a v e ( g ~ ) , ( M ) = Mo. O t h e r w i s e let j0 be the largest j such t h a t lj ~k d - 1. By p r o p o s i t i o n 3.3.2 we see t h a t ( g ~ ) , ( M ) = 0 if ljo < d - 1. So let ljo >_ d. By p r o p o s i t i o n 3.3.2 we can express M as a linear c o m b i n a t i o n w i t h 2g-coefficients of m o n o m i a l s N = N o P ~ ~ . .. p ~ n , w h e r e No is a m o n o m i a l in H, c l ( X ) , . . . , c d ( X ) , a n d we h a v e my = lj for j > j0 a n d mjo < ljo. So t h e result monomial.

follows by i n d u c t i o n .

[]

4. The Chow ring of relative Hilbert schemes of projective bundles In this chapter we treat the Chow rings of relative Hilbert schemes of projectivizations of vector bundles over smooth projective varieties. In section 4.1 we will first construct embeddings of relative Hilbert schemes into Grassmannian bundles and study them. The case of the relative Hilbert scheme of a Pl-bundle over a smooth variety is studied in more detail. From this we get the Chow ring of the variety AI~(Pe) parametrizing subschemes of length n of Pe which lie on a line in Pa. This variety has been used in [Le Barz (1),(2),(3),(4),(5),(8)] to obtain enumerative formulas for multisecants of curves and surfaces. ~ 3

In section 4.2 we compute the Chow ring of the variety Hilb (P2) parametrizing triangles in P2 with a marked side. This variety has been used in [Elencwajg-Le ~ 3

Barz (2),(3)] to compute the Chow ring of p~3]. The Chow ring of Hilb (P2) has a much simpler structure than that of p~31. In section 4.3 we generalize this result to a relative situation. We compute ~ 3

the Chow ring of the variety Hilb (P(E)/X) parametrizing triangles with a marked side in the fibres of the projectivization P ( E ) of a vector bundle E. We also consider the variety H3(p(E)/X) of complete triangles in the fibres of P ( E ) , which has been studied in [Collino-Fulton (1)]. We pull back the classes in the Chow ring ~ 3

(P(E)/X) to Br3(p(E)/X) to find some of the relations. The most imp3 portant case of our result is the variety Cop (Pc), parametrizing triangles with a marked side in Pd together with a plane containing them. A*(Hilb

In section 4.4 we finally treat the relative Hilbert scheme Hilba(p(E)/X) of subschemes of length 3 in the fibres of P ( E ) . Analogously to [Elencwajg-Le Barz (3)] in the case of P2 we define a system of generators for the Chow ring of

Hilba(p(E)/X) as A*(X)-alg~bra.

~ 3

By pulling these classes back to Hilb

(P(E)/X)

we determine their relations. To carry out the computations we have however to make use of a computer. The result is also quite complicated. The most important special case is again that of the variety Copa(pd), parametrizing pairs consisting of a subscheme of length 3 of Pd and a plane containing it. It can be obtained by blowing up p~3] along AI3(pe). The Betti numbers of this variety have been determined in [Rosselld (1)]. In the case d = 3 it has been used in [Rosselld (2)] to determine the Chow ring of p~3]. In a recent joint work with Fantechi [Fantechi-GSttsche (1)] we have computed the cohomology ring H * ( X [3], Q), for X an arbitrary smooth projective variety, by using an entirely different method.

146

4.1. n - v e r y a m p l e n e s s , e m b e d d i n g s o f t h e H i l b e r t s c h e m e a n d t h e structure of Aln(P(E)) Let X be a projective scheme over an algebraically closed field k.

In

[Beltrametti-Sommese (1)] the following definition was made:

D e f i n i t i o n 4.1.1. Let L: be an invertible sheaf on X. For every subscheme Z C X we study the restriction map rz, c : H ~ 1 6 3

, H~

s | Oz).

s is called n-very ample if rz, L is onto for every 0-dimensional subscheme Z C X of length fen(Z) <_ n + 1. R e m a r k 4.1.2. (1) We see that an invertible sheaf 1: is 0-very ample if and only if it is spanned by global sections and 1-very ample if and only if it is very ample. (2) Let s be an (n - 1)-very ample invertible sheaf on X. Then we can associate to each subscheme Z of length n on X the quotient H~

Oz @ s

= H~

C)/ker(rz, L )

of dimension n. This defines a m o r p h i s m . r 1 6 3 X ["] ~

Grass(n,H~

It is clear from the definition that an n-very ample invertible sheaf is also mvery ample for every m < n. In [Beltrametti-Sommese (1)] only the case of a smooth -surface S is considered. In this ease they show that r is injective if s is n-very ample and a closed embedding if/2 is 3n-very ample. In the appendix [Ghttsehe (3)] of [Beltrametti-Sommese (1)] the corresponding very ample invertible sheaf on S [nl is identified. In [Catanese-Ggttsche (1)] this result is sharpened and generalized to a general projective variety X. The main result is:

T h e o r e m 4.1.3. [Catanese-Ghttsche (1)] Let X be a projective scheme over an algebraically closed field k and f~ an (n - 1)-very ample invertible sheaf on X . The morphism Cn,,~ : X['q - - 4 G r a s s ( n , H ~ 1 6 3 is an embedding if and only if f_. is n-very ample.

4.1. Embeddings and the structure of AIn(p( E))

147

Now we want to generalize this result to a relative situation.

Let T be a

reduced projective variety a n d X a projective scheme over T. Let ~r : X ~

T be

the projection.

Definition 4.1.4. locally free.

Let s be an invertible sheaf on X for which also 7r.(s

For all n E zW let 7r~ : Hilbn(X/T)

is

~ T be the projection. Let

Zn (X/T) C X x T n i l b " (X/T) be the universal subscheme. We consider the d i a g r a m

Zn(X/T) r

P

x',~ qn

X

Hilb"(X/T) T,

in which p and qn are the projections. We get a n a t u r a l m o r p h i s m of locally free sheaves

on Hilbn(X/T) a follows: let

fa : r*~r.(s

~

7r*Tr.p.p*(s

h: be the n a t u r a l m o r p h i s m s of locally free sheaves on Hilbn(X/T). By the e o m m u t a tivity of the d i a g r a m we have

zc*r.p.p*(s = r * 0 r . ) . ( q . ) . p * ( / : ) , and r.,t: is given by ~r*~r.(s

r.,~

,

(q.).p*(Z)

\s, 7r*~r.p.p*(~) = 7r*0r.).(qn).p*(s ) For a fixed t E T let Xt be the fibre of X over t a n d p u t s

:= s

For a

fixed subscheme Z E Hilb"(X/T) lying in the fibre Hilbn(X/T)t = H i l b " ( X t ) of

Hilb'(X/T) over t the m a p rn,.~ between the fibres zc%r.(s (q.).p*(s

= H~

s

= H~163

|

Oz)

is just given by

rz, c, : H~163

----* H~163

|

Oz).

4. The Chow ring of relative Hilbert schemes of projective bundles

148

12 is called n-very ample on X relative to 7r, if rm,s is onto for all m _< n + 1 (in other words if for t E T and all subschemes Z C Xt of length Ien(Z) _< n + 1 the map rz,L~ is onto).

R e m a r k 4.1.5. Let/2 be an (n - 1) very ample invertible sheaf on X relative to 7r. T h e n (qn).p*(s

is a locally free quotient of rank n of ~ ' 7 r , ( s

By the universal

property of Grass(n, 7r.(E)) there is a morphism e L , n : H i l b n ( X / T ) - - ~ Grass(n, ~r, (E))

over T such that r

. . . . (t;)) : (qn)*P*(f--) 9

As an obvious corollary of theorem 4.1.3 we get :

R e m a r k 4.1.6. Let s be an n-very ample invertible sheaf on X relative to zr. T h e n

r

is one to one.

The question whether r

is an embedding we only want to consider in a very

simple case.

D e f i n i t i o n 4.1.7. Let X ~ ,T be a locally trivial fibre b u n d l e with fibre Xt and 12 an invertible sheaf on X. sheaf s

We call ~2 constant over T, if there is an invertible

on Xt a n d an open cover (Ui) of T such that ~r-l(Ui) ~ Ui • X t and

/:[~-l(v,) = P~(f-.t) with respect to the projection P2 : Ui x X t

~ Xt.

P r o p o s i t i o n 4.1.8. Let s be an (n - 1)-very ample invertibIe sheaf on X , constant

over T. Then r163 : H i l b n ( X / T ) ----+ Grass(n,~r,(E)) is an embedding if and only if f~ is n-very ample. P r o o f : As t; is constant over T we have with respect to a suitable local trivialisation

7r-l(gi) ~ gi x Xt: e n , c ] , r l ( u , ) = 1u~ x r

:Ui x (Xt) In]

The result follows by theorem 4.1.3.

, Ui x G r a s s ( n , H ~

[]

Now we want to consider the case of the projectivization of a vector bundle. X.

Let E be a vector bundle of rank d + 1 over a smooth projective variety Let P ( E ) p ~X be the bundle of one-dimensional linear subspaces of E and

Op(E)(--1) := T1,E the tautological s u b b u n d l e of p*(E).

Let P ( E ) - L ~ X

be the

4.1. Embeddings and the structure of AI'~(P(E))

149

bundle of one-dimensional quotients of E and Q1,E the universal quotient b u n d l e of 7r*(E). We note t h a t dualizing gives a n a t u r a l i s o m o r p h i s m d : P ( E ) ~

P(E*)

with d*(Ql,E*) = Op(E)(1). For Y = P ( E ) and Y = 15(E) respectively we again have the projections

z.(r/x) ~// p

NN q~

Y

Hilb~(Y/X).

Proposition 4.1.9.

(1)

QI,E is an m-very ample invertible sheaf on P ( E ) constant over X . For m >_ n - 1 it gives morphisms r over X

(2) r

:= CQtT,,~ : Hilb'~(O(E)/X) ---+ Grass(n, S y m ~ ( E ) ) =- (qn).p*(Q~'~).

with r

:= Cn,n iS an embedding.

Proof: W i t h respect to a suitable local trivialisation of E over X we have 7r-1 (Ui) = @n . Ui x Pd a n d Q1,EI,r~(u0 = P2(OPd(n)), where p2 : Ui • Pd ~ Pd is the projection. (1) follows by 7 r . ( Q ~ )

= S y m m ( E ) . (2) follows i m m e d i a t e l y from 4.1.8 a n d (1).

[]

Notation.

In future we will write Cn instead of CQ~,~,. and more generally Cm,n

for CQ~,~,~, if X a n d E are u n d e r s t o o d a n d m _> n - 1.

Now we specialize further to the case t h a t E is a vector bundle of rank 2 on X , i.e. P ( E ) is a P l - b u n d l e over X . We can express the class r

in a different way so t h a t its

geometric meaning is more visible.

Notation.

Remark

Let Hn : = (qn),p*(el((gp(E)(1))) C A I ( H i l b n ( p ( E ) / X ) .

4 . 1 . 1 0 . Let D := ~ aiDi be a divisor on P ( E ) (Di irreducible, ai 6 2g).

Then (q,),p*(D) = ~ ai (q,),p*(Di), and

(q.).p*(D,) = {Z

Hnb~

n D, r 0}.

4. The Chow ring of relative Hilbert schemes of projective bundles

150

Proposition

4.1.11.

r

: Hilbn(P(E)/X) ~

such that r

P ( S y m n ( E ) ) is an isomorphism

= Hn.

P r o o f : As P ( E ) is a locally trivial P l - b u n d l e over X , Hilbn(P(E)/X) has to be a locally trivial Pn-bundle over X. The same is true for P ( S y m n ( E ) ) . So the embedding Cn : Hilbn(P(E)/X) ----* P ( S y m n ( E ) ) over X must be an isomorphism. Let x C X and let u, v be a basis of the fibre E(x) of E over x. Then the polynomials of degree n in u, v are in a natural way a basis of the fibre Symn(E(x)) = Symn(E)(x). Let s be a (rational) section of Op(E)(1). The application

(alu + blv). . . . . (a,u + b,,v) ~

s(al u nt- b l y ) . . . . . 8(antt ~- bnv)

gives a (rational) section t of OP(Sym-(E))(i) with [div(t)] = Hn.

[]

As the Chern classes of symmetric powers of vector bundles of rank 2 are easy to compute, we know now the Chow ring of Hilb~(P(E)/X). In particular we obtain:

C o r o l l a r y 4.1.12.

If E is a vector bundle o.f rank 2 over X with Chern classes

Cl, C2, then

A*(X)[H2] A*(Hilb2(P(E)/X)) = (H~ + 3clH~ + (2c~ + 4c2)H2 + 4clc2)"

As a subscheme of length n of a fibre P1 of P ( E ) is just an effective zero cycle of degree n on this fibre, we see that Hilb'~(P(E)/X) is the n th symmetric power

S y m " ( P ( E ) / X ) of P ( E ) i.e. the quotient of ( P ( E ) / X ) " := P ( E ) x x P ( E ) x x . . .

xx P(E)

by the action of the symmetric group G(n) by permuting the factors. So we have

Sym"(P(E)/X) = e(Symn(E)). Let Z , ( P ( E ) / X ) C P ( E ) • Hilb'*(P(E)/X) be the universal subscheme. We see from the definitions that Z n ( P ( E ) / X ) is the reduced subscheme

Z n ( P ( E ) / X ) = { ( x , Z ) e P ( E ) x x Hilbn(P(E))

x e Z}.

We have a natural morphism r

P(E) xx Hilbn-I(p(E)/X) ~

Hilb~(P(E)/X).

4.1. Embeddings and the structure of AI'(P(E))

151

If we identify Hilbn(P(E)/X) with Sym~(P(E)/X), then this morphism is given by (x,~). , [x]+~. So we h a v e a m o r p h i s m

pl • r

P(E) •

Hilbn-l(P(E)/X)

>P(E) •

Hilb'~(P(E)/X),

and we see from the definitions that it is an isomorphism onto its image Zn(P(E)/X). If we identify Hilbn(P(E)/X) and P(Symn(E)) then r

P(E) •

P(Symn-I(E)) ~

P(Symn(E)),

is the morphism induced by the natural vector bundle morphism E | S y m n - l ( E ) ---+ Symn(E);

(~ | ( ~ " ~ ' . . . - ~ = - , ) ) ,

, (~' ~ " ~ ' . . . - ~ n - ~ )

So we get: L e m m a 4.1.13. Ip(E).X r

P ( E ) x • Hilb"(P(E)/X) ----* P ( E ) x x P ( S y m " ( E ) )

induces an isomorphism r Zn(P(E)/X) ~

P(E) x• P(Symn-l(E)).

We see that with respect to the projections Pl,P2 of P ( E ) x • P(Sym'~-I(E)) to P(E) and P(Sym"-l(E)) we have r ( P(Sym (E))(1))=p,(Op(E)(1))@p2((~P(Symn-t(E))(1)). Now let E be a vector bundle of arbitrary rank d + 1 over X. Definition 4.1.14. Let AIn(P(E)) be the reduced subvariety of Hilbn(P(E)/X), given by

AI'(P(E)))= {ZEHilb'~(P(E)/X)I Z is a subscheme of a line } in a fibre Pd Let Z~t(P(E)) be the universal subscheme over Aln(P(E)) and let

Z~t(P(E))

r

P(E)

\~.

AIn(P(E))

4. The Chow ring of relative Hilbert schemes of projective bundles

152

be the projections. In particular let Aln(Pd) C P~] be the subvariety given by

Al~(Pd)= { Z E P~]

ZisasubschemeofalineinP4}

and Z,~al(Pd) the universal subscheme over Aln(Pd). Let H, Ln-a, gn E AI(Z~t(P(E))) be the classes defined by H := i~*(q(Op(E)(1))), g ~ := ~ * ( ~ ) . ( g ) ,

Ln-1 := H , - H. We will also denote by H~ the class ( ~ ) . ( H ) E AI(AI"(P(E))). Let G := Grass(d - 1, E), which we view as the variety of lines in the fibres of P(E). Let T := T2,E be the tautological bundle of rank 2 over G. We can associate to each subscheme Z E AI'(P(E)) the line on which it lies. It is easy to see that this defines a morphism

axe: AIn(P(E)) Let F C P ( E ) •

~ G.

G be the incidence variety F::

{(x,/)EP(E)xxG

xEl}

with the projections F

\p2 P(E)

G.

Then we can identify F P~G with P ( T ) ~G, and with this identification we have Op(T)(1) ----p~(Op(E)(1)). Obviously the relative Hilbert scheme

Hilb'~(F/G) C Hilbn(p(E)/X) X x G is the closed reduced subscheme

Hilb~(F/G) = {(Z, I) E Hilbn(P(E)/X) x x G Z C l}, where we have now identified the points of G with the lines l in the fibres of E. We see that the projection Pl : Hilb"(F/G) --~ Hilb'(P(E)/X) defines an isomorphism of Hilb~(F/G) onto its image Aln(P(E))) C Hilb~(P(E)/X). (As a morphism to AIn(P(E))) it is obviously a bijection, and both nilbn(F/G) and AI~(P(E)) are smooth). Let Z,~(F/G) C P ( E ) • H i l b ~ ( P ( E ) / X ) • G

4.1. Embeddings and the structure of Aln(P( E))

153

be the universal subscheme. We see that the projection pl,2 : Zn(F/G) ) Zn(P(E)/X) gives an isomorphism of Z~(F/G) onto Z ~ l ( P ( E ) ) . So we get by lemma 4.1.13: Lemma

4.1.15.

Cn = CnoPl 1 : AU(P(E)) ----+P ( S y m ~ ( T ) )

(1)

is an isomorphism over G, such that r (2)

r

:= r

= Hn.

: Z ~ t ( P ( E ) ) ---+ P ( T ) x a P ( S y m n - l ( T ) )

is an isomorphism satisfying =H

r

"r

= Ln-1.

So by proposition 4.1.11 we now know the Chow ring of Aln(Pd). We keep in mind that by remark 4.1.10 we can write the class H~ E AI(AI~(Pd)) as H~=

[ { Z c Aln(Pd)

supp(Z)intersects a fixed hyperplane }].

So we get: Example

4.1.16.

A*(Aln(pd)) =

A*(Grass(d

- -

1, d + 1))[Hn]

n+l

.~ "

Z ci(Sym'~(Tz,d+l)H'~+l-') i=0

/

In particular we have with P := c1(Q1,3):

A*(AIn(Pz)) = ([pa, H~+a Here

n(2n-}-l)(n-.}-l)

6 + = { n(2n+l)(n+l) 6

+

-(

2~[P, H . ] .+1 n

2 ) H . P + w ( n ) H~-IP2)

(3n2--2n)(n2--1)

24

(n--2)(n--1)n 24

Jr-

,

n odd;

na(n--1)

8

,

n even.

154

~ 3

4.2. C o m p u t a t i o n of t h e C h o w ring o f Hilb (P2) Now we want to use the results of the preceeding section to compute the Chow ~ 3

ring of the variety Hilb (P2) of triangles in P2 with a marked side. Remember that --3 p~2] p~3] Hilb (P2) C x is defined as the subvariety Hil'---~a(p2)

:_-

~ 3

Hilb (P2) was defined in [Elencwajg-Le Sarz (2),(3)] to compute the Chow ring of p~S]. The result is however quite complicated. In this section we shall see that the ~ 3

Chow ring of Hilb (P2) is relatively simple, so it might be more useful for some ~ 3

enumerative applications. If the ground field is C, then the Chow ring of Hilb (P2) ~ 3

coincides with the eohomology ring (Proposition 2.5.19). Let res : Hilb (P2) ----* P2 be the residual morphism (see lemma 2.5.3) and ~ 3

Hilb (P2)

/ P~

\ P~

p~2}

p~3]

the projections.By proposition2.5.19we get A~(Hil~--b3(p2))= As(H~Ib3(P2))= 2~4 A2(Hil~---b3(p2)) = A4(H~]~b3(p2))= 2g 9 ~ 3

A3(Hilb (P2)) = 2g n. ~ 3

Now we define some elements of Al(Hilb (P2)), which will generate the Chow ring ~ 3

of Hilb (P2). D e f i n i t i o n 4.2.1. Let Z2(P2) C P2 • p~2] be the universal subscheme and let Z~(P2)

\q2

/p

P2

p~2]

be the projections. Let H := rcs*(el(Op~(1))) and let axe:

= Al2(e:)

be the axial morphism of 4.1.14. We put P := p~axe*(cl(Q1,3)), H2 := p~(q2),p*(cl(Oi%(1))).

~ 3

4.2. Computation of the Chow ring of Hilb (P2) N

155

~ 3

Let A C Hilb (Pc) be the subvariety

{ (Z2, Za) 9 Hilb --3

.4 :=

(P2)

Z3 is a subscheme of a line

~ 3

}

~ 3

and A := [.4] 9 Al(Hilb (P2)). Let P2 C Hilb (P2) be the closed subvariety /~2 :=

-- 3 (Z2, Za) 9 Hilb (P2)

the line through one of the subschemes Z1 C Z3 / of length 2 containing res(Z2, Z3) passes through a fixed point

J

and P2 := [P2IR e m a r k 4.2.2. Geometrically H2, H, P can also be described as H2 = n =

a point of Z2 }] lies on a fixed line '

(Z2,Za) 9 Hilb (P2)

[{(Z2, Z3) 9 Hilb -' (P2) L res(Z2,Z3) lies on a fixed line }],

P =

T h e o r e m 4.2.3.

Z2, Z3) 9 Hilb (P~)

l

the line through Z2 passesthroughafixedpoint }] "

3 2g[H, H2, P, A] A*(Hilb ( P 2 ) ) = ( I i , h , I 3 , I 4 , I s , I s )

with

/1 := H 3, I 2 : : P 3, /3 := H i - 3H22P + 6H2P 2, /4 :-- A ( H 2 - H P + p2), Is := A ( A -

3P + H + H2),

Is := d g ~ - ( H i P - H 2 P 2 + H H ~ - 3 H H 2 P + 2 g P 2 - 2H2H2 + 2 H 2 P + AH2P + 2AHH2 - AHP).

~ 3

Proof: By example 4.1.16 the subring of A*(Hilb (P2)) generated by H, P, H2 is (res • pl)*(A*(P2 x p~2])) _

3 As the morphism res x Pl : Hilb (P2)

2g[H, P, H2] (/-~,/2:/---~

p~2l is birational, the orientation cycle of Hilb (P2) is the class [*] := H 2 H ~ P 2. The restriction of res x t52 to the ~ 3

' P2 x

4. The Chow ring of relative Hilbert schemes of projective bundle~

156

subvariety A gives an isomorphism r .4 ----* Z~t(P2) C Z3(P2). By lemma 4.1.15 we have Z3al (P2) = P(T2,3 • Hilb2(p(T2,3)/152), where T2,3 is the tautological bundle over 152 = Grass(l, 3). So we get ~'[H, H2, P] A*(A) = (p3,H2 _ H P + P2,H~ - 3H~P + 6H2p2) ' and the orientation cycle of A is P2HH~. So relation/4 = 0 holds in A (Hilb (P2)), and for the orientation cycle we get [*] = AHH22P 2. To show Is = 0 we use the class P2 ff AI(H~]~b3(p2)). L e m m a 4.2.4. P + P2 = A + H + H2. P r o o f : Let

H-:= [ { Z E P~3] I Z intersects a fixed line}I, ~:=

[{ZEp~3]

asnbschemeZ2oflength2ofZ }] lies on a line passing through a fixed point

So we have by definition H = (/)2).(8), P = (152),(P), A = / ~ ( A ) , that the relations

and we see

p ~ ( n ) = H + H2,

f ~ ( P ) = P + Pc, (t52). (A) = 3A 3 p~3] hold, as the projection f2 : Hilb (P2) ~ is generically finite of degree 3. In [Elencwajg-Le Barz (3)] it has been shown that the relation P = A + H holds in A 1(p~31). We briefly repeat the elementary argument: we put

r := (P2),(HH~p2), r

:= (P2),(H2H2P 2) e AS(p~3]).

These classes can be geometrically described as follows: [{ r

=

Z consists of two distinct fixed points } ] Z E p~3] x l, x2 and another point x3 moving on a , fixed line containing neither xl nor x2 9

,2 [/z P J

Z consists of a fixed point x and ~ ] a subscheme Z2 of length 2 on a fixed line l not containing x; Z2 contains a fixed point x2 C l.

~ 3

4.2. Computation of the Chow ring of Hilb (P2)

157

Using this description we can easily compute the intersection table:

r

1

2

r

1

1

As the group A 1(P~3]) = A5 (p~a]) is free of rank 2, we see that H, A and r r form bases of Ax(p~ 3]) and As(P~ a]) respectively and the relation P = A + H holds. The result follows. []

Lemma 4.2.5. AP2 =

2AP.

P r o o f : We have to show the relation P21J~ = 2P]x. We have ~ P(T2,3) x~, 2 P(Sym2(T2,3)). Let 71"1 : P(T2,3) ----* I52 ~r2 : P(Sym2(T2,3)) -----* 152 Pl : P(T2,3) Xl~2 P(Sym2(T2,3)) ----* P(T2,3) P2 : P(T2,3) xp2 P(Sym2(T2,3)) ~

P(Sym2(T2,3))

be the projections. Then we have P = p~(~r~(cl(Q1,3))). Let .4 := P(T2,3) x~, 2 P(T2,3) xt, 2 P(T2,3), where Pl, P2, P3 : -4 ~ P(T2,3) are the projections. morphism r .4 ---+ P(T2,3) x p : P(Sym2(T2,3)). Let

We consider the natural

:P(T2,3) xp2 P(T2,3) - - ~ 152

:P(T~,~) •

P(T~,~) •

P(T2,~) ----, P~

be the projections. Then we see r

=

(t)2 • p3)*(~*(cx(Q~,3))) ~- (c~(Q~,3)),

r

= (151 •

(

* ~'*

( C 1(Q1,3)))+(pl •

2~ (c1(Q1,3)), N.

r

(c,(Q1,3)))--

The l e m m a follows.

[]

2PIz.

4. The Chow ring of relative Hilbert schemes of projective bundles

158

From lemmas 4.2.4 and 4.2.5 we get the relation I5 = 0:

A 2=A(P+P2-H-H2) = 3 A P - A H - AH2.

The information we have obtained until now is already enough to determine the ring ~ 3

structure of A*(Hilb (P2)). We use relations I1,. 9 9

to compute the intersection tables. We also use that

the orientation class is [*] = A P 2 H H ~ = H2H22P 2. We get the following tables: A 1 • A5

A H H~ P 2

1

H2H~p

3

H 2 H~ P~

1

1

1

A H H 2 p2

-1

A2 • A4

AH It~ P 2

1

HH2p

3

HH~P 2

1

1

3

1

1

1

H2H~

3

H2 H2 P

1

1

H~p 2 AH2P ~

1

-1

AHH~PI

1

-1

AHP ~

-1 -1

-1

~ 3

4.2. Computation of the Chow ring of Hilb (P2)

159

A3 x A3

H~P

3

H2P 2

1

HH~

3

3

HH2P!

3

1

Hp 2

1

H2H2

3

H2p

1

1

1

3

1

1

1

-1

-1

AH2P Ap 2

-1

-1

AHH2

3

AHP

1

-1

1

-1

-1

-1 -1

AH~

-1

We see that the intersection matrices are all invertible over 2~. By solving the system of equations given by the last intersection matrix we get 16 = 0. ~ 3

E n d o f t h e p r o o f o f t h e o r e m 4.2.3: As we have found a N-basis of A*(Hilb (P2)) ~ 3

consisting of monomials in H 2 , P , H , A the ring A*(Hilb (P2)) is generated by H 2 , P , H , A . We also have seen that the relations I~ = 0 , . . . , I s = 0 hold. We have to show that these generate all the relations. For this it is enough to show that every monomial in H2, P, H, A can be expressed in terms of the elements of the basis by making use of 11,..., Is. Let M be such a monomial. By I 1 , . . . , / 6 it can be expressed as a linear combination of monomials A~HhpPHh2 2 satisfying

h <2

(i~),

p < 2

(h),

h2 -< 2

(h),

a< 1

(h),

h+a<_2

(14),

a + h2 <_ 2

(I6).

We see that these conditions are only satisfied by the elements of the basis oceuring in the above intersection matrices.

[]

160

~ 3

4.3. T h e C h o w ring o f Hilb (P(E)/X) Now we want to generalize the result of the last section. Let X be a smooth variety and E a vector bundle of rank 3 on X. ~ 3

D e f i n i t i o n 4.3.1. Let Hilb (P(E)/X) C Hilbe(P(E)/X) xx Hilb3(p(E)/X) be the subvariety defined by Hilb (P(E)/X) := (Z1,Z) E Hilb2(P(E)/X) xx Hilb3(p(E)/X)

Zl C Z

Let

Hilb2(P(E)/X) xx Hilb2(p(E)/X) xx HilbZ(P(E)/X) xx HilbZ(e(E)/X)

V ( P ( E ) ) := P ( E ) x x P ( E ) x x P ( E ) x x

and s

C V(P(E)) be the subvariety defined by

xi,xj C Zk; Zi C Z; H3(P(E)/X) := { (xl, x2, x3, Z1, Z2, Z3, Z) C V(P(E))

]

x~ = r ~ ( x ~ , z~) = r ~ ( z k , z )

for all permutations (i,j, k) of (1,2,3)

As Hilb~(P(E)/X) is a locally trivial fibre bundle over X with fibre P~nl, we see easily: R e m a r k 4.3.2. ~ 3

(1) Hilb (P(E)/X) is a locally trivial fibre bundle (2) s

~ 3

over

X with fibre Hilb (P2).

is a locally trivial fibre bundle over X with fibre ~r3(P2).

N 3

Hilb (P(E)/X) parametrizes the triangles with a marked side and Ha(P(E)/X) the complete triangles in the fibres F ~ P2 of P ( E ) over X. We want to use results from [Collino-Fulton (1)] on the Chow ring of H3(p(E)/X), to com~ 3

pute A*(Hilb (P(E)/X)). In [Collino-Fulton (1)] another definitionof~r3(p(E)/X) is used, which we will denote by W(P(E)/X). First we give the definition of

W(P(E)/X). D e f i n i t i o n 4.3.3. Let

U(P(E)) := P(E)xxP(E) •

P(E) •

P(E) xx P(E)•

P(E) xx Grass(3, Sym2(E))

~3

4.3. The Chow ring of Hilb ( P ( E ) / X )

and let s : U ( P ( E ) )

161

, X be the projection. Let x E X. y = (~,~,~,~,,&,~,r)

~

~-~(~)

is called a honest triangle if xa, x2, x3 are three distinct points of a fibre P ( E ( x ) ) and ~k is the line connecting x,, xj (for all permutations (i,j, k) of (1,2, 3)) and F is the linear system of conics passing through xl, x2, x3, viewed as an element of the fibre ara~43, Sym2(E(x))). Let W o ( P ( E ) ) C U ( P ( E ) ) be the set of honest triangles and W ( P ( E ) ) the closure of W o ( P ( E ) ) in U ( P ( E ) ) . Now we want to construct an embedding of h r a ( p ( z ) / x ) into a product of bundles of Grassmannians. By the results of section 4.1 we get that the morphism r x r x Cz]~3(p(E)/X ) is a closed embedding of Br3(p(E)/X) into

P ( E ) x x P ( E ) X x P ( E ) x x Grass(4, Sym2(E)) x x Grass(4, Sym2(E))• x ara~s(4, Sym2(E)) x x a t ( 7 , Sym3(E)). On the other hand in [Le Sarz (10)] ~r3(P2) was shown to be a closed subscheme of p3 x p3 • Grass(3, 6), and we can see from the proof that the embedding/~3 (P2) p~ • p3 x Grass(3, 6) is given by the morphism (I) : : r

Op~(i),i x r o~(I),2 x r

We have the morphisms 1p(E) = r

: P(E)

, P(E),

axe := Cop(E)(1),2 : Hilb2(P(E)/X) r

:= r

: Hilb3(p(E)/X)

, Grass(l, E), Grass(3, Sym2(E)).

P r o p o s i t i o n 4.3.4. 3 := 1p(E) X axe 3 • r

H3(P(E)/X)

~U(P(E))

is a closed embedding with image W ( P ( E ) ) . P r o o f : Let U C X be an open subset over which E is trivial. Then with respect to suitable local trivialisations over U the restriction of ~ is the dosed embedding 1v x ~5: U • H 3 ( p 2 )

, V • P~ • 15~ x Grass(3,6).

So ~ is a closed embedding. We can see immediately that the image of the open subvariety hrgl'l'l)(P(E)) :=

e ~I3(p(E)/X)

the xi are distinct

162

4.

The Chow ring of relative Hilbert schemes of projective bundles

As/~(31,1,1)(X) lS

is the variety Wo(P(E)) C U ( P ( E ) ) of honest triangles in P ( E ) .

open and dense in ~r3(X) and W ( P ( E ) ) is defined as the closure of W0(P(E)) in U(P(E)), the result follows. [] In [Collino-Fulton (1)1 the Chow ring of W(P(E)/X) is computed as an algebra over A*(X). There the following classes are important: D e f i n i t i o n 4 . 3 . 5 . Let

151,f2,p3 : W ( P ( E ) )

P(E),

ql, q2, q3 : W ( P ( E ) )

, P(E),

Grass(3, Sym2(E))

4: W ( P ( E ) ) be the projections. We put a

:=

a := Then

p;(c,(Op(E)(1))), b : = p~(r

c = p~(~,(op(E)(1))),

~t~(c,(T~,E)), fl : : (t~(cl(T~,E)), "7:= ~t~(cl(T~,E))

a,b,c,a, fl,~f 9 AI(W(P(E))).

Let 7r: P ( E ) ~

X, ~ : W ( P ( E ) ) ~

X be

the projections. We write: ~, := ~*(c~(E*))

= -~*(cl(E)),

, 2 := ~ * ( c 2 ( E * ) ) = ~ * ( c 2 ( E ) ) , ~3 := ~ * ( c 3 ( E * ) ) = - ~ * ( c ~ ( E ) ) .

Let e 9

AI(W(P(E)))

be the class of the subvaxiety

"K:= { (xl'x2'x3'(l'(2'(3'F) I W ( P ( E )9)

F is the net of conics on the fibre P ( E ( ~ ( x i ) ) )

~ P~,

'

c o n t a i n i n g ~1

and

r 9 AI(w(P(E)))

the class of Xl = X2 = X3,

~:=

(Xl,X2,X3,~l,~2,~3,r)

e W(P(E))

F is the net of conics, on the fibre P(E(~r(xj))) ~ P2, having a singular point at Xl

By [Collino-Fulton (1)] we have: Lemma 4.3.6.

(1)

r=e+a+b+c+#l -~-fl-%

]

"

~ 3

4.3. The Chow ring o] Hilb ( P ( E ) / X )

(2) (3) (4)

a3 = # l a ~ - # 2 a + p a

163

(and similarly ]orb andc),

(~3 = 2#1a2 _ (p2 + #2)~ a~ : a 2 + ~2 - #1~ + #2

+

#1#2

--

#3

(and similarly ]or fl and 7),

(and similarly for a, 7; b, (~; b, 7; c, a; c, fl respectively),

(5) (6) (7)

Ta

Tb = TC~

=

6T = 0 ,

Now we want to describe the classes ~*(a), ~*(b), ~*(c), ~*((~), (~*(~), ~*(7), ~*(e), ~*(r)

9

AI(Ha(P(E)/X)).

Let Pl ,P2,P3 : H 3 ( P ( E ) / X ) ~

P(E),

ql, q2, q3: H a ( P ( E ) / X ) ~

Hilb2(p(E)/X),

q: H a ( P ( E ) / X ) ~

Hilba(P(E)/X)

be the projections. R e m a r k 4.3.7.

~*(~):

p~(ci(O~(~)(1))), ~*(b) : p~(ci(O~(E)(1))), ~*(c) = p~(e~(O~(E)(1))),

~*(a) = q~axe*(cl(T;,E)), %*(~) = q~axe*(o(T;,E)), ~*(7) = q~axe*(c,(T;,E)). Let A 9 AI(Hilb3(P(E)/X)) be the class of AIa(P(E)/X). Then we have ~*(e) -q*(A). ~*(v) is the class of the subvariety (xl, x~, xa, Z1, Z2, Z3, Z) 9 Ha(P(E)/X)

and with F = P ( E 0 r ( z l ))) m sF, x l is the ideal of Z in OF ~

.

P r o o f : The statements o n (~*(a),~*(b), ~*(c), ~*(a), ~*(~), ~*(7) follow easily from the definitions. By definition ~*(e) is the class of the subvariety

I

(xl, x2, z3, Z1, Z2, Z3, Z) 6 H3(P(E)/X)

the lines axe(Z1 ), axe(Z2), axe(Z3) ] through ZI, Z2, Z3 in the fibre F = P(E(r(Xl))) ~ P2 are equal and r is the net of conics in F, containing the line axe(Z1).

4. The Chow ring of relative Hilbert schemes of projective bundles

164

We consider this condition fibrewise. As r

rz : g ~

is the kernel of the restriction m a p

) H~ Oz| , the condition on 52,3(Z), means t h a t Z is a subseheme of the line axe(Z1) t h r o u g h Z1. So also Z2 and Z3 are subsehemes of axe(Z~), and the conditions on axe(Zj) and axe(Z3) are fulfilled automatically. Op2(2))

So we get ~*(e) = q*(A). By definition ~ * ( r ) is the class of the subvariety

Xl = x2 = x3 (xl,xj,x3,Z1,ZJ, Z3,Z) 9 H3(p(E)/X)

/

andr isthenetofconics in the fibre P ( E ( z r ( x l ) ) ) = P J , having a singular point at Xl

"

Let ( z l , x j , z3, ZI, Z j , Z3, Z) be a point of this subvariety. The condition on r means t h a t Z lies in the subscheme 5 C F = P ( E ( : r ( x l ) ) ) with s u p p o r t xa which is defined by m 2F,Xl in OF,~t. 2 is a subscheme of length 3 of P ( E ( T r ( x l ) ) ) , so we have Z = Z. As Xl, x j , x3 are subschemes of Z, the condition xl = x2 = x3 follows a u t o m a t i c a l l y fi'om the condition o n r

The result follows,

u

~3

Now we t u r n to the variety Hilb (P(E)/X) of triangles in the fibres of P ( E ) ~3

~3

with a m a r k e d side. Via res: Hilb (P(E)/X)

, P ( E ) we regard Hilb (P(E)/X)

as a subscheme of P ( E ) X x Hilb2(p(E)/X) x x Hilb3(p(E)/X): Hilb (P(E)/X) =

x,Z,,Z)

x c Z1 C Z, r e s ( Z 1 , Z ) = x

.

So we have a n a t u r a l m o r p h i s m

7n4z :-~3(P(E)/X) --~ ~lb3(P(E)/X); (Xl,Xj,x3,Z1,ZJ,Z3,Z)

Let

,

) (xl,Z1,Z)

~3

: = 7r147o~-a : W ( P ( E ) ) ----* Hilb (P(E)/X). Let

~3

p a : Hilb (P(E)/X) ~3

P2: Hilb (P(E)/X) ~3

~ : Hilb (P(E)/X) be the projections. Let r from lemma

HilbJ(P(E)/X)

, P(E),

) Hilbz(P(E)/X), , Hilba(p(E)/X) , P ( S y m J ( T j , E ) ) be the isomorphism

4.1.15 with ~~*2 ( O P ( S y m J ( T j , E ) ) ( 1 ) ) ---- (qj).p*(Op(E)(1)).

Zj(P(E)/X)

~/P P(E)

"~q2 Hilb 2 ( P ( E ) / X )

Here

~3

4.3. The Chow ring of Hilb (P(E)/X)

165

are the natural projections of the universal subscheme.

D e f i n i t i o n 4.3.8. We put H :=

p~(c,(Op(E)(1))),

~* *c I ( O p ( E ) (1) ) =P2r "* ~* H2 :=p2(q2).p P := p"~axe*(cx(T~,E)), A = ~'*(A).

~3

We want to show that H, H2, P, A generate A*(Hilb (P(E)/X)) as an A*(X)algebra and to determine the relations. For this we first determine the classes ~*(H), ~*(H2), ~*(P), ~ * ( A ) E AI(W(P(E))).

~*(H2)=b+c, ~ * ( P ) = a, ~-*(A) = e.

L e m m a 4.3.9. ~ * ( H ) = a ,

P r o o f : ~*(H) = a, ~*(P) = a, ~*(A) = e follow immediately from the definitions and remark 4.3.7. Now we show ~*(H2) = b + c. Let F(E) C P ( E ) • lb(E) be the incidence variety

F(E) : = {(x,/) ~ P(E) • and

C l}

F(E) ~/p,

\p2

P(E)

P(E)

the projections. It is easy to see that there is an isomorphism ~ : F(E) over P ( E ) with ~*(Op(T2.E)(1)) = p~(Op(E)(1)). Let r2,a : W ( P ( E ) ) ~

-----+

P(T2,E)

( P ( E ) x x ~'(E)) x x ( P ( E ) x x t)(E));

(Xl, X2, X3, El, ~2, ~3, r ) e------+((X2, ~2), (X3, (3)). We see from the definitions that the image r2,3(W(P(E))) lies in the subvariety

F(E) x x F(E) of ( P ( E ) x x 15(E)) • W(P(E))

Hilb (P(E)/X)

....

- - ~,2

,

( P ( E ) Xx 15(E)). The diagram

F(E) xx F(E)

Hilb2(P(E)/X)

*•

,

P(T2,E) x x P(T2,E)

~

P(Sym2(T2,E))

4. The Chow ring of relative Hilbert 3chemes of projective bundles

166

commutes. Here r] is the morphism defined by the natural map T2,E | T2,E Sym2(T2,E). With respect to the projections rl,r2

:

P(T2,E) •

P(T2,E) ~

P(T2,E)

we have:

rl*(Cl(OP(Sym2(T2,E))(1))) = r;(Ca(Op(T2,E)(1))) + r;(Cl(Op(T2.E)(1))). By r

= p~(Op(E)(1)) the result follows

[] ~ 3

Now we can give a first description of the Chow ring A*(Hilb (P(E)/X)). P r o p o s i t i o n 4.3.10.

~ 3

~-* : A*(Hilb (P(E)/X))

, A*(W(P(E))) is injective.

~ 3

~*(A*(Hilb (P(E)/X))) i~ the A*(X)-subalgebra of A*(W(P(E))) generated by

F*(H) = a, ~*(H2) = b + c, ~*(P) = a, ~*(A) = e. Proof." The classes which we called A, H, H2, P in section 4.2 will now be called Ap2 , ~ 3

Hp:, H2,P2, PP2" We see that the restrictions of A, H, H2, P to a fibre Hilb (Pc) are Ap2, Hp~, H2,p2, PP2' Then by the theorem of Leray-Hirsch for the Chow groups [Collino-Fulton (1)] the monomials in A, H, P, H2 occuring in the intersection tables ~ 3

at the end of section 4.2 form a basis of A*(Hilb (P(E)/X)) as a free A*(X)~ 3

module (as Hilb (P2) has a cell decomposition). So we only have to see that ~* is injective. Let 7rp ~ 3 (P2) be the restriction of ~ to a fibre A 2 : /~3(p2) , Hilb 03 (P2). The orientation classes [,] of Hil'-'-b3(P z) and [**] of HS(P2 ) fulfill ~b~ ([*]) = ~ 3

3[**], ~',([**]) = [*], as ~P2 is generically finite of degree 3. As both Hilb (P2) and Ha(P2) have a eell decomposition, the intersection product in complementary dimensions gives a nondegenerate pairing of free 2g-modules for both varieties. So ~*P ~ is injective. As a homomorphism of free A*(X)-modules ~ 3

--~rp, | 1A*(X) : A*(Hilb (P(E)/X)) ~ 3

= A*(Hilb (P2)) | A*(X) is one to one. So ~* is injective.

, A*(Hs(P2)) | A*(X) = A*(Hs(P(E)/X)) []

~ 3

We now describe A*(Hilb (P(E)/X))) directly by generators and relations. T h e o r e m 4.3.11.

3

A*(X)[H2, P, H, A]

A*(Hilb (P(E)/X)) = (I1,12, I3,14, I5,/6)

~ 3

4.3. The Chow ring of Hilb ( P ( E ) / X )

167

whert~

/1 : = H 3 - # 1 H 2 + p e H - # a , /2 := P ( P - #l) 2 + #2(P - #1) + #3, I3 := H i - 3VH~ + H~(GV ~ - 4 P r o + 4 ~ ) - 4 ( P 3 - P ~ m + P ~ ) ,

14

:

=

A(H 2 - PH + P(P

-

#1) 21- ~2),

15 := A ( A - 3P + H + H2 + # 1 ) ,

16 := - A H ~ + # I ( - H ~ + H2P + 2HH2 - 2 H P ) + H 2 p - H2P 2 + HH~ - 3 H H 2 P + 2 H P 2 -- H2H2 + 2 H 2 p + A ( H 2 P + 2HH2 - 2HP).

P r o o f : We have ~'~(A*(P(E))) = A*(X)[H]/(I1). Furthermore P = ~axe*(ca(Ql,E)) + #1

and thus ~ a x e * ( A * ( P ( E ) ) ) = ( ( p _ #1)3 + # I ( P

A*(X)[P] #1) 2 "[- #2(P - -

- -

#1) -~- /23)

= A*(X)[PI/(h).

We have ~2axe*cl ( T2,E ) = - P, ~axe*c2(T2,s) = P ( P - #1) + #2.

So we get by 3.1.9

~axe*c(Sym2(T2,E)) =

1 - 3 P + ( 6 P ~ - 4P#1 + 4#2) - 4 ( P 3 - P2#1 + P#2)

and thus Remark is

4.3.12.

The A*(X)-subalgebra of A * ( P ( E ) / X ) ) generated by H, P, H2

(P'I • ~2)*(A*(P(E) •

Hilb2(P(E)/X))) =

A*(X)[H, H2, P] (I1,/2, I3)

~3

Let A C Hilb ( P ( E ) / X ) be the subvariety defined by

.4:~

(x, Z1, Z)

~3

9 Hilb ( P ( E ) / X )

Z lies on a line ) in the fibre P ( E ( r ( x ) ) ) / " passing through x

Via ~~

= ax~~

: .~ - - ~ ~'(E)

168

4. The Chow ring of relative Hilbert schemes of projective bundles

,4 is a variety over 15(E).

~3

~ • ~ : Hilb ( P ( E ) / X )

~

P(E) xx Hilb3(p(E)/X)

m a p s fit isomorphically onto Z ~ t ( P ( E ) / X ) . phism "r : Z ~ t ( P ( E ) / X )

By l e m m a 4.1.15 there is an isomor-

----* P(T2,E) •

P(Sym2(T2,E))

over I~(E) satisfying

(51 • wl~') (P (CI(OP(%,E)(1))) = HIX, (Pl X 7i'1~") (~ (CI(OP(Sym2(T2,E))(1))):

H21~".

So we get A*(A) =

A * ( X ) [ H , P , H2] (I2, Iz, H 2 - P H + P ( P - #1 ) - / 2 2 ) . ~ 3

The r e l a t i o n / 4 = 0 in A*(Hilb ( P ( E ) / X ) )

follows by [A]-- A.

In order to prove the relations I5 = 0, /6 = 0, we want to compute in A * ( W ( P ( E ) ) ) and use the reations of Collino and Fulton from l e m m a 4.3.6. The proof of 15 = 0 is simple.

~*(A(A-

3 P + H + H2 +/21)) = e(e - 3a + a + b + c + / 2 1 ) = E(e-o~- f-7 z

+ a + b + c + /21)

s

zO.

So Ix = 0 holds. In order to proof I6 = 0, we write the relations in such a way that they can be applied formally (by substituting).

Remark

4 . 3 . 1 3 . In A * ( W ( P ( E ) ) ) the following relations hold:

(1)

a 3 = a2/21 - a/22 +/23 and similarly for b and c,

(2)

a 3 = 2#1 a2 -- (#12 + # 2 ) a +/21/22

(3)

~

(4)

3 2 = - a 2 + a/3 +

= - b 2 + ba +/21a - / 2 2 , #13

-- /A2,

(5)

72 = - a 2 + a'~ +/217 - #2,

(6)

ac = - b 2 + ba + c 2,

(7)

/ 3 c = - - a 2 + a/3 + c 2,

(8)

7b = - a 2 + a7 + b2,

--

/23 and similarly for 3 and 7,

N 3

4.3. The Chow ring o f Hilb ( P ( E ) / X )

169

(9)

(10)

eb =- ea + (a - b)(c + #1 ,c =

~, + (,

-

c)(b

- -

+ ~,1 -

Ol

- -

~ -

~), "y).

Now we just a p p l y these relations formally. We get 0 = ~*(A(H 2 - HP + P(P-

#1) + #2))

= ~(a ~ -- a,~ + ,~(,~ - # 1 ) + # 2 )

= --a2c -- a2#1 + a20! -t- a2/3 + ac 2 + a#2 -- aa/~ + bec + b2#1 - bea - b2/3 - bc 2 - b#2 -t- b a ~ .

F u r t h e r m o r e we get

~*(n~P) = (b +

~)~

= - b 2 c - 3b2#1 -4- 4b2a + 3bc 2 + 3b#2 + c2#1 - c#2 - 2#3, ~*( n 2 P ~) = a2(b + c) = - b 2 c - 3b2#1 -t- 2b2a + bc 2 + 2 b a # l + b#2 + c2#1 - c#2 - 2#3, ~*(HH22) = a(b 2 + 2bc + c2), ~*(HH2P)

= a a ( b + c) = a ( - b 2 + 2ba -4- c~),

~ * ( H p 2) = a a 2 = a ( - b ~ + ba + # l a

-

#2),

~ * ( H 2 H 2 ) = a2(b + c), ~*(H2p)

= a2a,

"~*(AHP) = aae, ~*(AHH2)

= a(ab + ac + 2 a p l - 2 a a - a/3 - a 7 + 2ae - b2 - 2bc - b#l + 2ba + b/3 + c 2 - c # 1 + c7),

~*(AH2P)

= ea(b + c) = a 2 b - a2c + a2/3 - a27 + 2ac 2 + 2a#2 - 2aa/~ + 2 a a e - b2/~ - 2bc 2 - b#2 4- 2ba/3 + c27 - c#2,

~ * ( A H ~ ) = e(b + c) 2 = a 2 b + a2c + 6a2#1 - 4 a 2 a - 3 a 2 ~ -

3a27 + 4 a 2 e -

2abc

+ 2ab~ + 2ac 7 - 2a#2 - 262c - 4b2#1 + 4b2a + b2~ + 2bc 2 - 2bc#l + 3b#2 + c27 - c#2,

170

4. The Chow ring of relative Hilbert schemes of projective bundles

^7r*

2 (H2Pl) = #l(b 2 + 2 b c + c 2 ) ,

~*(H2P#I)

---- ~ l ( - b 2 +

2b(~ + c2),

~ * ( H H 2 # I ) = a#l(b + c), "~*(HP#I) = aplc~. Thus we have ~ * ( - A H 2 + ~ I ( - H 2 -4- H 2 P + 2HH2 - 2 H P ) + H 2 P - H 2 P 2 + HH~ -

3 H H 2 P + 2 H P 2 - H2H2 + 2 H 2 p + A ( H 2 P + 2HH2 - 2 H P ) )

---- 2(--a2c-- a2 pl + a2 0~ + a2 /3 + ac 2 + a#2 -- ao~t3 + b2c + b2#1 - b2c~ - b2/3 - bc2 - b#2 + bo~/3) =0.

As ~* is injective, the relation/6 = 0 holds in A*(H~[-lb3(p(E)/X)). E n d o f the proof of t h e o r e m 4.3.11 The monomials in A, H, H2, P occuring in the intersection tables at the end of ~3

4.2 form a basis of Hilb ( P ( E ) / X ) ) as a free A*(X)-module. On the other hand using the relations I1,. 9 16 we can express any monomial M in A, H, P, H2 as an A*(X)-linear combination of monomials of the form AaHhPVHh22 with h~2,

p<2,

h2<2, a
h+a~2,

a+h2<2,

i.e. as a linear combination of these monomials. The result follows. In the rest of this section we look at

an important

[]

special case of

~3

Hilb ( P ( E ) / X ) ) . We put G := Grass(d - 2, d + 1) and let T := T3,d+l be the tautological bundle over G. D e f i n i t i o n 4.3.14. Let ~Cop3(pd) C Hilb3(pd)-- • G be the subvariety Cop ( P a ) : =

Let F C P a x ptojections

( ( ( Z a , Z ) , E ) E H i__3 l b (Pa)

xG

ZcE

)

G be the incidence variety F := { ( x , E ) e Pd • G I x C E} with F

Pa

G.

4.3.

~3 The Chowring of Hilb (P(E)/X)

There is an isomorphism r

: F

171 , P ( T ) over G with r

=

~3

p~(Or, d(1)). We see immediately from the definitions that Cop (Pd) is the sub~3

variety Hilb

~3

(F/G) C Hilb r

(Pd) x G. So we get an isomorphism ~3

~3

Cop (Pd) ----* Hilb

(P(T)/G).

The projection/51 : Coop3(Pd) ----* Hilb - - 3 (Pd) is a birational morphism (a general subscheme of length 3 lies on exactly one plane). It is an isomorphism outside -40:=

{ (Z1,Z) EHilb --3

(Pd)

Z lies on a line

}.

Over a point (Z1, Z) E A0, lying on a line l its fibre is /511(Z1,Z)

=

{E E a

E ~) l} ~- Pd-2

The exceptional locus of 151 is ~i ~' P ( T 2 , T )

X15(T ) P ( S y m 2 ( T 2 , T ) ) ,

in particular it is an irreducible divisor. So we get: ~3

N3

R e m a r k 4.3.15. Cop (Pc) is obtained by blowing up Hilb (Pd) along Z~t(Pd).

Definition 4.3.16. Let

A',H',H2,P'',#1,# 3#2, t,

~3

t

E A*(Cop (Pal)) be the classes

A' :=

[{(E, (Z1, Z)) 9 Cop (Pd)

Z lies on a line

H' :=

[{(E,(Za,Z))c

res(Z1,Z)lies on a fixed hyperplane }],

H; :=

[{(E, (Z1, Z)) e Cop (Pd)

pt :z

[{(E,(Z1,Z))~3

'E{,

#a :=

E Cop (Pa)

Coop3(pd)

}],

supp(Z1)intersects a fixed hyperplane }] ,

the line passing through Za intersects a fixed }] 2-codimensional linear subspace '

E, (Z1, Z)) E Cop (Pa)

linear subspace

A:=[{(E,(Z~,Z))eUoop~(Pd)

E has a one-dimensional intersection } ] with a fixed 2-codimensional linear subspace

#~:=

[{(E,(Z1,Z))E~op3(pa)

E l i e s o n a f i x e d h y p e r p l a n e }].

Then we see easily from the definitions :

4. The Chow ring of relative Hilbert schemes of projective bundles

172

R e m a r k 4.3.17.

r

= A', r

= H', r

= H;, r

= P',

g*(.1) = .'1, ;*(.2) -- . ; , g*(.3) = . ; ~ 3

So theorem 4.3.12 describes the Chow ring of Cop (Pd) in terms of classes determined by the position of subschemes relative to lines and planes in Pd.

173

4.4. T h e Chow ring of Hilba(P(E)/X) As in section 4.3 let E be a vector bundle of rank 3 over a smooth variety X. We ~ 3

want to use the results of the previous section about A*(Hilb ( P ( E ) / X ) ) , to com~ 3

pute the Chow ring A*(Hilba(p(E)/X)) of the relative Hilbert scheme. Hilb (P2) has been defined in [Elencwajg-Le Barz (3)] in order to determine the Chow ring of p~a] by generators and relations. There the following classes are introduced: 3

Definition 4.4.1. Let ~ : Hilb (P2)

--~ p~3]

be the projection. Let

H, ~ ~ a 1(e~l), ~,p,~ e A:(P~1), 5,/~ E A3(P~3]) be the classes defined by

:=

~,(H),

,i := [ { Z C P ~ a] Z l i e s o n a l i n e } ] , :=

~.(H~),

:= ~_.(p2), := [ { Z E p~a] Z lies on a line passing through a fixed point }], & := [{ZEP~3] Z lies on a fixed line }] , := ~.(HP2). ~ 3

Here H, P E Al(Hilb (P2)) are the classes from definition 4.2.1. [Elencwajg-Le Barz (3)1 get for instance: T h e o r e m 4.4.2. [Elencwajg-Le Barz (3)] (1) .fit, ft, h,~,5,~,~ generate A*(P~3]) as a ring. (2) Bases of the free 2~-rnodule~ A/(P~3]) are

i = O: 1; i=1: H,i;

i = 4: H25, tI&,tI2[z,[z2,[zD;

4. The Chow ring of relative Hilbert schemes of projective bundles

174

i = 5:

i = 6: Iz3.

Elencwajg and Le Barz determine all the relations between the generators. We will first define some classes in A*(Hilb3(P(E)/X)) as relative versions of the classes in [Elencwajg-Le Barz (3)]. ~ 3

D e f i n i t i o n 4.4.3. Let ~ : Hilb ( P ( E ) / X ) ~

Hilb3(p(E)/X) be the projection.

Let := ~ . ( H ) 6 AI(Hilba(P(E)/X)), := ~.(H2), /5 := ~ . ( p 2 ) C A2(Hilb3(p(E)/X)), := ~ . ( H P 2) 6 A3(Hilb3(P(E)/X)). ~ 3

Here H , P E Al(Hilb ( P ( E ) / X ) are the classes from definition 4.3.8. AI3(P(E)/X) ~ Hilba(P(E)/X) be the embedding and

Let i :

axe: AI3(p(E)/X) ----* P(E) the axial morphism from 4.1.14. Let again T2,E be the tautological subbundle on lb(E) a n d / ~ := axe*(Cl(T~*E) ). We put

i := [AIa(P(E)/X)] = i,(1) 6 AI(Hilb3(P(E)/X)), := i.(/~) 6 A2(Hilb3(P(E)/X)), (~ := i,(/32) 6 Aa(Hilb(P(E)/X)).

Proposition 4.4.4. (1) H, A, h,p, ?t, (~, ~ generate A*(Hilb3(P(E)/X))) as an A*(X)-algebra. (2) The Ai(Hilb3(P(E)/Z))) are free A*(X)-modules with basis

= O: 1; = 1: H, fi~; = 2: [-I2,/IA, a,[z,p; = 3: [t3,hH,[-I2A, H?z,~,~; = 4: [I2~,[-I~,[-I2h,[z2,hp; = 5: [-Ih2, [-Ihp, - - 6 : ~3.

4.4. The Chow ring ofHilb3(p(E)/X)

175

P r o o f i (1) follows from (2). Immediately from the definitions we get for the fibre F ~ p~a] of Hilb3(p(E)/X) over a point x E X:

Hit =~,

Air = i i,

hit = h, PIF = P, alF = ~, a i r = a, ~1~ = ~. As p~31 has a cell decomposition, we get (2) from the theorem of Leray-Hirsch for Chow groups [Collino-Fulton (1)] and 4.4.2. []

In order to be able to compute the image of these classes under ~*, we prove a result on the relations between ~*, ~., ~*, ~.. R e m e m b e r that ~ is defined by

~ : ~Ia(P(E)/X) (zl, x2, x3, Zl, Z2, Z3, Z),

-~3

Hilb (P(E)/X); (321, Z l , Z ) .

We also consider

~2 : ~I3(p(E)/X) (xl, x2, x3, Z1, Z2, Z3, Z)

~3

Hilb (P(E)/X);

(z2, Z~, Z )

Let ~3

, P ( E ) Xx Hilb2(P(E)/X);

Pl,2: Hilb (P(E)/X) (x, Z l , Z )

l

, (x, Zl).

Lemma 4.4.5. For W E ~,2(A*(P(E) x x H i l b 2 ( P ( E ) / X ) ) ) we have ~*~,(w) = w + ~ . ( ~ ( w ) ) .

P r o o f : Let W = ~ i ai[X/] be the representation of W as a linear combination of N,N~r.([ X i]). So it is classes of irreducible varieties. Then we have ~'*~.(W) }--~iaiTr enough to show the result for W = ~,2([Y]), where Y C P ( E ) x x Hilb2(p(E)/X) is an irreducible subvariety. By the definitions we get

Hilb3(p(E)/X )

[{ ~.p'~l,2([Y]) =

Z e

there is a subscheme Z1 C Z }] of length 2 with (res(Z1,Z),Z1) e Y "

176

4. The Chow ring of relative Hilbert schemes of projective bundles

So we also have ~*<~,2([Y])

}]

there is a subscheme ZI C Z

of lengm 2 wah (r~(Z~,Z), Zl) C Y

E Hilb ( P ( E ) / X )

[{ (x, Z 1 , Z ) E _3

~--

Hilb

+

{

(P(E)/X)

(x, Zl,Z) ~ 3

E Hilb ( P ( E ) / X )

= ;-, , : ( [ Y ] )

(TcN(Z1,Z),Z1) ~ r

}]

there is a subscheme ZI C Z ] 1 of length 2 with x C ZI and (res(Zl, Z), ZI) E Y

J

+ ^~ , ( ~ 2^(.[ y ] ) ) .

[]

~ 3

So we can obtain A * ( H i l b 3 ( p ( E ) / X ) ) as a subring of A*(Hilb ( P ( E ) / X ) ) . T h e o r e m 4.4.6. ~* : A * ( H i l b 3 ( p ( E ) / X ) ) , A*(Hil~b3(p(E)/X)) and "~*(A*(Hilba(P(E)/X))) is the A*(X)-subalgebra generated by

is injective,

~*([-I) = H + g2, ~*(A) = A, F*([z) = H 2 + H 2 P - 2P 2 + 2P#1 - 2#2, ~.(p) = p2 _ H P + HH2 _ H 2 + A H + # I ( - P + H2 + 2H + A) + pS - 2p2, ~*(a) = AP,

"~*((~) = d p 2, "~*(~) = g ( 3 P 2 - 2H2P + H~ + H P - HH2) + A ( P 2 - H P + HH2) +#1(2P 2-2H2P+H~-HP+HH2+H

2-AP+AH2-AH)

+ # 2 ( - 2 P + H2 - H) + # 2 ( - H 2 + g + A) + 2#1#2 + #3.

Proof." By proposition 4.4.4 A * ( H i l b 3 ( P ( E ) / X ) ) is as an A*(X)-algebra generated by H , A , h , fi,&,6,~. For each fibre F -~ P2 the m a p ~ 3

~P2 := ~*]FE31 : A*(P~ 3]) - - ~ A*(Hilb (P2)) is one to one. As a h o m o m o r p h i s m of A*(X)-modules ~* is just ~P2 | 1A*(X) : A * ( H i l b 3 ( p ( E ) / X ) ~ 3

= A*(P~ 3]) | A * ( X ) ---* A*(Hilb (P2)) | A * ( X ) = A * ( P ( E ) / X ) ;

4.4. The Ckow ring of H i l b 3 ( p ( E ) / X )

177

so it is one to one. We still have to determine the images of the generators under ~*. By definition 4.3.8 we have A = ~*(A). A is the class of (32, Z1, Z )

C Hilb ( P ( E ) / X )

Z lies on a line ] in the fibre P(E(Tr(z))) / ' passing through x

~3

Let 7r' := ~1~. Let again i~2 : Hilb ( P ( E ) / X ) , Hilb2(P(E)/X) = AI2(p(E)/X) be the projection and p~ := ~21X. Then we have by definition P -- ~ a x e * ( c l ( 2 , E ) ) , ? = axe*(Cl(T~,E) ) E A ~ ( A I 3 ( p ( E ) / X ) ) .

The diagram

'"N t

AI3(p(E)/X)

AI2(p(E)/X) axe NNa ~'(E)

~/

axe

commutes. So we get ( # ) . ( / 5 ) = PIX and thus ~-*(~) = AP, "~*(~) = A P 2. By l e m m a 4.4.5, l e m m a 4.3.9, remark 4.3.13 and the projection formula we have

,~*(/~)

=

~*(~)

=

H + ~.(b) H + H2,

H 2 + ~.(b 2) H 2 + ~,(bo~ - ~2 + # 1 ~ -- # 2 ) H 2 + H 2 P -- p2 + 2#1P - 2#2,

~*(~) =p2

p2 + ~ . . ( 3 ~ + ~2)

=

~_~ ' . ( a ( / ~ -4- 3') -- 2a 2 -t- #1(• -I- 3') -- 2#2)

=p2+

1^ ~r.((a+#l)(e+a+(b+c)-a-'r+#l)-aa2+2#2)

p2 _ H P + HH2

~*(f?) =

-

4.3.13(4), (5) 4.3.6(1)

H 2 + A H + # I ( - P + H2 + 2H + A) + #~ - 2p2,

H p 2 + 1~',(b/32 + c72). Z

Furthermore we have bfl 2 "4- c72 = b(afl - a 2 -4- # l ~

-

-

P2) "~ c(a7

-- a 2 -f- #17 -- #2)

4.3.13(4), (5)

178

4. The Chow ring of relative Hilbert schemes of projective bundles

= - ( a 2 + ~:)(b + c) + (a + ~l)(bZ + c7),

r

b9 + c7 = (b + e ) ( 9 + 3') - b3' -

= (b -4- c)(~ + 3') + 2a2 - a(/3 + 3') - b2 - c 2 = 2a 2 + (b + c - a)(/~ + 7) + 2~

4.3.13(7), (8)

- (b + c)c~ - 2#1o~ + 2#2

= 2a 2 + 2 a 2 - (b + c)(~ - 2 # 1 a + 2#2 + e(b + c) - ea

4.3.13(3) 4.3.6(1)

+ (b + c) 2 - a 2 - (b + c)(~ + ac~ - (b + c)7 + aT + (b + C)pl - a # l . So we get b/~2 + c3' 2 = a 3 + 2ae a + ae a - 2(b + c)aa - a2(b + c) + (b + c)e a - ea 2 + ea(b + c) + T(a(b + c) + a 2) + #l(--ac~ + (b + c)a - 2(b + c)c~ + (b + c) 2 + 2c~2 + e(b + c) - ea + T(--(b ~- c) -f- a))

+ #~(-2c~ + (b + c) - a) - #2((b + c) + 2a) + 2#1#2. Using the p r o j e c t i o n f o r m u l a we get ~*(~) = 3 H P e + H 3 + H e p + #I(-HP

2HHeP - Hell2 + HH 2 - AH 2 + AHHe

+ H H 2 - 2 H e P + H~ + 2 P e + A H e - A H )

+ #21(-2P + He - H ) + # 2 ( - H e + 2 H ) + 2#1#2.

T h e f o r m u l a for ~*(/~) is now o b t a i n e d by a p p l y i n g the relations H 3=#1H

e-#eH+#3,

A H e = A ( H P - p e + P S i - re).

~3

As we have d e t e r m i n e d A*(Hilb ( P ( E ) / X ) )

[]

in t h e o r e m 4.3.11, a n d its struc-

"ture is in fact r a t h e r simple, this gives us a simple d e s c r i p t i o n of A * ( H i l b 3 ( p ( E ) / X ) ) , which is also very useful for c o m p u t a t i o n s . We now also w a n t to describe this ring by g e n e r a t o r s a n d relations. Because the relations are very c o m p l i c a t e d , we d o n ' t want to s t a t e t h e m all, b u t r a t h e r refer to [Ghttsehe (6)] for the list of all relations.

Theorem

4.4.7. A* ( H i l b 3 ( p ( E ) / X ) )

for suitable classes R 1 , R e , R 3 , . . . , R 3 o listed in Satz ~.~.7 of [Ghttsche (6)].

= A * ( X ) [ H , A, h, 15, ~, 6,/~] (R1, Re, R3, 9 9 9 R30) in A*(X)[/~,fi~,h,/5,~,6~,~],

which are all

The relations in codimension at most three

4.4.

The Chow ring of Hilb3(p(E)/X)

179

are

R2 := - / ~ p + / ~ 3 + AH5 _ 4 H h - H ~ + 36 - 33 + #1(5H 5 + 4.~/t - 4]* - 6/5 - 3a) + #~(10/~ + 6A) + #5(-9/~r + 3.~) + 6#~ - 18#1#5 + 9#3, R3 : = - A h + / ~

- 36 + 3~#1 - 3fi-#5,

R4 : = - . 4 p + 36, R5 : = - - ~

- H ~ + 30 - ~#i.

S k e t c h o f proof." T h e d e t e r m i n a t i o n of the relations is a trivial b u t very extensive computation. We use theorem 4.4.6 and the relations I 1 , 1 2 , / 3 , / 4 , / 5 , / 6 of theorem 4.3.11, to express every element of the basis of K*(A*(Hilb3(p(E)/X))) over A*(X) from proposition 4.4.4 as an A * ( X ) - l i n e a r combination of elements of the basis of ~3 A*(Hilb (P(E)/X)) over A*(X) from the proof of proposition 4.3.10. For this we use the computer. Similarly we use proposition 4.4.6 a n d the relations 11,. 9 9 express the images of

to

2~2 ,

/~p, Ah, ~i/5,~i~, /~4 A/~3,/~3, As, 2,3, f,~,/52,pa, as, H36,/.~2 ~2 ~22j),6 2 , 6 3 , 32, B2~/5, Hh 3 ~3 under ~* as an A * ( X ) - l i n e a r combination of the basis of A*(Hilb

(P(E)/X)). Now

we only have to solve a system of linear equations in order to get the relations. For this we use again the computer. As a result we get relations R 1 , . . . , R30. We still have to show t h a t R 1 , . . . , R30 generate all relations. For this we have to show t h a t by using t h e m we can express any m o n o m i a l in H , A, h,/5,~, 6,3 as an A * ( X ) - l i n e a r combination of the elements of the basis from p r o p o s i t i o n 4.4.4. To show this we use arguments similar to those in the end p a r t of the proof of theorem 4.3.11. In the current case the arguments are however considerably more complicated and make use of the precise form of R 1 , . . . , R30. We refer to the proof of Satz 4.4.7 in [G6ttsche (6)] for the details. [] In the rest of this section we look at an i m p o r t a n t special case of ~3 Hilb ( P ( E ) / X ) ) . We p u t G := Grass(d - 2, d + 1) a n d let T := T3,u+I be the tautological bundle over G.

180

4.

The Chow ring of relative Hilbert ~cherne~of projective bundle~

D e f i n i t i o n 4.4.8. Let Cop3(Pd) C Hilb3(pd) x G be defined by Cop3(pd) :=

~(Z,E)E [

HilbZ(Pd) x G

Z C E~. 1

Let F C Pd x G be the incidence variety F := { ( x , E ) E Pd x G I z E E} with the projections F J

\p2

Pl

P~

G. ~3

In the same way as after definition 4.3.17 for Cop (Pd) we see that there is a natural isomorphism ~b: Cop3(pd) ----, Hilb3(p(T)/G) over G. The projection ibl : Copz(Pd) ----, Hilb3(pd) is a birational morphism, as every subschmeme Z of length 3 of Pd is a subscheme of a plane. This plane is uniquely determined if Z does not lie on a line. In the same way as in the proof of remark 4 . 3 . 1 5 w e see that the f i b r e / ~ - I ( Z ) over a point Z C Al3(pd) is isomorphic to Pd-2 and that the exceptional locus ~l(Al3(pd)) is AI3(P(T)/G) ~P(Sym3(T2,r)). Here again T2,T is the tautological bundle of rank 2 over P ( r ) . This shows analogously to remark 4.3.15: Remark

4.4.9. Cop3(Pa) is the blow up of Hilb3(pd) along

Al3(pd).

D e f i n i t i o n 4.4.10. L e t / t , A, h, ~, ~, 3, ~, #1, #2, #3 C A*(Cop3(pd)) be the classes

fi~:: [{(Z,E) CCop3(pd) s

Zliesonaline}],

[{(Z,E) ECopa(pd) Zintersectsafixedhyperplane}],

h := [{(Z,E) C Cop3(pd)

Z intersec sa0xed 2 codimensiooal}l linear subspace

P:= [{(Z,E) EC~

the line through one of the subschemes Z ~ C Z of length 2 intersects two different fixed 2-codimensional linear subspaces

3:= [{(Z,E) E Copa(pd)

Zliesonalineintersectingafixed 2-codimensional linear subspaee

}]

}]

4.4.

The Chow ring of Hilb3(p(E)/X)

F

:= /

::/'

E) c Cop3(Pd)

(Z,E) e Copz(Pd) 1-

181

Z lies on a line }] intersecting two different 2-codimensional linear subspaces the line through one of the subschemes Z' C Z of length 2 intersects two different fixed 2-codimensional linear subspaees, res(Z', Z) lies on a fixed hyperplane E intersects a fixed }] 3-codimensional linear subspace '

~:

[{,~,~, ~ ~op3,p~, ~ ~ ~ o~ d ~ m e n ~ ~ 1 7 6

}1

~ h ~ x e d }l

2-eodimensional linear subspace P3::

[{(Z,E) ECop3(pd)

E l i e s o n a f i x e d h y p e r p l a n e }].

From the definitions we get: R e m a r k 4.4.11.

r r

= :~, r = ~, r

= ~, = 3, r

r

= ~, r = 71, r

= 5, = ~2,r

r

= ~, = ~

So theorem 4.4.6 describes the Chow ring of Cop3(pd) in terms of classes describing the position of subschemes relative to linear subspaces of Pd. In the case of Cop3(p3) we get in particular: # : = / ~ 1 = [{(Z,E) ECop3(pa)

E contains a fixed point }],

]52 = #2, /~3 = 3 .

We can now use theorem 4.4.6 to compute the intersection tables with the help of the computer. We keep in mind that for u E Ai(Copa(p3)), v E A"-i(Cop3(p3)) the intersection numbers u - v and ~*(u). ~*(v) are related by ~*(u). ~*(v) = 3u-v and obtain the following tables:

4. The Chow ring of relative Hilbert schemes of projective bundles

182

A 1 x AS: h3#2

3

2

:Ih~u 3 I

i

1

A 2 x A7:

6

3

7

6

3

15

6

4

:/2h~3 3

3

1

/~2a~z 6

-3

h~pz

1

1

/:/&pz

1

-i

2

6

3

2

2

4

1

1

4

1

1

1

1

1 -1 I

h~u3 A 3 • A6:

~3 ]~3

6

/:/h2iz

20

6

13

6

/:/h)#

66

18

27

6

/~2]~#2 25

7

22

9

/?/2h/J2 40

9

~2~2

2

2

-10-7-2

6

6

3

6

7

6

3

13 15

6

4

5

3

3

1

1

2

6

-3

1

-1

1

7

3

6

2

1

1

/:/ 12

2

-7

-2

1

-I

]~i~#2

15

4

6

/:/3~3

15

3

15

6

]~/:/p3

3

1

3

i

3

3

i

-3

/~2j#3 15 /Ia~3

6

&pa

1

-1

2 l

3

-3 - I

I

-i

i

2

6

3

2

2

4

1

1

4

1

1

1

4.4.

The Chow ring of Hilba(P(E)/X)

183

A 4 • AS:

j~ 3 Hh2

6

/I]~/~

26

/f/2h]~

20

22

H2ap

22

-16

]~2~

6

/~/&p

13

6

24 66

18 ! 27

6

18 25

7

22

9

6 -8

6

40

9

-I0

6

2

6

7

3

6

9

-8

2 -2

12

2

-7

]~i~]J

18

6

6

8

15

4

6

/:/3/~2

25

40

7

12 15 ! 15

3

15

6

]~f/#2

7

9

3

2

4

3

1

3

1

-10

6 -7

6

15

3

3

-3

2

6

1

-3

-1

/;/2A#2 22

6

6

[/~#2

9

-7

&/~2

2

-2

3~2

5

2

if/2# 3

3

6

flap3

3

-3

]~p3

1

1

d#3

1

-I

6

9

-2

2 i

1

1 -i

1

1

20

6

I

1

-1

3

1

2

6

7

6

3

13

15

6

4

5

3

3

1

1

2

6

-3

1

-1

2

I

1

1

-2

I

-1

2

-7-2

2 1

3

-I

1

1

2

4

1

4

1

1

1

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Mumford, D. Fogarty, J. (1) Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, second edition, Springer-Verlag, New York Heidelberg Berlin 1982. Reider, I. (1) Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 127 (1988), 309-316. Roan, S. S. (1) On the generalization of Kummer surfaces, J. Diff. Geom. 30 (1989) 523-537. Roberts, J. (1) Old and new results about the triangle variety, Proc. Algebraic Geometry Sundance 1986, Holme, A. and Speiser, R., eds, Lecture Notes in Math. 1311, Springer-Verlag, Berlin Heidelberg 1988, 197-219. Roberts, J. Speiser, R. (1) Schubert's enumerative geometry of triangles from a modern viewpoint, Algebraic Geometry, Proc. Conf. Chicago Circle 1980, Lecture Notes in Math. 862, Springer Verlag, Berlin Heidelberg 1981, 272-281. (2) Enumerative geometry of triangles I, Comm in Alg. 12(9-10) (1984), 1213-1255. (3) Enumerative geometry of triangles II, Comm in Alg. 14(1) (1986), 155-191. (4) Enumerative geometry of triangles III, Comm in Alg. 15(9) (1987), 1929-1966. Rossell6 Llompart, F. (1) Les groupes de Chow de quelques sch@mas qui param@trisent des points coplanaires, C. R. Acad. Sci. Paris S~r. I Math. 303 (1986), 363-366. (2) The Chow-Ring of Hilb3p 3, Enumerative Geometry, Proc Sitjes 1987, S. Xamb6 Descamps, ed., Lecture Notes in Math. 1436, Springer-Verlag, Berlin Heidelberg 1990, 225-255. Rossell6 Llompart, F., Xamb6 Descambs, S. (1) Computing Chow groups, Proc. Algebraic Geometry Sundance 1986, Holme, A. and Speiser, R., eds, Lecture Notes in Math. 1311, Springer-Verlag, Berlin Heidelberg 1988, 220-234. (2) Chow groups and Horel-Moore schemes, Ann. Math. Pura Appl. (4) 160 (1991), 19-40. Schubert, H. (1) Kalkfil der abzghlenden Geometrie, Teubner, Leibzig 1879, reprinted by Springer-Verlag, Berlin 1979. (2) Anzahlgeometrische Behandlung des Dreiecks, Math. Ann. 17 (1880), 153-212. Speiser, R. (1) Enumerating contacts, Proceedings of Symposia in pure Mathematics of the AMS, Volume 46 part 2, Algebraic Geometry 1985, 401-418.

Bibliography

191

Semple, J. G. (1) Some investigations in the geometry of curve and surface elements, Proc. London Math. Soc. 4(3) (1954), 24-49. (2) The triangle as a geometric variable, Mathematica 1 (1954), 80-88. Steenbrink, J. H. M. (1) Mixed Hodge Structures on the vanishing cohomology, Nordic Summer School, Symposium in Mathematics, Oslo 1970, Sijthoff and Noordhoff, Alphen an den Rijn 1977, (525-563). Tyrrell, J. A. (1) On the enumerative geometry of triangles, Mathematica 6 (1959), 158-164. Tyurin, A. N. (1) Cycles, curves and vector bundles on an algebraic surface, Duke Math. J. 54 (1987), 1-26. Zagier, D. (1) Equivariant Pontrjagin classes and applications to orbit spaces, Lecture Notes in Math. 290, Springer-Verlag, Berlin Heidelberg New York 1972. (2) Note on the Landweber-Stong elliptic genus, Elliptic Curves and Modular Forms in Algebraic Topology, Proceedings Princeton 1986, P.S. Landweber (Ed.), Lecture Notes in Mathematics 1326, Springer-Verlag, Berlin Heidelberg New York 1988, 216-224.

Index

axial morphism 152, 154, 161, 163, 165, 167, 174, 177 Borel-Moore homology 19 cell decomposition 12, 19-28, 34, 79, t66, 175 Chern classes of symmetric powers 93, 114, 150 constant over T 148, 149 contact 81, 99 - bundle 88, 108, 115-118, 119, 121 with lines 124-125, 128-142 - of families of curves 85, 143 - with linear subspaces 119-125 second order 122, 126 cycle map 19 degeneracy - locus 98, 119, 123-124 - cycle 98, 119 evaluation morphism 115, 119, 122-124 formula of Macdonald 35, 49, 50, 79 geometric Frobenius 5, 7, 31, 43 good reduction 5, 6, 35, 49, 63, 78 higher-order Kummer varieties 12, 40-59 Hilbert-Chow morphism 4, 32, 40, 42, 54, 61 Hilbert function 9 strata 9-11, 16-18, 23-28, 64, 67, 74, 91, 97, II0, 131 Hilbert polynomial I Hilbert scheme I-4 - of subschemes of length n 2 punctual 9-11, 19, 29, 30, 33 of aligned subschemes 133, 138, 145, 151-153, 155, 156, 163, 168, 171,174, 177, 180 relative2, 147-149 relative - of projective bundles 145-184 for coplanar subschemes 145, 170, 180 stratification of by partitions 3, 14, 30, 60, 67 incidence variety 133-137, 140, 152, 165, 170, 180 initial - degree 9 form 9, 18 jet-bundle 14, 85, 90, 104, 110, 111,119 jumping index 10, 28 /-adic cohomology 5 Leray-Hirsch for Chow groups 166, 175 modular forms 35, 52 -

-

-

Index

193

multiplicative group 19m21 n-very a m p l e 146 relative 147-149 one-parameter subgroup 21 general 21, 25, 80 orbifold Euler n u m b e r 12, 54-56 partition 3, 20, 22, 26, 29, 42, 44 g r a p h of 23 dual 23 hook difference of 23 point geometric 4, 5, I0, 15, 29 k-valued 2 T-valued 1 division 43 Porteous formula 98, 99, 120, 123-124

representable functor I, 60, 63, 83, 101, 102, 109 residual morphism 61, 63, 154, 155, 160, 164, 171, 176, 181 Semple-bundle varieties 128-144 relative 142-144 Schubert cycle 135, 138, 139 Segre class 98, 124 Shintani-descent 41 symmetric power of a variety 3, 7, 32, 34, 54, 150 universal subscheme 2, 62, 65, 83, 88, Ii0, 116, 147, 149, 151,152,154, 156, 164 varieties of triangles 12, 60-80 complete triangles 63-73, 79, 85 with a chosen side 60-62 varieties of higher order d a t a 101-127, - of curves 85, 91, 94, 113, 114, 118 varieties of second order d a t a 81-100 relative 126-127 Weil conjectures 5-8, 29, 35, 41, 49, 74 zero-cycle 3, 40 primitive 8, 31-34, 45, 75 zeta-function 5 -

Index of notations

~ilb(X/T)

Hilbert functor relative Hilbert scheme Hilb(X/T) relative Hilbert scheme of subschemes of length n Hilb~(X/T) Hilbert scheme of subschemes of length n X["], Hilb"(X) universal family Z,.(X/T), Zn(X) a(,~) symmetric group on n letters X(n) symmetric power stratum of X (n) X (') stratum of X['] x~,q Hilbert-Chow morphism r n Hi(X, q t ) l-adic cohomology Poincard polynomial p(x, z) zeta function z~(x, t) b~(x) Betti number P~(X,F~) set of primitive cycles punctual Hilbert scheme Hilb'~(R/m"), Hilb"(R)~, t Hilbert function strata ZT, GT jumping index (~,)~>0 A, A '~ (thickened) diagonal J.(X), J~(X) jet bundles zT(x), a~(x) relative Hilbert function strata curvilinear subschemes concentrated in a point A.(X), A*(X) Chow ring el cycle map G.. multiplicative group p(n), p(~, l) number of partitions of n r(~) graph of partition .& dual partition ti(a), T(a) diagonal lengths of partition h~,v(~) hook difference Y. punctual Hilbert scheme *'(~) set of partitions of n ( n l , . . . , n ~ ) = (1~',2~2,...) partitions Gal(]c/ k ) Galois group length of subseheme fen(f) "length" of function P(X, •q) primitive 0-cycles Tn(X, Fq) set of admissible functions A(~-), ~(~-) Delta function, eta function

1 2 2 2 2 3 3 3 4 4 5 5 5 5 8 10 10 10 14 14 17 18 19 19 19 20 23 23 23 23 29 29 29 29 29 30 31 31 35

Index of notation3

hP'q(X) h(X,x,y) sign(X) xdx) G~

Is KAn-1 NH/H,

s[.] ~l(n) e(x, G) ~ n Hilb (X) res

~3(x) _~[3]

Hodge number Hodge polynomial signature xy(X)-genus modular forms higher order Kummer variaties Shintani descent set used for counting sum of numbers dividing n orbifold Euler number incidence variety of subschemes of lengths n - 1 and n residual morphism variety of complete triangles variety of complete unordered triangles

~ 3

Hilb (Z)

complete triangles with marked side Grassmannian bundle projection in Grassmannian bundle ";rm,E Grass(m, r) Grassmannian P(E), P ( E ) projectivized bundle Tm,~, Tm,r tautological subbundles universal quotient bundles Qm,E, O,,,,r scheme defined by product of ideals Z1 9Z2 D~(X) variety of second order data tautological and quotient bundle over Grass(m, T~ ) T1, Qi w~(x) bundle of second order data b~(x) other construction of D~(X) tautological and quotient bundle over D2(X) I'2, Q2 (ox)~ contact bundle Ar A~ (thickened) diagonals of morphism Ith-order datum of Y at x Dz,=(Y) Vk(~) degeneracy locus nn(x) variety of higher order data F.G "product" of sheaves W~(X) third order data sheaf 5~(x), bLl(x) variety of third order data tautological subbundle and quotient of W3(X) Ts, 03 contact bundle (E)~n evaluation map eVE D~(Y/T) relative data variety class of second order contact g(x, YT) Fn(X), an(x) Sample bundles

Grass(m, E)

195

37 37 37 37 38 40 42 42 51 54 60 61 63 67 71 82 82 82 82 82 82 82 83 85 86 87 87 88 88 97 98 101 105 105 107 107 108 115 126 126 128

Index of notations

196

AI~(PN) Kn(X), Kn,l(Z) Fn(X/T), Gn(X/T)

aligned n-tuples class of contact with lines relative Semple bundles r163 morphism of Hilbert scheme induced by/3 Hilbn(P(E)/X) relative Hilbert scheme of projective bundle era,n, Cn morphisms of Hilbert schemes of projective bundle AI~(P(E)), Al~(Pd) variety of aligned subschemes Z~'(P(E)), Z~t(Pd) universal subscheme of aligned subschemes axe ~ axial morphism ~ 3 H, A, H2, P, P2 classes in the Chow ring of Hilb (P2) ~ 3

Hilb (P(E)/X), W(P(E)), H3(P(E)/X) ~ 3

Cop (Pd) H, ,zl, fz, fi, 6,/~ Cop3(pd)

relative triangle varieties

variety of triangles in a plane in Pd classes in the Chow ring of Hilb3(P(E)/X) Hilbert scheme of subschemes in a plane in Pd

Printing: Weihert-Druck GmbH, Darmstadt Binding: Buchbinderei Schiiffer, Griinstadt

135 140 144 146 149 149 151 151 152 154 160 170 173 180

Bibliography

187

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classical algebraic Geometry, Nice 1981, Prog. in Math. 24, Birkhs 1982, 165-197. Platitude et non-platitude de certaines sous-sch6mas de Hilbkp N, Journal fiir die Reine und Angewandte Mathematik 348 (1984), 116-134. Contribution des droites d'une surface s ses multis~cantes, Bull. Soc. Math. France 112 (1984), 303-324. Quelques cMculs dans les vari~t~s d'alignements, Adv. in Math. 64 (1987), 87-117. Formules pour les tris6cantes des surfaces alg~briques. Enseign. Math. 33 no 1-2 (1987), 1-66. La vari~t~ des triplets complets, Duke Math. Journal 57 (1988), 925-946. Quelques formules multis~cantes pour les surfaces, Enumerative Geometry, Proc Sitjes 1987, S. XambS-Descamps, ed., Lecture Notes in Math. 1436, SpringerVerlag, Berlin Heidelberg 1990, 151-188.

Macdonald, I. G. (1) The Poincar6 polynomial of a symmetric product, Proc. Camb. Phil. Soc. 58 (1962), 563-568. Mallavibarrena, R., (1) Les groupes de Chow de Hilb4P 2, C. R. Acad. Sci. Paris 303, I13 (1986). (2) Validit~ de la formule classique des tris~cantes stationaires, C. R. Acad. Sci. Paris 303, I16 (1986). (3) E1 M~todo de las bases de los grupos de Chow de Hilbdp 2 en geometria enumerativa, Thesis 1987. Mallavibarrena, R., Sols, I. (1) Bases for the homology groups of the Hilbert scheme of points in the plane, Comp. Math. 74 (1990), 169-202. Matsumura, H. (1) Commutative Algebra, W. A. Benjamin, Inc., New York 1970. Mazur, B. (1) Eigenvalues of Frobenius acting on algebraic varieties over finite fields, Proceedings of Symposia in Pure Mathematics Vol. 29, Algebraic Geometry, Arcata 1974, 231-261. Milne, J. S. (1) Etale Cohomology, Princeton Math. Series 33, Princeton University Press, Princeton 1980. Mumford D. (1) Lectures on curves on an algebraic surface, Annals of Math. Studies vol. 59, Princeton 1966. (2) The Red Book on Varieties and Schemes, Lecture Notes in Mathematics 1358, Springer-Verlag, Berlin Heidelberg 1988.

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Mumford, D. Fogarty, J. (1) Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, second edition, Springer-Verlag, New York Heidelberg Berlin 1982. Reider, I. (1) Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 127 (1988), 309-316. Roan, S. S. (1) On the generalization of Kummer surfaces, J. Diff. Geom. 30 (1989) 523-537. Roberts, J. (1) Old and new results about the triangle variety, Proc. Algebraic Geometry Sundance 1986, Holme, A. and Speiser, R., eds, Lecture Notes in Math. 1311, Springer-Verlag, Berlin Heidelberg 1988, 197-219. Roberts, J. Speiser, R. (1) Schubert's enumerative geometry of triangles from a modern viewpoint, Algebraic Geometry, Proc. Conf. Chicago Circle 1980, Lecture Notes in Math. 862, Springer Verlag, Berlin Heidelberg 1981, 272-281. (2) Enumerative geometry of triangles I, Comm in Alg. 12(9-10) (1984), 1213-1255. (3) Enumerative geometry of triangles II, Comm in Alg. 14(1) (1986), 155-191. (4) Enumerative geometry of triangles III, Comm in Alg. 15(9) (1987), 1929-1966. Rossell6 Llompart, F. (1) Les groupes de Chow de quelques sch@mas qui param@trisent des points coplanaires, C. R. Acad. Sci. Paris S~r. I Math. 303 (1986), 363-366. (2) The Chow-Ring of Hilb3p 3, Enumerative Geometry, Proc Sitjes 1987, S. Xamb6 Descamps, ed., Lecture Notes in Math. 1436, Springer-Verlag, Berlin Heidelberg 1990, 225-255. Rossell6 Llompart, F., Xamb6 Descambs, S. (1) Computing Chow groups, Proc. Algebraic Geometry Sundance 1986, Holme, A. and Speiser, R., eds, Lecture Notes in Math. 1311, Springer-Verlag, Berlin Heidelberg 1988, 220-234. (2) Chow groups and Horel-Moore schemes, Ann. Math. Pura Appl. (4) 160 (1991), 19-40. Schubert, H. (1) Kalkfil der abzghlenden Geometrie, Teubner, Leibzig 1879, reprinted by Springer-Verlag, Berlin 1979. (2) Anzahlgeometrische Behandlung des Dreiecks, Math. Ann. 17 (1880), 153-212. Speiser, R. (1) Enumerating contacts, Proceedings of Symposia in pure Mathematics of the AMS, Volume 46 part 2, Algebraic Geometry 1985, 401-418.

Bibliography

191

Semple, J. G. (1) Some investigations in the geometry of curve and surface elements, Proc. London Math. Soc. 4(3) (1954), 24-49. (2) The triangle as a geometric variable, Mathematica 1 (1954), 80-88. Steenbrink, J. H. M. (1) Mixed Hodge Structures on the vanishing cohomology, Nordic Summer School, Symposium in Mathematics, Oslo 1970, Sijthoff and Noordhoff, Alphen an den Rijn 1977, (525-563). Tyrrell, J. A. (1) On the enumerative geometry of triangles, Mathematica 6 (1959), 158-164. Tyurin, A. N. (1) Cycles, curves and vector bundles on an algebraic surface, Duke Math. J. 54 (1987), 1-26. Zagier, D. (1) Equivariant Pontrjagin classes and applications to orbit spaces, Lecture Notes in Math. 290, Springer-Verlag, Berlin Heidelberg New York 1972. (2) Note on the Landweber-Stong elliptic genus, Elliptic Curves and Modular Forms in Algebraic Topology, Proceedings Princeton 1986, P.S. Landweber (Ed.), Lecture Notes in Mathematics 1326, Springer-Verlag, Berlin Heidelberg New York 1988, 216-224.

Index

axial morphism 152, 154, 161, 163, 165, 167, 174, 177 Borel-Moore homology 19 cell decomposition 12, 19-28, 34, 79, t66, 175 Chern classes of symmetric powers 93, 114, 150 constant over T 148, 149 contact 81, 99 - bundle 88, 108, 115-118, 119, 121 with lines 124-125, 128-142 - of families of curves 85, 143 - with linear subspaces 119-125 second order 122, 126 cycle map 19 degeneracy - locus 98, 119, 123-124 - cycle 98, 119 evaluation morphism 115, 119, 122-124 formula of Macdonald 35, 49, 50, 79 geometric Frobenius 5, 7, 31, 43 good reduction 5, 6, 35, 49, 63, 78 higher-order Kummer varieties 12, 40-59 Hilbert-Chow morphism 4, 32, 40, 42, 54, 61 Hilbert function 9 strata 9-11, 16-18, 23-28, 64, 67, 74, 91, 97, II0, 131 Hilbert polynomial I Hilbert scheme I-4 - of subschemes of length n 2 punctual 9-11, 19, 29, 30, 33 of aligned subschemes 133, 138, 145, 151-153, 155, 156, 163, 168, 171,174, 177, 180 relative2, 147-149 relative - of projective bundles 145-184 for coplanar subschemes 145, 170, 180 stratification of by partitions 3, 14, 30, 60, 67 incidence variety 133-137, 140, 152, 165, 170, 180 initial - degree 9 form 9, 18 jet-bundle 14, 85, 90, 104, 110, 111,119 jumping index 10, 28 /-adic cohomology 5 Leray-Hirsch for Chow groups 166, 175 modular forms 35, 52 -

-

-

Index

193

multiplicative group 19m21 n-very a m p l e 146 relative 147-149 one-parameter subgroup 21 general 21, 25, 80 orbifold Euler n u m b e r 12, 54-56 partition 3, 20, 22, 26, 29, 42, 44 g r a p h of 23 dual 23 hook difference of 23 point geometric 4, 5, I0, 15, 29 k-valued 2 T-valued 1 division 43 Porteous formula 98, 99, 120, 123-124

representable functor I, 60, 63, 83, 101, 102, 109 residual morphism 61, 63, 154, 155, 160, 164, 171, 176, 181 Semple-bundle varieties 128-144 relative 142-144 Schubert cycle 135, 138, 139 Segre class 98, 124 Shintani-descent 41 symmetric power of a variety 3, 7, 32, 34, 54, 150 universal subscheme 2, 62, 65, 83, 88, Ii0, 116, 147, 149, 151,152,154, 156, 164 varieties of triangles 12, 60-80 complete triangles 63-73, 79, 85 with a chosen side 60-62 varieties of higher order d a t a 101-127, - of curves 85, 91, 94, 113, 114, 118 varieties of second order d a t a 81-100 relative 126-127 Weil conjectures 5-8, 29, 35, 41, 49, 74 zero-cycle 3, 40 primitive 8, 31-34, 45, 75 zeta-function 5 -

Index of notations

~ilb(X/T)

Hilbert functor relative Hilbert scheme Hilb(X/T) relative Hilbert scheme of subschemes of length n Hilb~(X/T) Hilbert scheme of subschemes of length n X["], Hilb"(X) universal family Z,.(X/T), Zn(X) a(,~) symmetric group on n letters X(n) symmetric power stratum of X (n) X (') stratum of X['] x~,q Hilbert-Chow morphism r n Hi(X, q t ) l-adic cohomology Poincard polynomial p(x, z) zeta function z~(x, t) b~(x) Betti number P~(X,F~) set of primitive cycles punctual Hilbert scheme Hilb'~(R/m"), Hilb"(R)~, t Hilbert function strata ZT, GT jumping index (~,)~>0 A, A '~ (thickened) diagonal J.(X), J~(X) jet bundles zT(x), a~(x) relative Hilbert function strata curvilinear subschemes concentrated in a point A.(X), A*(X) Chow ring el cycle map G.. multiplicative group p(n), p(~, l) number of partitions of n r(~) graph of partition .& dual partition ti(a), T(a) diagonal lengths of partition h~,v(~) hook difference Y. punctual Hilbert scheme *'(~) set of partitions of n ( n l , . . . , n ~ ) = (1~',2~2,...) partitions Gal(]c/ k ) Galois group length of subseheme fen(f) "length" of function P(X, •q) primitive 0-cycles Tn(X, Fq) set of admissible functions A(~-), ~(~-) Delta function, eta function

1 2 2 2 2 3 3 3 4 4 5 5 5 5 8 10 10 10 14 14 17 18 19 19 19 20 23 23 23 23 29 29 29 29 29 30 31 31 35

Index of notation3

hP'q(X) h(X,x,y) sign(X) xdx) G~

Is KAn-1 NH/H,

s[.] ~l(n) e(x, G) ~ n Hilb (X) res

~3(x) _~[3]

Hodge number Hodge polynomial signature xy(X)-genus modular forms higher order Kummer variaties Shintani descent set used for counting sum of numbers dividing n orbifold Euler number incidence variety of subschemes of lengths n - 1 and n residual morphism variety of complete triangles variety of complete unordered triangles

~ 3

Hilb (Z)

complete triangles with marked side Grassmannian bundle projection in Grassmannian bundle ";rm,E Grass(m, r) Grassmannian P(E), P ( E ) projectivized bundle Tm,~, Tm,r tautological subbundles universal quotient bundles Qm,E, O,,,,r scheme defined by product of ideals Z1 9Z2 D~(X) variety of second order data tautological and quotient bundle over Grass(m, T~ ) T1, Qi w~(x) bundle of second order data b~(x) other construction of D~(X) tautological and quotient bundle over D2(X) I'2, Q2 (ox)~ contact bundle Ar A~ (thickened) diagonals of morphism Ith-order datum of Y at x Dz,=(Y) Vk(~) degeneracy locus nn(x) variety of higher order data F.G "product" of sheaves W~(X) third order data sheaf 5~(x), bLl(x) variety of third order data tautological subbundle and quotient of W3(X) Ts, 03 contact bundle (E)~n evaluation map eVE D~(Y/T) relative data variety class of second order contact g(x, YT) Fn(X), an(x) Sample bundles

Grass(m, E)

195

37 37 37 37 38 40 42 42 51 54 60 61 63 67 71 82 82 82 82 82 82 82 83 85 86 87 87 88 88 97 98 101 105 105 107 107 108 115 126 126 128

Index of notations

196

AI~(PN) Kn(X), Kn,l(Z) Fn(X/T), Gn(X/T)

aligned n-tuples class of contact with lines relative Semple bundles r163 morphism of Hilbert scheme induced by/3 Hilbn(P(E)/X) relative Hilbert scheme of projective bundle era,n, Cn morphisms of Hilbert schemes of projective bundle AI~(P(E)), Al~(Pd) variety of aligned subschemes Z~'(P(E)), Z~t(Pd) universal subscheme of aligned subschemes axe ~ axial morphism ~ 3 H, A, H2, P, P2 classes in the Chow ring of Hilb (P2) ~ 3

Hilb (P(E)/X), W(P(E)), H3(P(E)/X) ~ 3

Cop (Pd) H, ,zl, fz, fi, 6,/~ Cop3(pd)

relative triangle varieties

variety of triangles in a plane in Pd classes in the Chow ring of Hilb3(P(E)/X) Hilbert scheme of subschemes in a plane in Pd

Printing: Weihert-Druck GmbH, Darmstadt Binding: Buchbinderei Schiiffer, Griinstadt

135 140 144 146 149 149 151 151 152 154 160 170 173 180

Vol. 1478: M.-P. Malliavin (Ed.), Topics in Invariant Theory. Seminar 1989-1990. VI, 272 pages. 1991.

Vol. 1502: C. Simpson, Asymptotic Behavior of Monodromy. V, 139 pages. 1991.

Vol. 1479: S. Bloch, I. Dolgachev, W. Fulton (Eds.), Algebraic Geometry. Proceedings, 1989. VII, 300 pages. 1991.

Vol. 1503: S. Shokranian, The Selberg-Arthur Trace Formula (Lectures by J. Arthur). VII, 97 pages. 1991.

Vol. 1480: F. Dumortier, R. Roussarie, J. Sotomayor, H. Zoladek, Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals. VIII, 226 pages. 1991. Vol. 1481: D. Ferns, U. Pinkall, U. Simon, B. Wegner (Eds.), Global Differential Geometry and Global Analysis. Proceedings, 1991. VIII, 283 pages. 1991. Vol. 1482: J. Chabrowski, The Dirichlet Problem with L zBoundary Data for Elliptic Linear Equations. VI, 173 pages. 1991. Vol. 1483: E. Reithmeier, Periodic Solutions of Nonlinear Dynamical Systems. VI, 171 pages. 1991. Vol. 1484: H. Delfs, Homology of Locally Semialgebraic Spaces. IX, 136 pages. 1991. Vol. 1485: J. Az6ma, P. A. Meyer, M. Yor (Eds.), S6minaire de Probabilit6s XXV. VIII, 440 pages. 1991. Vol. 1486: L. Arnold, H. Crauel, J.-P. Eckmann (Eds.), Lyapunov Exponents. Proceedings, 1990. VIII, 365 pages. 1991. Vol. 1487: E. Freitag, Singular Modular Forms and Theta Relations. VI, 172 pages. 1991. Vol. 1488: A. Carboni, M. C. Pedicchio, G. Rosolini (Eds.), Category Theory. Proceedings, 1990. VII, 494 pages. 1991. Vol. 1489: A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds. X, 140 pages. 1991. Vol. 1490: K. Metseh, Linear Spaces with Few Lines. XIII, 196 pages. 1991. Vol. 1491: E. Lluis-Puebla, J.-L. Loday, H. Gillet, C. Soul~, V. Snaith, Higher Algebraic K-Theory: an overview. IX, 164 pages. 1992. Vol. 1492: K. R. Wicks, Fractais and Hyperspaces. VIII, 168 pages. 1991.

Vol. 1504: J. Cheeger, M. Gromov, C. Okonek, P. Pansu, Geometric Topology: Recent Developments. Editors: P. de Bartolomeis, F. Tricerri. VII, 197 pages. 1991. Vol. 1505: K. Kajitani, T. Nishitani, The Hyperbolic Cauchy Problem. VII, 168 pages. 1991. Vol. 1506: A. Buium, Differential Algebraic Groups of Finite Dimension. XV, 145 pages. 1992. Vol. 1507: K. Hulek, T. Peternell, M. Schneider, F.-O. Schreyer (Eds.), Complex Algebraic Varieties. Proceedings, 1990. VII, 179 pages. 1992. Vol. 1508: M. Vuorinen (Ed.), Quasiconformal Space Mappings. A Collection of Surveys 1960-1990. IX, 148 pages. 1992. Vol. 1509: J. Aguad6, M. Castellet, F. R. Cohen (Eds.), Algebraic Topology - Homotopy and Group Cohomology. Proceedings, 1990. X, 330 pages. 1992. Vol. 1510: P. P. Kulish (Ed.), Quantum Groups. Proceedings, 1990. XII, 398 pages. 1992. Vol. 1511: B. S. Yadav, D. Singh (Eds.), Functional Analysis and Operator Theory. Proceedings, 1990. VIII, 223 pages. 1992. Vol. 1512: L. M. Adleman, M.-D. A. Huang, Primality Testing and Abelian Varieties Over Finite Fields. VII, 142 pages. 1992. Vol. 1513: L. S. Block, W. A. Coppel, Dynamics in One Dimension. VIII, 249 pages. 1992. Vol. 1514: U. Krengel, K. Richter, V. Warstat (Eds.), Ergodic Theory and Related Topics II1, Proceedings, 1990. VIII, 236 pages. 1992. Vol. 1515: E. Ballico, F. Catanese, C. Ciliberto (Eds.), Classification of Irregular Varieties. Proceedings, 1990. VII, 149 pages. 1992.

Vol. 1493: E. Beno~t (Ed.), Dynamic Bifurcations. Proceedings, Luminy 1990. VII, 219 pages. 1991.

Vol. 1516: R. A. Lorentz, Multivariate Birkhoff Interpolation. IX, 192 pages. 1992.

Vol. 1494: M.-T. Cheng, X.-W. Zhou, D.-G. Deng (Eds.), Harmonic Analysis. Proceedings, 1988. IX, 226 pages. 1991.

Vol. 1517: K. Keimel, W. Roth, Ordered Cones and Approximation. VI, 134 pages. 1992.

Vol. 1495: J. M. Bony, G. Grnbb, L. H6rmander, H. Komatsu, J. Sj0strand, Microlocal Analysis and Applications. Montecatini Terme, 1989. Editors: L. Cattabriga, L. Rodino. VII, 349 pages. 1991.

Vol. 1518: H. Stichtenoth, M. A. Tsfasman (Eds.), Coding Theory and Algebraic Geometry. Proceedings, 1991. VIII, 223 pages. 1992. Vol. 1519: M. W. Short, The Primitive Soluble Permutation Groups of Degree less than 256. IX, 145 pages. 1992.

Vol. 1496: C. Foias, B. Francis, J. W. Helton, H. Kwakernaak, J. B. Pearson, H| Theory. Como, 1990. Editors: E. Mosca, L. Pandolfi. VII, 336 pages. 1991.

Vol. 1520: Yu. G. Borisovich, Yu. E. Gliklikh (Eds.), Global Analysis - Studies and Applications V. VII, 284 pages. 1992.

Vol. 1497: G. T. Herman, A. K. Louis, F. Natterer (Eds.), Mathematical Methods in Tomography. Proceedings 1990. X, 268 pages. 1991.

Vol. 1521: S. Busenberg, B. Forte, H. K. Kuiken, Mathematical Modelling of Industrial Process. B ari, 1990. Editors: V. Capasso, A. Fasano. VII, 162 pages. 1992.

Vol. 1498: R. Lang, Spectral Theory of Random SchrOdinger Operators. X, 125 pages. 1991.

Vol. 1522: J.-M. Delort, F. B. I. Transformation. VII, 101 pages. 1992.

Vol. 1499: K. Taira, Boundary Value Problems and Markov Processes. IX, 132 pages. 1991.

Vol. 1523: W. Xue, Rings with Morita Duality. X, 168 pages. 1992.

Vol. 1500: J.-P. Serre, Lie Algebras and Lie Groups. VII, 168 pages. 1992.

Vol. 1524: M. Coste, L. Mah6, M.-F. Roy (Eds.), Real Algebraic Geometry. Proceedings, 1991. VIH, 418 pages. 1992.

Vol. 1501: A. De Masi, E. Presutti, Mathematical Methods for Hydrodynamic Limits. IX, 196 pages. 1991.

Vol. 1525: C. Casacuberta, M. Castellet (Eds.), Mathematical Research Today and Tomorrow. VII, 112 pages. 1992.

Vol. 1550: A. A. Gonchar, E. B. Saff (Eds.), Methods of Approximation Theory in Complex Analysis and Mathematical Physics IV, 222 pages, 1993.

Vol. 1526: J. Az6ma, P. A. Meyer, M. Yor (Eds.), S~minaire de Probabilit6s XXVI. X, 633 pages. 1992.

Vol. 1551: L. Arkeryd, P. L. Lions, P.A. Markowich, S.R. S. Varadhan. Nonequilibrium Problems in Many-Particle Systems. Montecatini, 1992. Editors: C. Cercignani, M. Pulvirenti. VII, 158 pages 1993.

Vol. 1527: M. I. Freidlin, J.-F. Le Gall, Ecole d'Et6 de Probabilit6s de Saint-Flour XX - 1990. Editor: P. L. Hennequin. VIII, 244 pages. 1992. Vol. 1528: G. Isac, Complementarity Problems. VI, 297 pages. 1992.

Vol. 1552: J. Hilgert, K.-H. Neeb, Lie Semigroups and their Applications. XII, 315 pages. 1993.

Vol. 1529: J. van Neerven, The Adjoint o f a Semigroup of Linear Operators. X, 195 pages. 1992.

Vol. 1553: J.-L- Colliot-Th61~ne, J. Kato, P. Vojta. Arithmetic Algebraic Geometry. Trento, 1991. Editor: E. Ballico. VII, 223 pages. 1993.

Vol. 1530: J. G. Heywood, K. Masuda, R. Rautmann, S. A. Solonnikov (Eds.), The Navier-Stokes Equations II Theory and Numerical Methods. IX, 322 pages. 1992.

Vol. 1554: A. K. Lenstra, H. W. Lenstra, Jr. (Eds.), The Development of the Number Field Sieve. VIII, 131 pages. 1993.

Vol. 1531: M. Stoer, Design of Survivable Networks. IV, 206 pages. 1992.

Vol. 1555: O. Liess, Conical Refraction and Higher Microlocalization. X, 389 pages. 1993.

Vol. 1532: J. F. Colombeau, Multiplication of Distributions. X, 184 pages. 1992.

Vol. 1556: S. B. Kuksin, Nearly Integrable InfiniteDimensional Hamiltonian Systems. XXVII, 101 pages. 1993.

Vol. 1533: P. Jipsen, H. Rose, Varieties of Lattices. X, 162 pages. 1992. Vol. 1534: C. Greither, Cyclic Galois Extensions of Commutative Rings. X, 145 pages. 1992. Vol. 1535: A. B. Evans, Orthomorphism Graphs of Groups. VIII, 114 pages_ 1992. Vol. 1536: M. K. Kwong, A. Zettl, Norm Inequalities for Derivatives and Differences. VII, 150 pages. 1992. Vol. 1537: P. Fitzpatrick, M. Martelli, J. Mawhin, R. Nussbanm, Topological Methods for Ordinary Differential Equations. Montecatini Terme, 1991. Editors: M. Furi, P. Zecca. VII, 218 pages. 1993. Vol. 1538: P.-A. Meyer, Quantum Probability for Probabilists. X, 287 pages. 1993. Vol. 1539: M. Coornaert, A. Papadopoulos, Symbolic Dynamics and Hyperbolic Groups. VIII, 138 pages. 1993. Vol. 1540: H. Komatsu (Ed.), Functional Analysis and Related Topics, 1991. Proceedings. XXI, 413 pages. 1993. Vol. 1541: D. A. Dawson, B. Maisonneuve, J. Spencer, Eeole d" Et6 de Probabilit6s de Saint-Flour XXI - 1991. Editor: P. L. Hennequin. VIII, 356 pages. 1993. Vol. 1542: J.FrOhlich, Th.Kerler, Quantum Groups, Quantum Categories and Quantum Field Theory. VII, 431 pages. 1993. Vol. 1543: A. L. Dontchev, T. Zolezzi, Well-Posed Optimization Problems. XII, 421 pages. 1993. Vol. 1544: M.Schtirmann, White Noise on Bialgebras. VII, 146 pages. 1993. Vol. 1545: J. Morgan, K. O'Grady, Differential Topology of Complex Surfaces. VIII, 224 pages. 1993. Vol. 1546: V. V. Kalashnikov, V. M. Zolotarev (Eds.), Stability Problems for Stochastic Models. Proceedings, 1991. VIII, 229 pages. 1993. Vol. 1547: P. Harmand, D. Warner, W. Wemer, M-ideals in Banaeh Spaces and Banaeh Algebras. VIII, 387 pages. 1993. Vol. 1548: T. Urabe, Dynkin Graphs and Quadrilateral Singularities. VI, 233 pages. 1993. Vol. 1549: G. Vainikko, MultidimensionalWeakly Singular Integral Equations. XI, 159 pages. 1993.

Vol. 1557: J. Az6ma, P. A. Meyer, M. Yor (Eds.), S6minaire de Probabilit6s XXVIL VI, 327 pages. 1993. Vol. 1558: T. J. Bridges, J. E. Furter, Singularity Theory and Equivariant Symplectic Maps. VI, 226 pages. 1993. Vol. 1559: V. G. Sprind[uk, Classical Diophantine Equations. XII, 228 pages. I993. Vol. 1560: T. Bartsch, Topological Methods for Variational Problems with Symmetries. X, 152 pages. 1993. Vol. 1561: I. S. Molchanov, Limit Theorems for Unions of Random Closed Sets. X, 157 pages. 1993. Vol. 1562: G. Harder, Eisensteinkohomologie und die Konstruktion gemischter Motive. XX, 184 pages. 1993. Vol. 1563: E. Fabes, M. Fukushima, L. Gross, C. Kenig, M. R6ckner, D. W. Stroock, Dirichlet Forms. Varenna, 1992. Editors: G. Dell'Antonio, U. Mosco. VII, 245 pages. 1993. Vol. 1564: J. Jorgenson, S. Lang, Basic Analysis of Regularized Series and Products. IX, 122 pages. 1993. Vol. 1565: L. Boutet de Monvel, C. De Concini, C. Procesi, P. Schapira, M. Vergne. D-modules, Representation Theory, and Quantum Groups. Venezia, 1992. Editors: G. Zampieri, A. D'Agnolo. VII, 217 pages. 1993. Vol. 1566: B. Edixhoven, J.-H. Evertse (Eds.), Diophantine Approximation and Abelian Varieties. XIII, 127 pages. 1993. Vol. 1567: R. L. Dobrushin, S. Kusuoka, Statistical Mechanics and Fractals. VII, 98 pages. 1993. Vol. 1568: F. Weisz, Martingale Hardy Spaces and their Application in Fourier Analysis. VIII, 217 pages. 1994. Vol. 1569: V. Totik, Weighted Approximation with Varying Weight. VI, 117 pages. 1994. Vol. 1570: R. deLanbenfels, Existence Families, Functional Calculi and Evolution Equations. XV, 234 pages. 1994. Vol. 157I: S. Yu. Pilyugin, The Space of Dynamical Systems with C~ X, 188 pages. 1994. Vol. 1572: L. G~ttsche, Hilbert Schemes of ZeroDimensional Subschemes of Smooth Varieties. IX, 196 pages. 1994.