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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

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Introduct ion

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 s tudy 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 map 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 enu-

merative algebraic geometry concerning contacts of families of subvarieties of pro-

jective 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 subvari-

eties 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 ge-

ometry. 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 s tudying 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-Insti tut 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

Introduct ion

1. F u n d a m e n t a l f a c t s

1.1. The Hilbert scheme

1.2. The Weft conjectures

1.3. The punc tua l Hilbert scheme . . . . . . . . . . . . . . . . . . . .

2. C o m p u t a t i o n o f the Betti n u m b e r s o f H i l b e r t s c h e m e s . . . . .

2.1. The local s t ruc ture of y[n] -~(n) . . . . . . . . . . . . . . . . . . . . .

2.2. A cell decomposi t ion of P[2 hI, Hilb~(R), ZT, G T . . . . . . . . . . .

2.3. Computa t ion of the Bet t i numbers of S In] for a smooth surface S . . . .

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

2.5. The Bet t i numbers of varieties of t r iangles . . . . . . . . . . . . . .

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

V

1

1

5

9

12

14

19

29

40

60

81

3.1. The varieties of second order da t a . . . . . . . . . . . . . . . . . 82

3.2. Varieties of higher order d a t a and appl ica t ions . . . . . . . . . . . 101

3.3. Semple bundles and the formula for contacts with lines . . . . . . . 128

4. The Chow r i n g o f r e l a t i v e H i l b e r t schemes

o f p r o j e c t i v e b u n d l e s . . . . . . . . . . . . . . . . . . . . . 145

4.1. n-very arapleness, embeddings of the Hilbert scheme and the

s t ruc ture of A I n ( P ( E ) ) . . . . . . . . . . . . . . . . . . . . . 146 ~ 3

4.2. Computa t ion of the Chow ring of Hilb (P2) . . . . . . . . . . . . 154

4.3. The Chow ring of Hw~-f lb3(P(E)/X) . . . . . . . . . . . . . . . . . 160

4.4. The Chow ring of H i l b 3 ( P ( E ) / X ) . . . . . . . . . . . . . . . . . 173

B i b l i o g r a p h y . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Index of notat ions . . . . . . . . . . . . . . . . . . . . . . . . 194

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

In this work we want to s tudy 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 and its existence proven. We re-

peat the definition in pa rag raph 1.1 and list some results about X[ n]. X['q is re la ted y["] X(n). to the symmetr ic power X (n) via the Hi lber t -Chow morph i sm wn :"'red ----*

We will use it to define a s trat i f icat ion of X [n]. In chapter 2 we want to compute the

Betti numbers of Hilbert schemes and varieties tha t can be const ructed from them

by counting their points over finite fields and applying the Well conjectures. There-

fore we give a review of the Well conjectures in 1.2. Then we count the points of the

symmetr ic powers X ('0 of a variety X , because we will use this result in chapter 2.

In 1.3 we s tudy the punctua l Hilbert scheme Hi lb" (k [ [Xl , . . . , x4 ] ] ) , paramet r iz ing

subschemes of length n of a smooth d-dimensional variety concentra ted in a fixed

point. In par t i cu la r we give the s t rat i f icat ion of Iar robino by the Hilber t function

of ideals.

1.1. The Hilbert scheme

Let T be a locally noether ian scheme, X a quasiproject ive scheme over T and

s a very ample invert ible 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 contravar iant functor

from the category Schln T of locally noether ian T-schemes to the category Ens of

sets, which for locally noether ian T-schemes U, V and a morph i sm 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 : ni lb(X/T)(U) ,7~ilb(X/T)(V); Z , ~ Z xu V.

Let U be a locally noether ian T-scheme, Z C X XT U a subseheme, flat over U. Let

p : Z ---* X , q : Z ~ U be the project ions and u E U. We lJut Z~ = q-a(u). The

Hilber t polynomial of Z in u is

P.(z)(m) := x(Oz.(m)) = x(o o p*bc") ) .

P,,(Z)(m) is a polynomial in m and independent of u E U, if U is connected. For

every polynomial P E Q[x] let 7"[ilbP(X/T) be the subfunctor of 7(ilb(X/T) defined

by

TlilbP(X/T)(U) = ( Z C X • U I Z is flat ~ U and } closed subscheme 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 T- scheme 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 Hi lbP(x ) instead

of HilbP(X/T). 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.

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 * U

Zn(X/T)(r : Z , (X/T)(U) ----+ Z,(X/T)(V);

( z , ~) , , ( z • v v , ~0r

(U, V locally noetherian schemes ff : V ~ U).

For the rest of section 1.1 let X be a smooth projective variety over the field

k.

De f in i t i on 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:

De f in i t i on 1.1.6. Let u = ( n l , . . . , nr) be a parti t ion of n. Let

i n l := { ( X l , . . . , X n , ) Xl ~.X2 . . . . . Xni} c X n'

be the diagonal and r r

x : := I I c I I x " ' = x " i = 1 i = 1

Then we set

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

and

:= x!")\ U

Here # > u means that # is a coarser part i t ion then u.


The geometric points of X (n) are

x ( n ) ( - k ) m ( Z n i [ x i ] E x ( 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["] X(n), CO n : ~ L r e d )

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"

Defin i t ion 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 r e d

For u = ( n l , . . . , nr) the geometric points of X In] are just the unions of sub-

schemes 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 vari-

eties, 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 Fq , 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 Fq 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 Fq , 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 , QI ) by

F~]Hr(~,Q~). The zeta-function of X over Fq 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 Xn • n C = X. For every prime ideal p of R

let Xp := Xn • 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.


T h e o r e m 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.

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

(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, s l , . . . ,sin) e Q[t, s l , . . . ,sin] be a polynomial. Let X and

S be smooth projective varieties over F q such that

IX(Fq.)l = F(q", [S(Fq,)[,..., I S ( F q - , - ) 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(z2 ,p (U, - z ) , . . . ,p(V,-zm)) .

P r o o f i Let a l , . . . , a s be pairwise distinct complex numbers and h i , . . . , hs E Q.

We put

Then we have $

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 ,h l ) , . . . (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 i l l , . . . , i t E C, l l , . . . ,lt E 2~ be the corresponding numbers

for S. Then we have for all n E ZW:

t t

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

Let ~ 5 1 , . . . , ~ r be the distinct complex numbers which appear as monomials in q and

the 7i in

(• • I m F q, l i f l i , . . . , i!i �9 " i=1 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 nj is the coefficient of z k in F(za ,p(S , - z ) , . . . ,p(S,--zm)). [3

We finish by showing how to compute the number of points of the symmetric

power X (n) for a variety X over Fq . The geometric Frobenius F := Fq acts on X(n)('Fq) by

F ( E n i [ x i ] ) = E n i [ F ( x i ) ] ,

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

under the action of F .


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 ~ ) i = 0 j[r

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).

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

R e m a r k 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

pe rmu ta t i ona 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

= I I ( 1 -- tr)-lP.(X,F,)l T_>I

= Zq(X, t).

So (3) holds. []

1.3. T h e p u n c t u a l H i l b e r t s c h e m e

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 maximal 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-negat ive integers given by

ti = d i m k ( m l / ( I A m i + mi+l ) ) .

If T = (ti)i>_o is a sequence of non-negat ive integers, of which only finitely many do

not vanish, we put IT I = ~2 ti. The initial degree do of T is the smallest i such tha t ti < (d+i-1) .

Let Ri := m i / r n i+1 and Ii := ( m I Cl [ ) / ( m i+1 (-I I). Then Ri is the space of

forms of degree i in R and Ii the space of init ial forms of I (i.e. the forms of minimal

degree among elements of I ) of degree i, and we have:

t i ( I ) = d imk(Ri / I i ) .

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

L e m m a 1.3 .2 .

(1) dim(mJ / 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 , and Zi the image of m i under the projec t ion R ~ Z. Then we have

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 i somorphism

Zj = m J / ( m j N I) ~- ~ i ~ ~v i=j -~t/

of k-vector spaces, and Ri / I i = 0 holds for i > i0. If we choose io to be minimal ,

then Ri / I i 7 s 0 holds for i < io. So we get (1). If t j = 0 for some j , then I D m j.

Thus (2) fo l lows f rom Irl = n.

10 1. Fundamental fact~

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 Hi lbn(R/m n) also parametrizes the

ideals of colength n in R. We also see that the reduced schemes (Hi lb~(R/mk) )~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 .

Def in i t i on 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)].

De f in i t i on 1.3.5. The jumping index (ei)i>o of (ti)i>_o is given by ei = max(t i -1 -

ti, 0).

T h e o r e m 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 o f t h e B e t t i n u m b e r s o f H i l b e r t s c h e m e s

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 s tudy 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 Hi lb"(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 s tudy 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 Kummer varieties

KA,,. These varieties have been defined in [Beauville (1)] as new examples of Calabi-

Yau 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 s tudy 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 algebraical ly closed) field and X a smooth quasipro-

jective variety of dimension d over k. In this section we s tudy the s t ructure of the

s t r a tum (X(n))~d which parametr izes subschemes of X which are concentra ted in

a (variable) point in X.

D e f i n i t i o n 2 .1 .1 . Let X be a smooth project ive 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 project ions and /51,/52 the restr ict ions 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 ) := (p2) , (O~o)

on X. More general ly let Z ~ V A . be the ideal sheaf of A i in A n and J / _ a ( X ) be

the vector bundle associated to

J n ' _ , ( X ) := (p2),(ZA,/A.)

for all i < n - 1.

We see tha t the fibre Jn_l(X)(x) of Jn-a (X) over a point x e X can be identi-

fied in a na tu ra l way with Ox,~/mnx,x and similar ly Jn_l(X)(x) with m x , J m x , x . ~ n

We have

Symi(T; ).

H i l b n ( A n / X ) is a locally closed subscheme of

Hi lbn(X • X / X ) = Hi lbn(X) ,

and there is a na tu ra l morphism

r : H i l b ~ ( A n / x ) ~ X.

L e m m a 2.1 .2 . Hilb'~(An/X)r~d YX In] ~ : ~ (n))r~d as subschemes of X In] and 7r :

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

in a point to this point.

2.1. The local structure of X ['q 15 (n)

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 ~ (cf. 1.3.2). X,x

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 quo-

tients of Jn -~(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

o v e r X .

P r o o f : Let

~ : Hilb"(A"/X)r~a , Grass(n, Jn-1 (X))

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 ) ~ Hi lbn(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 Hi lb~(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 : Hi lbn(An/X) , Grass(n, dn-l(X)).

So we also get a morphsim

3: Hi lb"(A"/X)red ~ 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, .)

16 2. Betti numbers of Hilbert schemes

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. Then we see

~(Si lbn(A~/X)r~d) C H.

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

152 : h " ~ H be the projection. Then we have (/32),(Os = #* ( Jn -~ (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 of /~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 addit ion 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).

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

topology with fibre Hilbn(R)~d .

P r o o f : Let U C X be an open subset and Yl , . . . Yd local parameters on U. For

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

Y : : g((f2),(~7(yl) - /5~(yl) ) , . . . , (P2),(PT(yd) -/5~(yd))) �9 r(Jn-~(U)).

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 x Hilb"(n)r~d C U x G r a s s ( n , n / m n)

under en is ~'-I(U) ~ { t u [ n ] , , : �9 n = t< (n))red)" So the restriction of en to U x Hs (R)~d is

H" n ,Ub ] an isomorphism r : U x Jb ( R ) r e d ~ < (n)]re d. []

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

( n ) ) r ed"

2.1. The local structure ~ Y[~] 17 ~J -~(~)

D e f i n i t i o n 2.1.5. For i = 1 , . . . , n - 1 let

~)i: J / _ l ( X ) ~ J~(X) ~- Symi (T*X)

be the canonical map. Let T = ( to , . . . , t ~ - l ) be a sequence of non-negative inte-

gers. 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 natura l

vector bundle morph ism

7r~(Symi(T~) | T~) ~ Tr~(Symi+a(T~:)).

Let

p T ( x ) : z v ( x ) , c v ( x )

be the morphism defined by the bundles Qi]ZT(X).

Analogously to the proof of l emma 2.1.4 we can easily see:

R e m a r k 2 . 1 . 6 .

18 2. Betti numbers of Hilbert ~cherne~

(1) ZT(X) and GT(X) are locally tr ivial fibre bundles over X with fibres ZT and

GT respectively.

(2) W i t h respect to local t r ivial isat ions

ZT(U) ~ U X ZT,

GT(U) ~- U X GT

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

R e m a r k 2 . 1 . 7 .

all s �9 JTV

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

))( ' . G(1, I , ( t+ I ) ..... (t-I-;-, X) = Grass(l, Tfc ).

with fibre A r over Z(1,/,(,+l ~ , . - . ........('+;-1]](X) is a locally tr ivial fibre bundle

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

P r o o f : By remark 2.1.6 we have to prove this only for

Hi lb( '+ ' ) (R) . a(,,, ,( ,+,) ..... ( ,+:_,)),Z(, , , ,( ,1,) ..... (,+;_,)) c

The assert ion 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. The init ial forms ui of the yi all have degree 1 and

are l inearly independent . We can assume tha t x 1, �9 �9 �9 xl, ul+ 1, � 9 u d are l inearly

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

Yi : Ui + f i (Xl , . . . ,X l ) .

The f i ( x l , . . . , xt) can be a rb i t r a ry polynomials in x l , . . . , xt of degrees < s, whose

in i t i a l forms hasve degree > 2. Thus the result follows. []

R e m a r k 2 .1 .8 . Of par t i cu la r impor tance is the s t r a t um Z(1 ..... 1)(X) C X ['q It (~)-

is an open subvariety of X ['q It is however in general not dense in X ["] if d > 3 (n)- (n) - a n d if n is large. By the definitions it parametr izes subschemes of X which are

concent ra ted in a point z and lie on (the germ of) a smooth curve through z. We

Y['q is a will therefore also wri te X I'q ins tead of Z(1 ..... 1)(X). By remark 2.1.7 ~'(n),r (n),c locally tr ivial A(d -O(n-2 ) -bund le over P(Tx) .

19

2.2. A cell d e c o m p o s t i o n of p~n], Hilbn(R), Z T ' 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) = rk (Hi (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 ) = Ad-m(X) .

Defini t ion 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 . . . DXo 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 clo-

sures 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 Bett i numbers of Hilbert schemes

fixed points is the finite se t {Xl, . . . , X m ) . For all i = 1 , . . . , m let

X i : = { x e X 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 part i t ions of

n and by p(n, l) the number of part i t ions of n into l parts. This number coincides

with the number of part i t ions of n - 1 into numbers smaller or equal to I.

The main result of [Ellingsrud-Str0mme (1)] is:

T h e o r e m 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, H2i+I (X) = O, and the H2i (X) are free abelean groups.

I f k = C the Betti numbers are

(2) 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 [Ell ingsrud-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 writ ten 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 many fixed points.

Let X be a smooth projective variety over k with an action of a torus H which

has only finitely many 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 one-

paramete r 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) =

d i a g ( t W ~ 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 decomposit ion of P2 into F0, F1, F2. Ellingsrud and S t rcmme apply theorem 2.2.3

to the induced Gm-act ion on P~] . We will modify their arguments in order to

obtain a cell decomposit ion of the s t ra ta Z T of Hilbn(R).

We denote by "." the action of Gm on p~nl induced by q~. As it has only

finitely many fixed points, it gives a cell decomposit ion of p~n]. Hi lbn(R)~d =

Hilbn(A 2, 0) C P~] is the subvariety parametr iz ing 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

z)) = upp(Z)) r (P0}.

So

Z e Hilbn(A 2, 0) r l im(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 decomposit ion of P~'~]

which belong to fixed points in Hilbn(A2,0). In part icular Hilb'~(A2,0) has a cell

decomposition.

Using the identification := T I lTo ,

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~

22 2. The Betti numbers of Hilbert schemes

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,. �9 �9 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 de-

note 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

T = aj -1

E E O<_i<j<_r s=ai +t

( )~ i--j--l #al--s--1 _]_ )~j--i #s--al ).

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 ( a j - a j + l ) = 2 E a i = 2 n = d i m ( T ) O<i<j<r O < i < r

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 = [x]r 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'] i f l _< i, r : I ----+ R / I ; x ly ~ ' ) 0 otherwise.

We can see immediately that the compatibility conditions are fulfilled and r is a common eigenvector of F to the eigenvalue A J - i# ~-a~.

L e t 0 < _ ~ < ~ _ r + l a n d a g - a ~ _ l + a T _ < ~ < a 5. We put

r ~ ~ : I - - ~ R / I ; x ' y ~' , , {~x3+'-~Y a + ~ - ~ ] i f />_~, ' ' otherwise.

r is an eigenvector to the eigenvalue ~)-~#~-a~. The eigenvectors constructed

this way are obviously linearly independent. The result follows by the substi tution

s : = ~ - a ~ +a~

j : = ~ - I

i:----~. []

We now formulate our result on the cell decompositions of ZT and GT 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 parti t ion 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 part i t ion & = (~ l , . . - ,~a0) is the partit ion, whose

graph is F(a) with the roles of rows and columns switched. The diagonal sequence

is T ( a ) = ( t o ( a ) , . . . , t l (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


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 part i t ion c~ = (6, 3, 2) we get for instance the diagram

for F(c~) and

The hu,v(c~) are given by

& = (3,3,2, 1, 1, 1),

T(c~) = (1,2,3,3, 1, 1).

- 1 0 --1 0 0 --3 --2 - 2 - 2 - 1 O.

T h e o r e m 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 ) ~ H . ( X ) is an

isomorphism and H . ( X ) is free.

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

(2) b 2 , ( Z r ) = a � 9 i { (u , v ) e V ( a ) l h u , v ( a ) e { o , 1 } } l = n _ i

(3) b2~(Gr )= { a E P ( n ) T ( a ) = T ; 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 non-

empty 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], Hi lb~(R) , ZT; GT 25

then Z T is an A n - 2 - b u n d l e over GT = P1. It is easy to see t h a t to(a) = 1 and

t i(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 decompos i t i on of ZT of t h e o r e m 2.2.7 consists of one cell of d imens ion n - 2

and one of d imens ion n - 1 and tha t of GT of one cell of d imens ion 0 and one cell

of d imens ion 1 as expec ted .

As above let �9 w o W l w 2 (P:Gm-----4G; t ~ d m g ( t ,t , t )

be a genera l o n e - p a r a m e t e r subgroup of G wi th w0 < wl < w~ and w0 + W l + w 2 = 0.

We also requi re the inequa l i ty

n(w l - wo) > (n - l)(w2 - wo).

We consider the induced G m - a c t i o n on Hi lb~(R) . We know a l ready t h a t it gives a

cell decompos i t i on of Hi lbn(R) . Let T = (t i) be a sequence of non -nega t ive in tegers

wi th ITI = n.

L e m m a 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.

P r o o f : Let I be an ideal in R wi th Hi lber t func t ion T. Let j E f g , s := j + 1 - tj. Let Ij be the space of ini t ia l forms of degree j in I . We pu t

J := l i m t . I . t ~ O

For all i let Ji be the space of in i t ia l forms of degree i in J . Let T I = (t~)j_>0 be the

Hi lber t func t ion of J . Choose f l , �9 �9 �9 f~ E I such tha t the i r ini t ia l forms g l , . �9 �9 g~

are a basis of Ij. By rep lac ing the fi by su i tab le l inear combina t ions we can as sume

tha t the gi are of the fo rm

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

rn>l(i)

wi th gi,m E k and t h a t l(1) > /(2) > . . . > l(s). By the choice of the weights

WO~ Wl ~ W 2 w e get

l im ~ ( t ) - (tt(i)(w~176 f i) = xl(i)y j-l(i). t~O


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 we 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 partit ion 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:

t j - - j + l - {(i , /)E2g~_ 0 l i + l = j , l>_ai}

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

= t j ( . ) . []

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

I .

L e m m a 2.2.11. The dimension of the subspace T + of T on which the weights of

the action are positive is

d im(T + ) = n - {(u,v) e V ( ~ ) ] h u , v ( ~ ) = 0 o r h ~ , 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 ) ) = t o ( , ~ , - ~ , o ) + b ( w , - w o ) .

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 ( { aJ+z <--s<aJ ' } ) dirnT + = aj - aj+ 1 - s E 2~ 0 < j + s - i - ai + 1 < 1 O<_i<_j<_r -- - -

= �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 homo-

morphisms in Hornn( 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

d i m ( T + ) = E I { s E 2 ~ [ a j + l < - s < a j ' i - J + a i - s - 2 = O } l O<_i<j<_r

= E { s E 2 g O < _ s < a i , i + a i - - s - - & , = l } o<_i<r


and

dim(T~ Z I{sc2ZlaJ+, <-s<aj, i + a i - - s - - j = O } l O<i<_j<r

O<:i<r

By putt ing 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 GT and ZT, as

they are both smooth. From lemma 2.2.12 we have:

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~ w i t h j + s - a i - i = 0 . So we get:

dim( ZT ) = dim(T1)

=dim(T+)+ Z { sE2~ a i + l < - s < a J ' } O<i<_j~r J ~- S -- ai -- i -F 1 ~ 1

=d/re(w+)+

Using theorem 1.3.8 we get for each partit ion 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 Bet t i numbers of 5:[~] for a s m o o t h surface 5:

We want to use the Weil conjectures to compute the Bett i numbers of S ["]

for a smooth project ive surface 5: over C. Let X be a smooth project ive variety of

dimension d over a field k. Let R = k [ [ x l , . . . , xd]]. We denote Vn := Hi lb~(R)~a .

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

will wri te 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

(z2,

we write (xl,Z1) C (xz,Z2), if :cl ( n l , . . . , n~) we also write

c (x •

(x •

= z2 and Z 1 C Z2. For a par t i t ion u =

I~,2~,...),

where ai is the number of summands i in u. Let

[a[ := E ai. i

Let P(n) be the set of par t i t ions of n.

U ~ We will assume for the following that there exist a finite open cover ( i)i=l of

X and local parameters on each of the Ui, defined over k.

R e m a r k 2 .3 .1 . There is a sequence of bijections r : X l : l ( k ) ~ (X • V,,)(k),

commut ing with the action of the Galois group Gal(k, k) such tha t

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

r /U ~[n] Proof: For i = 1 , . . . , s let 7ri : kt i)(n))~ea ----4 Ui be the restr ic t ion of the projec-

tX["] ~ X from lemma 2.1.2. By lemma 2.1.4 there are isomorphisms tion 7r : t (n ) ) red -----+

t /U d n] ~ t : ~,~ i)(n))red ----4 Ui x Vn

over Ui for all n E 2V. Thus we have the required bijeetions r := aS~(k). For all

j = l , . . . , s l e t

w :=vj\Uv i<j

--1 X ['q (k~ there is a unique index i(Z) such tha t W C 7ri(z)(W,(z)). We For e a c h Z E (n) J

put C n ( Z ) : = r The result follows, as all the r are bijective. []


Def in i t i on 2.3.2. For any parti t ion v = ( h i , . . . , n~) = (1 ~ , 2~%.. . ) of n let

. x I o , l . ~ H ( x l : l ) o , 2 2 : = ( ~ , , , • • ..... , ) ) c u,,) • x u , ) = i

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

a(o~) := a ( , ~ ) x . . . • C(o,~, )

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

L e m m a 2.3.3. There is a natural morphism r : 2 n ~ 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 22(T). We put

r z~)) := Zl u . . . u 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 Z1, . . . ,Z~ are the

connected components of Z and [] the class modulo G(n). []

Def in i t i on 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 f l C f2, if f l (x) C f2(x) for al l 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 partit ion of n, and let Z �9 X[n](k) with

~ J ~ - l ( z ) = [ Z l , . . . , Z r ] ,

where len(Zi) = hi. We put

O,(Z) := f : X(k) ---4 V(k);

f p2(O,,(Zi)) if x = (Zi)red, X o if x ~ supp(Z),

where P2 : X x V~

and lemma 2.3.3.

Vn is the projection. The result follows from remark 2.3.1

De f in i t i on 2.3.6. Now let k be a finite field Fq , 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

and

m (g) :=

~CP(X,Fq)

f "1 Tn(X, Fq) := / g : P(X,~'q) ---+ V(~q) g admissible with lea(g)= n~.

For gl e Tnt(X,~gq), g2 e Tn2(X,J~q) w e write gl C g2, if g l ( ( ) C g2(~) for all

�9 P(X, •q).


L e m m a 2.3.7. There is a sequence of bijections rn : X [~] ~ Tn(X, Fq) such that

for all subschemea Z1, Z2 of X of finite length

rten(zo(Z1) C "rlen(z2)(Z2) r 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 ~ ~ le~(g(~)) ~,

~[n] X (n) is the IIilbert-Chow morphism. and Wn : ~'red

Proof : Let

(

Let

f ' ' Z len(f(x))[x].

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

such that the diagram

g-n(fl) C +n(f2) r f l C f2,

x(~)(Fqfl

commutes. We choose a linear ordering < on the set X(•q). Let f e ;V~(X, Fq). Then Cn(f) is in a unique way a linear combination

8

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

2.3. The Betti numbers of SD] 33

of distinct primitive zero cycles ~i E Pr i (X ,~q ) with non-negative integer coeffi-

cients 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 (x i ) ) = f ( F ~' (xi)) = f(xi) ,

so f ( z i ) E V(Fq. , ). We put

r ~ ( f ) : P(X,~Cq) -----+ V(I~'q);

f (x i ) ~ = ~i for a suitable i, I [ o otherwise.

The inverse r~ -1 is given as follows: let g C T~(X, Fq). For r E fg 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(Fq) ;

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

L e m m a 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

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

Irn(X, Fq)l = ~ [ I N(~,~,) �9 n l + 2 n ~ + 3 n a + . . . = n s = l

On the other hand we have

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

Z len(f(~)) = j } .

Now let S be a smooth projective surface over s Let k := •q, R := k[[x, y]]

and V~ := Hilbn(R).


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

( ~-~ tm L~(~qM~)I~ t 1. ISt'q(rr - exp m 1 - qMmtm] modulo n = O r n = l

Proof." Let l E $V. There is an m0 E $V such that for all n G l the cell decomposi-

tion 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 )z i t n 1 - z i - ' t i - 2 - ,

i = 1 n = 0 i = 0

theorem 2.2.4 implies

1 E IVn(1FQ~)[ff~ = "1 - Q~(i 1 ) t r i n = 0 i = 1

By lemma 2.3.8 we have:

E Is[n](FQ)[tn = 1 - Or(i-1)tri n = O r = l i = 1

modulo t z.

modulo t l

\ i=1 r = l h = l

= e x p \ i = 1 m = 1 r~lmr]Pr(S'-~Q)[)Qm('-l)'--m)

exp ~ 1 - - 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) = ~ d i m ( H i ( S M ; 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 (SN, 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 ~ n~O

= exp m = l /32 ~ - - z2mtm/

f i (1 + z2m-]tm)bl(S)(1 + z2m+ltm) bS(S) (3) EP(S[~] , z ) t " = (1 - - z2rn~-27m~=z---2~mtm~b~=Z'~-~-m~2tm) b4(S)

n = O r n = 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

e x p m 1 - - qhmtm m--~l

Now (2) follows by remark 1.2.2. (3) follows from (2) by an easy computa t ion 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)" [3 n = O i = 0

C o r o l l a r y 2 .3 .11. For the Euler numbera we have

(1) ~ ~(sC~l)~. = f I (1 - ~)-~(~) n = O k = l

(2) In particular, 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(,)] r t ~ 0


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 -- Z 2 m + 2 )

Proof." Let

oo ((1 + z2m- l tm)(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 of 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) := ~ hp,~(X)x%~ P,q

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.

(1) h(S In], z, y) =

or equivalently

(2)

oo

Z (xy) ~-I~l I I h(s(~'), x, y) ( l ~ l , 2 ~ 2 , . . . ) E p ( n ) i=1

h ( S M , - x , - y ) < = exp -- ~ = ( - ; ~ y ~ , n=0 m = l ~

(3) n~O

oo 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 p,q

From this we get:

(4) x_~(sM)t ~ = exp x-~ (s)

(5) sign(SE~ = F_, (-1)"-I~l [I ~ig~(S("')) (1"1 ,2~ : , . . . ) 6 P ( n ) i=1


or equivalently

fI ( 6 ) E sigTt(s[n])tn = ( l -- tkx~ (--1)ksign(S)/2 n=0 k=l \ 1 + t k J (1 -- t2k) - ' ( s ) / : .

(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

~=-~-a z q~ n = l d I 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 sign( s N ) ( - q ) . = q A(T)2/3 A(2T)I /3

n=o q z

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 ) ( 1 - q~k)-e(s)/~ n = 0 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 ) = 4 0 9 6 ~ ( 6 2 - ~)~

~ (~ )16 _ 64(65 _ ' ) ,7(2,-)8

(cf. [I-Iirzebruch-Berger-Jung (1)]) we get

r/(w) "ig'~(s) = (64(62 _ s162 ~( 2T )( , ig.( s)+~( s) ) /~

40

2.4. T h e B e t t i n u m b e r s of h i g h e r o r d e r K u m m e r v a r i e t i e s

D e f i n i t i o n 2.4.1. 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 ' S n - 1 = a ) n l ( a n l ( g n l ( 0 ) ) ) .

In the following two cases we want to compute the Betti numbers 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 Kummer surface of A. So we can see the K A n - 1 as higher order Kummer

varieties of A. This is the more impor tant case.

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

L e m m a 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

have the cartesian diagram

A • K S , - I

l A

This is true because the fibre product is

A be the mult ipl icat ion by n. Then we

, A [n]

[] 1 (n)

~ n .

{(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 KAn- 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 A x K ( A • , ( A x P 1 ) [ ' q

[] l g n ~

(n) A ~ 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 P1) [~] K(A x Kn_ 1 ~--(Ui x CI P1)~-1. []

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)].

De f in i t i on 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 Fq . Then (G, F ) has the Lang property by the theorem of Lang.

T h e o r e m 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

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


(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.

Def in i t i on 2.4.5. Let S be a smooth projective surface over hTq and

v = ( n l , . . . , n ~ ) = (I~,2~,3~3...)

a parti t ion 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

and define

by 7~ls[ . l : = 7 . .

i=1

-y. : s [ . ] , s ( " ) ( F q ) ;

((~i),,v), , ~ ] i . ~i i

U s[~] , s(n)(Fq) vEP(n)

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

: ~ Z qdi(ni-'#i') i=1 #iEp(ni)

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

= ( m l , . . . , m l t ~ q )


be a par t i t ion of hi, and let

V = ( n l , . . . nl~l)

be the union of d /copies 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, . . . ) . Then 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

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

= []

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

let ei ther S = A be an abel ian surface over F q or let S - - ~ A be a geometr ical ly

ruled surface over an elliptic curve A over ~'q. In this case we assume tha t there

exist an open cover (Ui)i of A and isomorphisms a-l(Ui) ~- Ui • P1 over F q . In

both cases we assume that , for all l _< n, all t he / -d iv i s i on points of A are defined

over F q . All these condit ions can be obta ined by extending F q if necessary. Let F

be the geometric Frobenius over F q . We put

try: A(Fq,) ~ A ( F q ) ;

I

x Z F/(x) i = 0

for all l E SV.

L e m m a 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(Fq) F = (A(•q)), and A(~q) F' = (A(Fq,)) . Let x E A(Fq,) .

Choose y C A(Fq) satisfying x : F(y) - y (this is possible by the Lang proper ty) .

Then we have NF~ /F : y -- Fl(y)

l--1

= E Fi(y - F(y)) i = 0

= - - t r l ( x ) .


As F t acts a s the identity on A(Fq) , 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 map

z ~ y + z gives a bijection between tr[ -1 (0) and tr~ l(x) []

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

L e m m a 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 part i t ion t, = (n~ , . . . ,n~) = ( 1 ~ , 2 ~ , . . . ) of n let M(v) be the

conjugacy class of the symmetr ic group G(n) whose elements consist of disjoint

cycles of lengths nl . . . . , n~. Then we have

n~ l i a r / ~1

H ic~' c~i! i=1

and ~-~veP(,~) IM(t,)l = n!, as G(n) is the union of the M(v). For a smooth variety

X over F q we put

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 x~)

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

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

C l a i m (*) . Ir = n! for all ~ C X(n) (Fq) .


Proof of (*): Let ~ = ~ i~1 mini E X (n ) (Fq ) , where the {i are distinct primi-

tive zero-cycles of lengths di. The points of ~xl0(~) can be obtained as follows: for

any i let

# i i i i i = ( l l , ' ' ' , / ] . i l ) = ( " l ~ t ' 2 c ~ 2 ' ' ' ' )

be a partit ion of mi, and put # : = ( ~ 1 , . . - , # r ) . Let

/2(~) = ( n l , . , 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 r kZ~ = H H (jdi)~'}" k=l i=1 j

46 2. The Betti numbers of Hilbert ~chemes

Thus we get

(lr �9 IM(,- ' ( f f ) ) l ) pEP(ml)x...xP(mr)

c~ k = l ioo----1

#EP(ml)x...xP(rn~) ,~k, k~, I I H oz; , i = l j = l

d ~1

-- n!. ~ ~ 1 .

X i = l j----1 i=1 j = l /

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

A(n)

d~ S , S ( " ) (Fq)

an(Fq)

~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

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

Thus the lemma follows. []

2.4. The Betti number8 of higher order Kummer varietie~ 47

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

A(Fq) .

L e m m a 2.4.9. Let v = ( n l , . . . , n~) be a partition of a number m �9 SV.

a . : A ( F q ) ~ - - * A ( F q ) g c d ( . ) ;

( X l , . - . , X r ) ~ ~-~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

and the map

~.((m,~, , m ~ ) ) = x,

f , : ~-1(0) ~ ~- l (x ) ;

( Y l , . . . , Y r ) ~ (Yl -~- mly , . . . , y r -~ - t o r Y )

is a bijection. []

Observe that

bl(S) = 2dim(A) = 4 2 [

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

L e m m a 2 .4 .10.

(1)

[ K S " - I ( F q ) I - IA(Fq) I Z gcd(v)bl(S)qn-]"] IS(~')(Fq)l v----(lal,2a2,...)EP(n) i=l

1 gcd(v)b~(S)z2(,_l,l) (2) -LA(Fq)I

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 lemma 2.4.6 we have

IKS~_~(Fq) 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 part i t ion of n and let

# = ( m l , . . . , m t ) : = ( l a ~ , 2 ~ , . . . )

be defined by ~i = m i n ( 1 , a i ) for all i. Let

f , : S[u] ---* A(Fq) t ;

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

Then the d iagram

r 1~ ~ A(1Fq) t ~" , A ( F q )

commutes. By l emma 2.4.8 and l emma 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 mul-

tiplication 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 emma 2.4.6, and (2) follows from this by remark 1.2.4(3) and an

easy calculation.

T h e o r e m 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

1 p ( K S , _ I , z) - (1 + z) 2 Z (g~(')) ~z~r l ]v(s (~) , z) -

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


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

P ( K S , - 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

zn l tmlw 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

1 -'}- w k - -1 -'}- (1 - - z 2 k - 2 t k ) ( 1 - - z 2 k t k ) b = ( S ) ( 1 - - z2~+2tk)]]l k = l w = l

An equivalent formula is o o

T~,=O

_ ( 1 (wA)b' s) (1- z)b,(S) \ dw )


P r o o f i It is easy to see tha t the two formulas are equivalent. So we only have to

show the following identi ty:

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

C) ( ( (l + z2k--ltk)bl(S)(1-b z2k+ltk)b3(S) = 1 + w k - 1 + (1 - z 2 k - 2 t k ) ( 1 -- z2k t k )b2 (S ) (1 -- z2k+2t k)

k = l

This however follows immedia te ly from the formula of Macdonald. []

))

We can now compute the Bett i numbers of the KSn-1 for small n. We get the

following tables:

Betti numbers b~(KA~) for

n

v

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

2 3

1 1 0 0 7 7 8 8

108 51 8 56 7 458 0 56 1 51

8 7 0 1

1

0 7 8

36 64

168 288

1046 288 168

64 36

8 7 0 1

higher order Kummer varieties:

36 64

191 344 915 312 748 312 915 344 191

64 36

7 8

36 64

176 352 786

1528 2879 4496 7870 4496 2879 1528

786 8 352 7 176 0 64 1 36

6 7

1 1 1 0 0 0 7 7 7 8 8 8

36 36 64 64

176 176 352 352 809 794

1584 1592 3327 3278 6136 6360

11298 12202 16432 21704 25524 36440 16432 51640 11298 67049

6136 51640 3327 36440 1584 21704

809 12202

1 0 7 8

36 64

176 352 794

1592 3301 6416

12571 23456 43043 74040

118672 162808 198270 162808 118672

9 10

1 0 7 8

36 64

176 352 794

1592 3286 6424

12522 23680 44142 79920

140073 232368 354034 471712 538070


Betti numbers bv(KSn) for c u r v e :

6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21

n 1 I]

0 1 1 0 2 6 3 2 4 6 5 0

1

2 3 4

1 1 1 0 0 0 3 3 3 4 4 4 6 13 10 8 14 16 6 45 4 32 48 3 45 90 0 14 72 1 13 90

4 48 3 30 0 16 1

S a geometrically ruled surface over an elliptic

1 0 3 4

10 16

30 35 54

198 142 247 232 247 142

10 108 4 54 3 35 0 16 1 10

4 3 0

0 3 4

10 16 32 56 97

156 243 348 486 472 486 348 243 156

97 56 32 16

4 10 16 32 56

102 162 278 434 568 892

1206 1232 1206

892 568 434 278 162

1 0 3 4

10 16 32 56 99

164 275 448 711

1C56 1541 2C48 2557 2640 2557 2C48 1541 1C56

1 0 3 4

10 16 32 56 99

164 280 454 738

1146 1763 2590 3643 4704 5737 5984 5737 4704

10

1 0 3 4

10 16 32 56 99

164 277 456 735

1160 1811 2764 4089 5824 7903

10O28 11788 12288

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

r E H : = { a + b i E C b > O }

let q := e 2'~i'. Then 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 put O(r) := q - ' / 2 4 ~ ( r ) .

C o r o l l a r y 2 .4 .13 .

(1) For an abelian surface A over C we have

e ( K A n - l ) ---- n3•l(n).


(2) F o r 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 3 - bl(S)/2 ~ q d ~ bl(S)-I d n=i -- 41Ti k dqJ log(iT(T))

= 1 ( 3 - b1(S)/2) \ dq] E2.

P r o o f i As p ( S , - z i) is divisible by (1 - z) b~(s), we see tha 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 theorem 2.4.11(3) is divisible by

(1 - z)b'(s)((~,J ~j)-l)

Thus it does not contr ibute to the Euler number , except if u is (n~ ~ ) and #n~ is

(~ - ) for some divisor nl of n. So we get from theorem 2.4.11(3):

nbl(S) nl p(S, - z n/n1) ~(~<s._ , ) : Z - n, ln 1 n (1-z)b~(s) I~=1

(~,,(s) ---- E -bl(S) n l ( 3 - - b l (S) /2) n it 1

nl[n n \ nl /

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

Table of the Euler numbers (A abel ian surface, S geometr ical ly ruled surface

over an ell iptic curve):

v n 1 2 3 4 5 27,: 7 9 7 4 : 0 10 e(KAn) 24 108 448 750 2592 7680 18000 15972 e(KSn) 12 24 56 60 144 112 240 234 360 264

We can again see easily tha t the Bett i numbers bi(KSn-1) become stable for

i<_n.


C o r o l l a r y 2 . 4 . 1 4 .

(1 "~- Z 2m'+1 )2bl(S) p(KS._I,z) - ( i -2 ) II f::z-~-z

m=l modulo z n.

Proof:

have For any par t i t ion u of n satisfying [u[ > n/2 we see tha t gcd(u) = 1. So we

p(S [nl, z) p(KSn-1, z) = (1 + z) bds) modulo z ".

Thus we have by corollary 2.3.13

i fi (i + z~m-1)bl(s)(1 + z2m+l) bi(S)

p(KSn-a , z ) - (1 +z) bl(s) ( 1 - - z - - ~ m ~ ( - i - z 2m+2) m~--I

f i (1 + Z2m+l) 2b1(S) = ~=~ (i - ~ w ~ - T~+~)

oo = (I - z ~) I I (I + z~m+') ~b,(s)

m=l (1 -- z2m) b2(s)+2

modulo z n

The result follows. []

In par t icu la r we have ba(KSn-1) = 0 for all n E ZW. In fact the KAn-1 were

proven to be simply connected in [Beauville (1)].


T h e o rb i fo ld E u l e r n u m b e r f o r m u l a

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 ),

' ' g E G

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-Vafa-

Wit ten (1),(2)] the orbifold Euler number is defined by

1 ~(x, a) = ~ ~ ~(x. n x h) g h = h g

(the sum is over all commuting pairs of elements in G). Now let X be an algebraic

variety. We assume that the canonical divisor Kx/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 Kx/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 Ks, 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 Kummer 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 Kummer varieties

K A n - 1 . Let A be an abelian surface. Let

A ' ~ : = { ( X l , . . . , x ~ ) E A n E x i = O } c A ~

with the reduced induced structure. Then A~ is isomorphic to A '~-1. The G(n)

action by permutat ion 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

KKA._ , = O. So the conjecture says that e ( K A n - 1 ) = e(A n-a, G(n)) should hold.

For a permutat ion a of {1 , . . . ,n} let

be the parti t ion 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 . } = .. x~,, . . . . xv, for all cycles ( , 1 , . . . , ui) of a

or

= ~ 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

For

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

h = (hi , h 2 , . . . ) e 1-[ i

the fixed point set ((An)~') h consists of the ( z l , . . . , xn) �9 A n satisfying zl = x j 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 Bett i numbers of Hilbert schemes

if and only if p(~) = ((~/~)~),

and ha is a cycle of length a in G(a ) for a posit ive integer a dividing n.

R e m a r k 2 .4 .15 . Let ~r E G(n). Then we have

{n 4 p(~) = (n); e((A~)~) = 0 otherwise.

P r o o f : Let B be an abel ian variety and h : B ~ B an au tomorph i sm of B. Then

every connected component of B h is ei ther an isolated point or a t rans la t ion of an

abel ian subvariety of posit ive dimension of B. In par t icu la r e(B h) is the number of

isolated points in B h. For a cycle a of length n we have

= x E A n x = ,

and this has Euler number n 4. Let a E G(n) with

p(a) = ( n l , . . . , n r ) , r > 2 .

Then we get

(A~)a~ {(Xl,...,Xr) Znix i =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(A n-l, C(n)) = n 3 o ' l ( r ~ ) = e(I(An-1).

P r o o f i e(A~, C(n)) = Z e ( ( A ~ ) ~ / C ( a ) )

M

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

n 4

Mn

= n30" l ( n ) [2]


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 ( K S n - l , x , y ) 1

((1 + x)(1 + y))b,(S)/2

�9 1] h(S(~ i

or equivalently

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

(1 - x)~,(s)/:(1 - u)b,(s)/z

�9 0( , /~' =(I p ,

~_, (gcd(~'))b'(S)(xy) "-H v=(l~t ,2~2 , . . . )EP(n)

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

,2~2,...)ep(c~i) J J ~/%. ]

In the case of the KA,_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 ( K A , - 1 ) = n E nla(1 + Y"" + yn/nl--1)2yn--n/n,

(2) s i g n ( g A , _ l ) = ( - 1 ) ~ - l n E d3" din , n / d odd

We can again express the signatures of the KAn-1 in terms of modular forms (no-

tations as in 2.3.15).

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

58 2. The Betti number~ o] Hilbert scheme~

(1) follows by easy computa t ion and (2) by pu t t ing y = - 1 . (3) is obvious from the

definition of e. []

By applying the same argument to the case of a geometr ical ly ruled surface

over an elliptic curve we get that sign(KSn-1) = O. This was however clear from

the beginning as the dimension of KS~-I is not divisible by 4. It seems remarkable

tha t in all cases the signatures and the Euler numbers can be expressed in terms

of the coefficients of the q-development of modula 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 number of posit ive eigenvalues of the intersection form on the

middle cohomology and b_ the number of negative ones. Then we get the following

table:

1 2 3 4 5 6 7 8 9 10

b2n(KAn) s ign(KA,)

22 108 458

1046 3748 7870

25524 67O49

198270 538070

-16 84

-256 630

-1320 2408

-4096 6813

-10080 146521

b+(KAn)

3 96

101 838

1214 5139

10714 36931 94095

276361

b-(KAn)

19 12

357 208

2534 2731

14810 30118

104175 251709

We can also determine the Chern numbers of KA2:

C 4 = O, C~C 2 : O, ClC3 : O,

c4 = 108,

c~ = 756,

This is t rue because cl = --KtcA._I = 0 and

sign(KA,_l) = l (7p2(KA2) - p2(KA2)) = 84,


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

60

2.5. The Bet t i numbers of varieties of tr iangles

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.

Def in i t ion 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[~] Zn-1 C Z . }.

~ 3 ~ 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) parametr izes 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;

, xt l(T) • XE I(T) c

, ( ( z ~ , z ~ ) ~ ( ( i x • r • r

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 part i t ion • 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 ) is defined.

Def in i t ion 2.5.2. [Elencwajg-Le Barz (5)] Let

(Zn-1, Zn) ~ X [~-1] • X [~].

2.5. The Betti numbers of varieties of triangles 61

Let In-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 annihi la tor Ann(I,,-1, Oz.) of It ,-1 in Oz..

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

a morph i sm res : Hilb (X) ~ X, if the ground field is C. We show this for an

a rb i t r a ry field.

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 noether ian 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 annihi-

la tor 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

morphism of functors.

R e m a r k 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 t r iangles with a marked

side, then res maps such a t r iangle to the vertex opposi te to the marked side.

Via ~ 3

i l := 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 t r iangles with a side and the

opposing vertex marked. Let

~ 3 t51 : H i l b (X) , X,

~ 3 t52 : H i l b (X) ~ X [21,

~ 3 Pa : Hilb (X) , X [a],

~ 3 Pl,2 : I-Iilb (X) ~ X x X [21,

~ 3 /~1,3 : Hilb (X) ~ X x X [a]

62 2. The Betti number8 of Hilbert schemes

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) , Z3(X).

This morphism is birational, 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 : ni lb (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 P l , 2 : Hilb (X) , X • X [2]

is birational, as it gives an 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 sub-

scheme. 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

X x X [2] w 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~ 63

T h e var ie t ies of complete triangles on X.

Semple [Semple (1)] has constructed a variety of complete triangles on P2-

This 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)].

Def ini t ion 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

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

Xi,Xj C Zl; Zi C Z; 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:

x . xj c z~; z~ c z ; (x1,~2,z3,zl,z2,z3,z) ~t = r ~ ( x , , Z j ) = r ~ ( Z t , Z )

7~3(X)(T) = E (Z 3 x (X[2]) 3 X X[aI)(T) 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~ : f i - 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


We can view the xi as the vertices of the triangle Z and Zi as the side opposite to xi.

Thus s parametr izes 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 : Z j} 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 da ta on X , which we want to s tudy 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 natura l 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 mx,~ / 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 one-

dimensional linear subspaces of the fibre of the tautological subbundle T1 of 7r*(T~)

over the point V. Thus we get

R e m a r k 2.5.7. E ~ P(T1) • . . . . (2,T;r P(T1) XCrass(2,T;r P(T1).

P r o p o s i t i o n 2.5.8. Let X be a smooth projective variety over C. Then

Pl,2 : H~b3(X) ----+ X x Z [2] is the blow up along Z2(X).

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:

: {(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}. F

Let Pl,4,7 : / ~3 (X) , X x X [21 x X [31

( X l , X 2 , x 3 , Z l , Z 2 , Z 3 , Z ) , "' ( 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~,

x3 := res(xl,Z1). If supp(Z) consists of two points, we see that Xl # x3 and

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


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 (x l ,Z l ,g) = {(.TI,X2,x3,ZI,ZI,Z3,Z ) 23 C Z} "~ P l .

Putt ing 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 •

--1 OH_~b3(x ) X [2]. From p~(Z2(X) ) = F we get that plaZ. 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

N 3 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 proposi-

tion 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).


Proposi t ion 2.5.8 also follows from [Kleiman (3)] t hm 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 mapping surjectively to S[ n+l] (proposit ion 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 ) ~ .~[3] be the restriction of this morphism t o / I 3 ( X ) C X s x (X[2]) 3 x X [3].

The symmetr ic group G(3) acts on X 3 • (X [2])3 • x[3l by permut ing the factors

in X 3 and (X[2]) 3 simultaniously. 7rl : H3(X) ~ 2 [3] is the quotient morph ism

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] ) We can again give a ( 1 , 1 , 1 ) �9

stratification of )~[3] by put t ing

L31 := p-l(x 31)


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<l,2)(X); / / supp(Z) = x; Z1, Z2, Z3 C 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)).


(1) 2{ai is smooth.

(2) p : 213J ~ x{31 i~ the bto~ up aZong Z(~,2)(X).

Proof 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),

r[21 Z2 E "'(2),

Then we have

Now let

x E X, y := supp(Z2) 5~ x.

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

( .g[a} \ Z ~ \==(3) ~-Z(1,2)(x)) = Z(1,1,1)(X)

and x := supp(Z). Then the ideal of Z in Ox, , is given by

I z = ( * ~ , x 2 , . . . , x d )

for suitable local parameters Xl, x2 , . . . , 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)}.


As X [3] is smooth, p is an i semorphism over X [31 \ ZO,2)(X ) by Zariski 's main

theorem [Hartshorne (2), V. 5.2]. Now we show (1). As p is an i somorphism 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 analyt ic local coordinates a round any point e E E and so proved

the smoothness of J~a(X). To simplify nota t ions we will assume tha t the dimension

of X is 3. The argument for general dimension d is completely analogous, only more

difficult to wri te down. Now let

E = ( o , o , o , Z l , Z 2 , Z 3 , Z ) c: E

and g := 7r 1 (E). We choose local coordinates x, y, z on X centered at o. By choosing

x, y, z sui tably we can assume tha t

is the ideal of Z and tha t

Iz := (x 2, xy, y~, z)

Iz, := (z ~, y, z)

is the ideal of Z1. W'e have to dis t inguish 3 cases:

(a) Z1 = Z2 = Z3. Then Le Barz constructs the chart

( r l , 81, t l , (31, C2, C3, V, p, IT)

around e as follows: let

r := (ol, o~, o3, Zl, Z~', Z;, Z')

be a point of .~3(X) near e. The ideal Iz, of Z ' can be wri t ten as:

Iz, = (x~ + u x + v y + w , x y + u ~ x + v t y § + u " x + v ' y W w " , z + p x + a y + O )

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

(r l ,81 , t l ) , (r2,8~,t~), (r3,83,t3)

be the coordinates of the points Ol, o2, o3. The ideal Iz~ of Z~ can be wri t ten

Iz~ = (x 2 + aix + b i , - y + cix + di, - z + eix -I- f i )

for sui table ai, bi,ci,di, ei , f i . Now Le Barz shows tha t all other

constants can be computed from r l , s ~ , t l , c l , c 2 , c a , v , p , a , and tha t

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

symmet ry 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.


(b)

(c) The Zi are pairwise distict.

are of the form

So we get the local chart (r , s , t , c l , c2 , c3 ,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 ~.

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

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

So (r, 8, t , cl + C2,ClC2,7, v,p,o') form a local chart of )~[3] at ~.

We can assume that the ideals Iz2, Iz~ of Z2, Z3

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 H3(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 .


Then (r,s,t, cl ,7,7' ,v,p,a) is also a local chart of )~[31 at ~. Put t ing 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 pa r t i cu la 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 Ha(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 ~xE2~,2 x 1xE~I and let

:r2: H3(X) , Hil"'b3(X)

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). Let 7r2 : Ha (X) , Hil""ba(X) be the quotient morphism. ~rl

factorizes into ~ 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 ~ 3 ~1,3,~: Hilb (X) , Hilb (X).

~ 3 Let 10a : Hilb (X) , X [31 be the projection. We put:

~ 3 B(X) := ~;1(Z(I,2)(X)) C Hilb (X)

with the reduced induced structure. B(X) is a P l -bundle 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 ) ; "~ / s u p p ( Z ) = x; Z1, Z2, Z3 C Z' f


Then we have / ) = ~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:

R e m a r k 2 .5 .12 .

/) ~ (P(T1) x c . . . . (2,T7r P(T1) • . . . . (2,T~r P ( T 1 ) ) / ( 2 Z / 2 , ~ )

= P(T1) x c . . . . ( 2 , T } ) P(Sym2(T1)),

and the restriction 151,a,s : / ) ~ B ( X ) is the projection onto the first factor.


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

~ 3 ~ 3 (2) Pl,a,s : Hilb (X) - -~ Hilb ( X ) is the blow up along B ( X ) .

~ 3 Proof : /51,3,5 is obviously an isomorphism over Hilb (X) (1 j j ) . Let (x, Z2 ,Z) E ~ 3 Hilb (X)(2,1). Then there are two cases:

(c 0 Z2 = x U y for a point y :fix and Z = 1472 U 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 ] , x U y , [ x U y ] 4 - [ W 2 ] , W 2 U y �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

proposi t ion 2.5.11. So we see immedia te ly that (r, s, t, ^{, c2 + c3, c2c3, v, p, a)

form a local chart a round ~.

(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 dist inct . Then (r, s, t, c l , 7, 7 ' , v, p, a ) form a local chart

near ~.

We have proved (1). By remark 2.5.12 E - - ~ B(X) is a locally tr ivial 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 iagram for the tr iangle

varieties of a smooth project ive variety X.

x a , P~ ~ ( x )

X x X (2) , Z3(X) , p~,~ ~ 3 p ..... ~ 3 Hilb (X) , Hilb ( Z )

l l TM

X (a) ~ ~, X[3] ~ P ~[a]

Here the horizontal arrows are b i ra t ional morphisms.

74 2. The Betti numbers of HiIbert schemes

C o m p u t a t i o n o f t h e B e t t i n u m b e r s o f s o m e o f t h e s e v a r i e t i e s

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 rs t we look

at the local s i tuat ion. Let k be a field and R = k[[xl,..., Xd]]. As above Hi lb"(R)

parametr izes the ideals of colength n in R.

D e f i n i t i o n 2 .5 .14 . For all l E zW let W~ C (Hilb2(R)) z x Hilba(R) be the reduced

closed subscheme defined by

= ~(Ii , . , . , I t , .])E(HilbZ(R)) ' x H i l b a ( R ) I , , . . . , I t D J / . w; ( )

Now let k be a finite field F o.

L e m m a 2 .5 .15 . There is a finite field extension FQ of ~'O such that for all finite extensions ~'q of ~'Q:

( 1 - q d - 1 ) ( 1 - qd) (1 qd-1 1 -- qd Iw~(F~)I= il--q--~-q~ "-+q)'+ 1-q ( 1 )

In particular

(2) i w O ( F q ) l = (1 - q d ) ( 1 - qd+l) iT- l '

I w ~ ( F q ) 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 s trat i f icat ion

Hilba(R) = Z(1 ,1 ,1 ) [-J Z ( 1 , 2 ) .

Over the algebraic closure F q , the s t r a t u m Z(1 ,1 ,1 ) is a fibre bundle over Pd-1

with fibre A d-1. We choose the extension ~b-'Q in such a way tha t the fibre bundle

s t ructure and a tr ivializing open cover are a l ready defined over F Q . Now let ~'q be

a finite extension o f /FQ. Let m = ( z l , . . . , Zd) be the maximal ideal in R. An ideal d I E Z(1,2)(~Tq) corresponds to a 2-codimensional l inear subspace of ( m / m 2) = Lb~q.

So we have

Z(1,2)(Fq) ~ Grass(2, F~).

An 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 project ion. Then the


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 computat ion. G

From now on let /F'Q be as in 2.5.15 and let .~itPq be a finite extension of F Q .

Let X be a smooth projective variety over F q .

D e f i n i t i o n 2.5.16. We write V~ instead of Hi lbn(R) (Fq) and put

T 1 l= X ( F q ) ,

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

Ta := { M C X ( F q ) IMI = 3} u (X(Fq) x P2(X,~'q)) u Pa(X, F q )

[--J { { X l , ( x 2 , b ) } Xl r x2 ~ X ( ~ q ) , b ~ g2 } [.J (X(~i 'q) x g3) �9

Recall the notations from 2.3.6. We identify a map 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 : x I 2 l ( F q ) ~ T2,

r : X[3I (Fq) ----* T3,

respecting C.

L e m m a 2 . 5 . 1 7 .

d-1 (1) Ixt3](G)l = Ix(3)(Fq) I + q k ~ l X ( F q ) l 2

- q

4- q2 (1 -- qd-1)(1 -- qd) Y( ~ ,, ( ~ _~ q--~ -- q q , . t, .l~ q ) [ ,

~ 3 _ d-1 (2) IHilb (X)(Fe)I = I(X x X(2))(Fq)I 4- 2qllq~qlX(Fq)[ 2


(3)

q_ q2 (1 -- qd-1)2

t~ r3(x ) (Eq) l = l X 3 ( F q ) l + 3 q = l ~ q IX(Eq)l u

( 1 - qd-J)(1 + 3q _ 3qd _ qd+l + q ) [x (zv~) l .

( 1 - q)2

Proof : Immediately from the definitions we get

~ 3 (~2 • e3)(Hilb (X)(1Fq))=

{x],x2}, {Xl,X2,Xa}) pairwise distinct

v ( z ( ~ q ) • P ~ ( X , ~ ) )

U{({ZI,Z2},{(xl,b),x2}) X, # X2 e X(.~q) be V2}

and

((1X(E,)) 3 x Cz a • r =

{ Xl 'X2'xaeX(~q)} (Xl,X2,x3, {x],x2}, {x2,xa}, {xa,x,}, {Xl,X2,X3}) pairwise distinct

U { ( x l ,Xl,X2,{Xl,X2},{Xl,X2},(Xl,b),{(xl,b),x2}) t xl ~ x2b e V2 e X(Fq) , }

I

U{(x 2 , Xl , Xl,(Xl.,b),{Xl,X2},{xl , x2},{(Xl,b),x2}) x I ~X2ebe w2X(.~q),}

o ( . , x , x , ( x , ~ ) , ( . , < ) , ( x , ~ ) , ( x , c ) ) 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 77

~ 3 ( I X ( f q ) l ) IHilb ( X ) ( . ~ ? q ) l = 3 ~- I P 2 ( X , JFq)IIXOFq) I

1 a - d ( 1 - q d ) 2 + 2-;~:IX(.Fq)I(IX(JFa) I 1 _ ( / - I) + ~ 7 ~ I X ( ~ ) l ,

I~r~(x)(F~)l 1-- d

= 6 ( I X ( f q ) l ) + 3 ~ q q l X ( F q ) ] ( l x ( . ~ ' q ) l - 1) (1 - qa)(1 + 2q + q2 _ 3qd _ qd+l) + IX(Fq)l .

( 1 - q ) 2

By remark 1.2.4 we have

I X ( 2 ) ( F q ) l =

So we get

+ IP~(X, Fq)IIX(F~)I + IPz(x, Fa)l ,

+ IP~(X~)l.

1 - qd IX[3](Fq) I _- IX(~)(Fq) I + k -~ _-q

+ ( ( i - qd)(1 - qd+l) ~- _- q--~-i ~- q~-

- - - 1) Ix (F~) I z

1-r l - - q / l x (F~ ) l ,

- - 3 ( 2 1 - q d ) IHilb ( X ) ( F q ) I = IX(2)(Fq)IIX(Fq)I + k 1 ~ q 2 [X(Fq)I 2

( ( 1 : q d ) 2 21--qd~ + \ (1 - q)2 + 1 - 1 - q / IX(Eq)l ,

( 31 _ qd _ 3 ) IX(Fq)l 2 1Lr3(X)(Fq)l -- lx(l~q){3 '~ ~ 1 - q

+ ( ( 1 - qd)(1 + 2q + q2 _ 3qd _ qd+l)

( 1 - q)2

and the result follows by an easy calculation. []

31 - qd~ + 2 - 1_-77 / [x(F~)I,

T h e o r e m 2.5 .18. Let X be a smooth projective variety over C. Then we have:

(1) 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 1 a p(X E~1, - z ) = ~ p ( X , - z ) ~ + ~p( , - z2 )p(X ,z ) + ~ p ( X , - z )

78 2. The Be t t i numbers of Hilbert schemes

1 - z 2d-2 + z 2 - - p ( X , - z ) 2

1 - z 2

(1 - z Z d - 2 ) ( 1 - z 2~) + z 4 p ( X , - z ) ,

(2) 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 + z 2 - z 2~ - z 2 d + 2 ) p ( X ' z ) ,

(1 - z 2 ) 2

p ( )~ [a ] ,_z ) = 1 1 X l p ( x , - z a ) - ~ p ( X , - z ) 3 + ~p( , - z 2 ) p ( X , z) +

1 - - Z 2 d - 2 + z 2 - - p ( X , - z) 2

1 - z 2

+ z2 (1 - z ~ d - 2 ) ( 1 + z 2 - z 2d - z 2d+2) ( 1 - z 2 ) ~ p(X,-z),

3 Z 2 d - 2

(3) p (Hi lb ( X ) , z) = p ( X , z) • p ( X (2), z) + 2z 21 - 1 - z 2 p ( X , z ) 2

(1 - z 2 d - ~ ) 2

p(Hi lb ( X ) , - z ) = ~ ( p ( X , - z ) 3 + p ( X , - z 2 ) p ( X , - z ) )

+ 2z 2 1 - z 2d-2 (1 - z2d-2) 2 1 - z2 P(X,z) + z 4 . p ( X , - z ) ,

3 z 2 d - 2

(4) p (Hi lb ( 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 -- z2d+2)p(X ' z) , ( 1 - z2) 2

p (Hi lb ( X ) , - z ) = p ( X , - z ) 3 + p ( X , - z 2 ) 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 ~2d+2~ - ~ J p ( X , - z ) ,

(1 - z 2 ) 2

(5) p ( H 3 ( X ) , z ) = p ( X , z) 3 + 322 1 - z 2d-2 1 - z 2 p ( X , z ) 2

+ z2 (1 - z2d-2)(1 + 322 - 3z 2d - z2d+2)p(X ' z) .

(I - z ~ ) 2

P r o o f i X is def ined over a f in i te ly g e n e r a t e d r ing ex tens ion T of 2~, i.e. t he re

is an X T over spec (T ) sa t i s fy ing Z T X T C =- X . Let Y = X T X T ( T / m ) be a ~ 3 ~ 3

good r educ t i on of X m o d u l o q. T h e n y[3], Hi lb (Y) and / ~ 3 ( y ) X[3], Hi lb ( Z )

and H 3 ( X ) are also r educ t ions m o d u l o q, and we can choose the m a x i m a l ideal

m E spec (T ) in such a way t h a t t hey are all good r educ t ions (see the r e m a r k s


before theorem 1.2.1). Choose m in such a way that fur thermore l emma 2.~15

holds. Then (1), (3) and (5) follow immediately from lemma 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

(1~1 z2d-2)( 1 _ z 2d) p(z(1,~)(x)) = - -~)-5-z~ p ( X , z )

By proposit ion 2.5.11 w e get

(1 -- z2d-2) (1 -- z2d)l 2 Z4 z6)p(X,z) . p(2t31z)=p(xE~Jz)+ ( ~ : ~ - ~ / : z ~ ~ + +

So (2) follows from (1) by an easy computat ion. B ( X ) is a P l - b u n d l e over Z(1,2)(X). So we have by proposit ion 2.5.13

A ~ ~ : z~ -~ ) ( l_ -_z~ )~z~ z4 )p (X , z ) p(Hilb ( X ) , z ) = p(Hilb ( X ) , z ) + ( I + z2)(1(1 _ z2)( 1 _ z4 ) , +

(4) follows again by an easy computat ion. []

For a smooth projective surface S over C these formulas can be wri t ten 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(.f f~(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 Ai (P [3])

and Ai (H3(pd) ) have already been determined in [Rossell6-Xambo (2)].

Propos i t i on 2.5.19. p[3], ~[3], - - 3 d d Hilb (Pal), H~b3(pd) and _~3(pd) all have a cell

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 on Pd . For i = 0 , . . . n let Pi be the point for whichTi = 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. The fixed points are p0 , . - - ,Pd . 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[d hI. The same is true for a general one-parameter subgroup of F. We fix a one- p[2]

parameter subgroup �9 of F which has only finitely many fixed points on Pd, ~d and P[d 3]. The induced action of 4) on P(d 3) x (P[d2]) (a) x p~3] and P d x P[d 21 X p[a] and on the quotients Pd • p~2) x P[d 2] • (pd[2])(2) • Pd[3] and (pd)3 • (Pd[2])3 X p!3]

restricts to an action on the subvarieties ~3] - - 3 ~ 3 , Hilb (Pd), Hilb (Pd) and Ha(Pd) .

As the action on Pd, P[d 2] and P[d 31 has only finitely many fixed points, it has only

finitely many on Pd x P[d 2] • 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[d 2] x (p~2])(2) x p[a] and (Pd) 3 X (P[d2]) a x P[d 3] 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 ] ~ 3 ~ 3 Hilb (Pal), Br3(pd). As are , Hilb (Pd) and these smooth, they have

a cell decomposition. []

3. T h e variet ies o f s e c o n d and higher order da ta

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] Using (3)" 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] Then "~(4)' 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 con-

structed 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 variet ies o f s e c o n d o r d e r data .

Let X be a smooth project ive variety of dimension d over an algebraical ly closed

field k. In this section we want to define a variety D2m(X) of second order da t a of

m-dimens iona l subvarieties of X for any non-negat ive integer m < d. A general

point of D2m(X) will correspond to the second order da tum of the germ of a smooth

m-dimens iona l subvariety Y C X in a point x C X, i.e. to the quotient Oy,~/m3x,~ of Ox,~. Assume for the moment tha t the ground field is C and x 6 Y C X, X is a

smooth complex d-manifold and we have local coordinates z l , . . . , Zd at x. Then Y

is given by equations

f i (Z l , . . . , Zd)~-O i = 1 , . . . , d - m .

Then the second order d a t u m Oy,,/m3x,, is

C[zl , . . . , Zd]/((fl,..., fd-m) + m3),

and giving the second order da tum is equivalent to giving the derivatives

0f~ Ozj(X), i= l , . . . , d - m , j = l , . . . , d

02 fi , , O~jOzt~X), i = 1 , . . . , d - m , j , l = 1 , . . . ,d

N o t a t i o n . In chapter 3 and 4 we will often use the Grassmannian bundle associated

to a vector bundle. So we fix some notat ions 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 Grassmannian bundle of m-dimensional quotients of E. Let

7rm,E : Grass(m, E) -----* S be the project ion, Qm,E the universal quotient bundle of

~rm,E(E ) and T~-m,E the tautological subbundle. Then the project iv iza t ion of E is

P ( E ) = Grass(r-1, E) and Op(E)(1) = (T1,E)*. We also pu t 15(E) := Grass(l, E). We wri te Grass(re, r) for the Grassmann variety of m-dimensional quotients of

C ~. Let Qm,~ and T~-m,~ be the universal quotient bundle and the tautological

subbundle on Grass(m, r ).

N o t a t i o n . For subschemes Z1, Z2 of a scheme S with ideal sheaves :Z'zl, Iz2

respectively in Os, let Z1 �9 Z2 denote the subscheme Z of S whose ideal sheaf I z is

given by 2"z := Zzl " Iz2 .

As above we will wri te Z1 C Z2; to mean tha t Z1 is a subscheme of Z2. In this

case we will wri te Zzl/z2 for the ideal of Z1 in Z2.

3.1. The varieties of second order data 83

D e f i n i t i o n 3 .1 .1 . Let 7)2re(X) be the contravar iant functor from the category of

noether ian k-schemes to the category of sets which for noether ian k-schemes S, T

and a morphism r : S ~ T is given by:

"D2(X)(T) = { (Z~

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

V i ( X ) ( r : VI(X) (T) , ~ i ( x ) ( s ) (Zo,Zl,Z2) ~ (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 Zl (X) := A c X • X

Zm+l(X) C X • X [m+l]

z(,o:~)(x) c x • x('~t ~)

be the universal subschemes. To shorten nota t ions we wri te

w := x • xEm+'I • X [(mt~)]

For i = 1 ,2 ,3 let Pi be the projec t ion 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"~ und 2"2 + 2-0 �9 2"a respectively. Then we have obviously Ui C Wi for

i = 0 ,1 ,2 . As X is a closed subvariety of a project ive space PN, Wo,W1,W2, [To, U1, U2 are in a na tu ra 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 morph i sm g : T , W of a noether ian scheme to W we put

~ := <IPN • g)*(s)

84 3. The varieties of second and higher order data

on P N x T. Let rrT : PN • 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).org is 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) X" C Z1 C Z2' } Z 1 Cx.x,

E X x X [m+l] x X[(m~2)] 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

flatZ0,Z1 C X x T closed subschemes } ~I(X)(T) : = (No, Z1) 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 • [m+l] x C Z C x . x } .

We will also see that D~(X) is 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


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.

Def in i t ion 3.1.4. Let again JR(X) be the rt th 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 j l : #~(T}) , #~(JI(X)) be the canon-

ical inclusion and

it ~ , , 0 ----, T1 - - - . ~ 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 0

) T1 il ~ , , ' rrl (T/~) ~ Q1 ~ 0

) T1 i q~ ' 7]'• ( J1 ( X ) ) ----+ O1 ----+ 0

0 ~ Ob~(x ) O ~ ( x ) ~ 0

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

86 3. The varietie~ of second and higher order data

with exact rows and columns in which the right lower square is cartesian

0 - - ~

0 - - ~

0 0

T 0 ' (~)1 01 -----~ 0

T T T o~ ~ (Sym: (T~()) J~, ~ ( J 2 ( X ) ) - - ~ ~ ( J l ( X ) ) ~ 0

l ~1 [] l '

Sym2 (5; (T~()) ~2 T1 - - ~ T1 ---* 0

T T T 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 , ) ~ 0

1 l l ~t(Sym2(T])) ~ --~ T1 --~ T1 , 0

l.~(ql) [] IP Sym2(QI) ~ W2m(X) ~ T1 , 0

l 1 1 0 0 0

Obviously W2m(X) is a vector bundle of rank

rn + 1) r = 2 + d - r n

over Grass(m, T~). (TI" T~c ) is also the image of the subbundle T1 | h~'(T~) under

the na tura l vector bundle morph ism s2 : # t ( T ~ | T~() , ~-~(Sym2(T~)). We can

see easily tha t (T1 �9 T~c ) is a vector bundle, and from the d iagram we get

Sym2(Q1) -- Sym2(T~ )/(T1 �9 T~ ).


s 2 (q,) : Sym 2 (T~) - -~ Sym2(Q1)is the quotient map.

Defini t ion 3.1.5. We put

b~(x) : : ar~((~t'), W~(X)).

Let ~2 := D2(X) --~ Grass(m,T~c) be the projection. Let T2 := Td_m,w~(x). Let

i2 o - - ~ T2 ---~ ~ ; ( W s - ~ 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

0 0 0

T T T 0 ~ T2 i2 ~ r ~ ( W ~ ( X ) ) q: , , Q2 - - , o

- - ~ , Q 2 - - , 0

T T l 0 --~ ~;(T~. T~) ~ (T~ . T~) , 0

T T 0 0

cartesian.

The vector bundle Q2 is now defined by the following diagram :

0 0 0

l 1 l

1~;(;1) ~;2

Ir2(Q1 ) 0 ~ (Q1) _, N

1 1 0 0

0

- -* 0

---* 0


and

From these diagrams we can read off the exact sequences

0 ---* Sym2(Qi) ~ W ~ ( X ) 02 T1 ----+

on D I ( X ) and

on D ~ ( X ) .

-----+ 2 ( (~1) ' 0

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) J p . "N qn

X X In] .

Let

be the projections. We put

~1 : D ~ ( X ) ~ XI~+i l ,

r2 : D ~ ( X ) ---, X [(~2+2)]

(ox)~ : : q(q,.+~).(Oz,~+~(x)) = q(q, .+~) .v;~+~(ox) ,

* ) ) * ) ) * )( ( O x ) i : : ,'~(q(,,,+~ ,(Oz<t~)(x)) : ~(q(~+~ ,p(,?~ Ox).

( O x ) ~ is a vector bundle of rank m + 1 on D ~ ( X ) and (Ox ) 2 a vector bundle of

rank ( ' 2 +2) on D ~ ( X ) . Let A C X x X be the diagonal and Za C O x • its ideal

sheaf. For all n E iW let A n be the subscheme of X x X defined by (Za)n (which

has support A --~ X). Let sl,s2 : X x X ~ X be the projections. Then we have

y , (x) : (~2),(Ox• ) n+') : (~2),(o~o+~).

Let r : T ~ X be a morphism from a noetherian scheme. We define Ar C X • T

by A ---* X x X

T [ ] T l~x~

Ar ~ x xT.


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 '~ , X x X

T [] l lxxq~

A 2 , X x T

T h e o r e m 3 . 1 . 7 . There exist isomorphisms

~)1 : D I ( X ) ~ G r a s s ( m , T ~ ),

r D~(X) ~ D2m(X),

for which the diagram

commutes such that

D~(X)

D~(X) , Grass(m,T~()

X

a;(O,) = ( o x L ,

Proof." With the notations of definition 3.1.6 we can rewrite the functors D~(X), Z~L(X) as:

DI(X)(T) := { (r Z,)

r , X } Z1 C X x T closed subscheme flat of degree m + 1 over T with '

A s C Z1 C A~

~L(X)(T) := {(r Zl, Z2)

$ : T ----~ } Z1, Z2 C X x T closed subschemes

flat over(m+2T of degrees m + 1 and respectively with "

A o C Z1 C A~; Z1 C Z2 C z2kr �9 Z 1

90 3. The varietie~ of ~econd and higher order data

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 2 7r 1 �9

Let ql : Ar t ) 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

r Dtm(X) > 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 f r~(J l (X)) = (p,),(O/,~) defines a subscheme Zl C A2,~, satis-

fying A ~ C Z1. The pair (~'1, Z1) defines the required morph ism

~i : Grass(m, T~) ~ D I ( x )

over X. We see that r is the inverse of (~l-

To construct r r we 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.,.,.2 C Z1 C A ~ ,ri-i o .ri- 2 ,

Z l C Z 2 C A . l o r r 2 �9 Z 1 C A 3 7pl o 1T 2 .

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+ 2-

submodule with (P2).(I2) = T2. As T2 is a subbundle of ~ ( T a ) , we have /2 C

ZZ, IA~to, ,. As ~'~(T1 �9 T~:) is a subbundle of :F2, we have


So we have in particular

Oa~_ �9 12 C 12. ~1 o~-2

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

* 2 r = (#~r~(J2(X))/T2 = Q2.

Obviously we have ~b2 -1 ---- (~2 �9 U]

In future we want to identify /91(X) with D~(X) and/9~(X) with D~(X) via r and 42.

R e m a r k 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)] therewitharexXcE z1X'cZ1z EcZO,m)(X)x �9 Z1 } "

(2) The projection r2 : D2m(X) ~ Z(1,rn,(m:,))(X) is a natural resolution of

�9

y[a] It is the blow up along (3) r2 : D~(X) , y[S] is a natural resolution of "'(a)- --(a) z(1,2)(x).

P r o o f : D~(X) is closed and irreducible. By the d e f i n i t i o n s Z(1,ra,(m+,)) (X) is the

image of the projection r2: D~(X) ~ 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 ) o f - 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

. v[a] is an isomorphism see that "'(3)Y[a] is the closure of Z(1,1,1)(X). r2 : D21(X) ' "'(a)

over Z(1A0)(X ). Z(la)(X) has codimension 2 in y[3] "'(3), as

X [ 3 ] = Z ( 1 , 1 , 1 ) ( X ) I J Z(1,2)(X), (a)


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 bundle of rank r over X.

Then 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 bundle P ( E ) we get in part icular

A*(P(E)) --

where P = cl(Op(E)(1)).

A*(X)[P]

For the Chern classes of a symmetric power of a vector bundle we have the

well-known relation:


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)) = H ( l + y i , +.. .+Yim). il <.. .<im

If E has rank 2, we have

c(Sym2(E)) = (1 + 2e, + 4e2)(1 + e,)

= 1 + 3el + (2e~ +4e2) 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 + 3el 2 + 4e2)

= 1 + 10el + (35e~ + 20~2/+ (50e~ + 120e,e~)

-t- (24el 4 + 20Sere2 + 64e 2) + 96e31e2 + 128e, e~

c(SymS(E)) = (1 + 5 e I -~- 25e2)(1 + 5e I ~- 4e~ + 9el)(1 + 5 e l -~ 6e~ + e2)

= 1 + 15el + (85e~ + 35e2) + (225e~ + 350ele2)

+ (274el ~ + l lSae% + 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 + el 2 -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 , . . - , f r the elementary sym-

metric polynomials in the Yi. Let

c m ( 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 , . . . , f~) be the

par t of weight i in c m ( f l , . . . , f,.).

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,.)


holds. 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)).

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 ) ]

where

d I (1 +dl + . . . + rid-m)(1 + el + . . . + era) = ~ ( - 1 ) ' c i ( X ) ,

(1 q- f l -4-. -4- fd-m)(1 Jr- gl q-.. q- g(~+~))

= ( 1 + d l + . . . + d d - m ) C 2 ( e l , . . . , e m )

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) ~ T1 ~ 0

gives C(Wm(X)) = c(Sym2Q1)c(T1). The result follows. []

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 , 0 < i < d - 1. j<i

Then we have

"i ~, d 1, , , ~ _-- A*(X)[P, Q] (d ) E ( - 1 ) i p d - i c i ( X ) , i=O

(~) ~ _ , (Q - PI ~_, O (~) C?(q~(P),..., q~-~(PI)

i = 0


where P = cl(Op(T~)(1)), q = Cl(Ot(w~_,(X))(1)).

P r o o f : 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. [3

C o r o l l a r y 3 .1 .13.

A*(D~(X)) = H*(X)[P, Q]

) 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) - P)Q + 2c2(S)

w h e r e P = 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 ) _ p)Q2 + (c2(X) - c~(X)P - p2)Q + 2c3(X) ) '

w h e r e P = C l ( o p ( ~ x ) ( 1 ) ) , Q = c~(Op(w~(x).)(1)). A*(X)[P,Q]

A*(D2(X) = pa _ c~(X)P: + c2(X)P - c3(X),

(Q + p - cI(X))(Q 2 + 2(P - Cl(X))Q P) (Q ) + 4 (P 2 - ca(X)P + c2(X))


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 - c l ( X ) pa+c 2( 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 - c i (X)P 2 - p3) q_ 2c4(X)

where P = Cl((.QP(Tx)(1)), Q = ca(Op(w~(x).)(1)).

where

with

A.(D2(X) ) = A*(X)[pl, P2,7"1, r2] (R1, R2, R3, R4 ),

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~(-2plc l (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),

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 ) ( Q 3 _ 2 ( P - q ( X ) ) Q 2 + 4 ( P 2 - c l ( x ) P + c2(X))Q

+ 8 ( P 3 - e 1 ( X ) P 2 + c 2 ( X ) P - c a ( 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 ) -- c 3 ( X ) )

w h e r e P = ca(Op(T~)(1)), Q = Cl(Op(w~(x))(1)).


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 subva r i e t i e s

o f 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 I th order

data of germs of smooth subvarieties.

De f in i t i on 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 I th order da tum of Y at x is

the subscheme l+l DI, , (Y) := spec(Ox, , / ( Iy + m x , , ) ).

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 IZ = (Xm+l , . . . ,Xd ) -~ " '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) = Ox,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] (see remark 2.1.8). So D~(X)o parametrizes ~(n+l),c I th 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.

R e m a r k 3 .1 .16. Let Pl : Dlm(X)o , l-1 D m (X)o

DI, , (Y) , , D t - I , , ( Y )


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 I th 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 map 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.

T h e o r e m 3.1.18[Fulton (1) Thm. 14.4].


(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 ( (c f -k+i- j (F - E))l<_i,j<_e-k).

We consider the morph ism

r 7r~(Sym2(Q1)) ---+ Q2

of vector bundles on D ~ ( X ) which is defined by the d iagram

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 na tura l way.

Definition 3.1 .20. Let Y1,Y2 C X be smooth locally closed subschemes of di-

mensions dl, d2 > rn and x C X. We say Y1, Y2 have I th order contact along an


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 I th order contact along an m-dimensional subvariety at x. If m is

the min imum 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 .21. Y1 and Y2 have m-dimensional I th 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

x E X .

In case dl = m <_ d2, Yt and Y2 have second order contact at x if and only if

D2m(Y1) and D~(Y2)intersect 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 ry to generalize the definition of the varieties of second order

da t a to a definition of varieties of higher order data . We will however only have

par t ia l success. This means tha t we give a general definition of the variety D ~ (X) of

n th order da t a of m-dimensional subvarieties of a smooth variety X, which however

does not behave very well in general. The varieties of th i rd order da t a of curves

and hypersurfaces on a smooth variety X turn out to be project ive bundles over the

corresponding varieties of second order data . However the varieties of th i rd order

da t a of subvarieties which have bo th dimension and codimension greater or equal

to two are not locally tr ivial fibre bundles over the corresponding varieties of second

order data. Also, even if X is a surface, D~(X) is not a locally tr ivial fibre bundle

over D~(X). At the end of this section we give some enumerat ive appl icat ions of

our results.

As a s t ra ightforward general isat ion of the definition of the varieties of second

order da 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 project ive variety of dimension d over a field

k. Let n,m E 2~>_o with 1 < m < d. Let ~),~(X) be the contravar iant functor from

the category of noether ian k-schemes to the category of sets which for noether ian

k-schemes S, T and a morphism r : S ~ T is given by:

I)~(X)(T) : { (Zo,..., Z,~)

Zi C X x T closed subscheme flat of degree (re+i) over T (i = O , . . . , n ) } Zo C Z1 C . , . C Zn

and Zi " Zj D Zi+j+l for all i,j with i + j < n - 1

p~,(x)(r : ~ ( X ) ( T ) , ~ ( X ) ( S )

( Z o , . . . , Z n ) , , ( Z o •

Here we use again the notat ions we have in t roduced in definition 3.1.1. In the same

way as in l emma 3.1.2 we can show tha t : / )n(X) is represented by a closed subscheme

D ~ , ( X ) c X x XEm+,1 x . . . x X [(into)].

We call D~(X) the variety of n th order da t a of m-dimensional subvarieties of X.

D~(X) is as a subset o f X [m+l] x . . . x X [ ( '~"+n)] given by:

( z 0 . . . . . z n ) c D~m(X) :=

x • xEm+, l • • x [(~:")]

Z0 C Zl C Z2 C . . . C Zn / and Zi - Zj D Z i+ j+ l /

for all i,j with i + 3 _< n - 1


Obviously we have D~ = X. For i = 0 , . . . , n let

r i : D~(X) ~ X [(~'+')]

be the project ion. We also consider the projec t ion

~r. : D~(X) , Dnm-I(x)�9

It is not clear in which cases D,~(X) is reduced, i r reduble or smooth. In

cases in which it is reducible a be t te r candida t for the variety of higher order

d a t a is the closure the image of D~(X)o under the obvious embedding. Let

A C X x X be the diagonal. Then Hilb('+m)(Ai+l/X) is a closed subscheme

of Hi lb( '+m)((X x X ) / X ) = X[( '+m)] for all i. We can see immedia te ly tha t the

projec t ion ri: D~(X) ~ X [ ( '+ ' ) ] factors through Hilb('+~)(Ai+I/X), as we see

from the definitions tha t Hilb('+'~)(Ai+~/X) represents the functor

T I , {(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 tha t D~(X) and D3d_a(X) are again Grassmannian

bundles corresponding to vector bundles over D~(X) and D ~ _ I ( X ) respectively.

Before doing this we want to show tha t these two cases are the only ones in which

we can expect such a result (exept for the t r ivial case m = d).

R e m a r k 3.2.2.

(1) Let 2 < m _< d - 2. Then ~r3 : DO(X ) ~ D~(X) is not a locally t r ivial fibre

bundle�9

(2) Let S be a smooth surface. Then 7r4 : D4(S) , D~(S) is not a locally tr ivial

fibre bundle.

Proof.* (1) Let x E X. Let xa . . . . Xd be local pa ramete r s near x and let

mx,~ := (xl . . . . ,xa)

be the maximal ideal at x. Let Z1, Z2 be the subschemes of X with suppor t x

defined by the ideals I1 = (xm+l,.. ,xd) T m 2

�9 Xp~g~

I2----(Xm+l, . , X d ) T m 3 �9 �9 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 idea l /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 , . . Xd,X2,XlX2)-~-(XiXj[i > m ) + m 3 �9 ~ X,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 := <Zrn-t-3,... ,Xd, X2,XlX2> 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, Z l , Z2, Z3)) ~-~ P1.


Let Z~, Z~ be the subschemes of S with suppor t 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 suppor t s whose ideal is of the form

I'~ = (t) + ( x ~ , ~ > x y ~ , ~ ~)

for a two-dimensional l inear 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 project ive variety of dimension d over a field

k. Let m be a posit ive integer with m < d. We will again use the nota t ions from

the definitions 3.1.3, 3.1.4 and 3.1.5. Let ~2 := #loft2. We define the subbundle T2

of ~ ( J a ( X ) ) by the d iagram

0 0

T T 0 ' Q2 Q2 ~ o

T T l 0 ---+ ~ ( S y m a ( T ~ ) ) ~ ~-;(da(X)) - -~ ~ ( J : ( X ) ) ) 0

0 - - ~ ~ ( S y m a ( T ~ ) ) ~ T2 - - ~ T2 , 0

T T T 0 0 0

Let again A C X x X be the diagonal and Zt, C O x x x its ideal sheaf. Let

sl ,s2 : X x X ----* X

be the projections. For all non-negat ive 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

0 , J~(X) - -* J j ( X ) - - + J i -~ (X) , O.

3�9 Varieties of higher order data and applications 105

We see J~ = Jj(X) and Jj(X) = SymJ(T~). Let il <_ i2 < j2 _< j l be positive

integers�9 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 ) __~ ~+,2 : Ji~+j2 (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~)) = ~ ( J 3 ( X ) ) / ( ~ i ( r l ) - * 1 �9 ~ (J~ (x))) .

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

T - 2 / ( ~ ' ; ( T 1 ) " ~;(Jl (X))) = T2.

From T2 C ~r~(T1) C ~'~(J~(X)) we get

r ~ 2 - - * 1 ~ . �9 7r2(J2(X)) C ~;(T1) V;(J~(X)), ~ ~ * ~ --* 1 --* 1 +~(T1)" +~(T1) C rr2 (T1) �9 u2(J2(X)) C T2 C u2(Ja (X)).

a a b2~(X) by We define the coherent sheaves Uam, V~, W~m(X) on

u~ : = ( ~ ( ~ ) - * 1 ~* �9 71"2(,]2 ( X ) ) ) / ( T r 2 ( T i ) " ~ ' ; (5~1)),

7 3 : = ( # ; ( r l ) " 7c2(J~(X)))/(Tr2(T~)" 9~(T1) + T2 .~(J~(X))) ,

w ~ ( x ) : = ( T ~ ) / ( ~ ( ~ ) . #~(~) + (:~). ~ ( j l ( x ) ) ) .

Then we obviously have the exact sequence

o , v ~ ~ w s ~ T~ - - - , O.

L e m m a 3 .2 .4 . L e t m = 1 o r m = d - 1.


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"

P r o o f : By the exac t sequence

o , v ~ , w ~ ( x ) , T~ , 0

it is enough to show these resul ts for [/am and V~. It is enough to check t h e m

fibrewise. Let v 6 D ~ ( X ) be a poin t lying over x 6 X . Let x l , . . . , X d be local

p a r a m e t e r s near x. For i = 1 . . . . , d we denote by ~'i the class of xi in Ox ,~ /m~: ,~ .

We can as sume tha t the fibre ~;(Tl (v) ) is of the fo rm (Xm+l, . . . ,Xd)/m2x,x. T h e n

we have for the fibres:

( ~ 1 . . . . 1 �9 ~r2~r2J ~ ( x ) ) ( v ) = <~22j[i > m ) + ( '2 i~jx t[ i , j , l <_ d),

(e~(T~) e~(T1))(~) : <~,~ l i , j > .~> + <~{~j~, li > - J .

Let A0 : : (~iYcj I i < m , j > rn} + (Yzi~j2z [ i , j , l < m). T h e n the res t r ic t ion of the

na tu r a l p ro jec t ion

~ : ( ~ T I ~ J ~ ( x ) ) ( ~ ) , V ~ ( v )

to A0 is an i somorph i sm, and (1) follows.

T2(v) is an (m - d ) -d imens iona l l inear subspace of

~ ( w ~ , ( z ) ) ( v ) - * - . . . . . = (7c2T,/(rc2T, �9 7r2T}))(v ).

Let p : f r~(W~(X)) (v ) ~ fr~(T1)(v) be the pro jec t ion . As we have a s sumed tha t

rn = 1 or m = d - 1 holds, we have e i ther p(T2(v)) = ~r~(T1)(v), or p(T2(v)) has

cod imens ion 1 in ~r~(T~)(v). (~~(T1)(v) is one -d imens iona l in case m = d - 1, and

T2(v) has cod imens ion 1 in ~r~(W~(X))(v) in case m = 1.)

(a) p is onto. T h e n we have

f~(v) (ym+~,.. 3 = . , y d ) / m x , ~ ,

where x l , �9 �9 Xm, Y m + l , . - �9 , Yd are local p a r a m e t e r s near x. So we can as sume

tha t xi = yi for i = m + 1 , . . . , d . T h e n we have

(~. ~;(J~(x)))(v) : <},~ li > ~)+ <~,~j~ li > ~).

Let A1 := (~2i2.j24[i,j, 1 <_ m). T h e n the res t r ic t ion of the na tu r a l p ro j ec t i on

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 , x 3

for an f C (xixj l i , j < rn) \ m 3 Let f denote the class of f in Ox,x/m4x,~. - - X ~ : c "

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. []

Def in i t ion 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

ia 0 -----+ T3 ~ 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 ,

T 0 - - - , h ; ( K )

T 0

i 3

[3

~

i3 )

0 0

T 1

?r~(T2) ~ Qa

T T - ~ ; ( K ) , 0

T 0

, 0

, 0


and the vector bundle 03 o n / ) a ( X ) by

0 0 0

0 ----+ T3 is - - 4~ - -~ #~(T2) ' Q3 - - ~ 0

l 0 ~ eft 3 "--')" ~:~(J3 ( X ) ) ~ (~3 ;' 0

773 (Q2) ' 0

0 0

In part icular we get the exact sequence

0 ~ 03 )3 # 3 03 ~ . -- ' ' z(QJ) ----* 0.

We now generalize the definition of the bundles (Ox)~ and (Ox)2~ from 3.1.3:

D e f i n i t i o n 3.2.6. For any I C /V let again Zt(X) 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 := (qt).(p~(E)).

This is a vector bundle of rank rl 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 3.2.1. ( E ) ~ is a vector

bundle of rank r . (m+,~) on D~(X). We call it the contact bundle corresponding to

E,X and m.

3.2. Varietie, of higher order data and applications 109

T h e o r e m 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 Dam(X) r , Dam(X)

Dam(X) r , s

commute, such that r = (Ox)am.

Proof." We use the' nota t ions from definition 3.1.6. Then we can write 1)am(X) as

Let

1)~(X)(T) = { (r Z1, a2, Z3)

r : T ~ X morphism over k Z1, Z2, Z3 C X x T / 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 Ar �9 Z1,

Za c A r Z a c Z ~ - Z 1 .

Z1, Z2, Z3 C X x Dam(X)

(re+z) (m+3"~ Dam(X). Via r we be the universal families of degrees m + 1, t 2 J, ~ a J over

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 projec t ion 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 morph i sm Ca : D 3 ( X ) ~ D 3 ( X ) over D2(X) .


Let Za := ga l (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 Ob~(x ) - submodu le of #*(J4(X)) . Let Ia C Ozx~ be the Ozx~-submodule

with/5.(I3) = Ta. By the inclusions

rta(rr2(T1))" -* * - ra , ~ ( . : ( T , ) ) c

(see definition 3.2.5) we have

:r/,~lA~ �9 :rZ~lA, C Ia, $ 2

Z, la~ C Ia,

So we have in part icular 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 y[4] As also X [4] "'(4)" (4),c = Z(1,1,1,1)(X) is smooth, ra is an isomorphism over Z(1,1,1,1)(X ). The preimage D~(X)o parametr izes third order da ta of


(germs of) smooth curves on X. Here the n th order d a t u m of a smooth subvariety

O ~m ~+1 of Ox,x . In a s imilar way Y C X in a point x C Y is the quotient Y,x/ x,~

one can t rea 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 res t r ic t ion r3[D~_~(X)o is an isomorphism.

r31 (D~_ l(x)0 ) parametr izes th i rd order da t a of (germs of) smooth hypersurfaces

of X.

R e m a r k 3 .2 .8 . Let Y C X be a smooth 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 ) . From the definitions of the vector bundles

( O x ) ~ and ( O y ) n we see that (Ox)nlD,~(y) = (Oy) n. So/~2 ( y ) C / 9 ~ ( X ) and

/9~(Y) C / ) l a ( X ) are closed subvarieties with

Q,i(x)[y = Q.i(Y), Q i (X) ]v = Qi (Y) , (i = 1,2,3) .

Here we write Qi(X) , Q i ( X ) for the classes Qi, Qi on D i ( X ) and similar for Y.

In case m = d = d i m ( X ) we see immedia te ly tha t D ~ ( X ) is isomorphic to X

via its project ion. The universal families are Zi = A i+1 C X x X for i = 0 , . . . , n.

So we have (Ox )~ = Zn(X) ,

n n + l ( O X ) m / ( O x ) ~ = Sym"(T~) .

Now we can compute the Chow rings of D ~ ( X ) and D3_I (X) . For this we first

have to determine the Chern classes of WI (X) and wL (x).

L e m m a 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 --+ 0

on

(2) In case m = d - 1 there are exact sequences

0 ---+ T2| , V#_ 1

o - - ,

0 ----* Vda_l

on 5 L I ( X ).

~ Yd3_l

---+ U~_ 1 ~ f f~(T1) |

---4" W ~ _ I ( X ) ----4 r 2 ) 0


Proof :

(~) Let ~ : ~ ; (~ ) ~ ~ ( : J ( X ) ) --~ V? be the ~atural homomorphism. We see immedia te ly tha 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 na tu ra l map w : 9i( :F1) | ff~(j1 (X)) - - ~

# i ( Q i ) | Q: .

(2) We a l ready know the lower sequence. The middle sequence comes from the

d iagram

0 0 0

l 1 1 0 ~ ~ ; ( T 1 ) . ~ ( J ~ ( X ) ) ~ ~'~ ( r l ) "~'~ ('~1) ~ ~r~ (Sym2 (T1))

0 ~ ~ ( S y m a ( r ~ ) ) ~ ~r~ ( ' ~ 1 ) - ~ ( J1 ( X ) ) ~ ~-~ ( r l ) . ~ ( T ~ )

0 ~ ~r~ ( S y m a ( Q 1 ) ) - - U~_ 1 ~ (~r~ (T1 ) .~ (T~) ) /Sym~( f r~ (T~) )

0 0 0

if we use (T1 �9 #~(T~i))/Sym2(T1) ~= T1 | Q1.

Let w2 : U~_ 1 ) V)_ 1 be the na tu ra l homomorphism. ex~ ~ . ker(w2) = T2 | % (QI). We consider the exact sequence

~2+~(jl(X)) ~ uL~ - ~ v)_~

We have to show

O,

where w0 is the obvious map. We see tha t

(~';(T1) % ( T ~ ) ) - * 1 ~ ' ~ ( r l ] r 1 6 2 �9 - * * Q ~ ( J ~ ( z ) ) + ~ e ) c

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


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*(Di - ' (X) ) and ~r~'(ao). We formulate our results (proposition 3.2.10 and propo-

sition 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) - 2c i - l (X)P - E cj(X) i = 0 j=O

pQi-l-j i - -2-- j '_ '_ x

Here P = C l ( O P ( T x ) ( 1 ) ) , Q = C l ( O p ( w ~ ( x ) . ) ( 1 ) ) , R = c l ( O p ( w a ( x ) . ) ( 1 ) ) .

Proo 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*(X)[P, Q, R] A (Dd_a(X))=

d

Z ( - 1 ) i p " - i c i ( X ) , i = 0

(Q - P) qd-l(P)), i=O

(R-e ) E / = 0 n = O

\ i+ j<d-1 qi(P) j

- - / - - T t

114 3. The varictie~ of second and higher order data

tlere (.),, de~ote~ the part of degrer ,~ (P,Q,R have each degree 1 and ci(X) ha~ d~gree i). We have P = ~ ( O p ( 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 lemma 3.2.9(2) we get

e ( W 3 _ I ( X ) ) c 3 - , c #* -* = ( S y m (Tc:(Q~))) ( 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

( r - i ) c i ( E ) c l ( A ) J . [] e(E | A) = ~ j i+j=r

We will rewri te these formulas explici tely for d < 3.

If X is a surface, then

A*(X)[P, Q, R] A*(D~(X)) = p2 + Cl(X)P + c2(X),

Q2 + (c~(X) - P)Q + 2c2(X), ] ,

] R 2 + (c~(X) - 2P)R + c~(X) - 2c~(X)P - PQ

If d = d im(X) = 3, then

A*(X)[P, Q, R] A*(Ds(X)) = liPS + c l (X)p2 + c2(X)P + cs(X),

L Qa + (Cl(X) _ p)Q2 + (e2(X) - c~(X)P - p2)Q + 2ca(X),

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).

�9 and

where

A*(Ds(X)) = A*(X)[P, Q, R]

(Sl , $2, $3)

Sx : : P ~ - e ~ ( x ) P ~ + e : ( X ) P -- c~(X),

$2 : = ( Q - P)(Q + P - cl(X))(Q 2 + 2 ( P - c~(X))Q

+ 4 ( P 2 - cl(X)P + c2(X)),

$3 :=R 5 + (4P + Q - 6c l (X) )R 4 + (12P 2 + 2PQ - 23Pe1(X)

+ Q2 _ 3Qcl(X) + 11e 2 + 10c2)R 3


+ (4p2Q _ 35p2cl(X) _ pQ2 _ 2PQcl(X) + P(41c,(X): + 23c~(X))

q_ Q3 + 2Q2cl(X) + Q(_4Cl(X)2 + 8c2(X)) - 6c l (X) 3

- 30cl(X)c2(X) + l lc3(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

- - 21p2Q 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 smooth project ive variety X. Now

we want to s tudy the vector bundles (E),~ from definit ion 3.2.6. For this purpose

we first consider the bundles /~t on the Hilbert scheme X Ill. We can associate in a

na tu ra l way to each section s of E a section ~) of fist and thus also a section (s)" m of

(E)7.:

D e f i n i t i o n 3 .2 .12 . For any point Z E XM the fibre /~t(Z) of fist over Z is the

vector space H~ E | Oz). Let

evz : H~ ~ H~ | Oz)

be the evaluat ion morphism. For any section s E H ~ E) we define a section ~t of

fist by

~ t ( z ) := ~vz(s)

and put ( s )~ := r*(s~(,~+,)). This defines the evaluat ion morphism

eVE: H~ | O o ~ ( x ) ~ (E)~n.

R e m a r k 3 .2 .13 . Let s be a section of E and Y C X its zero locus. From definit ion

3.2.12 we see immedia te ly tha t Y['q C X In] is exact ly the zero locus of ~l, and thus

D~(Y) C D,~(X) is the zero locus of ( s )~ . To begin with this is only t rue set-

theoretically, i.e. wi thout considering the possible non-reduced structure. If rn = 1

and n < 3 or n < 2 one can however show by computa t ions in local coordinates on

D~(X) tha t the smooth subvariety D~,(Y) is the zero locus of (s),~ in D,~(X), if


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)).

(2) For all ~ c ~V, .~ _< d we have ( E ) ~ = r~(E) | (Ox)~.

P r o o f : (1) For all l E ZVV we put Zt(X)(,~) := Hi lb t (An+i /X) XxvJ ZI(X) . Let

zz(x)(.) / q ~ p

X Hi lb l (An+l /X)

be the projections. Then we have the commutative diagram

Hi lb t (An+l /X) ' P Zt(X)(n)

X X ,

and by the projection formula we get

Ez[Hilb,(A.+l/x)

So we get (1). The projection

= p,(q*(E))

= p.(p*Qr*(E)))

= 7r*(E) | (Ox)t]Hilb~(A.+l/x).

r . : D ~ ( X ) ~ X [(r.+.)]

factors through Hilb("+-:~)(A"+I/X) (see the remarks after definition 3.2.1). So (2)

follows from (1). []


Now we specialize to the case X = PN and to the hyperplane bundle H = 0(1).

P r o p o s i t i o n 3 . 2 . 1 5 .

(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 + PR + 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 D2(X) = Grass(~m+~ W 2 fX ~ Then we have \ \ 2 )~ m \ ] ) �9

i+j<_(%+:) k+~=i

(3) I fm = 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<C+~ ) 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)


P r o o f : This follows from lemma 3.2.14 and the exact sequences

0 , Q1 ---* ( O x ) L ~ Ox----~ 0

0 ' Q2 - - ~ (Ox)~ --* (Ox )k

0 ' Qa ~ (Ox),3,, -----+ (Ox)2,. --~ 0

0

[]

Now we want to compute the class [D2(C)] C A3N-3(D~(PN)) for a smooth

curve C C PN.

Proposi t ion 3.2.16.

[D12(C)] = deg( C)HN-1pN-1Q N-1

+ ((N + 1)deg(C) + 2g(C) - 2)(HNpN-2Q N-1 + HNpN-1Q 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 hand we can use the relations to compute the intersection table:

HNpN-1QN-2

H

P

Q 1

HNpN-2QN-1 HN-1pN-1QN-1

1

1 - N - 1

1 - 2 N - 2

This proves the result. []


Enumerative applications for contacts of projective varieties with linear subvarieties of P N

Now we want to apply our considerat ions to obta in formulas for the numbers

of higher order contacts of a smooth project ive variety X C P N of dimension d with

l inear subvarieties of P N of dimension m. We have to dist iguish two cases: m _> d

and m < d. We will see tha t the first case is the s impler one, as in this case we

have X = D n ( x ) , and so the computa t ions can be carried out direct ly 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 hyper-

plane bundle on PN. We will denote by the same le t ter its res t r ic t ion to X and its

first Chern class.

Contacts with linear subvarieties of higher dimension

Let X C PN be a smooth m-dimensional subvariety. We can in a na tu ra l way

identify D,~(X) with X for all n E W, and with this identif icat ion we get

(H)~ = H | ((gx)~ = H | Jn(X).

On X = D ~ ( X ) we consider the evaluat ion morphism

eVm : H ~ OpN(1)) | Ox ) (H)~.

This is the composi t ion of the res t r ic t ion

r : H ~ Ov~(1) ) ~ H~ H)

with the evaluat ion morphism

evil : H ~ H) | Ox , ( g ) ~

from definition 3.2.12. Over every point x E X the kernel of the induced map

evm(x) : H ~ OpN(1)) ~ H~ H | (Ox,,/m~x+a,))

on the fibres consists of the sections s e H ~ OvN (1)) for which the hyperp lane

P ( k e r ( s ) ) C P ( H ~ O p N (1))*) has n th order contact with X. A l inear subvari-

ety V of P N of dimension ml > m has n th order contact in x, if and only if each

hyperp lane of P N containing V has nthorder contact with X at x. So the locus

where X has n th order contact with an / - cod imens iona l l inear subvariety of P N is

the degeneracy locus

~)N+l--l(eVm) = {X E X r]g(evm(X)) ~ N -}- 1 - / } .


So we get by Porteous formula (see. theorem 3.1.18):

Propos i t ion 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 . I f its codimension is r, then its class is

det( (C(m+n)_N_l_t_l+i_j( ,Jn(X)@g)) l<i , j<<_l) ~ A r ( - 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 f('+") - i)

E E (-1)i ~ " j~ [ I c i ' (SymI(TX))Hj" i4_j=(ra+nn)_ N i l + . . . + i n = i /=1

(2) Let C C P , be a smooth curve�9 The number of n th order contacts of C with

hyperplanes in P~ is

2 ( 2 g ( C ) - 2 ) + ( n + l ) d e g ( C ) .

(3) Let S be a smooth surface in PN If N = { n + l ~ _ 1, then the class of the �9 ~ 2 ]

(n - 1) th order contacts with hyperpIanes in PN is

n--1

k = l

I f 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) r a = l O_<k<~

4 - ( ~ (2 ) (n -- i)2 + E i j ( n - - i ) ( n - - j ) ) c l ( ' ) 2 \ i = 2 l<i<j<n--1

t k = l 2


I f 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 ) -- ~ (rl -- i) 2 cl(S) 2 l<_i<_j<_n--1 i=2 n-1

- Z Z - 2k)2c2(S) m:l 0_< k< --~ n--X ( ( / ) / )

_ Z k ( n _ k ) n + l (n+ 1 k=l 2 + 1 c l (S )H + 1)2+ H 2.

(4) Let X be a smooth threefold in P9. The class of second order contacts of X

with hyperpIanes is

- 5c 1 (X) + 10H.

Let X be a smooth threefold in Ps. The class of second order contacts of X

with hyperplanes is

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

- 7C 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 aubva-

rieties is

-42c1(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 (-1)' ("+ ) - i iic,,(Sym,(Tx))g,. i + j < ( ~ + ~) i l+ . . .+ i~=i /=1


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 - # ) X l ) k

k = l

respectively, and the coefficient of c2(X) is the number

TZ--1

E E (m_2k 2 m = 2 0 < k <

The rest follows by an easy computation. (4) follows from (1) and remark 3.1.9 by

an easy computation. []

C o n t a c t s w i t h l inea r subva r i e t i e s o f lower d i m e n s i o n

Let X C PN be a smooth projective variety of dimension d. Now we want to

treat the second order contacts of X with linear subvarieties of PN of dimensions

m < d and also the third order contacts of X with lines. We first s tudy 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 ~ 0pN(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 PN 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 ~


is exactly the degeneracy locus

= D (x) _< m + 1}. ~)rn+l (eVm )

Let r0 : D2m(X) , X be the projection. From the above we get for the image of

the degeneracy locus

{ there is an m-plane } ro('Dm+l(eVm)) = x E X 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 compo-

nent 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

PN-

We can also determine this class in a dual way:

let

e v * : ( ( 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:

P r o p o s i t i o n 3 . 2 . 1 8 . 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

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),


if this locus has codimension N - 2d + 1.

In a similar way we can argue for third order contacts with lines. Let X C PN

be a smooth projective variety. On D~(X) we consider the evaluation morphism

Let

ev : H ~ Ov^,(1)) | O,9~(x) ' (H) a.

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 PN, 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. I f 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 a �9 A2N-ad+2(X). - - SN(((H)m ) )SN-2(((H),~) )) E

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 computat ion 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 ( -3c1 (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)).


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 ( -3c1(X) + 7H).

The class of second order contacts of a smooth threefold X C Pr with lines is

4(7c1(X) 2 - 5c2(X) - 2 4 c l ( X ) H + 28H2).

The number 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 - 15c2(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 ( -15c1(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 number of second order contacts of a smooth fourfold X C P11 with lines is

8(31c1(X) 4 - 79c l (X)2c2(X) + 21c2(X) 2 + 4 4 q ( X ) c a ( X ) - 17c4(X)

- 180c , (X)3H + 2 8 8 c , ( X ) c 2 ( X ) H - 108ca(X)H

+ 462Cl(X)2H 2 - 330c2(X)H 2 - 660c1(X)H 3 + 495deg(X)).

The class of third order contacts of a smooth threefold X C Ps 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

-575c1(X) a + 790c l (X)c2(X) - 251c3(X) + 3 4 0 0 q ( X ) 2 H

- 1960c2(X)H - 8228c 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.


D e f i n i t i o n 3 .2 .20 . Let T be a smooth project ive variety and Y ~ T a smooth

morphism of relat ive dimension m. Here we asume Y to be quasiproject ive over T.

We put 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 relat ive

versions relat ive to T. Then we put 5 ~ ( Y / T ) : = arass((W1), W~(Y/T)) .

It is obvious from the definitions, tha t both D~m(Y/T) and b~(Y/T) are iso-

morphic to Y.

D e f i n i t i o n 3 .2 .21 . Let T be a smooth variety and YT C P N • T a flat family of

m-dimensional subvarieties of PN, i.e. we have the project ions

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 addi t ion

we assume tha t YT is irreducible, and there is a dense open subset Yr,o C YT such

that the restr ic t ion Yr, o ~ T is a smooth morphism.

Then 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 we again identify D ~ ( P N ) a n d /~2m(PN) via r Let ]D2m(yT) be 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 ~ ( P N ) x T ~ D2m(PN)

be the project ion. Let X C PN be a smooth project ive variety of dimension d > m.

Let i : D~(X) ~ D ~ ( P N ) be the embedding and r0 : D~(X) ----* X be the

project ion. We put

K(X, YT) := (ro).(i*(p.([D~(VT)]) ~ A*(X).

Remark 3.2.22. K(X, YT) is a candida te for the class of the locus where X and

elements of the family YT have second order contact.

Proposition 3 .2 .23 . 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)

m+2 P r o o f : Let f := ( 2 ) ( N - m) - t. We will show more general ly tha t for every

class W E A I ( D 2 ( p N ) ) there are integers n~ for all par t i t ions a of numbers s < e

such tha t the above formula holds for ( r0 ) . ( i* (W)) . As A * ( D ~ ( P N ) ) i s generated

by H and the Chern classes of the universal quotient bundles Q1 and Q2, it is

enough to show the result for the monomials M in H and the Chern classes of

Q~ and Q2. Using our conventions we can wri te i*(H) = H, i*(Q1) = Q1 and

i*(Q2) = Q2. Let M = MoMIM2, where M0 is a monomia l in H and the Chern

classes of X , M1 a monomia] in the Chern classes of Q1 and M2 a monomia l in

the Chern classes of Q2. We assume tha t M1 e Adl(D2m(X)), M2 e Ad2(D2m(X)). If dl = r e ( d - m ) and d2 = (m+l ) (d - rn ) , then we have ( r 0 ) . ( M ) = aMo for a

sui table 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) and on Grass((~+'),("~+~) + d - m) respectively. Then a is the

p roduc t of the intersect ion numbers M1 ( q l , . . . , qm) and M ~ ( r l , . . . , r(~+~)) on these

Grassmannians . ) If d2 < (m+l) (d - m) or d2 = m+] ( 2 ) (d - m) and dl < r e ( d - m ) , r n + l then we have ( r o ) . ( M ) = 0. If d2 > ( 2 ) (d - m), then we use the relat ions

of proposi t ions 3.1.11 to express M as a l inear combinat ion 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 re la t ions of propos i t ion 3.1.11 to

express M as a l inear combinat ion with 2g-coefficients of monomials N = NoN1 M2,

where N1 C A~I(D2(X)) with el < dl. So the result follows by induct ion. []

128

3.3. S e m p l e bund le s and the formula for contac t s w i th l ines

In this section we int roduce the Semple bundle varieties Fn(X) of a smooth

variety X. They paramet r ize in a sl ightly different sense than D~(X) the n th order

da t a of curves on X. Like D~(X) and D31(X) they are smooth compactif icat ions

of y [~+l ] by a tower of Pal_l-bundles over X (d = dim(X)). Remember tha t "~(n+l),c

X[n+l] (n+l),~ parametr izes the rt th order da t a of germs of smooth curves on X. We will

use the F~(X) to ob ta in a general formula for the higher order contacts of a smooth

variety X C P N with lines in PN as a l inear combinat ion of monomials in the

hyperp lane section H and the Chern classes of X. We finish by considering more

general ly higher order contacts of X with a family of curves.

For s implici ty we will assume during the whole of section 3.3 that the ground

field is C.

Def in i t i on 3.3 .1 . Let X be a smooth variety of dimension d. We define induct ively

varieties F~(X) and vector bundles Gn(X) on F.(X) . Let

f 0 ( x ) := x, Go(X) := Tx.

Assume induct ively t hat F0 ( X ) , . . , F~_ 1 (X) and G0 ( X ) , . . . , G n - 1 (X) are a l ready

defined. Assume fur thermore tha t G n - I ( X ) is a subbundle of the tangent bundle

TF,_~(x) of rank d. Then we put

Let

Fn(X) :---- P ( G n - I ( X ) ) .

f~ ,x : P ( G n - I ( X ) ) ~ Yn- l (X )

be the project ion. Let 8n := Op(G,~_KX))(--1 ) be the tautological subbundle of

f* ,x(Gn_l(X)) . Let TF.(X)/F._dX) = (f~F.(X)/F~_,(X))* be the relat ive tangent

bundle. We define the subbundle Gn(X) of TF.(x) by the d iagram

0 ~ TFo(X)/F._I(X) ' TF, dx) df.,x . ,. f~,x(TF~_~(X)) , 0

T [] lJ ' T F . ( X ) / F , ~ _ K X ) ) G n ( X ) ' an ) O.

) * Sn ~ fn,x(TF._,(x)) is the composi t ion of the na tu ra l inclusions

0

Here j

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.

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).

P r o p o s i t i o n 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) Z ( d ; i ) f*(ei(Gn-l(X)))PJn i+j<_d--,

c(ao(X)) : 4X).

P r o o f i This follows immediately from the exact sequence

o ~ Trn(x)/F,,_~(x) ~ Gn(X)

and the Euler sequence

0 ~ OF,(X) ~ f*(Gn- l (X))@s;

[]

8n ~ 0

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

130 3. The varieties of second and higher order

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*).

Proof 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 ) , s,~ --~ 0

T T (*) 0 ~ T F , ~ ( y ) / F , ~ _ , ( y ) ) G,,(Y) , s,~ . , 0

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 induct ion

* 1 * * * G . (X ) IF . ( y ) /G . (Y ) ~- s . | @ (g. ,x) (~,) | g . , x ( T x l y / T r ) ,',., * 1 * * * = 8 n | | ( g n , x ) (81) |

~- **. |174 | []

In the case of a smooth curve C C X we want to describe the embedding

Fn(C) C F , ( X ) a l i t t le more precisely.

R e m a r k 3 .3 .4 . Let C C X be a smooth curve. As G~(C) has rank 1 over

F~(C), the projec t ion fn,c : Fn(C) ----* Fn-a(C) is an i somorphism and so also

g~,c : F~(C) ----* C is an isomorphism. The embedding

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 embedding of C into X and in,c the embedding

�9 g-~" , . , c : c .,c ,F . (C) . ,g:,~(C).

Then we obviously have

i~,c = j . , c . . . . . j l ,coic.

~X

Remember tha t Xl,~,c C X["I parametr izes subschemes of the form

s p e c ( O c / m ~ # ) for smooth locally closed curves C C X and 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)


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

y[~+l l and F,~(X)o. Let Z E ~'(n+l),o" local coordinates on ~(n+l) ,c y-[n+1] Let (Xl, . . ,xd) be

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

' / i = 2 , . . . , d / j=O

So al,0 and the ai,j ( i = 2 , . . . , d, j = 0,. , n) are local coordinates of ~(~+l),~Y[n+l] near

Z.

We want to suppress the pullback in the notation. We write x ~ := xi. Let

V C / I - I ( u ) be the open subset on which dx~ # 0 holds. Then

1 dx~ xi : = 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 induct ion 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 1 --1

~ F , ~ ( X ) / F . _ I ( X ) [ ( g ~ ) - I ( V ) o F n ( X ) o . T h e n we have dx~ 7s 0 on (gn+l ) (V) N

Fn+l (X)o, and the functions

X n d / I~.+~ x~ +1 ._ dxOl~,+~

J ( i = 2 , d; j = 0 , . . . , n + are regular on (g1+1)- l (V) N Fn+a(X)0. x ~ and the x i . . . , 1 1) are local coordinates on ( g n + 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 pa-

rameter on C. From the definitions we get that in our coordinates the map in,c is

given by

So the map

spec( Oc /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

k! k - j = jIai,j + E (k - j)! a''kal'~ "

k>j

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 In+l] (n+l),c"

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 nA-1 p~+a] Aln+I(PN) C be the closed subvariety

~+1 { p~+a l Z is subscheme of a line, } AIn+I(PN ) := Z C and the supppor t of Z is one point

with the reduced induced structure.

Obviously ,~+1 subvanety of (PN)(n+l),c" AIn+I(PN ) is a " In+l]

Now we want to describe ~+1 AIn+I(PN). By definition it parametr izes subschemes

of the form spec(Ol/m~x+,~) for lines l C P N and points x E l. Let A ( N ) C PN • G(1, N) be the incidence variety

A ( N ) : = {(x , I ) E P N • x E l } .

L e m m a 3.3.7. Let n > 1. The application

spec Ol m ~ +1 (x, l) Pn G ( 1 , N ) ( / _ , _ ) , , c •

gives an isomorphism

n+l e~ : AIn+I(PN ) ~ A ( N ) C P N x G(1 ,N) .


P r o o f : Let X l , . . . , x N be the s t anda rd coordinates on A N C PN. Let Z E nq-1 Aln+a(PN ). We can assume tha t Z C A N, and that the ideal of Z is of the

n + l form I z := ( x ] + l , x 2 , . . . ,XN). A subscheme Z' E AIn+I(PN ) near Z has an ideal

of the form

I Z , : = ( ( X l - - a l , 0 ) n + l , x 2 - - a 2 , 1 X l - - a 2 , o , . . . , X N - - a N , o X l - - a N , o ) ,

n + l and al,0 and the ai,o, ai,a, (i = 2 , . . . , N) are local coordinates on A l n + 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 - - a N , o X l - - a N , 0 ) ,

and the ai,o,ai,1, (i = 2, . . . ,N ) are local coordinates on G ( 1 , N ) near I. So the

appl ica t ion n + l en : Aln+I(PN ) ---* P N • G ( 1 , N )

is given in our local coordinates by

( a l , o ~ a 2 , o , . . . , a N , o ~ a 2 , 1 . . . , a N , l ) ~

( ( a l , 0 , a 2 , 0 , . . . , a N , 0 ) , ( a 2 , 0 , . . . , a N , 0 , a 2 , 1 . . . , a N , 1 ) ) ,

and this defines an i somorphism with the subvariety

A(N) C PN • G ( 1 , N ) []

R e m a r k 3 .3 .8 . Let X C P N be a smooth subvariety of codimension r. From n + l the definitions we can see tha t the intersection points of X ['~+1] and A / n + l ( P n ) (n+l),c

-[,+a] in (PN)(n+l), c correspond exact ly to the n th order contacts of X with lines in PN.

More precisely we have: the image - t v [ n + q n+l ~ n l A ( n + l ) , r ('] AI,.,+I(PN)) is

(x , l ) E X • G ( 1 , N ) 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 project ions

A(N)

~/ Pl ~ P2

P N G ( 1 , N )


@N+I or,, /OpN(-1). natural projections

R e m a r k 3.3.9. Let OPN(--1 ) be the tautological line bundle on PN and T2 :----

T2,N+I the tautological subbundleon G(1, N) = Gr(2, N + I ) . Let Q1 := QN,N+I :-- Then we can see easily that Pl and P2 can be described as the

P (Qa) = A(N) = P(T2)

PN

(see also [Fulton (1)1 Ex 14.7.12). We put

/~ := p~(OpN (1)),

t5 := Op(Q~)(1),

H := c1(/~),

p :_-- c1(/5 ).

P2

G(1, N)

Then we can see easily t h a t / t = Op(T2)(1), and/5* is the universal quotient bundle

= * * ' T " / t * /5* QI,T~ = P2(T2)/Op(T~)(--1) =P2( 2)/ �9

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

( 0 GN+I /T ~ be the universal quotient bundle on G(1, N). Let Q2 : = Q N - 1 , N + I := ~ GO,N)/ 2)

p~(ck(Q2)) is the pullback of a Schubert cycle

I intersects a fixed } (x, l) E A(N) (k + 1)-codimensional linear subspace "

/~* = Op(T2)(--1) a n d / 5 . = Q1,T~ imply p~(Q2) = p~(Q1)/.P* and so

k * C P2( k(Q2)) = ~ -~HJP k- j .

j=0

The relative tangent bundle is

* 1 /5*. TA(N)/G(1,N ) = Op(T2)(1 ) ~ (P2(T2) /Op(T2)(- - )) = ~I (~


Now we want to describe the restr ict ion

inl.N+lz n : ~ N + l / n

For this we give embeddings an : A(N) --~ F,(PN with " IAN+~(pN ) Z n Nq-1 ~--- O/nO~ n .

D e f i n i t i o n 3 .3 .10 . We want to define a , : A(N) , F n ( P N ) for n _> 1 inductively.

We have

T P N = O P N ( 1 ) @ Q I .

So there is a na tura l i somorphism a l : A(N) = P(Q1) ------4 P(ZpN ) with

a~(sl) = a~(Op(TPN)(--1))

= TA(N)/G(I,N ).

We put AI := FI(PN). A1 is mapped to G(I,N) by p2oa~ I. TA,/G(I,N) is a sub line bundle of TF~ (PN)' and the diagram

TA1/G(],N ) ' ) TFI(pN)

~1 , , I ; ( T I , N)

commutes. So T&/G(I,N) is a sub line bundle of G I ( P N ) C TFI(PN)"

We assume by induct ion tha t a,~ : A(N) , F n ( P N ) is an embedding. Let

An C F n ( P N ) be its image. Am is mapped to G ( 1 , N ) by p2oa~ 1. We also assume

tha t TA~/G(1,N) is a sub line bundle of Gn(PN)[Ao. Let

f l n + l : An - 1 fn+l(An) C F n + I ( P N )

be the embedding defined by the sub line bundle TAn/G(1,N ) of 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 iagram

TAN+I/G(1,N ) ) TFu+I(pN)

1 ~dfN+l

~N+~ ' f ; ! + l C r P N )

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 put OLn+ 1 : = ~n+100~n- This is a closed embedding. We get induct ively for all n:

a*(sn) = TA(N)/G(1,N ) = !ft @ P*.

L e m m a 3 . 3 . 1 1 . i . l ~ + . . . . = a~0e~. ~,N+Ik-C~N)

P r o o f : We only have to show tha t

inltw+11 = C*.oenl/[N+~] (N+I) (N+I)

holds for every line. Here fin+l] i8 the closed subvariety of l[ "+1] paramet r iz ing ~(n-l- 1) subsehemes of length n + 1 which are concentra ted in a point of I. The projec t ion

][N+I] P : "(g+l) ~ l mapping such a subseheme to its suppor t is an i somorphism and

6 :~- enl l tN+l ] op--1 is the m a p (N+l)

So we have to show tha t a . oe = in3 holds for the embedding i~ 3 : I~-----+g21pN(I) C F , ( X ) .

By definition 3.3.10 a l : A ( N ) ---. P(TpN) is defined by the sub line bundle

TA(N)/G(1,N) C T p N. So the sub line bundle Tl C Tp NII defines the embedding

~10~: l - - - , f v l ( l ) c F I ( P N ) .

By remark 3.3.4 this also defines il,i. By induct ion we assume tha t anoe = in,l. In

par t icu la r we have

(a~oe)(l) = r~(1) C g~,~,N(l).

The embedding ~+1 : A,, ~ f ~ a ( A ~ ) is given by the sub line bundle

TA./a(1,N) C G~(PN)IA. , i.e.

/~-+llF,~O) : r , (1) ~ f211(F,(1))

is given by TF~(0 C Gn(PN)IY.(0 . By remark 3.3.4 this also defines the embedding

jn+l , l : Fn(1) ----+ f j~ l (Fn( l ) ) .

So we have/3n+lIF~(0 = j~+lj , and thus by remark 3.3.4

a n + l O e = Jn+l , l o in , l = i n + l , l . []

Now we can show a general formula for the numbers of higher order contacts

of X with lines in PN,


Def in i t i on 3.3.12. Let X C PN be a smooth projective subvariety. n + l A l n + l ( P g ) ~ PN be the projection. We put

Let p :

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 ~ Fn(PN) to Aln+l,X. Let [Fn(X)] be the class of Fn(X) in A*(g:,IpN(X)). The class of n th order contacts of

X with lines PN 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 PN 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 PN 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 PN 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

{ there is a l ine / with which } ql(a '~,~(Fn(X))) = x E X X has n *h order contact at x

3.3. Semple-bundtes and the formula for contacts with Iine~ 139

and ql(,~lx(F.(X)) n q;~ (W(h)))

there is a line l } = x 6 X intersecting h and having n th order .

contact with X at 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 part icular 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 many n th

order contacts of X with lines intersecting h in points of Y. Then the number of

these contacts counted with multiplicities is the intersection number Kn, t (X) " [Y].

T h e o r e m 3.3 .15. Let n be a positive integer. Let X C PN be a smooth projective

variety of codimension r, let NX/pN be the normal bundle of X in PN 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

k=O s=max(O,k-l)

n

E A e ( X ) .

In particular we have in the case l = O:

, ( ) K n ( X ) = E ( - 1 ) k ( N + k ) k E l~j~-~ /_/k II%(Nx/p,,) . k=O i l+. . .+in=d-k j = l 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. I f 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 E (_1) 8 N +k-I .

k=O s=max(O,k--I) S

( )


I f 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 i l + . . . + i ~ = N - r - k

We first show the following lemma:

L e m m a 3.3 .16.

(I) 0, k < O ,

( q l ) , ( p N - , + k ) = 1, k = 0," - H , k = 1; O, otherwise.

(2) l

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

(q l )* (pN- l+k) = - E g~ (q l )* (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 - I

s=max(O,k - - l )

Hk (-1)= s s = m a x ( O , k - l )

P r o o f of t h e o r e m 3.3.15: 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 ~ )) . . . .

�9 c,-(qr(Nx/p= ) @ a~,x(s~) | | a~,,x(s*,)))

j=1

n(• ) = 3 ql ( c i ( N x / P N ) ) ( P - H) r-z j= l i=1

So we get by the projection formula and remark 3.3.9:

K=,~(x)

= (ql).(a*,x([Fn(X)]) �9 q~(cdQ2)))

( i -/ )) = * 3 ql ( C i ( J Y x / P N ) ) ( P - H ) r - '

j= l i=1

' ( )n = Z E (ql)* E H t P t - t ( P - H ) n'-d+k jr-i '%(NX/PN)" k=0 i l + . . . + i n = d - k t=0 j= l

By the definitions we have nr - 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 PN with lines in PN as

linear combinations of monomials in H and the Chern classes ci(Nx/pN ). Using the

formula

c(Nx/PN ) = (1 + H)N+I/c(Tx)


we can replace the Chern classes of Nx/pN by those of X if we want. The result

will however be more complicated this way. It is easy to check tha t the formulas

after proposi t ion 3.2.19 can be obta ined as special cases.

Now we want to show that more generally the class in A*(X) of the locus

where a smooth project ive variety X C P N has rt th order contact with a given

family of curves is proper ly in terpre ted a l inear combinat ion of monomials in H and

the Chern classes of X. This l inear combinat ion will depend on the familyCT. We

will not t rea t here the much more difficult question how to determine this l inear

combinat ion for a given family CT. The argument is s imilar to tha t at the end of

section 3.2. Fi rs t we will generalize the Semple bundles to a relat ive s i tuat ion.

D e f i n i t i o n 3 .3 .17 . Let T be an irreducible algebraic variety. Let X - - ~ T

be a smooth morphism of relat ive dimension d. We will induct ively define va-

rieties Fn(X/T) and vector bundles Gn(X/T) on Fn(X/T). Let Fo(X/T) :=

X, Go(X/T) := Tx/T. By induct ion assume that Fo(X/T), . . . ,F ,_ I (X/T) are

a l ready defined. Assume tha 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 project ion. Let sn be the tautological subbundle of f*,X/T(G,~_I(X/T)). Now we define the subbundle Gn(X) of TF,(X)/T by the d iagram

0 ~ TFn(X/T)/Fn_I(X/T ) ~ TFn(X/T)/T ---+ f * , x / T T F n _ I ( X / T ) / T ~ 0

0--"* TF,,(X/T)/F,~_I(X/T ) ~ Gn(X/T) --* s,, ~ O.

If Y C X is a (locally) closed subvariety such tha t the restr ic t ion to Y of the

project ion X ) T is a smooth morphism of relat ive dimension m, then we see

in a similar way as in the proof of lemma 3.3.3 tha 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 .18 . Let T be a smooth project ive variety of dimension m - 1 and

CT C P N x T a flat family of curves, i.e. we have the project ions

CT

P N T,


p2 is flat and for all t E T the fibre C, = p~-I (t) is a curve. We assume in addi t ion

tha t there is a dense open subset CT, O C CT such tha t the res t r ic t ion CT, O ~ T

is a smooth morphism.

Then 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 and [Fn(CT)] its class in

A(N-1)(n+I)(Fn(PN) X T). Let

p : F ~ ( P N ) x T ~ F ~ ( P N )

be the project ion. We put

K~(CT) := p.([Fn(CT)]) E Ar (F~(PN) ) ,

where r := N + ( N - 1 ) n - r e . Let X C P N be a smooth project ive variety of

dimension d. Let i,~,x : F , , (X) ~ F n ( P N ) be the embedding. We put

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 .19 . K n ( X , CT) is a candida te 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 (Fn(CT,o /T) M ( F ~ ( X ) • T)))

there is a t E T such tha t x is a smooth point of } = x E X Ct and Ct has n th order contact with X in x "

Assume in par t icu la r e = d, i.e. m = (n + 1)(N - d), and assume the subset

Fn(CT,o/T) N (Fn (X) • T) C F n ( P N ) • T

to be finite and to coincide with fi',~(CT) f) (F,~(X) • T). Then the number of n th

order contacts of X with curves in the family CT counted with mult ipl ic i t ies is

K.(X, CT).

P r o p o s i t i o n 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 im(CT) = m. Assume e = N + ( N - d)n - m

and 0 < e < d. For all partitions

(~) = (1~, 2~ , . . . )


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 )

P r o o f : We show more genera l ly tha t for any W G Ae+n(d-1)(Fn(PN)) and for all

pa r t i t i ons a of s < e there are integers n~, sa t is fying

(g,,,,:i,(i:,.,.(w)) = Z s = 0 c~CP(s)

As A * ( F n - I ( P N ) ) is gene ra t ed by H, P1 := C l ( S l ) , . . . , P , , - 1 := c1(~,~_1), it is

enough to prove the resul t for m o n o m i a l s in H, P1, �9 - �9 P,,. We will now suppress

i*,x in the no t a t i on and wri te g~ ins tead of g~,x. Let M = MoP~ 1 . . .P~" be a

monomia l . 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 hen we have (g~) , (M) = Mo. Othe rwi se let j0 be the largest j such

tha t lj ~k d - 1. By p ropos i t ion 3.3.2 we see tha t (g~) , (M) = 0 if ljo < d - 1.

So let ljo >_ d. By p ropos i t ion 3.3.2 we can express M as a l inear c o m b i n a t i o n

wi th 2g-coefficients of m o n o m i a l s N = N o P ~ ~ . .. p ~ n , where No is a m o n o m i a l in

H, c l ( X ) , . . . , c d ( X ) , and we have my = lj for j > j0 and mjo < ljo. So the resul t

follows by induc t ion . []

4. The Chow ring of relative Hilbert schemes of projective bundles

In this chapter we treat the Chow rings of relative Hilbert schemes of projec-

tivizations 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 va-

riety 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 con-

sider 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

A*(Hilb (P(E)/X) to Br3(p(E)/X) to find some of the relations. The most im- p 3

portant case of our result is the variety Cop (Pc), parametrizing triangles with a

marked side in Pd together with a plane containing them.

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 ~ 3 Hilba(p(E)/X) as A*(X)-alg~bra. 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 -very a m p l e n e s s , e m b e d d i n g s o f t h e Hi lbert s c h e m e and the s truc- ture o f A l n ( P ( E ) )

Let X be a projective scheme over an algebraically closed field k. In

[Beltrametti-Sommese (1)] the following definition was made:

Def in i t i on 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~163 , 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~ O z @ s = H ~ C) /ker ( r z , L )

of dimension n. This defines a morph i sm.

r 1 6 3 X ["] ~ Grass(n ,H~

It is clear from the definition that an n-very ample invertible sheaf is also m-

very 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 relat ive s i tuat ion. Let T be a

reduced project ive variety and X a project ive scheme over T. Let ~r : X ~ T be

the project ion.

D e f i n i t i o n 4 .1 .4 . Let s be an invert ible sheaf on X for which also 7r.(s is

locally free. For all n E zW let 7r~ : Hilbn(X/T) ~ T be the project ion. Let

Zn (X/T) C X x T n i l b " (X/T) be the universal subscheme. We consider the d iagram

Zn(X/T)

X

r P x',~ qn

Hilb"(X/T)

T,

in which p and qn are the projections. We get a na tu ra l morphism of locally free

sheaves

on Hilbn(X/T) a follows: let

fa : r*~r.(s ~ 7r*Tr.p.p*(s

h :

be the na tu ra l morphisms of locally free sheaves on Hilbn(X/T). By the eommuta-

t ivi ty of the d iagram 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 and put s := s For a

fixed subscheme Z E Hilb"(X/T) lying in the fibre Hilbn(X/T)t = Hi lb" (Xt ) of

Hilb'(X/T) over t the map rn,.~ between the fibres

zc%r.(s = H~ s

(q.).p*(s = H~163 | Oz)

is jus t given by

rz, c, : H~163 ----* H~163 | Oz).

148 4. The Chow ring of relative Hilbert schemes of projective bundles

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.

Then (qn).p*(s is a locally free quotient of rank n of ~ '7r , (s By the universal

property of Grass(n, 7r.(E)) there is a morphism

eL,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. Then

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 bundle with fibre Xt and

12 an invertible sheaf on X. We call ~2 constant over T, if there is an invertible

sheaf s on Xt and 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 tr ivialisation

7r - l (g i ) ~ g i x X t :

en ,c ] , r l (u , ) = 1u~ x r :Ui x (Xt) In] , Ui x G r a s s ( n , H ~

The result follows by theorem 4.1.3. []

Now we want to consider the case of the projectivization of a vector bun-

dle. Let E be a vector bundle of rank d + 1 over a smooth projective variety

X. Let P ( E ) p ~X be the bundle of one-dimensional linear subspaces of E and

Op(E)(--1) := T1,E the tautological subbundle 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 bundle

of 7r*(E). We note tha t dualizing gives a na tura l i somorphism d : P ( E ) ~ P ( E * )

with d*(Ql,E*) = Op(E)(1). For Y = P ( E ) and Y = 15(E) respectively we again

z.(r/x)

have the project ions

~// p NN q~

Y Hi lb~(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 := CQtT,,~ : Hilb '~(O(E)/X) ---+ Grass(n, S y m ~ ( E ) )

o v e r X w i t h r =- (qn).p*(Q~'~).

(2) r := Cn,n iS an embedding.

Proof: W i t h respect to a sui table local t r iv ia l isa t ion of E over X we have 7r -1 (Ui) = @ n . Ui x Pd and Q1,EI,r~(u0 = P2(OPd(n)), where p2 : Ui • Pd ~ Pd is the project ion.

(1) follows by 7 r . ( Q ~ ) = Symm(E) . (2) follows immedia te ly from 4.1.8 and (1). []

N o t a t i o n . In future we will wri te Cn ins tead of CQ~,~,. and more general ly Cm,n

for CQ~,~,~, if X and E are unders tood and m _> n - 1.

Now we specialize fur ther to the case tha 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 tha t its

geometric meaning is more visible.

N o t a t i o n . Let Hn := (qn),p*(el((gp(E)(1))) C A I ( H i l b n ( p ( E ) / X ) .

R e m a r k 4 .1 .10 . 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}.


P r o p o s i t i o n 4 .1 .11. r : Hilbn(P(E)/X) ~ P(Symn(E) ) is an isomorphism

such that r = Hn.

P r o o f : As P ( E ) is a locally trivial P l -bund le over X, Hilbn(P(E)/X) has to be a

locally trivial Pn-bundle over X. The same is true for P (Symn(E) ) . So the embed-

ding Cn : Hilbn(P(E)/X) ----* P(Symn(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 n t- 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 . . . x x 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 Zn(P(E) /X ) is the reduced subscheme

Zn(P(E) /X ) = { (x ,Z) e P ( E ) x x Hi lbn(P(E)) x e Z} .

We have a natural morphism

r P ( E ) x x Hilbn- I (p (E) /X) ~ Hi lb~(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 haveamorphism

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 | Symn-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(Sym"(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.

Def ini t ion 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))


be the projections. In particular let Aln(Pd) C P~] be the subvariety given by

Al~(Pd)= { Z E P~] Z i s a s u b s c h e m e o f a l i n e i n P 4 }

and al Z,~ (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)) ~ G.

Let F C P (E) • G be the incidence variety

F : : { ( x , / ) E P ( E ) x x G x E l }

with the projections

P (E)

Then we can identify F P~G with P(T)

F

\p2

G.

~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) --~ Hi lb ' (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) • Hi lb~(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:

L e m m a 4 . 1 . 1 5 .

(1) Cn = CnoPl 1 : AU(P(E)) ----+ P(Sym~(T) )

is an isomorphism over G, such that r = Hn.

(2) r := r : Z~ t (P(E) ) ---+ P ( T ) x a P ( S y m n - l ( T ) )

is an isomorphism satisfying

r = H

"r = Ln-1.

So by proposit ion 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:

E x a m p l e 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 part icular we have with P := c1(Q1,3):

2~[P, H . ] A*(AIn(Pz)) = ([pa, H~+a .+1 n -( 2 ) H . P + w ( n ) H~-IP2)

Here

= { n(2n-}-l)(n-.}-l) (3n2--2n)(n2--1) 6 + 24 , n odd;

n(2n+l)(n+l) (n--2)(n--1)n n a ( n - - 1 ) 6 + 24 Jr- 8 , n e v e n .

154

~ 3 4.2. C o m p u t a t i o n of the Chow ring of 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 proposition 2.5.19 we 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).

Def in i t ion 4.2.1. Let Z2(P2) C P2 • p~2] be the universal subscheme and let

Z~(P2)

/p \q2

P2 p~2]

be the projections. Let H := rcs*(el(Op~(1))) and let

a x e : = A l 2 ( 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) 155

N ~ 3

Let A C Hilb (Pc) be the subvariety

{ - -3 } .4 := (Z2, Za) �9 Hilb (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

- - 3 the line through one of the subschemes Z1 C Z3 / /~2 := (Z2, Za) �9 Hilb (P2) of length 2 containing res(Z2, Z3)

J passes through a fixed point

and P2 := [P2I-

R e m a r k 4.2.2. Geometrically H2, H, P can also be described as

H2 = (Z2,Za) �9 Hilb (P2)

[{ - ' L n = (Z2, Z3) �9 Hilb (P2)

P = Z2, Z3) �9 Hilb (P~)

a point of Z2 }] lies on a fixed line '

res(Z2,Z3) lies on a fixed line }],

l the line through Z2 passesthroughafixedpoint }] "

T h e o r e m 4.2.3.

with

3 2g[H, H2, P, A] A*(Hilb (P2) )= ( I i , h , I 3 , I 4 , I s , I s )

/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 - H2P 2 + HH~ - 3 H H 2 P + 2 g P 2 - 2H2H2 + 2 H 2 P

+ A H 2 P + 2AHH2 - A H P ) .

~ 3 Proof : By example 4.1.16 the subring of A*(Hilb (P2)) generated by H, P, H2 is

2g[H, P, H2] (res • pl)*(A*(P2 x p~2])) _ (/-~,/2:/---~

3 p~2l As the morphism res x Pl : Hilb (P2) ' P2 x is birational, the orientation ~ 3

cycle of Hilb (P2) is the class [*] := H 2 H ~ P 2. The restriction of res x t52 to the

156 4. The Chow ring of relative Hilbert schemes of projective bundle~

subvariety A gives an isomorphism r .4 ----* Z~t(P2) C Z3(P2). By lemma 4.1.15 we have

al Z3 (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 relat ion/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.

Proof : Let

H- := [ { Z E P~3] I Z intersects a fixed l ine}I,

~ : = [ { Z E p ~ 3 ] a s n b s c h e m e Z 2 o f l e n g t h 2 o f Z }] lies on a line passing through a fixed point

So we have by definition H = ( /)2) .(8) , P = (152),(P), A = / ~ ( A ) , and we see that the relations

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:

[ { Z consists of two distinct fixed points } ] r = 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.


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. []

L e m m a 4 .2 .5 . A P 2 = 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. We consider the natural

morphism r .4 ---+ P(T2,3) x p : P(Sym2(T2,3)). Let

:P(T2,3) xp2 P(T2,3) - - ~ 152

:P(T~,~) • P(T~,~) • P(T2,~) ----, P~

be the projections. Then we see

r =

r =

N .

r (c , (Q1,3)) ) - -

(t)2 • p3)*(~*(cx(Q~,3)))

~- (c~(Q~,3)),

* ~ ' * C (151 • ( ( 1 (Q1,3) ) )+(p l •

2~ (c1(Q1,3)),

2PIz.

The lemma follows. []


From lemmas 4.2.4 and 4.2.5 we get the relation I5 = 0:

A 2 = A ( P + P 2 - H - H 2 )

= 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 • A 5

H H~ P 2

H 2 H ~ p

H 2 H~ P~

AHH2 p2

3 1

1

A

1

1

- 1

A 2 • A 4

It~ P 2

H H 2 p

H H ~ P 2

H2H~

H2 H2 P

H ~ p 2

A H 2 P ~

AHH~PI

A H P ~

3 1

1

1

1

3 1

1

3 1

1

- 1

- 1 - 1

A H

1

1

- 1

- 1


A 3 x A 3

H~P

H2P 2

HH~

HH2P!

H p 2

H2H2

H 2 p

AH2P

A p 2

AHH2

A H P

AH~

3 1

1

3 1

1

3 3

3 1

1

- 1

- 1 - 1

- 1

3 1

1

1 3 1

1

- 1 - 1

- 1

- 1

- 1

- 1

By solving the

We see that these conditions are only satisfied by the elements of the basis oceuring

in the above intersection matrices. []

h < 2 (i~),

p < 2 (h) ,

h2 -< 2 (h ) ,

a < 1 (h ) ,

h + a < _ 2 (14),

a + h2 <_ 2 (I6).

We see that the intersection matrices are all invertible over 2~.

system of equations given by the last intersection matrix we get 16 = 0.

~ 3 End of the 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 I1 , . . . , / 6 it can

be expressed as a linear combination of monomials A~HhpPHh2 2 satisfying

160

~ 3 4.3. T h e Chow ring of 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 Defin i t ion 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

V ( P ( E ) ) := P ( E ) x x P ( E ) x x P ( E ) x x Hilb2(P(E)/X) xx Hilb2(p(E)/X)

xx HilbZ(P(E)/X) xx HilbZ(e(E)/X)

and s C V(P(E)) be the subvariety defined by

H3(P(E)/X) := { (xl, x2, x3, Z1, Z2, Z3, Z) C V(P(E))

xi,xj C Zk; Zi C Z; ] 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 ~ 3

(1) Hilb (P(E)/X) is a locally trivial fibre bundle o v e r X with fibre Hilb (P2).

(2) s 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).

Def in i t i on 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 ) 161

and let s : U(P (E ) ) , 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 Wo(P(E)) C U(P (E) ) be the set of honest

triangles and W ( P ( E ) ) the closure of Wo(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 r Op~(i),i x 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 • H3(p2) , 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


is the variety Wo(P(E)) C U(P(E)) of honest triangles in P(E) . As/~(31,1,1)(X ) lS

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:

Def in i t ion 4 .3 .5 . Let

151,f2,p3 : W(P(E) )

ql, q2, q3 : W(P(E) )

4: W(P(E) )

be the projections. We put

P ( E ) ,

, P ( E ) ,

Grass(3, Sym2(E))

a : = p;(c,(Op(E)(1))), b : = p~(r c = p~(~,(op(E)(1))),

a := ~t~(c,(T~,E)), fl : : (t~(cl(T~,E)), "7 := ~t~(cl(T~,E) )

Then 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 * ) ) = - ~ * ( c l ( 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 � 9 W(P(E) ) F is the net of conics

on the fibre P ( E ( ~ ( x i ) ) ) ~ P~, ' con ta in ing ~1

and r �9 AI(w(P(E))) the class of

~ : = ( X l , X 2 , X 3 , ~ l , ~ 2 , ~ 3 , r ) e W(P(E) )

Xl = X2 = X3, ] 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:

L e m m a 4 .3 .6 .

(1) r = e + a + b + c + # l - ~ - f l - %

~ 3 4.3. The Chow ring o] Hilb ( P ( E ) / X ) 163

(2) (3) (4)

(5)

(6) (7)

a 3 =#la ~ - # 2 a + p a (and similarly ]orb andc),

(~3 = 2#1a2 _ (p2 + #2)~ + #1#2 -- #3 (and similarly ]or fl and 7),

a~ : a 2 + ~2 - #1~ + #2

(and similarly for a, 7; b, (~; b, 7; c, a; c, fl respectively),

Ta = Tb = TC~

6T = 0 ,

Let

Now we want to describe the classes

~*(a), ~*(b), ~*(c), ~*((~), (~*(~), ~*(7), ~*(e), ~*(r) �9 AI(Ha(P(E) /X)) .

Pl ,P2,P3 : H3(P(E) /X) ~ P(E) ,

ql, q2, q3: Ha(P(E) /X) ~ Hilb2(p(E)/X),

q: Ha(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) and with F = P(E0r (z l ))) . �9 Ha(P(E) /X) m s is the ideal of Z in OF ~

F, x l

Proof : The statements on (~*(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).


We consider this condit ion fibrewise. As r is the kernel of the restr ic t ion map

rz : g ~ Op2(2)) ) H~ Oz| , the condit ion on 52,3(Z), means tha t

Z is a subseheme of the line axe(Z1) through Z1. So also Z2 and Z3 are subsehemes

of axe(Z~), and the condit ions on axe(Zj) and axe(Z3) are fulfilled automatical ly .

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) and r i s t h e n e t o f c o n i c s

�9 H3(p(E) /X) in the fibre P (E(z r (x l ) ) ) = PJ , " having a singular point at Xl

Let ( z l , x j , z3, ZI , Zj , Z3, Z) be a point of this subvariety. The condit ion on r

means tha t Z lies in the subscheme 5 C F = P ( E ( : r ( x l ) ) ) with suppor t xa which

is defined by m 2 in OF,~t. 2 is a subscheme of length 3 of P(E(Tr(x l ) ) ) , so we F, Xl have Z = Z. As Xl, x j , x3 are subschemes of Z, the condit ion xl = x2 = x3 follows

au tomat ica l ly fi'om the condit ion o n r The result follows, u

~ 3 Now we turn to the variety Hilb (P(E)/X) of tr iangles in the fibres of P ( E )

~ 3 ~ 3 with a marked side. Via res: Hilb (P(E)/X) , P ( E ) we regard Hilb (P(E)/X) as a subscheme of P ( E ) Xx Hilb2(p(E)/X) xx 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 na tu ra l morphism

7n4z :-~3(P(E)/X) --~ ~lb3(P(E)/X);

( X l , X j , x 3 , Z 1 , Z J , Z 3 , Z ) , ) ( x l , Z 1 , Z )

Let

Let

~ 3 := 7r147o~ -a : W ( P ( E ) ) ----* Hilb (P(E)/X).

~ 3 p a : Hilb (P(E)/X)

~ 3 P2: Hilb (P(E)/X)

~ 3 ~ : Hilb (P(E)/X)

be the projections. Let r HilbJ(P(E)/X)

, P ( E ) ,

) Hilbz(P(E)/X),

, Hilba(p(E)/X)

, P (SymJ(T j ,E) ) be the i somorphism

from lemma 4.1.15 with ~* ~ 2 ( O P ( S y m J ( T j , E ) ) ( 1 ) ) ---- (qj) .p*(Op(E)(1)) . H e r e

Zj(P(E) /X)

~/P "~q2

P ( E ) Hilb 2 ( P ( E ) / X )

~ 3 4.3. The Chow ring of Hilb (P(E)/X) 165

are the natural projections of the universal subscheme.

Def in i t i on 4.3.8. We put

H := p~(c,(Op(E)(1))), ~* *c 1 "* ~* H2 :=p2(q2) .p I ( O p ( E ) ( ) ) =P2r

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))).

L e m m a 4.3.9. ~ * ( H ) = a , ~*(H2)=b+c, ~ * ( P ) = a, ~-*(A) = e.

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) • C l}

and F(E)

~/p, \p2

P ( E ) P ( E )

the projections. It is easy to see that there is an isomorphism ~ : F(E) -----+ P(T2,E) over P ( E ) with ~*(Op(T2.E)(1)) = p~(Op(E)(1)). Let

r2,a : W ( P ( 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) xx F(E) of ( P ( E ) x x 15(E)) • ( P ( E ) Xx 15(E)). The diagram

W ( P ( E ) ) .... , F(E) xx F(E) *• , P(T2,E) x x P(T2,E)

Hilb (P(E)/X) ,2 - - ~ Hilb2(P(E)/X) ~ P(Sym2(T2,E))

166 4. The Chow ring of relative Hilbert 3chemes of projective bundles

commutes. Here r] is the morphism defined by the natural map T2,E | T2,E Sym2(T2,E). With respect to the projections

r l , 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)).

~ 3 P r o p o s i t i o n 4.3.10. ~-* : 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 2 : /~3(p2) ~ 3 A , Hilb (P2) be the restriction of ~ to a fibre

03 (P2). The orientation classes [,] of Hil'-'-b 3 (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

~* is injective. As a homomorphism of free A*(X)-modules P ~

~ 3 --~rp, | 1A*(X) : A*(Hilb (P(E)/X))

~ 3 = A*(Hilb (P2)) | A*(X) , A*(Hs(P2)) | A*(X) = A*(Hs(P(E)/X))

is one to one. So ~* is injective. []

~ 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

w h e r t ~

/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 ~ - 4Pro + 4 ~ ) - 4 ( P 3 - P ~ m + P ~ ) ,

14 : = A(H 2 - P H + P ( P - #1) 21- ~2),

15 := A ( A - 3P + H + H2 + # 1 ) ,

16 := - A H ~ + # I ( - H ~ + H2P + 2HH2 - 2HP) + H2p - H2P 2 + HH~

- 3HH2P + 2HP 2 -- H2H2 + 2H2p + A(H2P + 2HH2 - 2HP).

P r o o f : We have ~'~(A*(P(E))) = A*(X)[H]/(I1). Fur thermore

P = ~axe*(ca(Ql,E)) + #1

and thus

A*(X)[P] ~axe*(A*(P(E) ) ) = ( ( p _ #1)3 + # I ( P - - #1) 2 "[- #2(P - - #1) -~- /23)

= A * ( X ) [ P I / ( 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 - 3P + (6P ~ - 4P#1 + 4#2) - 4 (P 3 - P2#1 + P#2)

and thus

R e m a r k 4 .3 .12. The A*(X)-subalgebra of A * ( P ( E ) / X ) ) generated by H, P, H2

is A*(X)[H, H2, P]

(P'I • ~2)*(A*(P(E) • Hilb2(P(E)/X))) = (I1, /2, I3)

Via

~ 3 Let A C Hilb ( P ( E ) / X ) be the subvariety defined by

(x, Z1, Z) . 4 : ~ ~ 3

�9 Hilb ( P ( E ) / X )

Z lies on a line ) in the fibre P ( E ( r ( x ) ) ) / "

passing through x

~ ~ = ax~~ : .~ - - ~ ~ ' (E)


,4 is a variety over 15(E).

~ 3 ~ • ~ : Hilb ( P ( E ) / X ) ~ P ( E ) x x H i l b 3 ( p ( E ) / X )

maps fit isomorphical ly onto Z ~ t ( P ( E ) / X ) . By lemma 4.1.15 there is an isomor-

phism

"r : Z ~ t ( P ( E ) / X ) ----* P(T2,E) • P(Sym2(T2,E))

over I~(E) satisfying

(51 • wl~') (P (CI(OP(%,E)(1))) = HI X,

(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 ) - /22) .

~ 3 The r e l a t i on /4 = 0 in A*(Hilb ( P ( E ) / X ) ) follows by [A]-- A.

In order to prove the relat ions I5 = 0, /6 = 0, we want to compute in

A * ( W ( P ( E ) ) ) and use the reations of Collino and Fulton from lemma 4.3.6. The

proof of 15 = 0 is simple.

~ * ( A ( A - 3P + H + H2 +/21)) = e(e - 3a + a + b+ c+/21)

= E ( e - o ~ - f - 7 + a + b+ c + /21)

z s

z O .

So Ix = 0 holds. In order to proof I6 = 0, we write the relat ions in such a way that

they can be appl ied formally (by subst i tut ing) .

R e m a r k 4 .3 .13 . In A * ( W ( P ( E ) ) ) the following relat ions hold:

(1) a 3 = a2/21 - a/22 +/23 and similarly for b and c,

( 2 ) a 3 =

( 3 ) ~ =

(4) 3 2 = - a 2 + a/3 + # 1 3 - - / A 2 ,

(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 + b 2,

2#1 a2 -- (#12 + #2)a +/21/22 - - /23 and similarly for 3 and 7,

- b 2 + ba +/21a - /22 ,

N 3 4.3. The Chow ring o f Hilb ( P ( E ) / X ) 169

(9)

(10) eb =- ea + (a - b)(c + #1 - - O l - - ~),

, c = ~ , + ( , - c ) ( b + ~,1 - ~ - "y ) .

Now we jus t a p p l y these re la t ions formal ly . We get

0 = ~ * ( A ( H 2 - H P + 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- ba~.

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),

~ * ( H H 2 P ) = aa (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),

~ * ( H 2 p ) = a2a ,

" ~ * ( A H P ) = aae ,

~ * ( A H H 2 ) = a(ab + ac + 2 a p l - 2aa - a/3 - a7 + 2ae

- b 2 - 2bc - b#l + 2ba + b/3 + c 2 - c # 1 + c7),

~ * ( A H 2 P ) = ea(b + c)

= a 2 b - a2c + a2/3 - a27 + 2ac 2 + 2a#2 - 2aa/~

+ 2aae - b2/~ - 2bc 2 - b#2 4- 2ba/3 + c27 - c#2,

~ * ( A H ~ ) = e(b + c) 2

= a 2 b + a2c + 6a2#1 - 4a2a - 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,


^* 2 7r (H2Pl) = #l(b 2 +2bc+c2) ,

~ * ( H 2 P # I ) ---- ~ l ( - b 2 + 2b(~ + c2),

~*(HH2#I) = a#l(b + c),

"~*(HP#I) = aplc~.

Thus we have

~ * ( - A H 2 + ~ I ( - H 2 -4- 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 - 2 H P ) )

---- 2(--a2c-- a2 pl + a2 0~ + a2 /3 + ac 2 + a#2 -- ao~t3

+ b2c + b2#1 - b2c~ - b2/3 - bc 2 - b#2 + bo~/3)

= 0 .

As ~* is injective, the relat ion/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 , h 2 < 2 , a < l , h + a ~ 2 , a + h 2 < 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.

De f in i t i on 4 .3 .14. Let ~Cop3(pd) C Hilb3(pd)-- • G be the subvariety

( __3 ) Cop ( P a ) : = ( ( Z a , Z ) , E ) E H i l b (Pa) x G Z c E

Let F C P a x G be the incidence variety F := { (x ,E) e Pd • G I x C E} with

ptojections F

Pa G.

~ 3 4.3. The Chow ring of Hilb (P(E)/X) 171

There is an isomorphism r : F , P(T) over G with r = ~ 3

p~(Or, d(1)). We see immediately from the definitions that Cop (Pd) is the sub- ~ 3 ~ 3

variety Hilb (F/G) C Hilb (Pd) x G. So we get an isomorphism

~ 3 ~ 3 r Cop (Pd) ----* Hilb (P(T)/G).

The projection/51 : Coop3(Pd) - - 3 ----* Hilb (Pd) is a birational morphism (a general

subscheme of length 3 lies on exactly one plane). It is an isomorphism outside

{ - -3 } -40:= (Z1,Z) EHilb (Pd) Z lies on a line .

Over a point (Z1, Z) E A0, lying on a line l its fibre is

/ 5 1 1 ( Z 1 , 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 N 3 R e m a r k 4.3.15. Cop (Pc) is obtained by blowing up Hilb (Pd) along Z~t(Pd).

~ 3 Defini t ion 4.3.16. Let A',H',H2,P' ' ,#1, #2, # 3 t , 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 Coop3(pd) res(Z1,Z)lies on a fixed hyperplane }],

[{ }] H; := (E, (Z1, Z)) e Cop (Pd) supp(Z1) intersects a fixed hyperplane ,

[{(E,(Z1,Z))~3 the line passing through Za intersects a fixed }] E Cop (Pa) 2-codimensional linear subspace '

pt :z

'E{, #a := E, (Z1, Z)) E Cop (Pa)

A:=[{(E,(Z~,Z))eUoop~(Pd) linear subspace

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 :


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 deter-

mined by the position of subschemes relative to lines and planes in Pd.

173

4.4. The 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 - - ~ p~3] Defini t ion 4.4.1. Let ~ : Hilb (P2) be the projection. Let

H, ~ ~ a 1 (e~l),

~,p,~ e A:(P~1),

5,/~ E A3(P~ 3])

be the classes defined by

,i

&

:= ~,(H), := [ { Z C P ~ a] Z l i e s o n a l i n e } ] ,

:= ~.(H~), := ~_. (p2),

:= [{ZE 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;


i = 5 :

i = 6: Iz 3.

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 Def in i t i on 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 := ~ . (p2) C A2(Hilb3(p(E)/X)),

:= ~ . (HP 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. Let i :

AI3(P(E) /X) ~ Hilba(P(E)/X) be the embedding and

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) and /~ := 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: [-Ih 2, [-Ihp,

- - 6 : ~3.

4.4. The Chow ring ofHilb3(p(E)/X) 175

P r o o f i (1) follows from (2). Immediate ly from the definitions we get for the fibre

F ~ p~a] of Hilb3(p(E)/X) over a point x E X:

H i t = ~ , Ai r = i i,

h i t = 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 ~*, ~. , ~*, ~.. Remember tha t ~ 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

Pl,2: Hilb (P(E)/X)

(x , Z l , Z ) l

, P ( E ) Xx Hilb2(P(E)/X);

, (x, Zl) .

Lemma 4.4.5. For W E ~,2(A*(P(E) x x Hilb2(P(E) /X) ) ) we have

~*~,(w) = w + ~ . ( ~ ( w ) ) .

P r o o f : Let W = ~ i ai[X/] be the representat ion of W as a linear combinat ion of N,N X classes of irreducible varieties. Then we have ~ '*~.(W) }--~i aiTr ~r.([ i]). So it is

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 ) there is a subscheme Z1 C Z }] ~.p'~l,2([Y]) = Z e of length 2 with (res(Z1,Z),Z1) e Y "


So we also have

~*<~,2 ( [Y] )

E Hilb ( P ( E ) / X )

there is a subscheme ZI C Z }] of lengm 2 wah (r~(Z~, Z), Zl) C Y

[{ _ 3 }] ~-- (x, Z1 ,Z) E H i l b ( P ( E ) / X ) (TcN(Z1,Z),Z1) ~ r

+ { (x, Zl,Z)

~ 3 E Hilb ( P ( E ) / X )

there is a subscheme ZI C Z ] 1 of length 2 with x C ZI J and ( res (Z l , Z), ZI) E Y

= - , ~ , ( ~ 2 ( [ y ] ) ) . [] ; , , : ( [ 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)) is injective,

and "~*(A*(Hilba(P(E)/X))) is the A*(X)-subalgebra generated by

~*([-I) = H + g2,

~*(A) = A,

F*([z) = H 2 + H2P - 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 ( 2 P 2 - 2 H 2 P + H ~ - H P + H H 2 + H 2 - A P + A H 2 - A H )

+ # 2 ( - 2 P + H2 - H) + #2( -H2 + g + A) + 2#1#2 + #3.

Proof." By proposit ion 4.4.4 A * ( H i l b 3 ( P ( E ) / X ) ) is as an A*(X)-a lgebra generated

by H , A , h , f i ,&,6,~. For each fibre F -~ P2 the map

~ 3 ~P2 := ~*]FE31 : A*(P~ 3]) - - ~ A*(Hilb (P2))

is one to one. As a homomorphism 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 ) , H i l b 2 ( P ( E ) / X ) = A I 2 ( p ( E ) / X )

be the projection and p~ := ~21X. Then we have by definition

The diagram

P -- ~ a x e * ( c l ( 2 , E ) ) ,

? = axe*(Cl(T~,E) ) E A ~ ( A I 3 ( p ( E ) / X ) ) .

A I 2 ( p ( E ) / X )

NNa axe

~'(E)

'"N t

A I 3 ( p ( E ) / X )

~ / axe

commutes. So we get ( # ) . ( / 5 ) = PIX and thus ~-*(~) = AP, "~*(~) = A P 2. By

l emma 4.4.5, l emma 4.3.9, remark 4.3.13 and the projection formula we have

,~*(/~) =

~*(~) =

~*(~) =

= p 2 ~_

= p 2 +

p2 _

~*(f?) =

H + ~.(b)

H + H2,

H 2 + ~.(b 2)

H 2 + ~,(bo~ - ~2 + # 1 ~ -- # 2 )

H 2 + H2P -- p2 + 2#1P - 2#2,

+ ~ . . ( 3 ~ + ~2) p2

~ ' . ( a ( / ~ -4- 3') -- -t- #1(• -I- 3') -- 2#2) 4.3.13(4), (5) 2a 2

1^ ~ r . ( ( a + # l ) ( e + a + ( b + c ) - a - ' r + # l ) - a a 2 + 2 # 2 ) 4.3.6(1)

H P + HH2 - H 2 + A H + # I ( - P + H2 + 2H + A) + #~ - 2p2,

+ 1~',(b/32 + c72). H p 2 Z

Fur thermore 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)


= - ( a 2 + ~: ) (b + c) + (a + ~ l ) (bZ + c7) ,

b9 + c7 = (b + e ) (9 + 3') - b3' - r = (b -4- c)(~ + 3') + 2a2 - a(/3 + 3') - b 2 - c 2 4.3.13(7), (8)

= 2a 2 + (b + c - a)(/~ + 7) + 2~ - (b + c)c~ - 2#1o~ + 2#2 4.3.13(3)

= 2a 2 + 2 a 2 - (b + c)(~ - 2#1a + 2#2 + e(b + c) - ea 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 ro j ec t i on fo rmula we get

~*(~) = 3 H P e + H 3 + H e p - 2 H H e P - He l l2 + H H 2 - A H 2 + A H H e

+ # I ( - H P + HH2 - 2 H e P + H~ + 2 P e + AHe - A H )

+ #21(-2P + He - H ) + # 2 ( - H e + 2 H ) + 2#1#2.

The fo rm u la for ~*(/~) is now o b t a i n e d by a p p l y i n g the re la t ions

H 3 = # 1 H e - # e H + # 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*(Hi lb ( P ( E ) / X ) ) in t h e o r e m 4.3.11, and i ts s t ruc-

"ture is in fact r a t h e r s imple , this gives us a s imple desc r ip t ion of A * ( H i l b 3 ( p ( E ) / X ) ) ,

which is also very useful for c o mp u ta t i o n s . We now also wan t to descr ibe th is r ing

by gene ra to r s and re la t ions . Because the re la t ions are very compl i ca t ed , 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 [Ght tsehe (6)] for the list of all re la t ions .

T h e o r e m 4 .4 .7 .

A* ( H i l b 3 ( p ( E ) / X ) ) = A * ( X ) [ H , A, h, 15, ~, 6,/~] (R1, Re, R3, �9 �9 �9 R30)

for suitable classes R 1 , R e , R 3 , . . . , R 3 o in A*(X)[ /~ , f i~ ,h , /5 ,~ ,6~,~] , which are all

listed in Satz ~.~.7 of [Ghttsche (6)]. The relations in codimension at most three

4.4. The Chow ring of Hilb3(p(E)/X) 179

are

R2 := - / ~ p + / ~ 3 + AH5 _ 4Hh - 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." The de te rmina t ion of the relat ions is a t r ivial but very extensive

computa t ion . We use theorem 4.4.6 and the relat ions 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 proposi t ion 4.4.4 as an A*(X)- l inear combinat ion of elements of the basis of

~ 3 A*(Hilb (P(E)/X)) over A*(X) from the proof of proposi t ion 4.3.10. For this we

use the computer . Similar ly we use proposi t ion 4.4.6 and the relat ions 11,. �9 �9 to

express the images of

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,

H h 3

~ 3 under ~* as an A*(X)- l inear combinat ion of the basis of A*(Hilb (P(E)/X)). Now

we only have to solve a system of l inear equations in order to get the relations. For

this we use again the computer . As a result we get relat ions R 1 , . . . , R30.

We still have to show tha t R 1 , . . . , R30 generate all relations. For this we have

to show tha t by using them we can express any monomial in H , A, h,/5,~, 6,3 as

an A*(X)- l inear combinat ion of the elements of the basis from propos i t ion 4.4.4.

To show this we use arguments similar to those in the end par 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 impor t an t special case of ~ 3 Hilb ( P ( E ) / X ) ) . We put G := Grass(d - 2, d + 1) and 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 Pl

P~

\ p 2

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 .15 we 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:

R e m a r k 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) Z l i e s o n a l i n e } ] ,

s [{(Z,E) ECopa(pd) Zintersectsafixedhyperplane}],

h := [{(Z,E) C Cop3(pd)

P:= [{(Z,E) EC~

3:= [{(Z,E) E Copa(pd)

Z intersec sa0xed 2 codimensiooal}l linear subspace

the line through one of the subschemes Z ~ C Z of length 2 intersects two different

fixed 2-codimensional linear subspaces

Z l i e s o n a l i n e i n t e r s e c t i n g a f i x e d }] 2-codimensional linear subspaee

}]

4.4. The Chow ring of Hilb3(p(E)/X) 181

F := / E) c Cop3(Pd)

: : / ' (Z,E) e Copz(Pd)

1-

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 '

}1 ~ : [ { ,~ ,~ , ~ ~op3,p~, ~ ~ ~ o ~ d ~ m e n ~ ~ 1 7 6 ~ h ~ x e d }l

2-eodimensional linear subspace

P 3 : : [{(Z,E) ECop3(pd) El i e sona f ixedhype rp l ane }].

From the definitions we get:

R e m a r k 4.4.11.

r = :~, r = ~ , r = ~, r = 3, r = ~,

r = ~, r = 5,

r = 71, r = ~2 , 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:

1 8 2 4. The Chow ring of relative Hilbert schemes of projective bundles

A 1 x AS:

h3#2 3 2 1

:Ih~u 3 I i

A 2 x A7:

A 3 • A6:

:/2h~3 /~2a~z

h~p z

/:/&pz

h~u 3

6 3

7 6

15 6 4

3 3 1

6 -3 1

1 1 I

1 - i

2 6

3 2 4

4

1 1

- 1

3 2 1

1 1

1

~3

]~3 6 6 6 3 2 6 3 2 1

/:/h2iz 20 6 13 6 2 6 7 6 3 2 4 1 1

/ : /h)# 66 18 27 6 13 15 6 4 4 1

/~2]~#2 25 7 22 9 2 5 3 3 1 1 1

/?/2h/J2 40 9 - 1 0 - 7 - 2 2 6 - 3 1 - 1

~2~2 7 3 6 2 1 1 1

/:/&#2 12 2 -7 -2 1 - I

]~i~# 2 15 4 6 2 i

/:/3~3 15 3 15 6 l 3

]~/:/p 3 3 1 3 i

/~2j#3 15 3 3 -3 - I I

/Ia~ 3 6 i -3 - i

&pa 1 - 1

4.4. The Chow ring of Hilba(P(E)/X) 1 8 3

A 4 • AS:

Hh2 6 6 20 6 13 6 2 6 7 6 3 2 4 1 I

/I]~/~ 26 6 24 66 18 ! 27 6 13 15 6 4 4 1

/f/2h]~ 20 22 6 9 18 25 7 22 9 2 5 3 3 1 1 1

H2ap 22 - 1 6 6 - 8 6 40 9 - I 0 - 7 - 2 2 6 - 3 1 - 1

]~2~ 6 6 2 6 7 3 6 2 I 1 1

/~/&p 9 - 8 2 - 2 12 2 - 7 - 2 I - 1

]~i~]J 18 6 6 8 15 4 6 2 1

/:/3/~2 25 40 7 12 15 ! 15 3 15 6 1 3

]~ f/#2 7 9 3 2 4 3 1 3 1

/;/2A#2 22 -10 6 - 7 6 15 3 3 - 3 - I 1

[/~#2 9 - 7 2 - 2 6 1 - 3 - 1

&/~2 2 - 2 1

3~2 5 2 2 3

if/2# 3 3 6 i 1 1

flap 3 3 -3 1 - i

]~p3 1 1 1

d# 3 1 - I

- 1

1

j ~ 3

1

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I n d e x

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

relative- 2, 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

mult ipl icat ive group 19m21 n-very ample 146

relat ive - 147-149

one-paramete r subgroup 21

general - 21, 25, 80 orbifold Euler number 12, 54-56

par t i t ion 3, 20, 22, 26, 29, 42, 44

graph of - 23

dual - 23

hook difference of 23 point

geometric -

k-valued -

T-valued -

division -

Por teous formula

4, 5, I0, 15, 29 2 1

43 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 t r iangles 12, 60-80

complete tr iangles 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 da t a 81-100

relat ive - 126-127

Weil conjectures 5-8, 29, 35, 41, 49, 74 zero-cycle 3, 40

pr imit ive - 8, 31-34, 45, 75

zeta-funct ion 5

Index of notat ions

~ilb(X/T) Hilb(X/T) Hilb~(X/T) X["], Hilb"(X)

Z,.(X/T), Zn(X) a(,~) X(n) X (') x~,q r n

Hi(X, qt )

p(x, z) z~(x, t) b~(x) P~(X,F~) Hilb'~(R/m"), Hilb"(R)~, t

ZT, GT (~,)~>0 A, A '~

J.(X), J~(X) zT(x), a~(x)

A.(X), A*(X) el G.. p(n), p(~, l) r(~) .&

ti(a), T(a) h~,v(~) Y. *'(~) ( n l , . . . , n ~ ) = (1~ ' ,2~2, . . . ) Gal( ]c / k )

fen(f) P(X, •q) Tn(X, Fq) A(~-), ~(~-)

Hilbert functor

relative Hilbert scheme

relative Hilbert scheme of subschemes of length n

Hilbert scheme of subschemes of length n

universal family

symmetric group on n letters

symmetric power

stratum of X (n)

stratum of X[']

Hilbert-Chow morphism

l-adic cohomology

Poincard polynomial

zeta function

Betti number

set of primitive cycles

punctual Hilbert scheme

Hilbert function strata

jumping index

(thickened) diagonal

jet bundles

relative Hilbert function strata

curvilinear subschemes concentrated in a point

Chow ring

cycle map

multiplicative group

number of partitions of n

graph of partition

dual partition

diagonal lengths of partition

hook difference


set of partitions of n

partitions

Galois group

length of subseheme

"length" of function

primitive 0-cycles

set of admissible functions

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 195

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)

r e s

~3(x) _~[3] ~ 3

Hilb (Z) Grass(m, E)

";rm,E

Grass(m, r) P(E) , P ( E ) Tm,~, Tm,r Qm,E, O,,,,r Z1 �9 Z2 D~(X) T1, Qi w~(x) b~(x) I'2, Q2 (ox)~ Ar A~

Dz,=(Y) Vk(~) nn(x ) F .G W~(X) 5 ~ ( x ) , b L l ( x ) Ts, 03 (E)~n eVE

D~(Y/T) g(x , YT) Fn(X), an(x)

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

complete triangles with marked side

Grassmannian bundle projection in Grassmannian bundle

Grassmannian

projectivized bundle

tautological subbundles

universal quotient bundles

scheme defined by product of ideals

variety of second order data

tautological and quotient bundle over Grass(m, T~ ) bundle of second order data

other construction of D~(X) tautological and quotient bundle over D2(X)

contact bundle (thickened) diagonals of morphism

Ith-order datum of Y at x

degeneracy locus

variety of higher order data

"product" of sheaves

third order data sheaf

variety of third order data tautological subbundle and quotient of W3(X)

contact bundle

evaluation map relative data variety

class of second order contact

Sample bundles

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

196 Index of notations

AI~(PN) aligned n-tuples Kn(X), Kn,l(Z) class of contact with lines Fn(X/T), Gn(X/T) 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) relative triangle varieties ~ 3 Cop (Pd) variety of triangles in a plane in Pd H, ,zl, fz, fi, 6,/~ classes in the Chow ring of Hilb3(P(E)/X) Cop3(pd) Hilbert scheme of subschemes in a plane in Pd

135 140 144 146 149 149 151 151 152

154

160

170 173 180

Printing: Weihert-Druck GmbH, Darmstadt Binding: Buchbinderei Schiiffer, Griinstadt

Bibliography 187

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I n d e x

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

relative- 2, 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

mult ipl icat ive group 19m21 n-very ample 146

relat ive - 147-149

one-paramete r subgroup 21

general - 21, 25, 80 orbifold Euler number 12, 54-56

par t i t ion 3, 20, 22, 26, 29, 42, 44

graph of - 23

dual - 23

hook difference of 23 point

geometric -

k-valued -

T-valued -

division -

Por teous formula

4, 5, I0, 15, 29 2 1

43 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 t r iangles 12, 60-80

complete tr iangles 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 da t a 81-100

relat ive - 126-127

Weil conjectures 5-8, 29, 35, 41, 49, 74 zero-cycle 3, 40

pr imit ive - 8, 31-34, 45, 75

zeta-funct ion 5

Index of notat ions

~ilb(X/T) Hilb(X/T) Hilb~(X/T) X["], Hilb"(X)

Z,.(X/T), Zn(X) a(,~) X(n) X (') x~,q r n

Hi(X, qt )

p(x, z) z~(x, t) b~(x) P~(X,F~) Hilb'~(R/m"), Hilb"(R)~, t

ZT, GT (~,)~>0 A, A '~

J.(X), J~(X) zT(x), a~(x)

A.(X), A*(X) el G.. p(n), p(~, l) r(~) .&

ti(a), T(a) h~,v(~) Y. *'(~) ( n l , . . . , n ~ ) = (1~ ' ,2~2, . . . ) Gal( ]c / k )

fen(f) P(X, •q) Tn(X, Fq) A(~-), ~(~-)

Hilbert functor

relative Hilbert scheme

relative Hilbert scheme of subschemes of length n

Hilbert scheme of subschemes of length n

universal family

symmetric group on n letters

symmetric power

stratum of X (n)

stratum of X[']

Hilbert-Chow morphism

l-adic cohomology

Poincard polynomial

zeta function

Betti number

set of primitive cycles


Hilbert function strata

jumping index

(thickened) diagonal

jet bundles

relative Hilbert function strata

curvilinear subschemes concentrated in a point

Chow ring

cycle map

multiplicative group

number of partitions of n

graph of partition

dual partition

diagonal lengths of partition

hook difference


set of partitions of n

partitions

Galois group

length of subseheme

"length" of function

primitive 0-cycles

set of admissible functions

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 195

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)

r e s

~3(x) _~[3] ~ 3

Hilb (Z) Grass(m, E)

";rm,E

Grass(m, r) P(E) , P ( E ) Tm,~, Tm,r Qm,E, O,,,,r Z1 �9 Z2 D~(X) T1, Qi w~(x) b~(x) I'2, Q2 (ox)~ Ar A~

Dz,=(Y) Vk(~) nn(x ) F .G W~(X) 5 ~ ( x ) , b L l ( x ) Ts, 03 (E)~n eVE

D~(Y/T) g(x , YT) Fn(X), an(x)

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

complete triangles with marked side

Grassmannian bundle projection in Grassmannian bundle

Grassmannian

projectivized bundle

tautological subbundles

universal quotient bundles

scheme defined by product of ideals

variety of second order data

tautological and quotient bundle over Grass(m, T~ ) bundle of second order data

other construction of D~(X) tautological and quotient bundle over D2(X)

contact bundle (thickened) diagonals of morphism

Ith-order datum of Y at x

degeneracy locus

variety of higher order data

"product" of sheaves

third order data sheaf

variety of third order data tautological subbundle and quotient of W3(X)

contact bundle

evaluation map relative data variety

class of second order contact

Sample bundles

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

196 Index of notations

AI~(PN) aligned n-tuples Kn(X), Kn,l(Z) class of contact with lines Fn(X/T), Gn(X/T) 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) relative triangle varieties ~ 3 Cop (Pd) variety of triangles in a plane in Pd H, ,zl, fz, fi, 6,/~ classes in the Chow ring of Hilb3(P(E)/X) Cop3(pd) Hilbert scheme of subschemes in a plane in Pd

135 140 144 146 149 149 151 151 152

154

160

170 173 180

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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. 1522: J.-M. Delort, F. B. I. Transformation. VII, 101 pages. 1992.

Vol. 1523: W. Xue, Rings with Morita Duality. X, 168 pages. 1992.

Vol. 1524: M. Coste, L. Mah6, M.-F. Roy (Eds.), Real Algebraic Geometry. Proceedings, 1991. VIH, 418 pages. 1992.

Vol. 1525: C. Casacuberta, M. Castel let (Eds.), Mathematical Research Today and Tomorrow. VII, 112 pages. 1992.

Vol. 1526: J. Az6ma, P. A. Meyer, M. Yor (Eds.), S~minaire de Probabilit6s XXVI. X, 633 pages. 1992.

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. 1529: J. van Neerven, The Adjoint o fa Semigroup of Linear Operators. X, 195 pages. 1992.

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. 1531: M. Stoer, Design of Survivable Networks. IV, 206 pages. 1992.

Vol. 1532: J. F. Colombeau, Multiplication of Distributions. X, 184 pages. 1992.

Vol. 1533: P. Jipsen, H. Rose, Varieties of Lattices. X, 162 pages. 1992.

Vol. 1534: C. Greither, Cyclic Galois Extensions of Com- mutative 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 Differenti- al Equations. Montecatini Terme, 1991. Editors: M. Furi, P. Zecca. VII, 218 pages. 1993.

Vol. 1538: P.-A. Meyer, Quantum Probabil i ty 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, Quan- tum 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, Multidimensional Weakly Singular Integral Equations. XI, 159 pages. 1993.

Vol. 1550: A. A. Gonchar, E. B. Saff (Eds.), Methods of Approximation Theory in Complex Analysis and Mathe- matical Physics IV, 222 pages, 1993.

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. 1552: J. Hilgert, K.-H. Neeb, Lie Semigroups and their Applications. XII, 315 pages. 1993.

Vol. 1553: J.-L- Colliot-Th61~ne, J. Kato, P. Vojta. Arithmetic Algebraic Geometry. Trento, 1991. Editor: E. Ballico. VII, 223 pages. 1993.

Vol. 1554: A. K. Lenstra, H. W. Lenstra, Jr. (Eds.), The Development of the Number Field Sieve. VIII, 131 pages. 1993.

Vol. 1555: O. Liess, Conical Refraction and Higher Microlocalization. X, 389 pages. 1993.

Vol. 1556: S. B. Kuksin, Nearly Integrable Infinite- Dimensional Hamiltonian Systems. XXVII, 101 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 Regu- larized 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 Sy- stems with C~ X, 188 pages. 1994.

Vol. 1572: L. G~ttsche, Hilbert Schemes of Zero- Dimensional Subschemes of Smooth Varieties. IX, 196 pages. 1994.

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