Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen Subseries: Mathematisc...
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen Subseries: Mathematisches Institut der Universit~it und Max-Planck-Insitut ftir Mathematik, Bonn - vol. 19 Advisor: E Hirzebruch
1572
Lothar G6ttsche
Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest
Author Lothar GSttsche Max-Planck-Institut fiir Mathematik Gottfried-Claren-Str. 26 53225 Bonn, Germany
Mathematics Subject Classification (1991): 14C05, 14N10, 14D22
ISBN 3-540-57814-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57814-5 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10078819
46/3140-543210 - Printed on acid-free paper
Introduction Let X be a smooth projective variety over an algebraically closed field k. The easiest examples of zero-dimensional subschemes of X are the sets of n distinct points on X. These have of course length n, where the length of a zero-dimensional subscheme Z is dimkH~ Oz). On the other hand these points can also partially coincide and then the scheme structure becomes important. For instance subschemes of length 2 are either two distinct points or can be viewed as pairs (p, t), where p is a point of X and t is a tangent direction to X at p. The main theme of this book is the study of the Hilbert scheme X In] := Hilbn(X) of subschemes of length n of X; this is a projective scheme paraxnetrizing zero-dimensional subschemes of length n on X. For n = 1, 2 the Hilbert scheme X In] is easy to describe; X [1] is just X itself and X [2] can be obtained by blowing up X x X along the diagonal and taking the quotient by the obvious involution, induced by exchanging factors in X x X. We will often be interested in the case where X In] is smooth; this happens precisely if n < 3 or dim X < 2. If X is a curve, X In] coincides with the n th symmetric power of X, X(n); more generally, the natural set-theoretic m a p X ['t] --~ X (n) associating to each subscheme its support (with multiplicities) gives a natural desingularization of X (n) whenever X In] is smooth. The case dim X -- 2 is particularly important as this desingularization turns out to be crepant; that is, the canonical bundle on X In] is the pullback of the dualizing sheaf oi X (~) (in particular X (n) has Gorenstein singularities). In this case, X In] is an interesting 2n-dimensional smooth variety in its own right. For instance, Beanville [Beauville (1),(2),(3)] used the Hilbert scheme of a K3-surface to construct examples of higher-dimensional symplectic manifolds. One of the main aims of the book is to understand the cohomology and Chow rings of Hilbert schemes of zero-dimensional subschemes. In chapter 2 we compute Betti numbers of Hilbert schemes and related varieties in a rather general context using the Weil conjectures; in chapter 3 and 4 the attention is focussed on easier and more special cases, in which one can also understand the ring structure of Chow and cohomology rings and give some enumerative applications. In chapter 1 we recall some fundamental facts, that will be used in the rest of the book. First in section 1.1, we give the definition and the most important properties of X[n]; then in section 1.2 we explain the Well conjectures in the form in which we are later going to use them in order to compute Betti numbers of Hilbert schemes, and finally in section 1.3 we introduce the punctual Hilbert scheme, which parametrizes subsehemes concentrated in a point of a smooth variety. We hope that the non-expert reader will find in particular sections 1.1 and 1.2 useful as a quick reference. In chapter 2 we compute the Betti numbers of S In] for S a surface, and of
vi
Introduction
KAn-1 for A an abelian surface, using the Well conjectures. Here KAn-1 is a symplectic manifold, defined as the kernel of the map A [nl --* A given by composing the natural map A In] ~ A (n) with the sum A (n) --* A; it was introduced by Beauville [Beanville (1),(2),(3)1. We obtain quite simple power series expressions for the Betti numbers of all the S[n] in terms of the Betti numbers of S. Similar results hold for the KAn-1. The formulas specialize to particularly simple expressions for the Euler numbers of S[ n] and KAn-1. It is noteworthy that the Euler numbers can also be identified as the coefficients in the q-development of certain modular functions and coincide with the predictions of the orbifold Euler number formula about the Euler numbers of crepant resolutions of orbifolds conjectured by the physicists. The formulas for the Betti numbers of the S [~] and KAn-1 lead to the conjecture of similar formulas for the Hodge numbers. These have in the meantime been proven in a joint work with Wolfgang Soergel [Ghttsche-Soergel (1)]. One sees that also the signatures of S [nl and KAn-1 can be expressed in terms of the q-development of modular functions. The formulas for the Hodge numbers of S[ ~l have also recently been obtained independently by Cheah [Cheah (1)] using a different technique. Computing the Betti numbers of X[ nl can be viewed as a first step towards understanding the cohomology ring. A detailed knowledge of this ring or of the Chow ring of X[ nl would be very useful, for instance in classical problems in enumerative geometry or in computing Donaldson polynomials for the surface X. In section 2.5 various triangle varieties are introduced; by triangle variety we mean a variety parametrizing length 3 subschemes together with some additional structure. We then compute the Betti numbers of X[ 3] and of these triangle varieties for X smooth of arbitrary dimension, again by using the Well conjectures. The Well conjectures are a powerful tool whose use is not as widely spread as it could be; we hope that the applications given in chapter 2 will convince the reader that they are not only important theoretically, but also quite useful in many concrete cases. Chapters 3 and 4 are more classical in nature and approach then chapter 2. Chapter 3 uses Hilbert schemes of zero-dimensional subschemes to construct and study varieties of higher order data of subvarieties of smooth varieties. Varieties of higher order data are needed to give precise solutions to classical problems in enumerative algebraic geometry concerning contacts of families of subvarieties of projective space. The case that the subvarieties are curves has already been studied for a while in the literature [Roberts-Speiser (1),(2),(3)], [Collino (1)], [Colley-Kennedy (1)]. We will deal with subvarieties of arbitrary dimension and construct varieties of second and third order data. As a first application we compute formulas for the numbers of higher order contacts of a smooth projective variety with linear subvarieties in the ambient projective space. For a different and more general construction,
Introduction
vii
which is however also more difficult to treat, as well as for examples of the type of problem that can be dealt with, we also refer the reader to [Arrondo-Sols-Speiser
(I)]. The last chapter is the most elementary and classical of the book. We describe the Chow ring of the relative Hilbert scheme of three points of a p2 bundle. The main example one has in mind is the tautological p2-bundle over the Grassmannian of two-planes in pn. In this case it turns out hat our variety is a blow up of (p,,)[3]. This fact has been used in [Rossell5 (2)] to determine the Chow ring of (p3)[3]. The techniques we use are mostly elementary, for instance a study of the relative Hilbert scheme of finite length subschemes in a Pl-bundle; I do however hope that the reader will find them useful in applications. For a more detailed description of their contents the reader can consult the introductions of the chapters. The various chapters are reasonably independent from each other; chapters 2, 3 and 4 are independent of each other, chapter 2 uses all of chapter 1, chapter 3 uses only the sections 1.1 and 1.3 of chapter 1 and chapter 4 uses only section 1.1. To read this book the reader only needs to know the basics of algebraic geometry. For instance the knowledge of [Hartshorne (1)], is certainly enough, but also that of [Eisenbud-Harris (1)] suffices for reading most parts of the book. At some points a certain familiarity with the functor of points (like in the last chapter of [Eisenbud-narris (1)]) will be useful. Of course we expect the reader to accept some results without proof, like the existence of the Hilbert scheme and obviously the Weil conjectures. The book should therefore be of interest not only to experts but also to graduate students and researchers in algebraic geometry not familiar with Hilbert schemes of points.
viii
Introduction
Acknowledgements I want to thank Professor Andrew Sommese, who has made me interested in Hilbert schemes of points. While I was still studying for my Diplom he proposed the problem on Betti numbers of Hilbert schemes of points on a surface, with which my work in this field has begun. He also suggested that I might try to use the Weil conjectures. After my Diplom I studied a year with him at Notre Dame University and had many interesting conversations. During most of the time in which I worked on the results of this book I was at the Max-Planck-Institut fiir Mathematik in Bonn. I am very grateful to Professor Hirzebruch for his interest and helpful remarks. For instance he has made me interested in the orbifold Euler number formula. Of course I am also very grateful for having had the possibility of working in the inspiring atmosphere of the Max-Planck-Institut. I also want to thank Professor Iarrobino, who made me interested in the Hilbert function stratification of Hilbn(k[[x, y]]). Finally I am very thankful to Professor Ellingsrud, with whom I had several very inspiring conversations.
Contents
Introduction 1.
V
Fundamental
facts
1
1.1. T h e Hilbert scheme
1
1.2. T h e Weft conjectures
5
1.3. T h e p u n c t u a l Hilbert scheme . . . . . . . . . . . . . . . . . . . .
9
2.
Computation
o f the Betti n u m b e r s
of Hilbert schemes
.....
2.1. T h e local s t r u c t u r e of y[n] -~(n) . . . . . . . . . . . . . . . . . . . . . 2.2. A cell decomposition of P[2hI, Hilb~(R),
ZT, G T
. . . . . . . . . . .
2.3. C o m p u t a t i o n of the Betti n u m b e r s of S In] for a s m o o t h surface S 2.4. T h e Betti numbers of higher order K u m m e r varieties 2.5. T h e Betti n u m b e r s of varieties of triangles 3.
. . . . . . . . .
. . . . . . . . . . . . . .
The varieties o f s e c o n d a n d higher order d a t a . . . . . . . . . .
3.1. T h e varieties of second order d a t a
. . . . . . . . . . . . . . . . .
3.2. Varieties of higher o r d e r d a t a a n d applications
. . . . . . . . . . .
3.3. Semple bundles a n d the formula for contacts with lines 4.
....
. . . . . . .
12 14 19 29 40 60 81 82 101 128
The Chow r i n g o f r e l a t i v e H i l b e r t schemes of projective bundles
. . . . . . . . . . . . . . . . . . . . .
145
4.1. n-very arapleness, embeddings of the Hilbert scheme a n d the s t r u c t u r e of A I n ( P ( E ) )
. . . . . . . . . . . . . . . . . . . . . ~ 3
4.2. C o m p u t a t i o n of the Chow ring of Hilb (P2)
. . . . . . . . . . . .
146 154
4.3. T h e Chow ring of Hw~-flb3(P(E)/X) . . . . . . . . . . . . . . . . .
160
4.4. T h e Chow ring of H i l b 3 ( P ( E ) / X )
173
Bibliography
Index
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
184
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
192
Index o f notations
. . . . . . . . . . . . . . . . . . . . . . . .
194
1. F u n d a m e n t a l
facts
In this work we want to s t u d y the Hilbert scheme X In] of subschemes of length n on a smooth variety. For this we have to review some concepts and results. In [Grothendieck (1)] the Hilbert scheme was defined a n d its existence proven. We repeat the definition in p a r a g r a p h 1.1 a n d list some results a b o u t X[ n]. X['q is related y["] ----* X(n). to the s y m m e t r i c power X (n) via the Hilbert-Chow m o r p h i s m wn :"'red We will use it to define a stratification of X [n]. In chapter 2 we want to c o m p u t e the Betti numbers of Hilbert schemes a n d varieties t h a t can be constructed from t h e m by counting their points over finite fields a n d applying the Well conjectures. Therefore we give a review of the Well conjectures in 1.2. T h e n we count the points of the s y m m e t r i c powers X ('0 of a variety X , because we will use this result in chapter 2. In 1.3 we s t u d y the p u n c t u a l Hilbert scheme H i l b " ( k [ [ X l , . . . , x 4 ] ] ) , p a r a m e t r i z i n g subschemes of length n of a s m o o t h d-dimensional variety concentrated in a fixed point. In p a r t i c u l a r we give the stratification of Iarrobino by the Hilbert function of ideals.
1.1. T h e Hilbert s c h e m e Let T be a locally noetherian scheme, X a quasiprojective scheme over T a n d s a very a m p l e invertible sheaf on X over T.
D e f i n i t i o n 1.1.1. [Grothendieck (1)] Let
7"liIb(X/T) be the contravariant functor
from the category Schln T of locally noetherian T-schemes to the category Ens of sets, which for locally noetherian T-schemes U, V and a m o r p h i s m r : V -----+U is given by f
7-lilb(X/T)(U) = I Z C X XTU closed subscheme, flat over U ) "Hilb(X/T)(r
: nilb(X/T)(U)
,7~ilb(X/T)(V); Z , ~ Z xu V.
Let U b e a locally noetherian T-scheme, Z C X XT U a subseheme, flat over U. Let p : Z ---* X , q : Z ~ U be the projections a n d u E U. We lJut Z~ = Hilbert p o l y n o m i a l of Z in u is
P.(z)(m) := x(Oz.(m)) =
x(o
o
q-a(u). T h e
p*bc")).
P,,(Z)(m) is a polynomial in m a n d independent of u E U, if U is connected. For 7"[ilbP(X/T) be the subfunctor of 7(ilb(X/T) defined
every p o l y n o m i a l P E Q[x] let by
TlilbP(X/T)(U) = (
Z CX •
U
closed subscheme
I
Z is flat ~ U and } P~(Z) = P for all u E U "
2
1. Fundamental facts
T h e o r e m 1.1.2 [Grothendieck (1)]. Let X be projective over T. Then for every polynomial P E Q[x] the functor 7-lilbP(X/T) is representable by a projective Tscheme HilbP(X/T). 7-lilb(X/T) is represented by
Hilb(X/T) : =
U HilbP(X/T)" PEQ[x]
For an open subscheme Y C X the functor 7"lilbP(Y/T) is represented by an open subscheme HilDP(Y/T) C HilDP(X/T).
D e f i n i t i o n 1.1.3. Hilb(X/T) is the Hilbert scheme of X over T. If T is spec(k) for a field k, we will write Hilb(X) instead of Hilb(X/T) and H i l b P ( x ) instead of If P is the constant polynomial P = n, then Hilbn(X/T) is the relative Hilbert scheme of subschemes of length n on X over T. If T is the spectrum of a field, we will write X In] for Hilbn(X) = Hilbn(X/spec(k)). X["] is the Hilbert scheme of subschemes of length n on X.
HilbP(X/T).
If U is a locally noetherian T-scheme, then Tlilbn(X/T)(U) is the set closed subschemes Z C X XT U
Z is flat of degree n over U}.
In particular we can identify the set X['q(k) of k-valued points of X In] with the set of closed zero-dimensional subschemes of length n of X which are defined over k. In the simplest case such a subscheme is just a set of n distinct points of X with the reduced induced structure. The length of a zero-dimensional subscheme Z C X is dim~H~ Oz). The fact that Hilbn(X/T) represents the funetor 7-lilbn(X/T) means that there is a universal subscheme
Zn(X/T) C X XT Hilbn(X/T), which is fiat of degree n over Hilbn(X/T) and fulfills the following universal property: for every locally noetherian T-scheme U and every subscheme Z C X XT U which is flat of degree n over U there is a unique morphism
f z : U -----* Hilbn(X/T) such that Z = ( l x XT f z ) - I ( Z . ( X / T ) ) . For T = spee(k) we will again write Z,,(X) instead of Zn(X/T).
1.1. The ttilbert scheme
3
R e m a r k 1.1.4. It is easy to see from the definitions that Zn(X/T) represents the functor Zn(X/T) from the category of locally noetherian schemes to the category of sets which is given by
Z,(X/T)(U) { (Z, a)
Z closed subschemes of X x T U, flat of degree n over U, a : U ----+ Z a section of the projection Z
Zn(X/T)(r
] / *U
: Z,(X/T)(U) ----+Z,(X/T)(V); ( z , ~),
(U, V locally noetherian schemes ff : V ~
, ( z • v v,
~0r
U).
For the rest of section 1.1 let X be a smooth projective variety over the field k. D e f i n i t i o n 1.1.5. Let G(n) be the symmetric group in n letters acting on X n by permuting the factors. The geometric quotient X (n) : = X"/G(n) exists and is called the n-fold symmetric power of X. Let ~ . : X n _ _ , X(") be the quotient map.
X (n) parametrizes effective zero-cycles of degree n on X, i.e. formal linear combinations ~ ni[xi] of points xi in X with coefficients ni E *W fulfilling ~ ni = n. X (~) has a natural stratification into locally closed subschemes: D e f i n i t i o n 1.1.6. Let u = ( n l , . . . , nr) be a partition of n. Let
i n l := {(Xl,...,Xn,)
Xl ~.X2 . . . . .
Xni} c X n'
be the diagonal and r
r
x : := I I
c II x"' = x"
i=1
i=1
Then we set
x~(") := + . ( x"~ ) and
:= x!")\ U Here # > u means that # is a coarser partition then u.
4
1. Fundamental facts
The geometric points of X (n) are
x(n)(-k)m(Zni[xi]Ex(n)(-k )
the points xi axe pairwise distinct }.
The X (~) form a stratification of X (n) into locally closed subschemes, i.e they axe locally closed subschemes, and every point of X (n) lies in a unique X (~). The relation between X [~] and X (n) is given by: T h e o r e m 1.1.7 [Mumford-Fogarty (1) 5.4]. There is a canonical morphism (the
Hilbert Chow morphism) y["]
COn : ~ L r e d
)
X(n),
which as a map of points is given by
z
Z xEX
~r y[n]. So the above stratification of X (n) induces a stratification . . . . red" D e f i n i t i o n 1.1.8. For every partition u of n let X In]
: : conl(x(n)).
Then the X[~n] form a stratification of y[n] into locally closed subschemes. .L red For u = ( n l , . . . , nr) the geometric points of X In] are just the unions of subschemes Z1 , . . . , Zr, where each Zi is a subscheme of length ni of X concentrated in a point xi and the xi are distinct.
1.2. The Weil conjectures We will use the Weil conjectures to compute the Betti numbers of Hilbert schemes. They have been used before to compute Betti numbers of algebraic varieties, at least since in [Harder-Narasimhan (1)] they were applied for moduli spaces of vector bundles on smooth curves. Let X be a projective scheme over a finite field F q , let J~'q be an algebraic closure of s and X := X x Fq ~'q" The geometric Frobenius
Fx : X - - + X is the morphism of X to itself which as a map of points is the identity and the map a ~-~ a q on the structure sheaf Ox. The geometric Frobenius of X over F q is Fq := Fx
x
l~q.
The action of Fq on the geometric points X ( F q ) is the inverse of the action of the Frobenius of Fq. As this is a topological generator of the Galois group Gal(F~, Fq), a point x E X ( F r is defined over F q , if and only if x = Fq(x). For a prime I which does not divide q let Hi(X, Q~) be the i th l-adic cohomology group of X and
bi(--Z) := dimq,(Hi(-x, Ql)),
p(Y, z) :=
b,(X)z
e(X) := b~(X) is independent of I. We will denote the action of Fq* on H ~ ( X , Q I ) by F~]Hr(~,Q~). The zeta-function of X over F q is the power series Zq(X't) := exp (n~>o 'X(Fq" )'tn / Here IMI denotes the number of elements in a finite set M. Let X be a smooth projective variety over the complex numbers C. Then X is already defined over a finitely generated extension ring R of 2~, i.e. there is a variety XR defined over R such that X n • n C = X. For every prime ideal p of R let Xp := X n • n R/p. There is a nonempty open subset U C spec(R) such that Xp is smooth for all p E U, and the l-adic Betti-numbers of Xp coincide with those of X for all primes l different from the characteristic of Alp (cf. [Kirwan (1) 15.], [Bialynicki-Birula, Sommese (1) 2.]. If m C R is a maximal ideal lying in U for which R / m is a finite field ~'q of characteristic p r l, we call Xm a good reduction of X modulo q.
6
1. Fundamental facts
Theorem
1.2.1. (Well conjectures [Deligne (1)], c]. [Milne (1)1 , [Mazur (1)1)
(1) z ~ ( x , t ) is a rational ]unction 2d
Zq(X, t) = ~ I Q~(X, t) (-1Y+' r~0
with Q~(X, t) = det(1 - tFr [Hr(~,q,))(2) Q~(X,t) e 2g[t]. (3) The eigenvalues ai,r of Fq*[Hr(~-,q,) have the absolute value tail[ = at~2 with
respect to any embedding into the complex numbers.
Zq(X, 1/qdt) = 4-qe(-x)/2t~(~) Zq(X, t).
(4)
(5) If X is a good reduction of a smooth projective variety Y over C, then we have
bi(Y) = bi(X) = deg(Qi(X, t)).
R e m a r k 1.2.2. Let F(t, sl,... ,sin) e Q[t, sl,... ,sin] be a polynomial. Let X and S be smooth projective varieties over F q such that
IX(Fq.)l
=
F(q", [S(Fq,)[,...,
IS(Fq-,-)t)
holds for all n E ~N'. Then we have
p(X, - z ) = F(z 2, p(-S, - z ) , . . . , p(-S, _zm)). If X and S are good reductions of smooth varieties Y and U over C, we have:
p ( V , - z ) = F ( z 2 , p ( U , - z ) , . . . ,p(V,-zm)).
P r o o f i Let a l , . . . , We put
Then we have
as
be pairwise distinct complex numbers and h i , . . . , hs E Q.
$
z ( ( a , , h,),) = I I ( 1 - a,) -h, i=1
1.2.
The Well conjectures
7
So we can read off the set of pairs {(al,hl),...(as,hs)} from the function Z((ai,hi)i). For each c 9 C let r(c) := 21ogq(Icl). By theorem 1.2.1 we have: for a smooth projective variety W over F q there are distinct complex numbers
(ii)~=l 9 C and integers (li)~=l 9 2g such that t
IW(Fq-)l = ~ li!~ i=1
for all n E / V . Furthermore we have r ( t i ) E ~_>0 and
(-1)%(w)= ~
l,
~(~)=k for all k E 2~_>o. Let ill,... ,it E C, for S. Then we have for all n E ZW:
ll,... ,lt
E 2~ be the corresponding numbers
t
IX(Fc)I
"
Let
~51,... , ~r
t
K"~l~n = F [kq n ,~...~ iPi ," "', E l i t i mn) 9 i=1
i=1
be the distinct complex numbers which appear as monomials in q and
the 7i in
(•
F q, "
lifli,...,
i=1
•
Ii!im
9
i=1
Then there are rational numbers h i , . . . , n~ such that
IX(Fqo)l = ~ n,e~ i=1
for all n E SV and
(-i)%(X)= ~
ni
r(~j )=k for all k E 2g>0. We see from the definitions that ~r(6~)=k z k in F(za,p(S,-z),... ,p(S,--zm)). [3
nj is the coefficient of
We finish by showing how to compute the number of points of the symmetric power X (n) for a variety X over F q . The geometric Frobenius F := Fq acts on X(n)('Fq) by
F(Eni[xi])
=Eni[F(xi)],
axtd X ( " ) ( F q ) is the set of effective zero-cycles of degree n on X which are invariant under the action of F.
8
1. Fundamental facts
D e f i n i t i o n 1 . 2 . 3 . A zero-cycle of the form r
E[Fi(x)]
with x 9 X(~b-'q. ) \ U Z(ZWq~ ) j[r
i=0
is called a primitive zero-cycle of degree r on X over Z~'q. The set of primitive zero-cycles of degree r on X over hrq will be denoted by Pr(X, ~'q).
Remark
1.2.4.
(1) Each element ( E X ('0 (~'q) has a unique representation as a linear combination of distinct primitive zero-cycles over F q with positive integer coefficients.
(2) IX(Fq.)l = y ] r . IP~(X, Fq)I tin (3) Zq(X,t) = ~
Ix(")(z~q)l~",
n>O
i.e. Zq(X, t) is the generating function for the numbers of effective zero-cycles of X over s r
P r o o f : (1) Let ( = ~ i = x ni[xi] E X(n)(lFq), where z l , . . . , xr are distinct elements of X(-~q). For all j let {j := En,kj[xi] 9 X(")(Fq). Then we have ( = y ] j (j, and it suffices to prove the result for the {j. So we can assume that ( is of the form ( = ~i~=l [xi] with pairwise distinct xi E X(-~q). As we have F({) = {, there is a p e r m u t a t i o n a of { 1 , . . . , r } with F(zi) = x~(i) for alli. Let M s , . . . , M s C { 1 , . . . , r } be the distinct orbits under the action of g. Then we set r/j := E [xi] iEMj for j = 1 , . . . s . Then ~ = ~j=ls r/j is the unique representation o f ~ as a sum of primitive zero-cycles. (2) follows immediately from the definitions. From (1) we have
Ix(")(Fq)lt" n>O
= I I ( 1 -T_>I
tr)-lP.(X,F,)l
= Zq(X, t). So (3) holds.
[]
1.3. T h e p u n c t u a l
Hilbert scheme
Let R := k [ [ x l , . . . , Xd]] be the field of formal power series in d variables over a field k. Let m = (Xl . . . . ,Xd) be the m a x i m a l ideal of R. D e f i n i t i o n 1.3.1. Let I C R be an ideal of colength n. The Hilbert function T ( I ) of I is the sequence T ( I ) = (ti(I))i>o of non-negative integers given by
ti = d i m k ( m l / ( I A m i + m i + l ) ) . If T = (ti)i>_o is a sequence of non-negative integers, of which only finitely m a n y do not vanish, we p u t ti < (d+i-1).
IT I =
~2 ti. The initial degree do of T is the smallest i such t h a t
Let Ri := m i / r n i+1 and Ii : = ( m I Cl [ ) / ( m i+1 (-I I). T h e n Ri is the space of forms of degree i in R and Ii the space of initial forms of I (i.e. the forms of m i n i m a l degree among elements of I ) of degree i, and we have:
ti(I) = d i m k ( R i / I i ) .
Let I C R be an ideal of colength n a n d T = (ti)i>_o the Hilbert function of I. Lemma
1.3.2.
(1) d i m ( m J / I N m / ) = E ti i>_j holds for all j > O. In particular we have IT] = n.
(2)
m".
P r o o f : Let Z : = R / I , a n d Zi the image of m i under the projection R ~ we have
Z. Then
N Zi = 0 .
i>0
As Z is finite dimensional, there exists an i0 with Zio = O. For such an i0 we have I D m i~ There is an isomorphism
Zj = m J / ( m j N I) ~- ~~vii=j ~ -~t/ of k-vector spaces, and R i / I i = 0 holds for i > i0. If we choose io to be minimal, then R i / I i 7s 0 holds for i < io. So we get (1). If t j = 0 for some j , then I D m j. T h u s (2) follows from Irl = n.
1. Fundamental fact~
10
In a similar way one can prove: Let X be a smooth projective variety over an algebraically closed field k. Let x E X be a point and Z C X a subscheme of length n with supp(Z) = x. Let Iz,, be the stalk of the ideal of Z at X. Then we have n
Iz,, D m x , ~. (Just replace R by Ox,~ in the proof above.)
R e m a r k 1.3.3. As every ideal of colength n in R contains m n, we can regard it as an ideal in R / m ~. Thus the Hilbert scheme H i l b n ( R / m n) also parametrizes the ideals of colength n in R. We also see that the reduced schemes ( H i l b ~ ( R / m k ) ) ~ d are naturally isomorphic for k _> n. We will therefore denote these schemes also by Hilbn(R)~d . Hilb~(R)~d is the closed subscheme with the reduced induced structure of the Grassmannian Grass(n, R / m ~) of n dimensional quotients of R / m ~ whose geometric points are the ideals of colength n of k [ [ x l , . . . , Xd]]/m ~.
Using the Hilbert function we get a stratification of Hilbn(R)red .
D e f i n i t i o n 1.3.4. Let T = (ti)i>_o be a sequence of non-negative integers with ITI = n. Let Z T C Hilbn(R)red be the locally closed subseheme (with the reduced induced structure) parametrizing ideals I C R with Hilbert function T. Let GT C ZT be the closed subscheme (with the reduced induced structure) parametrizing homogeneous ideals I C R with Hilbert function T. Let
PT : ZT
) GT
be the morphism which maps an ideal I to the associated homogeneous ideal (i.e. the ideal generated by the initial forms of elements of I). The embedding GT C ZT is a natural section of PT.
In the case d = 2 i.e. R = k[[x, y]] many results about these varieties have been obtained in [Iarrobino (2), (4)].
D e f i n i t i o n 1.3.5. The jumping index (ei)i>o of (ti)i>_o is given by ei = max(ti-1 ti, 0).
Theorem
1.3.6. [Iarrobino (4), prop. 1.6, thm. 2.11, thin. 2.12, thm. 3.13]
(1) ZT are GT non-empty if and only if to = 1 and ti do (here
again do is the initial degree of T).
1.3. The punctual Hilbert scheme
11
(2) GT and ZT are smooth, GT is projective of dimension
dim(GT) = ~ ( e i + 1)e~+1. (3) PT : ZT ~
GT is a locally trivial fibre bundle in the Zariski topology, whose fibre is an aj~ne space A n(T) of dimension n(T) = n - E (ei + 1)(ej+l + ej/2). j>_do
2. C o m p u t a t i o n
of the Betti numbers of Hilbert schemes
The second chapter is devoted to computing the Betti numbers of Hitbert schemes of points. The main tool we want to use are the Well conjectures. In section 2.1 we will study the structure of the closed subscheme X (-) ['] of X["] which parametrizes subschemes of length n on X concentrated in a variable point of X. We will show that (X(n))r,d is a locally trivial fibre bundle over X in the Zariski topology with fibre Hilb"(k[[xl,... xd]]). We will then also gtobalize the stratification of Hilbr'(k[[xl , ..., Xd]]) from section 1.3 to a stratification of X (~,). ["] Some of the strata parametrize higher order data of smooth m-dimensional subvarieties Y C X for m _< d. In chapter 3 we will study natural smooth compactifications of these strata. In section 2.2 we consider the punctual Hilbert schemes Hilbn(k[[x, y]]). We give a cell decomposition of the strata and so determine their Betti numbers. I have published most of the results of this section in a different form in [G6ttsche (3)]. They have afterwards been used in [Iarrobino-Yameogo (1)] to study the structure of the cohomology ring of the GT. We also recall the results of [Ellingsrud-Stromme (1),(2)] on a cell decomposition of Hilb"(k[[x, y]]) and p~n]. In section 2.3 we compute the Betti numbers of S ['~1 for an arbitrary smooth projective surface S using the Weil conjectures. This section gives a simplified version of my diplom paper [G6ttsche (1),(2)]. The auxiliary results that we prove here will be used several times in the rest of the chapter. We also formulate a conjecture for the Hodge numbers of the S In]. In a joint work with Wolfgang Soergel [G6ttsche-Soergel (1)] it has in the meantime been proved. Independently Cheah [Cheah (1)] has recently obtained a proof using a different method. One can see that the Euler numbers of the S In] can be expressed in terms of modular forms. By the conjecture on the Hodge numbers this is also true for the signatures. In section 2.4 we compute the Betti numbers of higher order K u m m e r varieties KA,,. These varieties have been defined in [Beauville (1)] as new examples of CalabiYau manifolds. While for a general surface S only the symmetric group G(n) in n letters acts on S n in a natural way by commuting the factors, there is also a natural action of G(n + 1) on An. KAn can be seen as a natural desingularisation of the quotient An/G(n + 1). To determine the Betti numbers we again use the Well conjectures. One can easily see from the formulas that the Euler numbers of the K A , can be expressed in terms of modular forms. It was shown in [Hirzebruch-HSfer (1)] that the formula for the Euler numbers of the S In] from section 2.3 coincides with the orbifold Euler number e(S", G(n)) of the action of G(n). We show that the Euler number of KA,, coincides with the orbifold Euler number e(A'*, G(n + 1)). As in section 2.3 we formulate a conjecture for the Hodge numbers. From this we also get an expression for the signatures of the KAn in terms of modular forms. In section 2.5 we study varieties of triangles.
As mentioned above X [3] is
smooth for an arbitrary smooth projective variety X.
So we can use the Weil
2. Betti numbers of Hilbert schemes
13
conjectures to compute its Betti numbers. We can view X [3] as a variety of unordered triangles on X. From X [3] we can construct several other varieties of triangles on ~ 3
X. The variety Hilb (X) of triangles on X with a marked side has been used in [Elencwajg-Le Barz (3)] in the case of Z = P2 to compute the Chow ring of p~3], and the variety H 3 ( X ) of complete triangles on X has been studied in detail in [Roberts-Speiser (1),(2),(3)], [Collino-Fulton (1)] for X = P2. For general X it has been constructed in [Le Barz (10)]. There is also a new functorial construction by Keel [Keel (1)]. We will construct two additional varieties of triangles. We show that they are smooth and study maps and relations among the triangle varieties. Then we use the Well conjectures to compute their Betti numbers.
14
2.1. T h e l o c a l s t r u c t u r e
o f X In] (,0
Let k be a (not necessarily algebraically closed) field and X a smooth quasiprojective variety of dimension d over k. In this section we s t u d y the structure of the s t r a t u m (X(n))~d which parametrizes subschemes of X which are concentrated in a (variable) point in X .
D e f i n i t i o n 2.1.1.
Let X be a smooth projective variety over a field k. Let A C
X x X be the diagonal a n d Z A / x x x
its ideal. Let A n C X x X be the closed
subscheme which is defined by Z~x/x xX" Let
pl,P2 : X • X
~X
be the projections and /51,/52 the restrictions to A n.
The (n - 1) th jet-bundle
Jn-1 ( X ) of X is the vector bundle associated to the locally free sheaf J,_,(X)
:= ( p 2 ) , ( O ~ o )
on X . More generally let Z ~ V A . be the ideal sheaf of A i in A n and J / _ a ( X ) be the vector bundle associated to Jn'_,(X) := for all i < n -
(p2),(ZA,/A.)
1.
We see t h a t the fibre
Jn_l(X)(x)
of J n - a ( X ) over a point x e X can be identi-
fied in a n a t u r a l way with Ox,~/mnx,x and similarly
Jn_l(X)(x)
with m xx, J m . x~, n
We have
Symi(T; ). H i l b n ( A n / X ) is a locally closed subscheme of Hilbn(X • X / X )
= Hilbn(X),
and there is a n a t u r a l m o r p h i s m r : Hilb~(An/x) ~
Lemma
2.1.2.
Hilb'~(An/X)r~d :
X.
YX (n))r~d ~ In] ~ as subschemes of X In] and 7r :
(n))~d ---* X is given by mapping a subscheme of length n which concentrated is (X['q in a point to this point.
2.1.
The local structure of X (n) ['q
15
P r o o f i Let k be an algebraic closure of k a n d X - := X x k k . Let Z C X be a subscheme of length n of X concentrated in a point, Iz its ideal in the local ring Ox,~ and m x , , the maximal ideal of Ox,~. Then we have Iz D m ~X , x (cf. 1.3.2). So we see that Hilb"(An/X)red and ~X ['q' t (n))r~d are closed subschemes of X [~] with the reduced induced structure, which have the same geometric points. Thus they are equal. The assertion on 7r follows directly from the definitions. D Let Grass(n, Jn-l(X)) be the Grassmannian bundle of n-dimensional quotients of J n - ~ ( X ) let and # : Grass(n, J n - l ( X ) ) ~ X be the projection.
L e m m a 2.1.3.
There is a closed embedding ~ : Hilb"(A"/X)r~a
, Grass(n, Jn-1 (X))
over X.
P r o o f : Let
Z,(A"/X) C A " x x H i l b " ( A " / X ) be the universal family (cf. 1.1.3) and let t52 : A " x x H i l b " ( A n / X ) ~
Hilbn(An/X)
be the projection. Then we have
(#2),(Oa.x,:Hilb.(a./x)) -- ~r*(Jn-l(X)). As Zn(An/X) is flat of degree n over Hilb~(A~/X), ([~2),(Oz.(A./x)) is a locally free quotient of rank n of 7r*(d~_a(X)). Thus it defines a morphism i: Hilbn(An/X)
,
Grass(n, dn-l(X)).
So we also get a morphsim 3: H i l b " ( A " / X ) r e d ~
Grass(n, Jn-l(X)).
Let T be the tautological subbundle of corank n of fr*(dn_l(X)). We abreviate Grass(n, Jn-l(X)) by Y. r is in a natural way an Oy-algebra. Let Q be the quotient of ~c*(ffn_l(X)) by the subalgebra generated by T. Q is a coherent sheaf on Y. For all x in Y let
q(x)
:=
dimk(G •
x Oy,./my,.)
2. Betti numbers of Hilbert schemes
16
be the rank of Q at x. From the definitions we see that q(x) _~, otherwise.
r is an eigenvector to the eigenvalue ~)-~#~-a~. The eigenvectors constructed this way are obviously linearly independent. The result follows by the substitution s:=~-a~
+a~
j:=~-I i:----~.
[]
We now formulate our result on the cell decompositions of Z T and G T in a form which has been influenced by [Iarrobino-Yameogo (1)]. In particular the formula for the Betti numbers of GT does not follow immediately from my original formulation. In [Iarrobino-Yameogo (1)] two combinatorical formulas are shown in order to derive this formula from my original one in [G6ttsche (4)]. Here we will give a direct proof. D e f i n i t i o n 2.2.6. Let (~ = (a0,... ,aT) be a partition of n. The graph of o~ is the set F ( a ) = {(i,/)E2g~_0 i ~ r , l < a i } . Picturally we can represent F(a) as a set of points, one point in position (i,j) for each ( i , j ) E F(a). The dual partition & = ( ~ l , . . - , ~ a 0 ) is the partition, whose graph is F(a) with the roles of rows and columns switched. The diagonal sequence is T ( a ) = ( t o ( a ) , . . . ,tl(a)), where
So it is the sequence of numbers of points on the diagonals of F(a). Let (u, v) E F(a). Then the hook difference h~,v(a) is
24
2. The Betti numbers of Hilbert schemes
I.e. hu,v(a) is the difference of the number of points in F(a) in the same column above (u, v) and the number of points in the same row to the left of (u, v). So we have h~,~(~) = (i~, + v - a~ - u.
For the partition c~ = (6, 3, 2) we get for instance the diagram
for F(c~) and & = (3,3,2, 1, 1, 1), T(c~) = (1,2,3,3, 1, 1). The hu,v(c~) are given by -1 --1 --3
0 0 --2
0 -2
-2
-1
O.
Theorem
2.2.7. Let T = (ti)i>_o be a sequence of non-negative integers with ITI = n. Then we have for X = GT and X = Z r : (1) X has a ceil decomposition.
I f k = C, then cl : A . ( X ) isomorphism and H . ( X ) is free.
~
H.(X)
is an
In case k = C we have for the Betti numbers:
(2)
b2,(Zr)=
(3)
b2~(Gr)=
a9
i{(u,v) eV(a)lhu,v(a)e{o,
{aEP(n)
T(a)=T;
1}}l=n_i
I{(u,v) E F ( a ) l h , ~ , ~ ( a ) = l } I = i } .
In particular the Euler numbers are =
=
9 P(n) I
: T}.
R e m a r k 2.2.8. In [Iarrobino (2),(4)] it has been shown that ZT and GT are none m p t y if and only if to = 1 and ti (n -
l)(w2 - wo).
We consider the i n d u c e d G m - a c t i o n on H i l b ~ ( R ) . We k n o w a l r e a d y t h a t it gives a cell d e c o m p o s i t i o n of H i l b n ( R ) . Let T = (ti) be a s e q u e n c e of n o n - n e g a t i v e integers w i t h ITI = n.
Lemma
2.2.9.
( l ) ZT is it union of cells of the cell decomposition of H i l b ' ( R ) .
(2) PT : ZT
~ GT is equivariant with respect to the Gin-action.
(3) The Gin-action induces a cell decomposition of GT. Its cells are the intersec-
tions of the cells of ZT with GT. Proof:
Let I be an ideal in R w i t h Hilbert f u n c t i o n T. Let j E f g , s : = j + 1 - tj.
Let Ij be t h e space of initial f o r m s of degree j in I. We p u t J : = l i m t . I. t~O
For all i let Ji be t h e space of initial forms of degree i in J . Let T I = (t~)j_>0 be t h e H i l b e r t f u n c t i o n of J . C h o o s e f l , 99 9 f~ E I such t h a t t h e i r initial f o r m s g l , . 9 9 g~ are a basis of Ij. By r e p l a c i n g t h e fi by s u i t a b l e linear c o m b i n a t i o n s we can a s s u m e t h a t t h e gi are of t h e f o r m
gi = xl(i)Y j-l(i) ~- E
gi,mxmy j - m
rn>l(i) w i t h gi,m E k a n d t h a t l(1) > /(2) > . . . > l(s).
WO~Wl ~W 2
we
By t h e choice of t h e weights
get lim ~ ( t ) - (tt(i)(w~176 t~O
fi) = xl(i)y j-l(i).
2. The Betti numbers of Hilbert schemes
26
So the span of the xl(i)y j-l(i) is contained in Jj. So we have
t'i = i + 1 - dim(Ji) 0. The monomials xiy I with i + l = j and l > ai form a basis of the space Ij of homogeneous polynomials of degree j in I. So we have:
tj--j+l=
{(i,/)E2g~_ 0 l i + l = j ,
l>_ai}
{(i,j) e r ( ~ ) l i + j = l}
= tj(.).
[]
Let again T be the tangent space of Hilbn(A ~) in the point corresponding to I.
The dimension of the subspace T + of T on which the weights of the action are positive is
L e m m a 2.2.11.
dim(T + )=n-
{(u,v) e V ( ~ ) ] h u , v ( ~ ) = 0
orh~,v(a)=l}
Proof." We apply lemma 2.2.5 to r -- G and AI
A2
Then we have for every character A~# b of G: (),a~b)(,~(t))
= to(,~,-~,o)+b(w,-wo).
.
2.2. A cell decompostion of P~'q, Hilb~(R), ZT, GT
27
By the choice of w0, wl, w2 the action of Gm has a positive weight on ~ # b , if and only if a + b > 0 or a + b = 0 and b > 0. Let i , j be integers satisfying
O < i < j < r , aj+l < s < aj. The weight of (,V-3-1 # ~ - ~ - 1 ) o ~ is positive, if and only if i + ai > j + s + 1, and the weight of ~ j - i ~ s - a i is positive, if and only if i + ai < j + s. From the definition we see that/z, is the smallest j satisfying s > a j, so/z, - 1 is the smallest j satisfying S > a j + l . So we have
E
dirnT + =
( aj - aj+ 1 -
{ s E 2~
OO}. Let A ( r ) be the cusp form of weight 12 for Sl2(2g) and r/(r) := A ( r ) '/2a the rkfunction. Then
ql/24 ~ e( S)
~ e(st-l)q- = \ 7(,)] rt~0
2. Betti numbers of Hilbert schemes
36
For a K3-surface we get in particular q
Z e(s[.l)q~ A(~) =
n~O
The Betti numbers bi(S In]) become stable for n > i:
Corollary 2.3.13. Let S be a smoo~h irreducible surface over C. Then
p(S[n],z) _ f i
((1 + z 2 m - 1)(1 _+ z_2m_+_1)) bl(..____s) modulo z n+l.
m=l
(1 --
z2m)b:(S)+l(1 --
Z2 m + 2 )
Proof." Let oo ((1 + z2m-ltm)(1 + z2m+ltm)) bl(S) G(z,t) := (1 - t) ml-I1 (1 -- ~----2t~)(1 - " Z ' ~ - m ~ --Z-2~'-+2~rn)
"
We have to show
P(S[~],z) - a ( z , 1 )
modulo z '~+'.
For a power series f C q[[z,t]] we denote the coefficient of zit j by ai,j(f). We see that ai,j(G(z,t)) = 0 holds for i > j. Let i _< n. By theorem 2.3.10(3) we have:
bi(S In]) =
ai,n
~fi-~t j a ( z , t ) j=0
/
= ~ a,,AC(z,t)) j=0 oo
= Z a, j(a(z,,)) j=O
= ai,o(G(z, 1)).
2.3. The Betti numbers of S ['q
37
T h e H o d g e n u m b e r s o f S [~] One would expect that similar formulas as for the Betti numbers of Hilbert schemes of points also hold for their Hodge numbers. For a smooth projective variety X over C let hP'q(x) := dirnHq(X, ftPx) be the (p,q)th Hodge number and let
h(x, x, y) := ~
P,q
hp,~(X)x%~
The xy-genus of X is given by xy(X) = h ( X , y , - 1 ) . By Hodge theory we have for
the signature ~ig~(X) = ~ ( X ) . Together with WoKgang Soergel I have computed the Hodge numbers of S [hI using intersection homology, perverse sheaves and mixed Hodge modules (cf. [GSttsche-Soergel (1)].) Independently Cheah [Cheah (1)] has recently proven this result by using a different technique, the so-called virtual Hodge polynomials. The result is:
T h e o r e m 2.3.14. oo
(1)
h(S In], z, y) =
Z
(xy) ~-I~l I I h(s(~'), x, y)
(l~l,2~2,...)Ep(n)
i=1
or equivalently
(2)
h(SM,-x,-y)
n/2 we see t h a t gcd(u) = 1. So we
have p ( S [nl, z)
p(KSn-1, z) = (1 + z) bds)
m o d u l o z ".
Thus we have by corollary 2.3.13
i
fi (i +
p ( K S n - a , z ) - (1 + z ) bl(s)
m~--I
z~m-1)bl(s)(1 (1-- z--~m~(-i
+
z2m+l) bi(S)
-
z 2m+2)
modulo zn
(1 + Z2m+l)2b1(S)
fi
= ~=~ (i - ~ w ~ -
T~+~)
oo = (I - z ~) I I (I + z~m+') ~b,(s) m=l (1 -- z2m) b2(s)+2 The result follows.
[]
In p a r t i c u l a r we have ba(KSn-1) = 0 for all n E ZW. In fact the K A n - 1 were proven to be simply connected in [Beauville (1)].
2. The Betti numbers of Hilbert schemes
54
The orbifold Euler number formula Let G be a finite group acting on a compact differentiable manifold X. Then there exists the well known formula for the Euler number of the quotient 1
g
~(x/a) = -~, ~ ~(x ), '
' gEG
where X g denotes the set of fixed points of g E G. If the quotient X / G is viewed as an orbifold, it still carries information on the action of G. In [Dixon-Harvey-VafaWitten (1),(2)] the orbifold Euler number is defined by
~(x, a) = ~1 ~
~(x. n x h)
gh=hg
(the sum is over all commuting pairs of elements in G). Now let X be an algebraic variety. We assume that the canonical divisor K x / a of X / G exists as a Cartier divisor. Furthermore we assume that there is a resolution ~G--Z--~X/G satisfying K x~ / a = ~r* K x / c . Then it has been conjectured that A
e(x,a) = e(x/a). This formula we will call the orbifold Euler number formula. In the case that the group G is abelian this conjecture has been proved in [Roan (1)] under certain additional hypotheses. In [Hirzebruch-Hbfer (1)] some examples of this formula are studied. First they give a reformulation:
~(x, a) = ~ , e(xg/c(g)). [g] Here C(g) is the centralizer of g and [g] runs through the conjugacy classes of G. Hirzebruch and Hbfer consider in particular the action of the symmetric group G(n) on the n th power S n of a smooth projective surface S by permuting the factors. The quotient is the symmetric power S (~), and w,, : S ['q ~ S ('0 is a canonical resolution of S ('). The canonical divisor K s , is invariant under the G(n) action. Thus it gives a canionical Cartier divisor Ks(,) on S ( ' ) , and it is easy to show that
~ * ( K s ( . ) ) = Kst.~. So the assumptions of the conjecture are fulfilled, and in fact Hirzebruch and Hbfer use my formulas (corollary 2.3.11) to prove that
~(s~-]) = e(s -, o(~)).
2.4. The Betti numbers of higher order K u m m e r varieties
55
Another case in which they check the formula is that of the K u m m e r surface KA1 of an abelian surface as a resolution of the quotient of A by G(2) = 2g/2 acting by x ~ - x . We will now generalize this result to the higher order K u m m e r varieties K A n - 1 . Let A be an abelian surface. Let
A'~:={(Xl,...,x~)EA
n
Exi=O}
cA~
with the reduced induced structure. Then A~ is isomorphic to A '~-1. The G(n) action by permutation of the factors of A n maps A~ to itself. So we can restrict it to A~ and the quotient is A~n). Let w := COn[Ix'AN_,. Then w : K A n - 1 ~ A~ '0 is a canonical desingularisation of A~n). The canonical divisor of A~n) is trivial, and by [Beauville (1)] KA,~-I is a symplectic variety; in particular we also have K K A . _ , = O. So the conjecture says that e ( K A n - 1 ) = e(A n-a, G(n)) should hold. For a permutation a of {1,... ,n} let
be the partition of n which consists of the lengths of the cycles of a. It determines the conjugacy class of a. The fixed point set is given by ( A n ) a = { ( X l , . .. ,Xn) 9 d n or
x~,, . . . .
xv, for all cycles ( , 1 , . . . , ui) of a }
~ I I A~i (~) = i=1 The centralizer C(a) acts by permuting the cycles of a of the same lengths. So we get
- 1-I i
= HA~'(~)/G(c~i(a)). For
h = (hi, h2,...) e 1-[ i the fixed point set ((An)~') h consists of the ( z l , . . . , xn) 9 A n satisfying zl = xj for all i , j for which the following holds: either i and j occur in the same cycle of a, or they occur in two different cycles of the same length l, and these are permuted by hr. So we get that ((An)~) h = (An) ~ for some 7 9 G(n) and
((An)~') h = (An) (1 ...... ) --"~A,
56
2. The Betti numbers of Hilbert schemes
if and only if
p(~) = ((~/~)~), and ha is a cycle of length a in G ( a ) for a positive integer a dividing n. Remark
2 . 4 . 1 5 . Let ~r E G(n). Then we have
{n 4 p(~) = e((A~)~) =
0
(n);
otherwise.
P r o o f : Let B be an abelian variety and h : B
~ B an a u t o m o r p h i s m of B. T h e n
every connected component of B h is either an isolated point or a t r a n s l a t i o n of an abelian subvariety of positive dimension of B. In p a r t i c u l a r e(B h) is the n u m b e r of isolated points in B h. For a cycle a of length n we have
=
xEA
nx=
,
and this has Euler n u m b e r n 4. Let a E G(n) with p(a) = ( n l , . . . , n r ) ,
r >2.
Then we get
(A~)a~ {(Xl,...,Xr) Znixi =0}. Let ( z ~ , . . . ,x~) C (A~) ~. For every y e A the point (Xl + n2y, x2 - n l y , x 3 , . . . , x r ) lies in the same connected component of (A~) a as ( x a , . . . ,x~). By the above we have e((A~) ~) = 0.
T h e o r e m 2.4.16.
[]
e(An-l, C(n)) = n 3 o ' l ( r ~ )
e(I(An-1).
=
Proofi e(A~, C ( n ) ) = Z e ( ( A ~ ) ~ / C ( a ) ) M
Ili ~(~)! ( ( ( 0 ) ) ) n4
Mn = n30"l ( n )
[2]
2.4. The Betti numbers of higher order Kummer varieties
57
Conjectures on the Hodge numbers of the KSn-1 Similar to the results of theorem 2.3.14 we can formulate conjectures on the Hodge numbers of the KSn-1.
Conjecture 2.4.17. h(KSn-l,x,y) 1 ((1 + x)(1 + y))b,(S)/2
~_,
(gcd(~'))b'(S)(xy) "-H
v=(l~t ,2~2 ,...)EP(n)
91] h(S(~ i
or equivalently h(KSn-a, - x , - y ) 1
(1 x)~,(s)/:(1 -
90(
u)b,(s)/z
-
g~d(v)~l(s)(x~,) " - H v=(l':' 1,2c'2 ,...)EP(n)
,
/~'=(I p , ,2~2,...)ep(c~i)
J ~/%.
J
]
In the case of the K A , _ a the conjecture has been verified in [GSttsche-Soergel
(1)]. R e m a r k 2.4.18. From the proven part of conjecture 2.4.17 we get for the )/y-genus and the signature: (1)
X-y(KA,-1) = n E
nla(1 + Y " " + yn/nl--1)2yn--n/n,
(2)
sign(gA,_l) = (-1)~-ln
E
d3"
din , n / d odd
We can again express the signatures of the KAn-1 in terms of modular forms (notations as in 2.3.15).
sign(KAn)(-q) n = ~q.
(3) rt~O
Proof." As in 2.4.12 only the terms with u = (n~/n'), #,~In, = (n/nx) give a contribution to the xu-genus. So we get
x-y(gA._l) = ~ ~ ( 1 - x"/"')(1 - ~"/",) ~--1 -,I-
(1
-
x)(1
y)
2. The Betti number~ o] Hilbert scheme~
58
(1) follows by easy c o m p u t a t i o n and (2) by p u t t i n g y = - 1 . (3) is obvious from the definition of e. [] By applying the same a r g u m e n t to the case of a geometrically ruled surface over an elliptic curve we get that sign(KSn-1) = O. This was however clear from the beginning as the dimension of K S~ - I is not divisible by 4. It seems r e m a r k a b l e t h a t in all cases the signatures and the Euler numbers can be expressed in terms of the coefficients of the q-development of m o d u l a r forms. For the first few of the X - y ( K A ~ - I ) we get: X - y ( K A 1 ) = 2 + 20y + 2y 2,
X-v(KA2) = 3 + 6y + 90y 2 + 6y 3 + 3y 4, X-y(KA3) = 4 + 8y + 44y 2 + 336y 3 + 44y 4 + 8y ~ + 4y 6, X-y(KA4) = 5 + 10y + 15y 2 + 20y 3 + 650y 4 + 20y 5 + 15y 6 + 10y 7 + 5y s, X-y(KA~) = 6 + 12y + 18y 2 + 72y 3 + 288y 4 + 1800y ~ + 288y 6 + 72y 7 + 18y s, + 12y 9 + 6y 1~ Let b+ be the n u m b e r of positive eigenvalues of the intersection form on the middle cohomology and b_ the n u m b e r of negative ones. T h e n we get the following table:
b2n(KAn) sign(KA,) b+(KAn) b-(KAn) 1 2 3 4 5 6 7 8 9 10
22 108 458 1046 3748 7870 25524 67O49 198270 538070
-16 84 -256 630 -1320 2408 -4096 6813 -10080 146521
3 96 101 838 1214 5139 10714 36931 94095 276361
19 12 357 208 2534 2731 14810 30118 104175 251709
We can also determine the Chern numbers of KA2: C4 = O, C~C2 :
O, ClC3 :
O,
c4 = 108, c~ = 756, This is true because cl = --KtcA._I = 0 and
s i gn( K A , _l ) = l (7p2(KA2) - p2(KA2)) = 84,
2.4. The Betti numbers of higher order Kummer varieties
p~(KA~) = ( ~ - 2~)(IZA~) = - 2 ~ ( K A ~ ) ,
59
60
2.5. T h e B e t t i n u m b e r s of varieties of triangles Let X be a smooth projective variety of dimension d over a field k. For d _> 3 and n > 4 the Hilbert scheme X['q is singular. However X [3] is smooth for all d E ~W. In this section we want to compute the Betti numbers of X [3]. X [3] can be viewed as a variety of unordered triangles on X. We also consider a number of other varieties of triangles on X, some of which have not yet appeared in the literature. As far as this is not yet known, we show that all these varieties are smooth. We study the relations between these varieties and compute their Betti numbers using the Weil conjectures.
Definition 2.5.1. [Elencwajg-Le Barz (5)] Let Hil'---b'~(X) C X [~-1] x X In] be the reduced subvariety defined by Hil--'bn(X) = { ( Z . _ I , Zn) C X [~-1] • X[~]
~ 3
Zn-1 C Z . }.
~ 3
Here we will be interested in Hilb (X). Let i : Hilb (X) ~ X [2] • X [3] be the embedding. If one interprets X[ 3] as a variety of unordered triangles on X, then ~ 3
Hilb ( X ) parametrizes triangles Z3 with a marked side Z2. In the case k = C it was ~ 3
~ 3
shown in [Elencwajg-Le Barz (5)] that Hilb (X) is smooth. Hilb (X) represents the contravaria~t functor from the category of Schln k locally noetherian k-schemes to the category Ens of sets ~ 3
7-lilb (X) : Schln k T, (r
---~ T2),
Ens; ,
, ((z~,z~)
~
xt l(T) • XE I(T)
c
r
• r
((ix
•
So for a smooth variety X over C and a reduction X0 of X modulo q the variety ~ 3
~ 3
Hilb (X0) is a reduction of Hilb (X) modulo q. Let P2 : Hii-b3(X) ----* X [31 be the projection. For any partition • of 3 ( i . e . . = (1, 1, 1), J v = (2, 1), J ~ = (3) ) we put ~ 3
Hilb~(Z) := p~-l(x[~3l). In [Elencwajg-Le Sarz (5)] a residual point of a pair (Z2, Z3) E ~ b 3 ( X )
Definition 2.5.2. [Elencwajg-Le Barz (5)] Let (Zn-1, Zn) ~ X [~-1] • X [~].
is defined.
2.5. The Betti numbers of varieties of triangles
61
Let I n - 1 be the ideal of Zn_ 1 in Oz.. Then the residual point r e s ( Z n - 1 , Zn) E X is the point whose ideal in Oz. is the annihilator Ann(I,,-1, Oz.) of It,-1 in Oz..
Elencwajg and Le Barz show t h a t the m a p (Zn-a, Zn) ~-* res(Z,_a, Z,) gives ~ n
a m o r p h i s m res : Hilb ( X ) ~ a r b i t r a r y field.
X , if the ground field is C. We show this for an
L e m m a 2.5.3. The map (Z,~-a,Zn) ~ res(Zn-l,Zn) defines a morphism res : ~ n Hilb ( X ) ~ X. ~ n
P r o o f i Let T be an integral noetherian scheme and (Zn-1, Zn) E 7-filb (X)(T). Let I be the ideal sheaf of Z,,-1 in Oz.. Then for all t C T the dimension of the annihilator Ann(It, Oz.,t)is 1, so Ann(I, Ozn) defines a subscheme res( Zn-1, Zn) C Zn, which is flat of degree 1 over T, i.e a T-valued point of X . So res is given by a m o r p h i s m of functors.
Remark
2.5.4. We can also describe the residual point as follows: for ( Z n - a , Zn) E
~ 3
Hilb ( X ) the zero-cycle wn(Zn) - w n - a ( Z , - a ) is Ix] for some point x e Z and ~ n
r e s ( Z , _ l , Zn) = x. If we consider Hilb ( X ) as a variety of triangles with a m a r k e d side, then res m a p s such a triangle to the vertex opposite to the m a r k e d side.
Via ~ 3
il : = res x i : Hilb ( X ) ~
X x X [2] x X [a]
~ 3
we will in future consider Hilb ( X ) as a subvariety of X x X [21 x X[3]:
~ 3
This means we consider Hilb ( X ) as a variety of triangles with a side a n d the opposing vertex marked. Let ~ 3
t51 : H i l b ( X ) ~ 3
t52 : H i l b ( X )
, X, ~ X [21,
~ 3
Pa : Hilb ( X ) ~ 3
Pl,2 : I-Iilb ( X ) ~ ~ 3
/~1,3 : Hilb ( X ) ~
, X [a], X x X [21, X x X [a]
2. The Betti number8 of Hilbert schemes
62
be the projections. From the definitions we can see that the support of the image ~ 3
of/51,3 coincides with the support of the universal subscheme Z3(X). As Hilb (X) is reduced, this defines a morphism ~ 3
/~1,3 : Hilb (X) This morphism is birational,
, Z3(X).
as its restriction gives an isomorphism from
~ 3
~ 3
(Hilb (X))(1,1,1) to a dense open subset of Z3(X). So/51,3 : nilb (X) ----* Z3(X) is a canonical resolution of Z3(X). We can consider Z3(X) as the variety of triangles with a marked vertex. Then i51,3 is given by forgetting the marked side. ~ 3
Pl,2 : Hilb
is birational,
as
it
gives
an
(X)
, X • X [2]
isomorphism
of the
dense
open
subvariety
~ 3
(Hilb (X))(1,1,1) onto ints image. Let Z2(X) C X • X [2] be the universal subscheme. As a set Z2(X) is given by
z (x)
=
x • xc l
x c z}.
One can also verify easily that it carries the reduced induced structure and that it can be described as X x X blown up allong the diagonal. Let w : X • X [21 ~
Z3(X)
be the rational map which is defined on the open dense subvariety (X • X [2]) \ Z2 (X) by w((x, Z)) := (x, x t2 Z). Then obviously the diagram N 3
Hilb (X)
l
lbl,2
w
X x X [2]
4.
z (x)
~ 3
commutes.
So Pl,3 : Hilb (X) ~
Z3(X) is a natural resolution of the indeter~ 3
minacy of w. We will see later that Hilb (X) is the blow up of X • X [2] along
2.5. The Betti numbers of varieties of triangle~
T h e varieties o f
complete triangles
63
on X.
Semple [Semple (1)] has constructed a variety of complete triangles on P2This variety has been studied and its Chow ring was determined in [Roberts (1)], [Roberts-Speiser (1),(2),(3),(4)], [Collino-Fulton (1)]. (The Chow ring coincides with the cohomology ring in case k = C). Le Barz has generalized this construction in [Le Barz (10)] to general projective varieties and shown that the resulting varieties of complete triangles are smooth. Keel [Keel (1)] also gave a functorial construction of these varieties. Let X be a smooth projective variety of dimension d over a field k. We want to define other varieties of complete triangles. Because of this we call the variety defined by Le Barz the variety of complete ordered triangles on X. We also want to show that our varieties of complete triangles are smooth by using results fl'om [Le naxz (10)]. Definition 2.5.5. [Le Barz (10)] Let X be a smooth projective variety over a field k. The variety H3(X) of complete ordered triangles on X is the closed subvariety of X 3 x (X[2]) 3 • X [3] defined by Xi,Xj C Zl; Zi C Z; ~(x)
:
(Xl,X2,z3,Z1,Z2,Z3, Z) ~ ( x ~ • (xI~l)~ • xI~l)
x, : r ~ ( ~ , , Z j ) = r ~ ( Z t , Z )
for all permutations (i,j,l) of (1, 2, 3)
In [Le Barz (10)] /t3(X) is shown to be smooth for X a smooth variety over C. H3(X) represents the obvious functor 7~3(X) : Schln k > Ens:
(x1,~2,z3,zl,z2,z3,z) 7~3(X)(T) =
E (Z 3 x (X[2]) 3 X X[aI)(T)
x . xj c z~; z~ c z ; ~t = r ~ ( x , , Z j ) = r ~ ( Z t , Z ) for all permutations (i,j,l) of (1, 2,3)
(see also [Collino-Fulton (1) rem. (5)]). So if X is a smooth projective variety over C and X0 is a good reduction of X modulo q, then H3(X0) is a reduction of H3(X) modulo q. Let j : 9~(x) , x ~ • (x~21) ~ • xE31 be the embedding. Let :~1 : ~q3(X) - - ~ X 3, p~ : 9 ~ ( x )
- - , (x[~]) ~,
b~ : fi-z(x) __~ xC~l, be the projections. From the stratification of X [3] we get one of H3(X). Let u be a partition of 3. Then we put
2. The Betti numbers of Hilbert schemes
64
We can view the xi as the vertices of the triangle Z and Zi as the side opposite to xi. Thus s parametrizes the complete ordered triangles on X (i.e. together with a triangle we are given all its vertices and all its sides together with an ordering). The projection t51 : H 3 ( X ) -----+X 3 is birational. D e f i n i t i o n 2.5.6. [Le Barz (10)] For a pair (i,j) satisfying 1 < i < j < 3 let
Z~i,j := {(ZI,N2,Z3 ) E (X[2]) 3 Zi : Zj} c be the diagonal between the i th and jth factors. Let
be the small diagonal in X 3, and ~2 the small diagonal in (X[2I) 3. Then we put
E~,j(X) :=/5;~(/x~,~), D~(X) := (/~1 x/52)-1((~1 X (~2).
In [Le Barz (10)] these varieties are shown to be smooth for X a smooth variety over C. The Ei,j(X) are irreducible divisors in H3(X). D~(X) is the variety of second order data on X , which we want to study in more detail in chapter 3. For x 6 X let m x , . be the maximal ideal in the local ring Ox,x and
q. : m x , .
~ mx,~:/m2x,.
'the natural projection. We can describe the subscheme Z(1,2)(X) C X Is] (cf. section 2.1) as the closed reduced subvariety given by
Z 6 X [3]
supp(Z) = x for an x E X, and there is a ] 2-codimensional linear subspace V C rnx,x/m2x,, such that / " the ideal Iz of Z in Ox,. is of the form Iz = q[l(V)
Obviously Z(1,2)(X) is isomorphic to the Grassmannian bundle Graas(2, T } ) of two-dimensional quotients of the cotangent bundle of X. We put E
:=/5;~(z(i,2)(x)) [
I
supp(Z) -= x; Z1, Z2, Z3 C Z
j "
2.5. The Betti numbers of varieties of triangle8
65
Let Z E Z(1,2)(X), x : - supp(Z). Then the ideal Iz of Z in Ox,~ is of the form Iz = q~-l(V) for a suitable 2-eodimensional linear subspace V of m x , x / m ~ , z. Let
qz : mx,~: ---+ m x , ~ / I z be the natural projection. The ideals Iz2 of subschemes Z2 of length 2 of Z are given exactly by the qz I ( W ) for the one-dimensionM linear subspaces W of m x , ~ / I z . Let 7r: Z(1,2)(X) = Grass(2, T~) ~
X
be the projection. Then the subschemes Z2 C Z of length 2 are given by the onedimensional linear subspaces of the fibre of the tautological subbundle T1 of 7r*(T~) over the point V. Thus we get
Remark
2.5.7. E ~ P(T1) •
. . . . (2,T;r P(T1) XCrass(2,T;r P(T1).
Let X be a smooth projective variety over C. Pl,2 : H~b3(X) ----+ X x Z [2] is the blow up along Z2(X). Proposition
2.5.8.
Then
Proof: ~3 /51,2 : Hilb (X) , X x X[ 2] is an isomorphism over (X x Z [2]) \ Z2(X). Let F :=/5~,1(Z2(X)). Then F can be described as the set:
F : {(X, Z l , g )
E X x X [2] x X [31 x 1 C Z1, Z1 C Z, r e s ( Z l , Z ) ~ - x 1}.
Let Pl,4,7 : / ~ 3 ( X ) (Xl,X2,x3,Zl,Z2,Z3,Z),
, X x X [21 x X [31 "' ( x l , Z l , Z )
~3 be the projection. We see immediately that the image of this morphism is Hilb (X) so we get a morphism
:~,,,~ : ~ ( x )
--~ n~i-b3(X).
Let
(xl, z2, z3, Z1, Z2, Z3, Z) E E1,2(X). Then we have Z1 = Z2 and thus Xl = x~. So we get Xl C Z1. We see that P1,4,7(E1,2(X)) C F.
So we get a m o r p h i s m q : E1,2(X) , F. Let ( x l , Z 1 , Z ) 6 F. We put x2 := x~, If supp(Z) consists of two points, we see that Xl # x3 and
x3 := res(xl,Z1).
Z = Z3 tJ x3 for a unique subscheme Z3 of length 2 with support xl. If supp(Z) is a
2. The Betti numbers of Hilbert schemes
66
point but Z does not lie in ZO,2)(X), then Z has a unique subscheme Z3 of length 2. In both cases we get
q--l(xl, Z1, Z) = {(Xl, g2, x3, Zl, Z1, Z3, Z)}, If Z lies in Z(1,2)(X), then it is given by a two-dimensional quotient W of the cotangent space T~;(xa) of X at xl, and the subschemes Za of Z are given by the one-dimensional quotients V of W. So we get
q-l(xl,Zl,g) = {(.TI,X2,x3,ZI,ZI,Z3,Z ) 23 C Z} "~ P l . Putting things together we see that q is onto and a bijection over the open set F\paa(Z(1,2)(X)). As Ea,z(X) is an irreducible divisor of s F is an irreducible ~3 divisor on Hilb (X). Let e : X x-'X[~] ---+ X • X [21 be the blow up of X • X [2] along Zz(X). Let Z be the ideal of Z2(X) in X • X [2]. From p ~ ( Z 2 ( X ) ) = F we get that plaZ. --1 OH_~b3(x ) is the invertible sheaf corresponding to F. By the universal property of the blow up (cf. e.g. [Hartshorne (2), II. prop.7.14]) there is a morphism
N3 g : Hilb (X) -----+X x-X[2] such that the diagram
~3 Hilb (X)
~
,
X x'--X[2]
X x X[ z] commutes, g is a birational mo..rphism. By [Hartshorne (1) II Thm. 7.17] g is the blow up of a subscheme of X • X[2]. g is an isomorphism outside F, F is irr__._~educible, and the image g(F) is the exceptional divisor of the blow up g : X • X[ 2] X • X [21. Thus g is an isomorphism and the result follows. D In a joint work with Barbara Fantechi [Fantechi-G6ttsche (1)] we use proposition 2.5.8 to compute the ring structure cohomology ring H*(X [3],Q) of the Hilbert scheme of three points on a smooth projective variety X of arbitrary dimension in terms of the cohomology ring of X. We also compute the cohomology ring of ~3 Hilb (X).
2.5. The Betti numbers of varieties of triangles
67
Proposition 2.5.8 also follows from [Kleiman (3)] t h m 2.8. I have learned that Ellingsrud [Ellingsrud (1)] has proven independently the following: if S is a smooth surface, the blow up of S x S In} along the universal family
zo(s)
=
s • stol
x
z}
is a smooth variety m a p p i n g surjectively to S[ n+l] (proposition 2.5.8 is essentially the case n = 2 of this). One can see easily that E1,2(X) is obtained from F by blowing up along
p31(zr D e f i n i t i o n 2.5.9. For all n E zW let ~ x , n : X n -----+ X (~),
excel,, : ( X [21)" ---+ (X [21)(") be the quotient morphisms. Then let )~[31 C X (3) • (X [2])(3) • X[3] be the image of H s ( X ) under ~X,3 • ~X[21,3 X ~X[Sl : X 3 • (X[2]) 3 • X [31 ---+ X (3) • (x[2l) (3) • X [3] with the reduced induced structure. Let ZCl : H 3 ( X ) ~ this morphism t o / I 3 ( X ) C X s x (X[2]) 3 x X [3].
.~[3] be the restriction of
The symmetric group G(3) acts on X 3 • (X [2])3 • x[3l by permuting the factors in X 3 and (X[2]) 3 simultaniously. 7rl : H 3 ( X ) ~ 2 [3] is the quotient m o r p h i s m with respect to the induced action on ~r3(X). We can consider )~[3] as a variety of complete unordered triangles on X, as together with a triangle Z E X [3] we axe given all its vertices [xl] + Ix2] + [x3] and all the sides [Z1] + [Z2] + [Z3] (however without an ordering). The projection P3 : X (3) • ( X ( 2 ) ) (3) x X [3] ~
X [31
induces a birational morphism p : )~[31
~ X[3]
(p is an isomorphism over the open dense subset Y ([3] ) 9 We can again give a 1,1,1) stratification of )~[3] by putting
L31 := p-l(x 31)
2. The Betti numbers of Hilbert schemes
68
for all partitions v of 3. We put
F_~:~. p-l(z(1,2)(X))
f(a[4,[z,]+[z,]+[z~l,z)
/
x ~ x; ZI,Z2,Z3 ~ X[2]; a E Zc, < > e
"
~3
We sum the numbers of elements of Ta, (r • e3)(Hilb (X)(~q) and (13(Eq) x r x ea)(/ta(X)(/Fq)) respectively. Then we use remark 1.2.4 and lemma 2.5.15 to get
IX[3](~q)[ = (]X(f q)l) -~-IP2(X, ff2q)l[X(.~q)I -}-[P3(X,-~q)[ 1 qd + ~ _ q X(Eq)I(IX(~'q) I - 1) -
(1 -
+
q~)(1 - q~+~)ix(E~)l,
(1 - q)(1 - q~)
2.5. The Betti numbers of varietiea of triangles
~3 IHilb
(X)(.~?q)l
= 3
(IX(fq)l)
+
~- I P 2 ( X ,
JFq)IIXOFq) I
1 a- d
2-;~:IX(.Fq)I(IX(JFa) I 1 _ ( / -
I) +
(1-qd)2 ~ 7~
1-- d
I~r~(x)(F~)l = 6 ( I X ( f q ) l ) +
+
77
3~qqlX(Fq)](lx(.~'q)l
-
IX(~)l,
1)
(1 - qa)(1 + 2q + q2 _ 3qd _ qd+l) (1
-
IX(Fq)l.
q)2
By remark 1.2.4 we have
IX(2)(Fq)l =
+
IP~(X, Fq)IIX(F~)I + IPz(x, Fa)l,
+
IP~(X~)l.
So we get
qd
1) Ix(F~)I z 1-r - _~qd)(1 q--~-i-~-qd+l) q~l--q/lx(F~)l, 1
-
IX[3](Fq) I _- IX(~)(Fq) I + k -~ _-q -
+((i
-
-
--3
(21-q d
IHilb ( X ) ( F q ) I = IX(2)(Fq)IIX(Fq)I + k
1~ q
2
)
[X(Fq)I 2
((1:qd) 2 21--qd~ + \ (1 - q)2 + 1 1 - q / IX(Eq)l,
1Lr3(X)(Fq)l -- lx(l~q){3 '~ (~3 1 1 _- qd q _ 3 ) IX(Fq)l 2 + ( ( 1 - qd)(1 + 2q + q2 _ 3qd _ qd+l) 31 - qd~ + 2 - 1_-77 / [x(F~)I, (1 - q)2 and the result follows by an easy calculation.
Theorem
(1)
[]
2.5.18. Let X be a smooth projective variety over C. Then we have: p ( X [a] , z) = p ( X (3), z) + z 21 - z 2d-2
1 - z 2 p(X,z) 2
+
__ z 2 d - 2 ~ l ~ __ 2d
-
)(-
)
)p(X,z),
1 X , - z 2 ) p ( X , z ) + ~1 p ( X , - z a ) p(X E~1,- z ) = ~ p ( X , - z ) ~ + ~p(
78
2. The B e t t i numbers of Hilbert schemes
1 + z2 -
+z 4
(2)
z 2d-2
-
-
1 - z2
p(X,
-z)
2
(1 - z Z d - 2 ) ( 1 - z 2~)
p(X,-z),
p ( Z [31, z) = p ( X (3), z) + z 2 1 - z 2d-2 1 - z 2 p(X,z) 2 + z2 (1 - Y - 2 ) ( 1
+ z2 -
z 2~ -
z2d+2)p(X
' z),
(1 - z 2 ) 2 p ( ) ~ [ a ] , _ z ) = -1~ p ( X , - z ) 3 + ~p( 1 X ,-z2)p(X,
z) + l p ( x , - z a )
1 -- Z 2d-2
+ z2 -
-
1 - z2
- z) 2
p(X,
+ z2 (1 - z ~ d - 2 ) ( 1 + z 2 - z 2d - z 2d+2) (1 -
p(X,-z),
z2) ~
3
(3)
Z 2d-2
p ( H i l b ( X ) , z) = p ( X , z) • p ( X (2), z) + 2z 21
-1 -
z2
p(X,z)
2
(1 - z 2 d - ~ ) 2
p(Hilb (X),-z)
= ~ (p(X,-z)
3 + p(X,-z2)p(X,-z))
+ 2z 2 1 - z 2d-2
(1 - z2d-2) 2
1 - z2 P(X,z)
+z4
3
(4)
.p(X,-z),
z 2d-2
p ( H i l b ( Z ) , z) = p ( X , z) x p ( X (2), z) + 2z 2 1 1 - z 2 p(X,z) 2 + z2 (1 - z2d-2)(1 + 2z z -- 2z 2d -- z 2 d + 2 ) p ( X ' z), ( 1 - z2) 2 p(Hilb (X),-z)
=
p(X,-z)
3 + p(X,-z2)p(X,-z))
+ 2z 2 1 -- z 2d-2 1- z 2 p(X,-z)
2
+ z2 (1 - z2d-2)(1 + 2z 2 -- 2z 2d (1 - z 2 ) 2 (5)
~2d+2~ -
~
Jp(X,
-z),
p ( H 3 ( X ) , z) = p ( X , z) 3 + 322 1 1- -z 2d-2 z 2 p(X,z) 2 + z2 (1 - z2d-2)(1 + 322 - 3z 2d - z 2 d + 2 ) p ( X ' z).
(I Proofi
-z~) 2
X is defined over a finitely g e n e r a t e d ring e x t e n s i o n T of 2~, i.e. t h e r e
is an X T over s p e c ( T ) satisfying Z T
X T
C
=-
X.
~ 3
Let Y = X T
XT
(T/m)
be a
~ 3
good r e d u c t i o n of X m o d u l o q. T h e n y[3], Hilb (Y) a n d / ~ 3 ( y ) X[3], Hilb ( Z ) and H 3 ( X )
are also r e d u c t i o n s m o d u l o q, a n d we can choose the m a x i m a l ideal
m E s p e c ( T ) in such a way t h a t t h e y are all g o o d r e d u c t i o n s (see the r e m a r k s
2.5. The Betti numbers of varieties of triangles
79
before theorem 1.2.1). Choose m in such a way that furthermore l e m m a 2.~15 holds. Then (1), (3) and (5) follow immediately from l e m m a 2.5.17, remark 1.2.2 and Macdonald's formula. Z(~,2)(X) is a Grass(2, d)-bundle over X. So we have
p(z(1,~)(x))
(1~1 z2d-2)( 1 _ z 2d)
--~)-5-z~
=
p(X,z)
By proposition 2.5.11 w e get
p(2t31z)=p(xE~Jz)+
(1 -- z2d-2)(1 -- z2d)l 2
(~:~-~/:z~
~ +
Z4
+
z6)p(X,z).
So (2) follows from (1) by an easy computation. B ( X ) is a P l - b u n d l e over Z(1,2)(X). So we have by proposition 2.5.13
A ~ ( X ) , z ) = p(Hilb ~ p(Hilb (X),z) +
(I
- ~ ) (1l __ - _z4 z ~)) ~ z,~ + z4)p(X,z) + z2)(1(1: _z ~ z2)(
(4) follows again by an easy computation.
[]
For a smooth projective surface S over C these formulas can be written as follows: p ( S [3] , Z) : p ( S (3), Z) -Jr-z2p(S, Z) 2 -4- z4p(S, Z) ~3 p(Hilb (S) = p(S • S (2), z) + 2z2p(S, z) 2 + z4p(S, z)
p(~[31,z)
= p(S (~), z) + z~p(S, z) ~ + (z ~ + 2z 4 + z~)p(S, ~)
~3
p(Hilb (S), z) = p(S x S (2), z) + 2z2p(S, z) 2 + (z 2 + 3z 4 -{- z6)p(S, z)
p(.ff~(s), ~) = p(S, ~)~ + 3z~p(S, z) ~ + (z ~ + 4z ~ + z~)p(S, z) Now we consider the case of projective space Pd. The Chow groups A i ( P [3]) and A i ( H 3 ( p d ) ) have already been determined in [Rossell6-Xambo (2)].
Proposition
2.5.19. p[3], --3 (Pal), H~b3(pd) and _~3(pd) all have a cell d ~[3], d Hilb decomposition. In particular for Y one of these varieties H2i+I(Y, 2Z) = O; the groups Ai(Y) = H2i(Y, 2~) are free, and their ranks can be computed by theorem 2.5.18. Proof." Let To,... ,Td be homogeneous coordinates o n P d . For i = 0 , . . . n let Pi be the point for w h i c h T i = l a n d T j = O f o r i # j . Let r C S l ( d + l , C ) be the maximal torus of diagonal matrices and let A0,..., Ad be the linearly independent characters o f t for which any g E I" is of the form g = diag(Ao(g),..., Ad(g)). Then F acts on Pd by g. Ti := Ai(g)Ti. T h e fixed points are p 0 , . - - , P d . We have an induced
80
2. The Betti numbers of HiIbert scheme~
action of P on p~n] for all n, as F acts on the homogeneous ideals in To,... ,Td. A subscheme Z E p~n] is a fixed point of this action, if and only if its ideal is generated by monomials in To,..., Td. So the action of F has only finitely many fixed points on P[dhI. The same is true for a general one-parameter subgroup of F. We fix a onep[2] parameter subgroup 9 of F which has only finitely many fixed points on Pd, ~d and P[d3]. The induced action of 4) on P(d3) x (P[d2])(a) x p~3] and P d x P[d21 X p[a] and on the quotients Pd • p~2) x P[d2] • (pd[2])(2) • Pd[3] and (pd)3 • (Pd[2])3 X p!3] restricts to an action on the subvarieties ~3] , Hilb - - 3 (Pd), Hilb 3 (Pd) and Ha(Pd). As the action on Pd, P[d2] and P[d31 has only finitely many fixed points, it has only finitely many on Pd x P[d2] • p~3] and (Pd) 3 x (P[d2])3 x p~3]. The fixed points on the quotients P d X p~2) x P[d2] x (p~2])(2) x p[a] and (Pd) 3 X (P[d2])a x P[d3] are the images of the fixed points on (Pd) 3 X (p~2])a • p~a] under the quotient map. So there are also only finitely many. In particular the action of 9 has only finitely many fixed points o n ~ 3 ] , Hilb ~ 3 (Pal), Hilb ~ 3 (Pd) and Br3(pd). As these are smooth, they have a cell decomposition. [] ~
3. T h e varieties o f s e c o n d a n d higher o r d e r d a t a The second part of this work (chapters 3 and 4) is devoted to the computation of the cohomology and Chow rings of Hilbert schemes. In chapter 3 we define varieties of second and higher order data on a smooth variety X and study them. In section 3.1 we consider the varieties D~(X) of second order data of m-dimensional subvarieties of X. We define D~(X) as a subvariety of a product of Hilbert schemes of zero-dimensional subschemes of X. Then we show that D~(X) can be described as a Grassmanian bundle over the Grassmannian bundle of m-dimensional subspaces of the cotangent bundle of X. D~(X) is a natural desingularisation of X (3)" [3] Using the description as a bundle of Grassmanians we compute the ring structure of the cohomology ring of D~(X). Then we descibe in what sense D~(X) parametrizes the second order data of m-dimensional subvarieties of X and the relation to second order contacts of such subvarieties. In section 3.2 we consider the varieties of higher order data D~(X). Their definition is a generalisation of that of D~(X). We show that only the varieties of third order data of curves and hypersurfaces are well-behaved, i.e. they are locally trivial bundles over the corresponding varieties of second order data with fibre a projective space. In particular D3(X) is a natural desingularisation of ~c[4] "~(4)' Then we compute the Chow ring of these varieties. As an enumerative application of the results of chapter 3 we determine formulas for the numbers of second and third order contacts of a smooth projective variety X C PN with linear subspaces of PN. In section 3.3 we introduce the Semple bundle varieties Fn(X), which parametrize higher order data of curves on X in a slightly different sense. We use them to show a general formula for the number of higher order contacts of a smooth projective variety X C PN with lines in PN. Arrondo, Sols and Speiser [Arrondo-Sols-Speiser (1)] have independently constructed new contact varieties for m-dimensional subvarieties of a given variety X, for which they also give a number of applications. Their approach is different from the one of sections 3.1 and 3.2 and is in fact a generalization of the Semple bundle construction. This approach is more general then mine, as it gives varieties of arbitrary order. It has however the disadvantage of not taking the commutativity of higher order derivatives into account, and thus, except in the case m = 1, the actual data varieties are given as subvarieties (by requiring "symmetry") of considerably bigger varieties. The precise description of these subvarieties appears to be not a very easy task, and as far as I know has been carried out only in the case of second order data of surfaces in P3.
82
3.1. T h e varieties o f s e c o n d o r d e r data. Let X be a s m o o t h projective variety of dimension d over an algebraically closed field k. In this section we want to define a variety
D2m(X) of second order d a t a of
m - d i m e n s i o n a l subvarieties of X for any non-negative integer m < d. A general
D2m(X) will correspond to the second order d a t u m of the germ of a smooth m - d i m e n s i o n a l subvariety Y C X in a point x C X , i.e. to the quotient Oy,~/m3x,~ point of
of Ox,~. Assume for the m o m e n t t h a t the ground field is C and x 6 Y C X , X is a smooth complex d-manifold and we have local coordinates
zl,..., Zd at x. Then Y
is given by equations
f i ( Z l , . . . , Zd)~-O Then the second order d a t u m
i= 1,..., d-m.
Oy,,/m3x,, is
C[zl, . . . ,
Zd]/((fl,..., fd-m) + m3),
and giving the second order d a t u m is equivalent to giving the derivatives
0f~
Ozj(X), i = l , . . . , d - m , 02 fi , , O~jOzt~X), i = 1 , . . . , d - m ,
j= l,...,d j,l = 1,...,d
N o t a t i o n . In chapter 3 and 4 we will often use the G r a s s m a n n i a n bundle associated to a vector bundle. So we fix some notations for these. Let S be a scheme and E a vector bundle of rank r on X . For any m < r let
Grass(m, E) denote the G r a s s m a n n i a n bundle of m - d i m e n s i o n a l quotients of E. Let 7rm,E : Grass(m, E) -----* S be the projection, Qm,E the universal quotient bundle of ~rm,E(E) a n d T~-m,E the tautological subbundle. T h e n the projectivization of E is P ( E ) = Grass(r-1, E) and Op(E)(1) = (T1,E)*. We also p u t 15(E) : = Grass(l, E). We write Grass(re, r) for the G r a s s m a n n variety of m-dimensional quotients of C ~. Let Qm,~ and T~-m,~ be the universal quotient bundle a n d the tautological s u b b u n d l e on Grass(m, r ).
Notation.
For subschemes
respectively in
Z1, Z2 of a scheme S with ideal sheaves :Z'zl, Iz2
Os, let Z1 9 Z2 denote the subscheme Z of S whose ideal sheaf I z is
given by 2"z : = Zzl " I z 2 . As above we will write Z1 C Z2; to mean t h a t Z1 is a subscheme of Z2. In this case we will write
Zzl/z2 for the ideal of Z1 in Z2.
3.1. The varieties of second order data
Definition 3.1.1.
83
Let 7)2re(X) be the contravariant functor from the category of
noetherian k-schemes to the category of sets which for noetherian k-schemes S, T and a m o r p h i s m r : S ~
T is given by:
Zo,Z1,Z2 C X x T closed subschemes fiat of degrees 1, m + 1, (%+2) over T Z o C Z a c Z 2 , Z1 c Z 0 " Z 0 , Z2 C Z0 9 Z1
"D2(X)(T)= { (Z~ Vi(X)(r
:
VI(X)(T)
(Zo,Zl,Z2) ~
,
] ,
~i(x)(s)
(go XTS,Z1 XTS, Z2 XTS).
L e m m a 3.1.2. :D2(X) is representable by a closed subscheme D 2 ( X ) C X x Zlm+11 • X[C+2)]. P r o o f i Let
Z l ( X ) := A c X • X
Zm+l(X) C X z(,o:~)(x) c
x
• X [m+l] •
x('~t ~)
be the universal subschemes. To shorten notations we write
w := x • xEm+'I • X [(mt~)] For i = 1,2,3 let Pi be the projection of W to the i th factor. Let 2-0, 2-1, 2"2 be the ideals of
Wo := ( i x • p~)-~(Zl(X)), Wa : = (1x • p2)-l(Zm+a(X)),
w2 := ( i x • p3)-l(z(m+~)(x)) respectively in Ox • w. Let U0, U1,0"2 C W be the subschemes defined by 2-0 + 2"1, 2-1 + 2"2 + 2"~ u n d 2"2 + 2-0 92"a respectively. T h e n we have obviously Ui C Wi for i = 0,1,2. As X is a closed subvariety of a projective space P N , Wo,W1,W2, [To, U1, U2 are in a n a t u r a l way subschemes of P N • W . The Wi are flat of degree (i+m) o v e r W f o r i = O , 1,2. W e p u t ~" := Ouo @ Ou, G Ou2. For may m o r p h i s m g : T
, W of a noetherian scheme to W we p u t
~
:= r - m, l > m respectively. One can summarize these relations to (1 + dl + . . . +
dr-m)(1 +
el + . . .
"~ ern) =
c(E).
One has to note that the relation holds for every weight. We have
e(Tr-m,.) = (1 + dl + . . . + dr-m), c(0m,E) = (1 + el + . . . + ~m). In the case of a projective b u n d l e P ( E ) we get in particular A * ( P ( E ) ) --
A*(X)[P]
where P = cl(Op(E)(1)).
For the Chern classes of a symmetric power of a vector bundle we have the well-known relation:
3.1. The varieties of second order data
93
R e m a r k 3.1.9. Let E be a vector bundle of rank r over X with total Chern class c(E) = 1 + el + . . . er. Let c(E) = (1 + Yl)... (1 + y~) be a formal splitting of c(E). Then we have c(Symm(E)) =
( l + y i , +...+Yim).
H il 2 and d _> 2 however D ~ ( X ) is not compact. Remark
3.1.16. Let
Pl : Dlm(X)o DI,,(Y),
, Dm l-1 (X)o , Dt-I,,(Y)
3. The varieties of second and higher order data
98
Then DIm(X)o is via Pt a locally trivial fibre bundle o v e r D~-l(X)0 with fibre A(d-m)(m+t-1). This is only a reformulation of remark 2.1.7. Now a variety of Ith order data should be a natural smooth compactification of D~(X)o. This is for instance the case for D~(X), as this is given in a canonical way as a subscheme of a product of Hilbert schemes, it is smooth, compact and contains D2m(X)o as a dense open subvariety. There is a morphism r
D~m(X) ~
Dim(X) = Crass(m,7~ ),
extending P2- The fibres of r are obtained by compactifying the fibres of p2 to the Grassmannian Grass(("+'), (,~+1)+ (d - m)). Now we want to compute the class of the complement D~(X)oo := D~(X) \ D~(X)o. It parametrizes in a suitable sense the second order data of singular m-dimensional subvarieties of X. We will use a tool that will play a major role in the enumerative applications of higher order data in section 3.2, the Porteous formula. We will not quote the result in full generality but in the formulation in which we are going to use it. D e f i n i t i o n 3.1.17. Let X be a smooth variety and E and F vector bundles on X of ranks e and f respectively. Let c(E), c(F) E A*(X) be their total Chern classes. We write
c(F
-
E):= c(F)/c(E)
and ci(F - E) for the part of c(F - E) lying in AJ(X). The total Segre class s(E) of E is given by 4 E ) := c ( - E ) = 1/4E), and the jth Segre class s / ( E ) of E is the part of s(E) in AJ(X). Let a : E -----* F be a morphism of vector bundles on X. For all x r X let a(x) be the corresponding m a p on the fibres. Let :Dk(cr) C X be the subscheme
with its natural scheme structure, i.e. with respect to local trivialisations of E and F it is defined by the vanishing of minors of the matrix representing a. We call ~ k ( a ) the k th degeneracy locus of a. Let [~Pk(a)] r A*(X) be the class of ~Pk(a). We call [T)k(a)] the k th degeneracy cycle of a.
Theorem
3.1.18[Fulton (1) Thm. 14.4].
3.1. The varieties of second order data
99
(1) Each irreducible component of :Dk(rf) has codimension at most r := ( e - k ) ( f k) i n X .
(2) If the codimension of T)k(a) in X is r, then we have: [:/)k(a)] ----dct((cf-k+i-j(F - E))l rn and x C X. We say Y1, Y2 have Ith order contact along an
3. The varieties of second and higher order data
100
m-dimensional subvariety at x, if x C Y1 A Y2, and there is a germ of a smooth subvariety Z C X at x satisfying Dl,x(Z) C Y~ and Do:(Z ) C Y2. We say Y1, II2 have m-dimensional l th order contact, if there is an x C X such that they have Ith order contact along an m-dimensional subvariety at x. If m is the m i n i m u m rain(d1, d2), we say in this case that Y1 and Y2 have l th order contact
(at ~). From the definitions we get immediately:
R e m a r k 3 . 1 . 2 1 . Y1 and Y2 have m-dimensional Ith order contact at x, if and only if D~(Y1)o, and D~(Y2)o intersect as subvarieties of Dtm(X)o in points lying over
xEX. In case dl = m _owith 1 < m < d. Let ~),~(X) be the contravariant functor from the category of noetherian k-schemes to the category of sets which for noetherian k-schemes S, T and a m o r p h i s m r : S ~
T is given by:
Zi C X x T closed subscheme flat of degree (re+i) over T (i = O , . . . , n ) I)~(X)(T) : { (Zo,..., Z,~)
Zo C Z1 C . , . C Z n
and Zi " Zj D Zi+j+l for all i,j with i + j < n -
p~,(x)(r
:
~(X)(T) (Zo,...,Zn)
,
, ,
1
}
~(X)(S) (Zo•
Here we use again the notations we have introduced in definition 3.1.1. In the same way as in l e m m a 3.1.2 we can show t h a t : / ) n ( X ) is represented by a closed subscheme D ~ , ( X ) c X x XEm+,1 x . . . x X
We call
D~(X)
D~(X)
is as a subset o f X [m+l] x . . .
D~m(X) :=
the variety of
n th
order d a t a of m - d i m e n s i o n a l subvarieties of X. x X [ ( '~"+n)] given by:
(z0 ..... zn) c x • xEm+,l •
[(into)].
• x
[(~:")]
Z0 C Zl C Z2 C . . . C Zn / and Zi - Z j D Z i + j + l / for all i,j with i + 3 _< n - 1
3. The varieties of second and higher order data
102
Obviously we have D ~
= X. For i = 0 , . . . , n let r i : D~(X) ~
X [(~'+')]
be the projection. We also consider the projection
~r. : D~(X)
, Dnm-I(x) 9
It is not clear in which cases D,~(X) is reduced, irreduble or smooth. In cases in which it is reducible a b e t t e r c a n d i d a t for the variety of higher order d a t a is the closure the image of D~(X)o under the obvious embedding. A C X x X be the diagonal.
Let
T h e n Hilb('+m)(Ai+l/X) is a closed subscheme
of Hilb('+m)((X x X ) / X ) = X[('+m)] for all i. We can see i m m e d i a t e l y t h a t the projection ri: D~(X) ~
X [ ( ' + ' ) ] factors through Hilb('+~)(Ai+I/X), as we see
from the definitions t h a t Hilb('+'~)(Ai+~/X) represents the functor
TI
, {(Zo,zd
Z0, Zi C X • T closed subschemes ) fiat of degrees 1, (re+i) over T /
Zi C Zg"}-1 We now want to show t h a t D~(X) and D3d_a(X) are again G r a s s m a n n i a n bundles corresponding to vector bundles over D~(X) and D ~ _ I ( X ) respectively. Before doing this we want to show t h a t these two cases are the only ones in which we can expect such a result (exept for the trivial case m = d).
R e m a r k 3.2.2. (1) Let 2 < m _< d - 2. T h e n ~r3 : DO(X ) ~
D~(X) is not a locally trivial fibre
bundle 9 (2) Let S be a s m o o t h surface. T h e n 7r4 : D4(S)
, D~(S) is not a locally trivial
fibre bundle. Proof.* (1) Let x E X . Let xa . . . . Xd be local p a r a m e t e r s near x a n d let m x , ~ : = (xl . . . . , x a ) be the m a x i m a l ideal at x.
Let Z1, Z2 be the subschemes of X with s u p p o r t x
defined by the ideals
I1 = (xm+l,.. 9 ,xd) T m Xp~g~ 2 I2----(Xm+l,.
9 9
,Xd)Tm 3
X~x
3.2. Varieties of higher order data and applications
103
in Ox,,. Then we have (x, Z1,Z2) E D ~ ( X ) . The fibre rc31((x,Z,,Z2)) consists exactly of the subschemes Z3 C X with support x whose ideal/3 in Ox,, is of the form I3 = ( V ) -[- (Xm.t- 1 Xd)" m x , . + m 4 for some (d - m)-dimensional linear subspace V of
(Xm+l ...Xd> -t- <XiXjXl [ i,j,l m)+m
3X,x
in Ox,,. Then (z, Z1, Z ~ ) i s a point of D ~ ( X ) . The fibre rr31((x, Z1, Z~)) consists of exactly those subschemes Z~ with support x whose ideal I~ in Ox,, is of the form
/~ =(w) + ( z ~ z j l i >_ m + 3) + ( x ~ x i l i , j > m + 1) + ( x ~ z i , X l ~
i < d)
+ ( ~ z ~ x , l i > m + 1) + m~x,. for an (m+2)-codimensional linear subspace W of
V := 2[_<Xm+lXi,Xm.l_2Xi li < m> + (xixjxz[1 < i < j < l < m ; j > 2>. By dim(U) = d - m + (,,,+2) + 1 we have
% ' ( ( z , Z , , Z ' ~ ) ) ~- Grass((~+2),d - m + 1 + (~+~)). (2) Now let S be a smooth surface, s E S and x, y local parameters near s. Let Z1, Z2, Z3 be the subschemes of S with support s defined by I1 :-- (x, y2),
I2 := (~,y~), 13 := (x,y4). Then we have (s, Z1, Z2, Za) e D3(X). Thus r~-l((s, Z1, Z2, Za)) consists of the subschemes Z4 with support s whose ideal /4 in Os, s is of the form /4 = w + (x 2 , xy, yh) for a one-dimensional linear subspace w C (z, y4 ). So we have
7r41 ((s, Zl, Z2, Z3)) ~-~ P1.
3. The varieties of second and higher order data
104
Let Z~, Z~ be the subschemes of S with support s defined by I ; := (x2,xy,y2), /~ := (x2,xy, y3). Then (s, Z1, Z~, Z~) is a point of D~(S).
zr4~((s, Z1, Z~, Z~)) consists of the sub-
schemes Z~ with s u p p o r t s whose ideal is of the form
I'~ = (t) + ( x ~ , ~ > x y ~ , ~ ~) for a two-dimensional linear subspace t C (x 2, xy, y3>. So we have
7F41((W, Zl,Z2, Z3)) ~ P2.
[]
D e f i n i t i o n 3.2.3. Let X be a smooth projective variety of dimension d over a field k. Let m be a positive integer with m < d. We will again use the notations from the definitions 3.1.3, 3.1.4 and 3.1.5. Let ~2 := #loft2. We define the s u b b u n d l e T2 of ~ ( J a ( X ) ) by the d i a g r a m
0
0
'
0
T
T
Q2
Q2
T
T
~ o
l
0
---+
~(Syma(T~))
~ ~-;(da(X))
--~
0
--~
~(Syma(T~))
~
--~
T2
~(J:(X))
T2
T
T
T
0
0
0
Let again A C X x X be the diagonal and Zt, C O x x x
be the projections. For all non-negative integers i _< j let
J j ( X ) : = (S2).((Za)i/(zA )J+I). Then J~(X) is locally free, and we have the exact sequence
,
J~(X)
--*
Jj(X)
--+
0
,
0
its ideal sheaf. Let
sl,s2 : X x X ----* X
0
)
Ji-~(X)
,
O.
39
Varieties of higher order data and applications
105
We see J~ = Jj(X) and Jj(X) = SymJ(T~). Let il rn} + (Yzi~j2z [i,j,l < m). T h e n t h e r e s t r i c t i o n of the natural projection ~ : (~TI
~J~(x))(~)
, V~(v)
to A0 is an i s o m o r p h i s m , and (1) follows.
T2(v) is an ( m - d ) - d i m e n s i o n a l linear s u b s p a c e of ~(w~,(z))(v) Let p : fr~(W~(X))(v) ~
-*. . . 9. 7r2T}))(v . = (7c2T,/(rc2T, ).
fr~(T1)(v) be t h e p r o j e c t i o n . As we h a v e a s s u m e d t h a t
rn = 1 or m = d - 1 holds, we h a v e either p(T2(v)) = ~r~(T1)(v), or p(T2(v)) has c o d i m e n s i o n 1 in ~r~(T~)(v). (~~(T1)(v) is o n e - d i m e n s i o n a l in case m = d - 1, a n d
T2(v) has c o d i m e n s i o n 1 in ~r~(W~(X))(v) in case m = 1.) (a) p is onto. T h e n we h a v e
f~(v) = (ym+~,... , w h e r e x l , 9 9 Xm,
Y m + l ,
.
-
9 ,
y d ) / m x , ~3,
Yd are local p a r a m e t e r s n e a r x. So we can a s s u m e
t h a t xi = yi for i = m + 1 , . . . , d .
T h e n we h a v e
(~. ~;(J~(x)))(v) : ~)+ ~). Let A1 : = (~2i2.j24[i,j, 1 0 A, A '~ (thickened) diagonal J.(X), J~(X) jet bundles zT(x), a~(x) relative Hilbert function strata curvilinear subschemes concentrated in a point A.(X), A*(X) Chow ring el cycle map G.. multiplicative group p(n), p(~, l) number of partitions of n r(~) graph of partition .& dual partition ti(a), T(a) diagonal lengths of partition h~,v(~) hook difference Y. punctual Hilbert scheme *'(~) set of partitions of n ( n l , . . . , n ~ ) = (1~',2~2,...) partitions Gal(]c/ k ) Galois group length of subseheme fen(f) "length" of function P(X, •q) primitive 0-cycles Tn(X, Fq) set of admissible functions A(~-), ~(~-) Delta function, eta function
1 2 2 2 2 3 3 3 4 4 5 5 5 5 8 10 10 10 14 14 17 18 19 19 19 20 23 23 23 23 29 29 29 29 29 30 31 31 35
Index of notation3
hP'q(X) h(X,x,y) sign(X) xdx) G~
Is KAn-1 NH/H,
s[.] ~l(n) e(x, G) ~ n Hilb (X) res
~3(x) _~[3]
Hodge number Hodge polynomial signature xy(X)-genus modular forms higher order Kummer variaties Shintani descent set used for counting sum of numbers dividing n orbifold Euler number incidence variety of subschemes of lengths n - 1 and n residual morphism variety of complete triangles variety of complete unordered triangles
~ 3
Hilb (Z)
complete triangles with marked side Grassmannian bundle projection in Grassmannian bundle ";rm,E Grass(m, r) Grassmannian P(E), P ( E ) projectivized bundle Tm,~, Tm,r tautological subbundles universal quotient bundles Qm,E, O,,,,r scheme defined by product of ideals Z1 9Z2 D~(X) variety of second order data tautological and quotient bundle over Grass(m, T~ ) T1, Qi w~(x) bundle of second order data b~(x) other construction of D~(X) tautological and quotient bundle over D2(X) I'2, Q2 (ox)~ contact bundle Ar A~ (thickened) diagonals of morphism Ith-order datum of Y at x Dz,=(Y) Vk(~) degeneracy locus nn(x) variety of higher order data F.G "product" of sheaves W~(X) third order data sheaf 5~(x), bLl(x) variety of third order data tautological subbundle and quotient of W3(X) Ts, 03 contact bundle (E)~n evaluation map eVE D~(Y/T) relative data variety class of second order contact g(x, YT) Fn(X), an(x) Sample bundles
Grass(m, E)
195
37 37 37 37 38 40 42 42 51 54 60 61 63 67 71 82 82 82 82 82 82 82 83 85 86 87 87 88 88 97 98 101 105 105 107 107 108 115 126 126 128
Index of notations
196
AI~(PN) Kn(X), Kn,l(Z) Fn(X/T), Gn(X/T)
aligned n-tuples class of contact with lines relative Semple bundles r163 morphism of Hilbert scheme induced by/3 Hilbn(P(E)/X) relative Hilbert scheme of projective bundle era,n, Cn morphisms of Hilbert schemes of projective bundle AI~(P(E)), Al~(Pd) variety of aligned subschemes Z~'(P(E)), Z~t(Pd) universal subscheme of aligned subschemes axe ~ axial morphism ~ 3 H, A, H2, P, P2 classes in the Chow ring of Hilb (P2) ~ 3
Hilb (P(E)/X), W(P(E)), H3(P(E)/X) ~ 3
Cop (Pd) H, ,zl, fz, fi, 6,/~ Cop3(pd)
relative triangle varieties
variety of triangles in a plane in Pd classes in the Chow ring of Hilb3(P(E)/X) Hilbert scheme of subschemes in a plane in Pd
Printing: Weihert-Druck GmbH, Darmstadt Binding: Buchbinderei Schiiffer, Griinstadt
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