Table Of ContentDanielle Dias Patrick Le Barz
Configuration Spaces
over Hilbert Schemes
and Applications
Springer
Authors
Danielle Dias
Patrick Le Barz
Laboratoire de Math~matiques
Universit6 de Nice - Sophia Antipolis
Parc Valrose
F-06108 Nice, France
e-mail: ddias @math.unice.fr
lebarz@math.unice.fr
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Dtas, Danielle:
Configuration spaces over Hilbert schemes and applications /
Danielle Dias ; Patrick LeBarz. - Berlin ; Heidelberg ; New
York ; Barcelona ; Budapest ; I-long Kong ; London ; Milan ;
Paris ; Santa Clara ; Singapore ; Tokyo (cid:12)9 Springer, 1996
(Leclure notes in mathematics ; 1647)
ISBN 3-540-62050-8
NE: LeBarz, Patrick:; GT
Mathematics Subject Classification (1991): 14C05, 14C17
ISSN 0075- 8434
ISBN 3-540-62050-8 Springer-Verlag Berlin Heidelberg New York
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Table of Contents
Introduction 1
Part one : Double and triple points formula 9
Conventions and notation 11
1.1 Fundamental facts ............................. 11
1.2 Conventions ................................. 11
1.3 Notation ................................... 12
Double formula 13
2.1 The class of H2(X) in H2(Z) ....................... 13
2.2 Definition of the double class ........................ 16
2.3 Computation of the double class ...................... 18
2.3.1 Computation of 3//2 ......................... 18
2.3.2 .................................... 19
2.3.3 .................................... 20
3 Triple formula 22
3.1 The class of Ha(X) in H3(Z) ....................... 22
3.2 The triple formula ............................. 26
3.2.1 Some notation ............................ 26
3.2.2 Computation of M3 and r ................... 28
3.2.3 Computation of prl,Wll,Ul ..................... 31
3.2.4 Computation of {s(U) (cid:141) cW} m . ................. 32
3.2.5 Computation of prl,~,u2, first part ................ 35
3.2.6 Computation of prl,~,u2, second part .............. 38
3.2.7 Conclusion .............................. 42
Intermediate computations 44
4.1 ........................................ 44
4.2 Flatness of 7h and ~2 ............................ 44
vi
4.3 Proof of lemma 4.(iv) and of ~lg = 0 .................. 45
4.4 Proof of lemma 4.(iii) ........................... 46
4.5 Proof of lemma 4.(ii) and (v) ....................... 47
4.6 Proof of lemma 1 .............................. 48
4.7 Flatness of P12 and of P3 .......................... 49
4.8 Proof of lemma 4.(i) ............................ 51
4.9 Proof of lemma 3 .............................. 52
4.10 Transversality of~ and ~ .......................... 53
Application to the case where V is a surface and W a volume 55
5.1 Computation of c~(v) ............................ 56
5.2 Computation of cp(w)jl ........................... 59
5.3 Computation of the contribution of I ................... 63
Part two : Construction of a complete quadruples variety 65
6 Construction of the variety B(V) 67
6.1 Statement of the theorem ......................... 67
6.2 Definitions, drawing conventions ...................... 67
6.3 Irreducibility and dimension of B(V) ................... 68
6.3.1 General facts on Hilbert schemes : ................. 68
6.3.2 .................................... 69
6.4 Non-singularity of B(V) .......................... 77
6.4.0 Preliminaries ............................ 78
6.4.1 Non-singularity of B(V) at ~o where qo is a locally complete
intersection quadruple point .................... 83
6.4.2 Non-singularity of B(V) at q'o where qo is a non-locally complete
intersection quadruple point .................... 91
Construction of the variety H4(V) 102
7.1 Non-singularity of H4(V) at ~ where q is a locally complete intersection
quadruplet .................................. 104
7.1.1 Case of the curvilinear quadruplet ................. 104
7.1.2 Case of the square quadruplet ................... 108
7.2 The variety H4(V) at ~ where q is a non locally complete intersection
quadruplet .................................. 111
7.2.1 Case of the elongated quadruplet ................. 111
7.2.2 Case of the spherical quadruplet .................. 123
7.3 Irreducibility of H4(V) ........................... 127
vii
Appendix A 129
A.1 Local chart of H3(V) at t', where t is a curvilinear triple point ..... 129
A.2 Local chart of H3(~V) at t', where t is amorphous ............. 130
Appendix B 132
B.1 Local chart of H4(V) at an elongated quadruplet ............ 133
Ha(v)
B.2 Local equations of at a spherical quadruplet ........... 134
Bibliography 136
Index 139
Index of notation 141
Introduction
0.1
Let f : V > W be a morphism of non-singular varieties over C, with dimV < dimW.
Let d = cod(f) = dimW - dimV. The locus (cid:1)88 of elements x E V such that there
exists at least (k - 1) other points of V in the fiber f-if(x) is called k-uple locus of
f. When it exists, a class m~: in the Chow ring CH'(V) of V, which represents V~:, is
called k-uple class of f. Then a k-uple formula is a polynomial expression which gives
m~: in terms of the Chern classes c, of the virtual normal bundle u(f) = f*TW - TV.
0.2
When one deals with the double formula, one is interested in the set of elements
x E V such that the fiber f-if(x) contains at least one other point in addition to x.
A typical example is the imbedding with normal crossings f : C ~ ~2 of a smooth
curve, where one wishes to count the number of double points of f(C).
The case k = 2 was treated thoroughly by Laksov ([La]). The double formula was
also found by Ronga ([Ro]) in the C ~ case. The demonstration consists in looking
at the blowing-up V x V of V x V along the diagonal and applying the residual
intersection formula ([FU2], thm 9.2, pp 161-162) in order to remove the exceptional
divisor which corresponds to the solutions xl = x2 of f(xl) = f(x2) (which we do not
want) at the lifted double locus (see [FU2], pp 165-166). Then the double class m2 is
given in the Chow ring of V by the double formula :
~ : f*L[v] - cd,
where Ca is the d th Chern class of u(f), defined above.
0.3
When one deals with the triple formula, one is interested in the set of elements x C V
such that f-if(x) contains at least two points in addition to x. A typical example
2 Introduction
is the imbedding with normal crossings f : S ~ ~2 of a non-singular surface, where
one wishes to count the number of triple points of f(S).
One difficulty is to define a class m 3 E CH'(V) representing the set V3 and to
compute this class so that one has the trzple formula :
d
m3 = f* f.m2 - 2cdm2 + ~ 2JCd_jCd+j ,
j=l
where c~ is the ith Chern class of ,(f).
This was done by Kleiman [KL1, KL2], modulo some general hypotheses on the mor-
phism f. Kleiman even established a stronger formula [KL3]. So did Ronga [Ro] in
the C ~ case. (These are "refined" formulas in the sense that if f(xl) = f(x2) = f(x3),
they count the set of non ordered {xl, x2, x3} having the same image by f and not the
set of ordered (Xl, x2, x3) ; therefore there is a gain of 3! in the formulas. The present
work, despite the use of Hilbert schemes, will only deal with non refined formulas.
However, all the demonstrations of triple formulas use general hypotheses on the
morphism f, essentially the regularity of some "derivative" applications. See also the
paper by Colley [Co].
0.4
The goal of the first part of this book is to establish the triple formula without any
hypotheses on the genericity of f. Of course, one must immediately :
(i) make it clear that this requires to choose an ad hoc definition of m3,
(ii) emphasize that in the degenerate case where the triple locus is too big, the
formula does not mean much !
Looking for the triple locus of f means looking for the set of (xl, x2, x3) of V (cid:141) V x V
such that f(xl) = f(x2) = f(x3). Once again, one wishes to eliminate the solutions
with Xl = x2 or x2 = x3 or x3 = xl. One must find a "good" space of triples for
V : a space where the locus to be eliminated is a Cartier's divisor. In [KL1], Kleiman
uses the space 'gilb 2(V) x v'Hilb2(Y) where 'Hilb 2(V) denotes the universal two-
sheeted cover of Hilb2(V). In [Ro], Ronga blows-up in gilb2(V) x Y the tautological
~Hilb2(V). Our suggestion here is to use the space H3(V) of completely ordered triples
of V, introduced in [LB1], which is birational to V x V x V. Let us recall briefly the
construction of H3(V) :
An element t = (Pl,P2,P3, d12, d23, d31, t) in the product V 3 (cid:141) [Hilb2(V)] 3 (cid:141) Hilb3(V)
is a complete triple if it verifies the relations :
{ p~ C dii C t ( scheme-theoretic inclusions )
Pi = Res(p~, dii)
p~: = Res(d~i,t ) with {i, j, k} = {1, 2, 3}
where Res(7 h ~) denotes the residual closed point of the (h - 1)-uplet U contained in
the k-uplet ~.
The motivation is that this space appears to be more natural, in view of the action
of the symmetric group $3. However, one must realize that one ends up computing in
~Hilb2(V) x V in the process of the demonstration. In particular, the origin of the 2 j
that one finds in the triple formula stems from the computation (see w 3.2.4) of the
virtual normal bundle of the morphism 'Hilb2(V) --+ Hilb2(V), that one already finds
in [KL1] and [Ro].
0.5
Once the space of triples we work with has been chosen, we work along the same lines
as Ran [Ra] and Gaffney [Ga]:
(i) if X C Z is a non-singular subvariety of a non-singular variety Z, one gives the
A
fundamental class [H3(X)] in the Chow ring CH~ (theorem 3).
(ii) if f : V --+ W is a morphism, one defines the triple class m 3 e CH~ as the
direct image of the cycle
M3 = [HA 3(rs)]. [HA 3(V) x W] e CH'(H3(AV W))
x
where Ff is the graph of f. Tedious but straightforward computations lead to the
triple formula (theorem 4).
However, one must realize that in the case where the morphism f has S2-singularities,
the scheme-theoretic intersection H3(r/)n (H3(V) x W) c H3(V (cid:141) W) has automat-
ically excess components. This makes the interpretation of the formula tricky. In
enumerative geometry, one often gives a formula which is valid in general, even if one
must explain afterwards how many "improper" solutions must be removed in order
to find the number of "proper" solutions. We have chosen to follow this approach,
i.e. we provide a formula which is valid in general, but we are aware that the second
half of our task would be to interpret this formula in the degenerate cases, which one
cannot avoid. This is done in a preliminary way in chapter 5, where one considers
the simplest case where f : V ~ W is a morphism from a surface to a volume with
S2-singularity.
0.6
The second part of this book is devoted to the construction of a variety of complete
quadruples in order to define a class m4 in the Chow ring CH~ The goal is
to construct a "good" space of completely ordered quadruples of V, in which the
locus to be eliminated is a Cartier's divisor. To do so, one wishes to generalize the
4 Introduction
A
construction of the variety Ha(V) of the complete triples. Therefore, the question is to
construct naturally a variety H4(V) consisting of ordered quadruples of V, possessing
a birational morphism :
HA 4(V) --+ V (cid:141) V (cid:141) V (cid:141) V
compatible with the action of the symmetric group $4, and an order-forgetting mor-
phism :
H4(V) --+ Hilb4(V)
The construction of this variety must also be compatible with closed imbeddings :
if V C W is a subvariety of W, then H4(V) can be identified with a subvariety of
H4(W).
0.7
A naive generalization H~4Aa ~,,e(V) of the construction of HA 3(V) is not sufficient, as
was already pointed out by Fulton ([FU1]). The variety H~4,, ,~,,,~(V) is defined as a
subvariety of the product V 4 x [Hilb2(V)] 6 x [Hilba(V)] 4 x Hilb4(V). To do this, one
introduces the following notation :
Notation 1 : If ~ is a point in Hilbd(V), one will denote by Z~ the ideal sheaf of Ov
which defines the corresponding subscheme.
An element (PI, P2, P3, P4, d12, d13, d14, d23, d24, d34, tl, t2, t3, t4, q) in the above product
is a complete nai've quadruple if it verifies the relationships :
z,,, z,,, z<j z,,, z,,,
c c n
z,,~ Zt,, c z,, c z,,~ n z,t,, for{i,j,k,l}={1,2,3,4},
z,,, z,, c zq c zv, n ~, i<jandk<l
Unfortunately, H,4,A, , ~,,~.(V) is reducible and singular. We will see below that the condi-
tions (.) introduce excess components.
Recall [I2, F] that (for dimV = 3) the Hilbert scheme Hilb4(V) is irreducible and
singular at the quadruplets q of ideal A/I2, where 2~4v is the ideal of a closed point p
of V. Consider the subvariety R(V) of Hilb2(V) (cid:141) Hilb4(V) (cid:141) HilbZ(V) consisting of
elements (d, q, d') satisfying the relations :
The variety R(V) possesses a projection onto Hilb4(V), which we denote by II. We
will see (chapter 6) that R(V) is reducible : R(V) is the union of two irreducible
