Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1647
Springer
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Danielle 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
[email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Dtas, Danielle: Configuration spaces over Hilbert schemes and applications / Danielle Dias ; Patrick LeBarz. - Berlin ; Heidelberg ; New Y o r k ; B a r c e l o n a ; B u d a p e s t ; I-long K o n g ; L o n d o n ; M i l a n ; P a r i s ; S a n t a C l a r a ; S i n g a p o r e ; T o k y o 9S p r i n g e r , 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 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 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10520222 46/3142-543210 - Printed on acid-free paper
Table of Contents
Introduction
1
Part one : Double and triple points formula
9
Conventions and notation
11 11
1.1
Fundamental facts
1.2
Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.............................
11
1.3
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Double formula
3
13
2.1
The class of H 2 ( X ) in H 2 ( Z )
2.2
Definition of the double class . . . . . . . . . . . . . . . . . . . . . . . .
16
2.3
C o m p u t at i o n of the double class . . . . . . . . . . . . . . . . . . . . . .
18
2.3.1
18
13
.......................
Computation of 3//2 . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2
....................................
19
2.3.3
....................................
20 22
Triple formula 3.1
The class of H a ( X ) in H3(Z)
3.2
The triple formula
4.2
.............................
22 26
3.2.1
Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.2.2
C o m p u t a t io n of M3 and r
28
3.2.3
Computation of prl,Wll,Ul
3.2.4
Computation of {s(U) • c W } m
3.2.5
C o m p u ta ti o n of p r l , ~ , u 2 ,
first part . . . . . . . . . . . . . . . .
35
3.2.6
Computation of p r l , ~ , u 2 ,
second part
38
3.2.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Intermediate 4.1
.......................
................... ..................... . .................
..............
31 32
42
computations
44
........................................
44
Flatness of 7h and ~2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
vi 4.3
Proof of lemma 4.(iv) and of ~ l g
4.4
Proof of lemma 4.(iii)
4.5
Proof of lemma 4.(ii) and (v)
45 46
.......................
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 o f ~ and ~ . . . . . . . . . . . . . . . . . . . . . . . . . .
53
A p p l i c a t i o n t o t h e case w h e r e V is a s u r f a c e a n d W a v o l u m e
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 6
= 0 ..................
...........................
two
: Construction
of a complete
quadruples
variety
65
C o n s t r u c t i o n of the variety B(V)
67
6.1
67
Statement of the theorem
.........................
6.2
Definitions, drawing conventions . . . . . . . . . . . . . . . . . . . . . .
67
6.3
Irreducibility and dimension of B ( V )
68
6.3.1 6.3.2 6.4
General facts on Hilbert schemes : . . . . . . . . . . . . . . . . . ....................................
Non-singularity of B ( V ) 6.4.0 6.4.1
...................
Preliminaries
..........................
............................
C o n s t r u c t i o n o f t h e v a r i e t y H4(V)
7.2
83 91 102
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
The variety H4(V) at ~ where q is a non locally complete intersection quadruplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
78
Non-singularity of B ( V ) at q'o where qo is a non-locally complete intersection quadruple point . . . . . . . . . . . . . . . . . . . .
7.1
77
Non-singularity of B ( V ) at ~o where qo is a locally complete intersection quadruple point . . . . . . . . . . . . . . . . . . . .
6.4.2
68 69
111
7.2.1
Case of the elongated quadruplet
.................
7.2.2
Case of the spherical quadruplet . . . . . . . . . . . . . . . . . .
Irreducibility of H4(V) . . . . . . . . . . . . . . . . . . . . . . . . . . .
111 123 127
vii Appendix A
129
A.1 Local chart of H3(V) at t', where t is a curvilinear triple point A.2 Local chart of H3(~V) at t', where t is amorphous . . . . . . . . . . . . . Appendix B B.1 Local chart of H4(V) at an elongated quadruplet B.2 Local equations of
Ha(v) at
a spherical quadruplet
.....
129 130 132
............ ...........
133 134
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 < d i m W .
Let d = c o d ( f ) = d i m W - dimV. The locus 88 of elements x E V such t h a t there exists at least (k - 1) other points of V in the fiber f - i f ( x ) is called k-uple locus of f . W h e n it exists, a class m~: in the Chow ring C H ' ( 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 * T W - TV.
0.2
W h e n one deals with the double formula, one is interested in the set of elements x E V such t h a t the fiber f - i f ( 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], t h m 9.2, pp 161-162) in order to remove the exceptional divisor which corresponds to the solutions xl = x2 of f ( x l ) = 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
W h e n one deals with the triple formula, one is interested in the set of elements x C V such t h a t f - i f ( 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 C H ' ( V ) representing the set V3 and to compute this class so t h a t 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 morphism 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 t h a t this requires to choose an ad hoc definition of m3,
(ii) emphasize t h a t 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 • V x V such t h a t 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 • [Hilb2(V)] 3 • 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 2j 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 A
A
M3 = [H3(rs)]. [H3(V) x W]
A
e
CH'(H3(V
x
W))
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
• 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 : A
H4(V) --+ V • V • V • V compatible with the action of the symmetric group phism :
$4, and an order-forgetting mor-
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
A
A naive generalization H~a~,,e(V 4 ) of the construction of H3(V) is not sufficient, as was already pointed out by Fulton ([FU1]). The variety H~,,,~,,,~(V) 4 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 : N o t a t i o n 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,,,
c
z,,~ z,,,
Zt,, z,,
c z,, c zq
z<j
c
z,,,
n
z,,,
c z,,~ n z,t,, c zv, n ~,
for{i,j,k,l}={1,2,3,4}, i<jandk
A
Unfortunately, H,,,,~,,~.(V) 4 is reducible and singular. We will see below that the conditions (.) introduce excess components. Recall [I2, F] that (for d i m V = 3) the Hilbert scheme Hilb4(V) is irreducible and singular at the quadruplets q of ideal A/I,2 where 2~4v is the ideal of a closed point p of V. Consider the subvariety R(V) of Hilb2(V) • Hilb4(V) • 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
components, one of which dominates Hilb4(V) by II. When dim V = 3, for example, we will see t h a t the non dominant excess component is smooth of dimension 8. We will give a geometric description of this extra component (w 6.4.2, p. 100). The other component, of dimension 12 (= dim Hilb4(V)), is the closure of the graph of the
residual rational application Res, which is defined below. Let us first introduce the following definition : D e f i n i t i o n 1 : One defines the incidence variety I(V) as subvariety of the product
Hilb2(V)
x
gilb4(V) by the c o n d i t i o n : (d, q) E I(V) if and only if d is subscheme of q.
The second projection II2:
is generically a 6 = / 4 / - s h e e t e d \ 2]
I(V) --+ gilb4(V) (d,q) ~-~ q Cover.
D e f i n i t i o n 2 : A triplet t C V is said to be amorphous (cf. [KL4]) if its support is reduced to only one point p and if the ideal of t is the square of the ideal of p in a germ of a smooth surface containing p. In this variety I ( V ) , the complementary doublet of d in q is not always well defined. Indeed, let W be the locally closed subvariety of I(V) consisting of elements (d, q) such t h a t : -q
is the union of an amorphous triplet t of support p and a simple point m
distinct from p, -
d is the simple doublet p U m.
At such elements (d, q) = (pUm, tUrn) of I(V), one cannot define the complementary doublet d' because the closed point p does not define canonically a doublet d ~ in the amorphous triplet t (cf. [ELB]). However, outside of W, the closure of W in I(V), one can always define the complementary doublet d I of d in q. One gives the following definition of the residual rational application : D e f i n i t i o n 3 : Let Res be the residual rational application :
Res : I(V) ... ~ (d,q) ...-+
Hilb:(V) d'=q \ d
where d ~ denotes the "other" doublet, once the doublet d in q has been fixed. The ideal which defines the subscheme d' of V is given by the ideal Ann(Zd/Zq) of Or. Then this ideal verifies the inclusions :
6
Introduction
Let U denote the open subset of I(V) on which the residual rational application Res is well defined. The open set U contains the dense open subset (cf. proposition 3) consisting of elements (d, q) such that the quadruplet q is a simple quadruplet. D e f i n i t i o n 4 : If Resu denotes the restriction of the rational application Res to U, the graph of the regular application Resu : U --+ Hilb2(V) is called the graph of the rational application Res. Then, the inclusion
r~.(= rRo,u) c
u • Hilb2(V)
holds. Then the subvariety R(V) of Hilb2(V) • Hilb4(V) • Hilb2(V), whose elements (d, q, d') satisfy the inclusions
contains FR~, the graph of the application Res :
FR~., C R(V) C Hilb2(V) • Hilb4(y) • HiIb2(V) Therefore, this subvariety R(V) contains FR~.,, the closure of Pa~, in the product Hilbe(V) • Hilb4(V) • HiIb2(V). The following notation is then introduced : N o t a t i o n 2 : Let B(V) -- FR,s be the closure of the graph of the rational application Res. One has the inclusions B(V) C R(V) c Hilb2(y) • Hilb4(V) x Hilb2(V). The variety R(V) possesses a projection 1-I onto Hilb4(Y). Its restriction to B(V) will be denoted by zr. If q E Hilb4(V) is a quadruplet, ~ will denote a~ element of the fiber 7r-l(q) C B(V) :
B(V)
C R(V)
q Hilb4(V) It will be shown in chapter 6 that the variety B(V) is irreducible, smooth, of dimension 4 9dim V. The tool for the proof consists in using local coordinates for Hilb2(V) and Hilb4(V) in order to give local equations for B(V).
0.8 Let us see how a construction of the variety of complete quadruples as subvariety of the product [Ha(V)] 4 x [B(V)] 6 appeared to be natural to us. Consider the simple
quadruplet q E Hilb4(V), which is the union of four distinct points Pl,P2,P3,P4 of V. The element ~ in H4(V) constructed from the point (pl,p2,P3,P4) of V 4 consists in the data of : the four complete triples t1,/2,t3,t4 of H3(--~), constructed from the triplets tl, t2, t3, t4 contained in the quadruplet q. (The notation t~ means that the triplet t~ is disjoint from the simple point Pi.) The complete triple t, corresponds to the point -
, P i , ' "' ,P4) of V 3 ; - the six elements (d~j, q, d~j){i,j}c{1,2,3,4} of B(V) constructed from q, where d~j denotes the simple doublet which is the union of the two points p~,pj, and d'ij is the residual doublet of d~j in the quadruplet q. (Pl,'''
/t3
td24
tl
\
Pl
02
An element 0 of H4(I-V) is defined as an element (il, t2, t3,/4,012, q13, 014, 023, (~24,034) E [H3(V)] 4 x [B(V)] 6, where (P2, P3, P4, d2a, d34, d24, tl) (P1, P3, P4, D13, D34, D14, t2)
0~j =
(Pl, ~D2,~D4,~)12, ~)24, ~)14, t3) (Pl, P2, P3, d12, d2a, dla, t4) (3i:i,qij,5~j) for {i,j} C {1,2,3,4}
are such that : 1. All the quadruplets qli are equal to a same quadruplet q, 2. The doublets verify the equalities (f~j = 3kz, for {i,j, k, l} = {1, 2, 3, 4}
8
Introduction 3. T h e points verify the equalities : Pl
=
PI
192
=
~2
=
P2
P3
=
P3
=
P3
W4
=
P4
=
P4
4. T h e doublets verify the conditions : d12
=
d23
~__ d23
T)12
-~
(~12
z
~23
d13 =
D13
=
313
7:)24 =
d24
=
524
/)14
=
D14
=
~24
Da4
=
d34
=
834
5. T h e triplets verify the scheme-theoretic inclusions t, C q. For i = 1, 2, 3, the p o i n t s satisfy the conditions : p~ =
Res(t,, q)
a n d P4 =
Res(t4, q).
C h a p t e r 7 will be devoted to the construction of this variety. As before, the cons t r u c t i o n of this subvariety H4(V) C [H3(V)] 4 • [B(V)] 6 will be completely explicit. We will see t h a t such a construction enables us to remove the excess c o m p o n e n t s of 4 H~,,,i.,,,~(V) (defined
in 0.7). T h e variety H4(V) constructed in this way is irreducible
b u t u n f o r t u n a t e l y it is still singular. A geometric description of the singular locus of H4(V) will be given in section 7.2.1.
P a r t one Double and triple points formula
Chapter 1 Conventions and notation 1.1
F u n d a m e n t a l facts
If X is a non-singular C-variety, its Chow ring graded by the codimension is denoted by C H ' ( X ) .
If a E C H ' ( X ) , {a} k denotes the part of codimension k. Of course,
these two rules apply : (~1) If
X
is of pure codimension r, then
x . {a} k = {x. a} k+r
(T~2) If f : X -~ X ' is a proper morphism and if cod(f) denotes d i m X ' - dimX, then : f , { a ) k _The part of dimension k of a E CH~
{f,a)k+cod(l) graded by the dimension, will be also denoted
by {a}~:. Note that : (T~3) If f is a proper morphism,
1.2
{f,a}k
f,{a}k =
Conventions
From [FU2], p. 13, if ~ : Y ~ a E CH~
Y' is the canonical imbedding of a subvariety and
one will often write a in CH~
instead of ~ , a , which would be more
correct. The fundamental class of a variety X is denoted by IX] or sometimes by 1. Improperly but for the sake of simplicity, from w 3.2.4 on, if Y C Y' is a subvariety, one will denote by Y and not by [Y], the class of the associated cycle in CH~ Finally, one writes "from (FP)" each time one uses the projection formula : f , ( a . f*/3) = f , a . / 3
(FP)
12
1.3
Conventions and notation
Notation
One denotes by O : H 2 ( X ) -+ H2(X) the universal two-sheeted cover of H 2 ( X ) = Hilb 2(X). The scheme H 2(X) is non-singular and possesses two canonical morphisms,
which are submersions : 7rl, 7r2 : H2(X) ----+ X . An element d of H2(--"X) can be identified with a couple (Pl, d) consisting of a point and a doublet containing this point. One can denote by P2 point" (see [LB1]). One has a natural isomorphism:
=
Res(pl,d)
the "other
A
H2(X) ~
X x X
with the blowing-up X x X along the diagonal, which commutes with the projections onto X. This is why one will also write d = (Pl, P2, d). In H2(X), there is the "exceptional" divisor F, consisting of the couple (Pl, d) with supp(d) reduced to only one point ; it corresponds to the exceptional divisor of X • X via the above mentioned isomorphism. Finally, one checks that : F is the locus of ramification of the cover 0 : H2(--'-X)~
H2(X) .
(1.1)
Chapter 2 Double formula A
2.1
T h e class of
A
H2(X) in H2(Z)
Let Z be a non-singular variety of dimension z and j : X ~-+ Z the canonical imbedding of a subvariety of dimension x. One can identify in a natural way the variety H2(X) of dimension 2x with a subvariety of H2(Z), which is of dimension 2z. Our purpose is to compute the fundamental class [H2(X)] in CH'(H2(Z)). Let 111 and 112 : H2(Z) -~ Z denote the two canonical morphisms.
In I =
111-1(X) N l12-1(X) one has the excess c o m p o n e n t : 7) = F A I I I - I ( x ) = F N 112-1(X) where F C H2(Z) denotes, as said already, the exceptional divisor of H2(Z). The component 7) consists of the (d, pl) with supp(d) = pl E X .
Let us consider the commutative diagram where the restriction 11qn~-l(x) is denoted by H/I and the other arrows are the canonical imbeddings (the parentheses indicate the dimensions) :
14
Double formula
(2x)
(x+z-1)
~) c
H2(----f)
a ,
(
I
g2
+ z)
r] gl
(5+z)
n l(x)
(
il
nl
X
II1
(
,
Z
Diagram 1
From [FU2], theorem 9.2, p. 161 and corollary 9.2.2, p. 163, one has in CH2~(I) : [H2(---'X)] = [IIx-I(X)] 9[II2-1(X)] - { c i . a,s(I), H2-i(x))}2,~ , where g -- g[v(IIl-l(X), g2(--"~)) is the restriction of the normal bundle to I I I - I ( x ) in H2(Z). H y p o t h e s i s 1 : From now on, one assumes that the total Chern class cu(X, Z) of the normal bundle at X in Z can be written as j*d where d E CH~ Since H I and H2 are fiat morphisms (see (4.5) and (4.6) in w 4.2), one h a s :
[ I I , - i ( x ) ] = H*[X] Moreover, the normal bundle v ( I I l - l ( x ) , H ~ ) )
i = 1, 2
(2.1)
can by identified with Hi*u(X , Z).
Hence the total Chern class N is :
cN = g ~ c / 2 ( H I - i ( x ) , H2(Z)) = glrI1 . . clJ(X,Z) . . ~- g l n i ' . 3. . . c. . ._.= . g2z2Yi1cl
,
considering the commutativity of diagram 1. We also have the following lemma : L e m m a 1 The intersection 7) = F N II2-1(X) is transverse.
The class of He(X) in He(Z)
15
Proof : See (4.20) in section 4.6.
[]
As a result, one gets the equality between Segre classes (inverse of Chern classes) :
gu.a.s( 7), 112-1(X)) = i~s( F, He(Z))
Notation
a : One writes s(F) for s(F, H2(-'-'Z)) and s(7)) for s(7:), II2-1(X)).
Considering the above results, one can rewrite in CH2~(11e-1(X)) :
g e , { c N , a,s(~)}e,~ = g e , { g i i l n l e , a,s(~))e~ = {i~111c'. g2,a,s(:D)}2~ by (FP) and (7~3) = {i~11"S 9i~s(F)} =-:': ( in I12-1(X), dim = 2x is equivalent to codim = z - x) = i~{nle, s ( F ) p - * A fortiori in CH2:,:(H2(-'~)) = C H 2 * - 2 * ( H ~ ) ) , one gets
i2.g2. { cN " a.s(7))}2~
9
"*
*
I
=
,2.,2{Hlc 9s(F)} *-=
=
{ 1 1 V . s ( F ) } z-~
[11e-1(X)]
, by (FP).
Finally the formula in C H 2 Z - 2 ~ ' ( H ~ ) ) [g2(-~)] = [111-1(X)]. [I12-1(X)] - {111c' s ( F ) } *-~' 9[112-1(X)] follows. But for all cycles a with support in F, one has a
[III-1(X)]
= c~. [112-1(X)]
in
A
CH'(H2(Z));
one applies this to a = { H i d . s(F)} *-~.
Moreover, equation (2.1)
yields [11i-1(X)1 = 11"[X1. Therefore we have shown the following theorem (Ran IRa], p. 90, with k = 2) : 1 ( R a n ) Let j : X r Z be the canonical imbedding of a smooth subvariety of dimension x in a smooth variety of dimension z. Assume that the total Chern class cu(X, Z) of the normal bundle can be written as j ' d , where d E t i f f ( Z ) . If [I1, 1112 : H2(Z) ---+Z are the two natural morphisms, then one has in CH'( H2( Z) ) the equality :
Theorem
[He(X)] _- 111IX]- (11~[X] - { s ( F ) . 111c'F -~)
where s(F) denotes the Segre class s(F, He(Z)) of the exceptional divisor.
16
Double formula
2.2
Definition of the double class
From now on, let f:V ----+ W be a morphism of fixed proper smooth varieties, of respective dimensions n and m. Let k = m - n = cod(f).
Hypothesis 2 : (i) Assume rn > n (i.e. k > 0).
(ii) Assume 2n - m _> 0 (i.e. k _< n) , so that the classes defined below are meaningful. Let F be the graph of f and j : F ~-~ V • W be the canonical imbedding. As in A
w 2.1, H2(F) can be identified with a subvariety of codimension 2m of H2(V x W). Moreover, one has an imbedding
: H~(V) • W ~ H~(V • w ) defined canonically. Thus, if d C V is a doublet and w a point of W, then a(d, w) denotes the doublet image of d by the imbedding
V ~
VxW
These doublets might be called "horizontal".
VxW
:t
aid, ~I d
9
V
A
Let us denote by P~ : H2(V) • W ~ V the morphism ((vl,d),w) ~ diagram
H2(-'---f)~ H2(V'--~W) ~ H-Z-~) • W 5 V follows. D e f i n i t i o n 5 : Using the preceding notation, let : A
(a) M2 = a*[H2(r)] 'M. 2 (b) m2 = P~,
9 9
C H 2 ......
(H2(V) • W)
CH2..... (Y) = CHk(V).
vl. The
Definition of the double class
17
One says t h a t m2 is the "double class". Remark
1 : The cycle M2 corresponds to the idea of points vl, v2 of V having the
same image by f ; while the cycle m2 corresponds to the idea of points Vl E V such t h a t there exists v2 E V having the same image as vl. T h e q u e s t i o n is t o e v a l u a t e m2 in CH~:(V). To do so, one introduces some new notation : Notation 4 : (i) If g : X
~ Y is a morphism, one denotes by .q and g' the two following
morphisms : g'
=
gxidw
: XxW
~
=
g x idy
:
----+ Y x V
X x V
YxW
(ii) One denotes by prl and pr2 the projections from V x W onto V and W , respec-
tively. (iii) One denotes by Pl and P2 the two projections from V x V onto V. Note t h a t Pl :
idv x pr2 : V x V x W ---+ V x W A
(iv) Let us denote by 7ri : H2(V)
v~, f o r i = 1,2.
~ V the morphism which sends ((vl, v2), d) onto
Then one denotes by 7r : H 2 ( V ) ~
V x V the morphism
(Th, 7r2). Note t h a t rr can be identified with the blowing-up of the diagonal of VxV.
(v) One denotes by P w : H 2 ( V ) x W ~
W the natural projection.
(vi) Last, as said above, if F is the graph of f , one denotes by j : F ~-+ V x W the
canonical imbedding. One has inverse isomorphisms : 7:V-7-~F
and
cr:r
W i t h this notation, one has a commutative diagram :
~>V
18
Double formula A
H2(V x W)
~r
H2(V) • W
VxVxW
Pw pr2
W
VxW
J
prl
f J V
7 I"
q
cr
Diagram 2
W a r n i n g : We have represented p r l with a dotted line, since p r l does not make the di_ag,ram commute ? In fact, f o p r l 7~ pr2 ; however p r l o j = or. Remark
2 : W i t h t h i s n o t a t i o n , one h a s :
m2 = pr1,151,#,M2 in CHk(V)
2.3 2.3.1
Computation
of the
double
class
Computation of M2
Let us apply theorem 1 with X = F, Z = V x W and d = p r ~ c W , where c W denotes the total Chern class of the tangent bundle T W to W. Indeed, the normal bundle to the graph is identified with j * p r ~ T W . For x = n, z = n + m, the formula of theorem 1 gives in CH~
x W)) :
[H2(r)] = II~[r]. (rib[r] - {s(~) 9ri~c'}'-) where ~ C H 2 ( - V - ~ x W ) denotes the exceptional divisor. But one has the commutative diagram :
C o m p u t a t i o n of the double class
19 A
OL
H2(V) x W c
, H2(V • W)
Pw
1]1 pr2
W
V•
It follows t h a t : *
*
A
I
~
*
,
,
a HlC = a Hxpr2cW = p w c W = [H2(VI] x cW Moreover, one has the commutative diagram :
H2(V) x W r
a
, H2(V x W )
H~
73
P~
VxVxW
i=1,2
V•
hence a ' H * = 73"p-~*. Also, a* [~] = IF] • [W] where F denotes in this case the exceptional divisor of H2(V).
(This comes from
A
the fact t h a t F and H2(V) • W intersect transversally in H2(V • W), as shown by a computation on coordinates (4.32) performed in w 4.10). It finally leads to the equality in CH~
x W)) :
M2 = a*[H2(r)] = 73*~F[F]. (73"~2"[F] - { s ( F ) • cW} m)
(2.2)
Note t h a t a*s(~) = s(F) since the Segre classes are the inverse of the total Chern classes of the normal bundles. 2.3.2 Let us apply the projection formula to the morphism 73 : 73,M2 = ~,*[V l 973.(73%*[F l - { s ( F ) • c W } ' ) . Since 73 is birational, 73,1 = 1. Hence, since 73 = 7r x idw and from (FP) : 73,M2 =/51"[F]. (/52"[F] - {Tr,s ( F ) x c W } ' ) . (we have used ('R2) with cod(73) = 0). But 7 r - l ( A y ) = F where A v C V • V denotes the diagonal. From [FU2], proposition 4.2.5, one has 7c.s(F) = s ( A v ) . Hence the equality in C H ' ( V • Y • W) 73.M2 = / 3 , ' [ F ] . (/~2"[F] - { s ( A y ) x cW} m)
20
Double formula
holds. T h i s time let us apply the projection formula to the morphism/31. Since cod(~i) = - n , we get by (7/2) the equality in C H ' ( V x W ) ;6i,#,M2 = [C]. (i~i,/52*[F] - { ; i , s ( A y ) x c W } ...... ). But the c o m m u t a t i v e d i a g r a m
VxV
Av
'
V
yields p l , s ( A v ) = c(V) - i , the inverse total C h e r n class of V, since the n o r m a l b u n d l e to A v c a n be identified with the t a n g e n t b u n d l e to V.
Notation
5 : One writes
Notation
6 : Let
where
~ : c ( V ) - 1 X c W E C H ' ( V x W).
# = #1 - #2 in CHk(V x W ) ,
#i=/5i,/32"[F]
and
# 2 = { ~ } ~,
Using the previous n o t a t i o n and the fact t h a t m - n = k, one gets : p l , ~ , M 2 = [V]. (~1,/32"[F1 - {~}~) = [ r ] . ~
(2.3)
From r e m a r k 2, one has in CH~'(V) : m2 = prl,~l,~r, M2 = p r i , ( [ F ] . #). T h i s is also (from ( F F ) applied to j ) : m2 = p r a , ( j , j * # ) = 7*J*# car p r l , j , = a, = 7*. H e n c e : m2 = 7*J*#
in
CHk(V)
(2.4)
We will need the following l e m m a : Lemma
2 Let a E C I t ' ( V x W ) . Then one has the equality/31,/32"a = pr~pr2,a.
Proof : By the c o n s t r u c t i o n of 132, one has/52"a = [Y] x a E C H ' ( V x V x W).
/31 = idv x pr2. Hence pl,/52*a = [V] x pr2,a = pr~pr2,a.
2.3.3 We saw ( e q u a t i o n (2.4)) t h a t m2 = 7*J*# = 7*J*/zi - 7"J*#2 (using n o t a t i o n 6). Let us first compute
7"j*#1, i.e. 3,*j*igl,/52*[r ].
Also,
[]
C o m p u t a t i o n of the double class
21
F r o m the previous l e m m a , 7*J*#l =
7*j*pr~pr2,[r] =
f*pr2,[r], since f = pr2 o j o 7.
Moreover, since f , = p r 2 , j , 7 , , one has :
pr2,[r] = f , [ Y ] .
(2.5)
Therefore, we have shown t h a t : 7*J*#l = f * f , [ V ] Let us now compute
in
CH~:(V)
(2.6)
7*J*#z, i.e. 7*j*{~} ~:.
We have let ~ = c(V) -1 x c W i n
C H ' ( V • W).
But it can also be w r i t t e n a s ~ =
pr~c(V) - 1 . pr~cW. Hence j*~ = j*pr~c(V) - 1 . j*pr~cW. B u t prl o j
= foa.
= a andpr2oj
Hence j*~ = a*(c(V) - 1 . f * c W ) , and
consequently : =
Notation
-1
.
f*cW)
7 : In the Grothendieck's group K ( V ) , let us denote by
u(f) = f*TW-
TV
,
the v i r t u a l n o r m a l b u n d l e to f. One denotes by ci the C h e r n classes of u ( f ) .
Since c u ( f ) = f * c W . c(V) -1, one g e t s : =
(2.7)
7"j'#2 = 7"J'{~} k = 7*a'ek = ck.
(2.8)
Finally, 7*~r* = identity leads to
It follows t h a t :
m2 = 7*j*(#l - t22) = f ' f , [ V ] - ck. We have j u s t proved again the following t h e o r e m : Theorem
2 ( L a k s o v ) Let f
: V
~
W be a morphism of proper and smooth
varieties, with dim W = k + dim Y (k > 0). Let m2 E CHk(V) be the "double class", direct image of M2 = [g2(r)] 9[g:(Y) • W]
9
cI-r(H2(V x W))
where F is the graph of f . Then one has the double formula : m2 = f* f,[V] - ck where c~: is the U h Chern class of the virtual normal bundle f * T W - T V .
Chapter 3 Triple formula A
3.1
A
The class of H3(X) in H3(Z)
Let Z be a smooth variety of dimension z. One introduces in [LB1] the smooth variety H3(Z), of dimension 3z, of "complete triples". Below we recall briefly its construction. A
N o t a t i o n 8 : Let us denote by H'(Z) the Hilbert scheme Hilb'(Z) (cf. [G], [I1]). A complete triple of Z consists of the data t -- (p1,P2,pa, d12, d23, d31,t) E Z a x (H2(Z)) 3 x H3(Z) where
{ p~ C d~j C t ( scheme-theoretic inclusions ) pj = Res(p,, d{j) p~ = Res(d,j,t) with {i, j, k} = {1, 2, 3} A
Let us denote by H3(Z) the set of complete triples of Z. We show that it is a nonsingular variety, birational to Z x Z x Z. In the case where Z = 1~2, the variety H3(~ 2) is canonically isomorphic to the Semple's complete triangles variety IS], since studied A
by Roberts-Speiser [RS1]. One has in H3(Z) for Cartier's divisors : E12---- { t [ d23--= d31} E23=-{ttd12=d31}
and
E~
E31 : { t[ d23 -~ d12} Recall that (cf. definition 2) an amorphous triplet t is such that supp(t) is reduced to only one point p and t is defined by the square of the ideal of p in a germ of a smooth surface containing p. Then let j : X ~ Z be the canonical imbedding of a smooth subvariety of dimension x. The variety H3(X) can clearly be identified with a subvariety of H3(Z), A
as one sees by coordinate computations (see (4.30) in w 4.9).
A
A
The class of H3(X) in H3(Z)
23
The question is to evaluate the fundamental class of [H3(X)] E CH~ with the same hypothesis as in w 2.1, i.e. c v ( X , Z) = j * d where d e CH~ Let P~ : H3(Z)
--+ Z ~ p~
be the three natural morphisms. A computation (see (4.25) in w 4.7) shows that the P~ are flat. Similarly, one has the morphism A
P12 : H3(Z) [
~
H2(Z)
~
(Pl, d12)
and a computation (see (4.23)) shows that P12 is flat. Then the intersection I -= P~21(H-~((X)) N p a l ( x )
in H3(-'-~) possesses three
excess components, respectively C23 = E23 VI p I ~ I ( H ' ~ ) ) ,
C31 = E31 VI P ~ I ( H " ~ ) ) ,
E ~ = E ~ V~ P ~ I ( H - - ~ ) ) .
In the following drawing, a doublet of support one point (therefore isomorphic to Spec(C[T]/(T2)) is represented by:
.X
X
Z
Z
E31
s
X
Z
Note that in the last drawing, d12 is drawn "tangent" to X, because we have the scheme-theoretic inclusion d12 C X.
24
Triple formula A
N o t a t i o n 9 : The sum of the three divisors E23 + E3x + E ~ of H3(Z) is denoted by andg23+E31+E" Clbys Thus one has the commutative diagram where the parentheses indicate the dimensions and where the imbeddings are the canonical imbeddings : A
(3x)
(2x+z-1) E
r
a
H3(X)
'
I
i'
r
, P~I(H~(-X))
(2x + z)
g (2z + =)
A
c
,
H3(Z)
(3z)
P3 X
c
J
.
Z
Diagram 3
The restriction of P3 to P31(X) is denoted by _P~. From IFU2], corollary 9.2.2, one has the equality in CH'(I) : [H3('-'~)] ----[P~t(H--~))]- [P~-t(X)]- { c N . a,s(E, p1~1(H'5~)))}3~, where g = g*v(P31(X), H-~(Z)). But P3 is flat (see (4.25)), so g = g*P~*v(X, Z). Then, since by hypothesis cv(X, Z) = j ' d , one has : eN = g*P~*j*d = i'*h*P~d. Moreover, P12 and/)3 are flat morphisms (see (4.23) and (4.25)), therefore { [P~I(H--~))] = [P3-'(X)] =
P12*[H2(~-X)] P3*[X]
(3.1)
Moreover, one has the following lemma A
L e m m a 3 The intersection s = E N P ~ I ( H - ~ ) ) is transverse in H3(Z).
A
A
The class of H3(X) in Ha(Z)
25
Proof : See (4.31) in w 4.9.
[]
This implies the equality of Segre classes (inverse of Chern classes) : z,a,s(s
--
'!
--1
(H (X))) = h*s(E,H-~(Z)) .
N o t a t i o n 10 : One writes s(E) for s(E, H3(--~)) and also s(s for s(s P121(H-~))). Using what is above, one writes in CH3~,(PZ*(H-~"X))) : i ~fi,*h*p*,v = {h*P~d.i',a,s(s from (FP) and (743) = {h*P~c'. h*s(E)} .... (in P ~ I ( H ' ~ ) ) , the dimension 3x is equivalent to the codimension z - x.) =
=
h*{P;d,
s(E)}
....
Afortiori, we get in CH3~(H3(--~)) = CH 3~ 3*'(H--~(Z)) : h, il,{cg 9a,s(E)}3,
= =
h,h*{P~d, s(E)} ~-* { P i e ' . s(E)} "-~. [px~I(H'~))]
from
(FP).
Finally the formula in CHa~-a~'(H-~(Z)) [H3(--~)] = [ P ~ I ( H - - ~ ) ) ] . ( [ p 3 - I ( x ) ] - {P~e'. s(E)} z-*) follows. But we noticed that P12 and P3 are flat ; with the help of (3.1), we have therefore proved the following theorem (which generalizes Ran [Ra], p. 90 for k = 3) : T h e o r e m 3 Let j : X ~-+ Z be the canonical imbedding of a smooth subvariety of dimension x in a smooth variety of dimension z. Suppose that the total Chern class cv(X, Z) of the normal bundle can be written as j*d where d E CH'(Z). Let P12 and P3 be the morphisms defined in the following way : if { = (Pl,P2,P3, d12, d23, d31, t) is a complete triplet of Z, let A
P12:
~
H3(Z)
~
H2(Z)
~
A
and
(Pl,d12)
P3:
H3(Z)
-+
Z
t
~
Pz 9
A
Then the equality in CI-I"( H3 ( Z ) ) follows : [g3(-'~)]
=
P12*[H2(--~)] 9(P3*[X] - { s ( E ) . P~c'} .... )
(3.2)
where s(-E) denotes the Segre class s(E, H~(Z)) of the divisor -E = E23 + E31 + E ~ of H3(Z). A
Triple formula
26
3.2
The triple formula
3.2.1
Some
notation
The notation of w 2.2 for a morphism f : V ----+ W of smooth varieties is used again : n=dimV,
m=dimW
andm=k+n,k>0.
H y p o t h e s i s 3 : Suppose that 3n - 2m _> 0, i.e. k < n/2, so that the classes defined below are meaningful. One has a natural imbedding A
: H3(V) • W ~ H3(V • W )
similar to the imbedding a : H2(V) • W ,--+ H2(V • W) seen in w 2.2. Its image consists of the "horizontal" triplets of V x W.
9
~.tt, ~)
V Let us denote by P~' : H3(---~) x W ~
V the morphism which takes (i, w) to vl,
where [ = (t, d12, d23, d31, vl, v2, v3) is in H3(~"V). If F is the graph of f, the diagram H3(-~) ~ H3(V-""~W) ~ H-~(V) x W ~ V follows. D e f i n i t i o n 6 : With the above notation, let (a) M3 = j3*[H3(F)] (b) ~-~3= P "1. M 3
e 9
CH3,,_2,,(H3(V) x W) CH3,,_2,,(V) = CH2k(V).
m-'-~is said to be the "triple class". R e m a r k 3 : As in remark 2, the cycle M3 corresponds intuitively to points Vl, v2, v3 of V having the same image by f ; while the cycle m33 corresponds intuitively to points Vl 9 V such that there exists v2, v3 in V having the same image as vl.
The triple formula
27
T h e q u e s t i o n is to e v a l u a t e ~
in CH2k(V). To do so, we introduce some addi-
tional notation besides the morphisms considered in w 2.2 :
Notation
11 : A
(i) Let q : H2(V) x V ----+ H2(V) be the natural projection ; (ii) Let w~ : H2(V) x V
--+ V
i=1,2,3
where d = (vl,v2, d) e H2(--"V). Note the asymmetric role of 3 in comparison with 1 and 2 in this notation. (iii) Let us denote by Pw : H3(V) x W ----+ W the natural projection. (iv) Let r be the birational morphism
r
H3(V)
~
H~(V) • V
i
~
((vl, v2, d~), v3)
if [ = (t, d12, d23, d31, Vl, v2, v3). (v) The imbedding/3 : H3(V) x W '-~ H3(V x W) has been defined above. (vi) Recall (see notation 4.(i)) that .~ denotes g • idw, where g is a morphism. With this notation (in addition to the notation of w 2.2), one has a commutative diagram :
28
Triple formula
P~2
A
H3(V x W)
A
H2(V •
A
W)
A
H2(V) x
H3(V) X W
Fw
H2(~-~) • W
V x W
CO 1
pr2
W
f J
J
7
D
V
VxVxW
J
J
Prl j J
J
pl
VxW
r
Diagram 4
~Varning : p r l is represented by a dotted line, since it does not make the diagram commute ; however, p r l o j = a. R e m a r k 4 : With this notation, one has
3.2.2
Computation
m~-'~= p r l , ~ l l , r
C
CH2k(V).
o f M3 a n d r
In the same way as in w 2.3 with theorem 1, one applies theorem 3 with X = F, Z = V • W and d = pr~cW. Theorem 3 yields the equality in C H ' ( H 3 ( ~ •
[H3(F)] = P~*2[H2(F)] 9(P~[r]- {s(~). P~c'}'")
W)) :
,
where ~ is E23 + E31 + E'. The double bar denotes the divisors related to H 3 ( ~ x W). The divisors related to H3(V) are denoted by E = E 2 3 + E 3 1 + E ' , so that ;3"(~) = E • [W] by transversality (cf. (4.33) in w 4.10). The definition of M3 (see definition 6.(a)) yields : A
M3 = ;3*P~*2[H2(P)]
9( ; 3 * P 3 * [ r ] -
{(s(~) x [w])./3*p~c'}")
On the other hand, as it can be checked easily, pr2 o P3 o ;3 = Pw. Therefore A
9
*
t
*
*
*
--*
;3 P~c = ;3 P ~ p r 2 c W = P w C W = [H3(V)] • c W .
The triple formula
29
Furthermore, as it can be seen on the previous diagram 4,/3"P1" 2 = r
Moreover,
Therefore fl*P~ = r
one verifies easily t h a t Pa o/3 = o., a or Finally, one has in CH*(Ha(V) x W) : M3 = ~*(]*o~'[H2(-~)]
-
"
• eWF)
Let us then apply the projection formula to the birational morphism r = r x idw. Since cod(qS) = 0 and (~,r
&M3
= id, one has from (7~2) :
=
-
x cW}")
(3.3)
Let us then introduce some additional notation.
Notation
12 :
(a) Let |
: U
) H2(V) be the universal two-sheeted covering and R C U its
ramification locus.
One has U C H2(V) x V.
Of course, it is the same as
0 : H2(V) ---+ H2(V), but we denote it differently in order to avoid any confusion. Similarly, R C U corresponds to F C H2(V) and it is the ramification locus of 0 (see w 1.3). (b) Let 0' = 0 x idv. One constructs the cartesian diagram (where 0" is the restriction of 0' and u, ~ are the canonical imbeddings) : 0 r
u
, H2(-"-'V)• V
0"
U c
0' U
, H2(V) • V
(c) Let Gla and G2a be the graphs in H2(V) • V of the natural morphisms : Pi:
H2(V)
--+
((vl,v2),d)
(d) Let t3 C F • V be the graph of PIIF : F ~
V v,.
V.
(e) We denote by D C H2(V) the set consisting of the doublets of support one point.
Triple formula
3O
(f) The Segre class s(U, H ~ ) (resp. s(u)).
x V) (resp. s(U, H2(V) • V)) is denoted by s(U)
See drawing in w 3.2.4, page 43.
Lemma 4 (i) One has the inverse scheme theoretic image
r
= E in H ~ ) .
(ii) One has the equality of schemes
U = G13 U G23 in H2(---"V) x V.
(iii) One has the equality of schemes
B = G13 N G23 in H2(V) x V.
(iv) One has the equality
0*[D] = 2[F] in C H I ( H - ~ ) ) .
(v) One also has the equality
0'*[R] = 2[B] in CI-F(H2(V) x V).
Proof: The proof consists in computing coordinates ; the results are shown in chapter 4 (see resp. (4.28), (4.15), (4.13), (4.9) and (4.16)). [] Lemma 4.(i) in conjunction with prop. 4.2.a, p. 74 in [FU2] shows that the direct image by r of the Segre class s(E) is given by : r
=
s(U)
in
CH'(H2(~-V) x V).
(3.4)
Equation (3.3) implies the final result in CH'(H2(V) x V x W):
~,M3 =
~*~*[H2(~-~)]. ( ~ ' [ r ] - {s(0) •
cwy").
(3.5)
Let us introduce some more notation.
Notation
13
:
Let r
A
= v = vl - ~2
C
CH'(H2(V) x V x W )
where
vl = ~*~*[H2(r)]. ~*[r] ,2 : ~*~*[H2(~-~)]. { 4 0 ) •
cWy"
Remark 4 implies :
-~ = prl.~,,
(3.6)
The triple formula
3.2.3
31
Computation of prl,~,ul
As it can be seen on diagram 4, one has ~ = Pl o 7r o q ; one first calculates ~ , U l = PI,~,~,vl.
The definition of ul (notation 13) and the application of (FP) to ~ yield : q,//1
A a*[H2(r)]
:
9 4,~,[r]
But W~3 : H2(V) x V x W ----+ V x W simply is the natural projection. Therefore : ~ * [ P ] = [H2(V)] x [F]
9
C H ' ( H 2 ( V ) x V x W)
,
A
) H2(V) • W, one has :
and since ~ is the natural projection H 2 ( V ) x V • W A
~,w"~'[s
x pr2,[F] .
But we have seen that pr2,[F] = f,[V] (see equality (2.5)). Therefore ~,W~*[F] can also be written as : A
[H2(V)]
x
f,[V]
=
7r*p-~*(iV] x / , I V ] )
=
~*~* l r Pi
*
pr2f,[V]
We finally obtain c/,., = a * [ H 2 ( r ) ]
9~ p l i'DT2f*[ v ]
Applying (FP) to ~ o # yields in C H ' ( V x W) : A
~"i,Vl = ~ , # , ~ , U l
= p~,#,a*[H2(r)]
pr;f,[Y].
9
A
But we let (definition 5): M2 = a*[H2(r)]. The equalities (2.3) yield: p'i,7?,[m2] = [s
9
C H ' ( V x W)
Therefore w~,,l = IF]. ff.pr~f,[V]. But, for all a 9 C H ' ( V x W), (FP) gives:
j,j*a = a. IF] Therefore
(3.7)
w~,,l = j,j*(t~" pr~f,[Y]). Since a = prl o j, one h a s : prl,w~,,l = a,j*(t~' pr~f,[Y])
But 7" = a, (since 7 (resp. or) is the inverse isomorphism of cr (resp. 7)). It follows that : prl,U~l,Vl
=
7*J*#" 7*J*Pr]f*[ V]
On one hand, 7*J*# = m2 (see (2.4)) ; on the other hand, f = pr2 o j o 7. Thus we have obtained what we were looking for : p/'l,~,b'
1 ~ - 77/, 2 9
f*f,[V]
(3.8)
32
Triple formula
3.2.4
of {s(U) • cW} m
Computation
We already mentioned in the introduction that the notation is abused in the following way : if Y C Y' is a subvariety, the class in CH*(Y') is denoted by Y and not by [Y]. a) In the following calculations, we need to know s(U) • c W
e
OH'(H2(I-V) x V x W)
,
or more exactly (see w 1.2): ~2,s(U) x c W , where t2 : s(&) r H2(----~)x V is the canonical imbedding. But one has (notation 12) a commutative diagram : A
O
(
I
9
0"
U
H2(V) • Y
V U
C
O' ,
H2(V) x V
H2(V) where w3 and ~-~ are the natural projections on the second factor and i5, p are their restrictions; ~ is the projection on the first factor. | is the two-sheeted universal covering. (We denote it by (9 : U ~
H2(V) and not by 0 : H2(V) ~
H2(V),
A
since U is another copy of H2(V), in order to avoid any confusion. Same thing for and q). N o t a t i o n 14 : In order to simplify the notation, one writes H for H2(V).
b) L e m m a 5 (Kleiman, Ronga) : Let v be the normal bundle to U in H • V. Then one has in C H ' ( U ) the total Chern class : c(v) = p * c ( T Y ) .
l+2R 1+ R
where T V is the tangent bundle to V and R is the ramification locus of U on H. Proof: From [HI, II, prop. 8.12, one has the exact sequence of sheaves on U :
sis ~
> a~,i~
| o ~ - ~ r ~ i . --+ 0
The triple formula where I = O u ( - R )
33 is the ideal of R. Then
I / P = I e o u / I -~ o ~ ( - n ) Furthermore, (see (4.10) in w 4.3), the projection R ----9 H is not ramified, i.e. ~ I / H = 0. Thus one obtains the isomorphism ~ / H
"~ OR(--R). Let us apply again [H], II, prop. 8.12 but this time to the morphism ~. One obtains the exact sequence of sheaves onU:
O----+ v* -"~ q ~V [ U ---+ fl~/H
)0
~* denoting the conormal bundle. It follows that in the Grothendieck group K ( U ) of coherent sheaves (or vector bundles), one has the equality : 1.1" : p*~]V1 -- O R ( _ R )
.
From the exact sequence of sheaves on U : 0 ~
Ou(-F~ ) ~
OU(12R) ~
OR(--Jt~) ~
0
one obtains in K ( U ) : .* = ;*~vl _ o u ( - R )
+ o~:(-2R)
.
The equality of Chern polynomials follows : c,(~,*) =
;*c,(~)
1 - 2tR 9i =
For t = - 1 , it yields the equality between total Chern classes of duals :
c(~,) = p*c(TV). 1 + 2R 1+ R
E
CH'(U) .
Thus, lemma 5 has been shown. c) From temma 5, we get inverse Chern classes (by denoting c(TV) by c(V) ) :
c(~,)_ 1 = p * c ( V ) - 1 I+R But 1 + 2 ~ which gives |
1-R(I+2R)
-1.
9
1+ R l+2R
e
CH~
.
Lemma 4.(iv) gives 0*D = 2F in C H I ( H - - ~ ) ) ;
= 2R in CHI(U), if rewritten in the other copy U of H2(V). Thus : lq-R l+2R
_ 1 - R u * ~ * ( l q - D ) -1
One denotes (notation 12.(f)) by s(U) e CH~
E
CH~
.
the Segre class of U C H2(V) x Y.
From ([FU2], chapter 4), one obtains:
s(U) = c(~,) -1 = p*c(V) -1. (1 - Ru*~*(1 + D ) - ' )
e
CH'(U) .
34
Triple formula
Let us apply (FP) to the morphism u; the equality in CH~
x V)
u,s(U) : ~3*c(V) - 1 . (U - u , R . ~*(1 + D) -1) follows since p* = u*~aa*. Let us lift by 0'. From ([FU2], prop. 4.2.5) the equality in C H ' ( H 2 ( V ) x V)
~,~(0)
=
~ , 0 " * 4 u ) : o'*~,~(u)
---- co~c(V) - 1 ' (U - O'*R. 0'*~*(1 + 0 ) -1) . follows since 0' is flat (0 is flat). But one has the commutative diagram (q and ~ are the two natural projections) : q
H2(V) x V
,
H2(V)
O'
H~(V) • V
~
H~(V)
Thus, one has O'*q*D = q*O*D; but O*D = 2F and O'*R = 2B (cf. lemma 4). Therefore, in C H ' ( H 2 ( V ) x V), one has
u,s(U) -- co~c(Y) - 1 . (U - 2B(1 Jr. 2q'F) -1) .
(3.9)
d) Let us then look at {~2,s(0) • c W } " 9 CH'(H2('~) • V x W). One can formally expand ~2,s(U) given by (3.9) to obtain: ~ , s ( O ) = ~o~c(V)-' . ( 0 + B E ( - 2 ) " q * s ' - 1 ) h_~l Thus, the equality in C H ' ( H 2 ( V ) x V • W)
{(o3~c(V) - 1 . U) x
c W } m --[-
E ( - 2 ) h { ( o 3 ~ c ( V ) -1. B q ' F h-l) x c W } m h> 1
follows. For the first term in this sum, one has in C H ' ( H 2 ( V ) x V x W) :
(03~c(V) -1. 8 ) x
c W = (co~e(V)
1 x cW) . (U x W)
.
But codim(U x W) = n. Applying (T~I) yields: { ( ~ o ( v ) -1. 0 ) x c W } "
:
{ ~ c ( v ) -1 x c w } ...... 9( 0 • w )
= ~*{~}~. (O x w )
35
The triple formula (recall t h a t ~ = c(V) -1 x c W - see n o t a t i o n 5). For the other terms of this sum, one has : (od~c(V) - 1 . J~q*F h-l) x c W = (od~c(V) -1 X c W ) . ( ( B q ' F h-l) x W ) But c o d i m ( B q * F h-l) = (n + 1) + (h - 1) = n + h. From (741) again, one h a s : {(0J~c(V) - 1 . B q * F h - l ) x
cW}
TM
= {w~c(V) -1 x
cW} ...... h, ((~q, Fh 1) X W )
= ~3"{~} ~:-h. ( ( B q * S h - l ) • W ) . We have therefore shown the following proposition :
Proposition
1 In C H ' ( H 2 ( V ) x V x W ) , one has the forvnula :
{~2,s(f)) x c W } m = ~3"{~} k. (U x W) + ~--~(-2)h~33"{~} k - h ' ( ( B q * F h-l) x W ) , h_>l
where ~
=
3.2.5
c(V) -1 x c W and k = m - n.
Computation
of prl,wl,P2, first p a r t
a) F r o m n o t a t i o n 13, one has in C H ' ( H 2 ( V ) x V x W) : L,2 = ~*a'H2('---F) 9{ ~ , s ( P ) x c W } m .
But one knows t h a t (see (2.2)) : A
a*H~(r) = #*pi*r. (#*p2*r - {s(F)
x cW}"*)
.
Moreover, for i = 1, 2, one has
~ =p;o~o~.
(3.10)
Since ~ = q x i d w , it follows t h a t
~2 = ~ * r . ( ~ * r - {q*s(y) x c W F ) . { a , s ( 8 ) •
cWy"
Let us then d e c o m p o s e u2, by using the following n o t a t i o n : A
Notation
15 : Let u2 = ~'~ - u2' e C H ' ( H 2 ( V ) x V x W ) , where
.; = ~l*r. ~-~*r. { ~ , s ( 0 ) • c w ) ' - . u~' =
~*F.
{q*s(F) x c W } m . {z2,s(U) x c W } " .
36
Triple formula w e a r e g o i n g t o c a l c u l a t e p r l . W2l..~V '
In this paragraph,
F r o m p r o p o s i t i o n 1, one has :
~'
=
+
~ * r.03~ ~ * {c} - *: 9( 0 • ~i*r-03~ ~(--2)hUI*F
W)
. . . . .lcl~'-h 9( ( B . q*F ' - 1 ) • W ) , 9~2*F "033
(3.11)
h=l
which we shorten as follows :
k //2 ~
~ //2 h=O
(3.12)
'
b) L e t u s c o n s i d e r t h e f i r s t t e r m in t h e s u m (3.12) :
4 0 = ~q*r. ~ * r . ~*{~}~. (0 x w) (see l e m m a 4.(ii)) is the union of G13 and G23, the two g r a p h s of the m o r p h i s m s
H2(V) (va,v2, d)
--+ V w+ v,
(i=1,2).
Let us use some a d d i t i o n a l n o t a t i o n :
Notation
16 : Let u~ ~ = 31 + 32, where
Let us first study
31 = ~ * r . ~ * r .
-~* { C- } 033
a~ = ~H*r. ~ * r .
~*{~}~
*: 9 ( G 1 3 x W) 9( c ~
• w).
al a n d r
Since on (G13 • W ) the restrictions W'il and 033] are equal, one has : 31 = ~ * ( r { ~ } ' : ) . ~ ' r .
(c13 • w ) .
A p p l y i n g ( F P ) to w-i yields :
W~.al = F{~}*:. w~'i.(W-~*F 9(G13 Since (see (3.10))
w~ = / 7 / o 7? o ~
X W))
.
for i =1, 2, one h a s :
~ i , ( ~ * r - (a13 • w)) = p5,~,~,(~*~*p~*r- (G13 • W ) ) .
T h e triple formula
37 i
But ~,(G13 x W ) = [H2(V) • W] = 1, since Gla is a graph of a m o r p h i s m from H z ( V ) to V. A p p l y i n g (FP) to ~ yields : WI,(~2*F ' (G13 • W)) ~- p~,~,(~-*p'2*F 91) = p~,p~*r. 1 (from (FP) a n d the fact t h a t # is birational), or pr~pr2,F by l e m m a 2. But one has the equality p r 2 , F = f , V (see (2.5)). By (3.7), it follows t h a t : .
.
.
.
J,3 ({c} . p r z f , V )
It r e m a i n s to c o m p u t e
p r l , w l , a l = prl,3,(3 { c } ' . J pr2f, V ) 9 Since p r l . j . = or. = 7*, one has :
p~,~,al O n one h a n d , one has (2.8):
= ~*j*{~}~:' ~*j*PrU, V .
7*j*{~} k = ck , on the other h a n d , 7*j*pr~ = f*. T h e
result
p r l , ~ { , a l = c~:f* f, V
(3.13)
follows. L e t u s n o w s t u d y a2 a n d w l , a 2 . Since on G23 • W , one has the equality w3l = w2I, one o b t a i n s : a2 = W'il*F" ~ * ( F { ~ } k ) 9(G23
X
W)
Therefore one gets from (FP) :
w~'-{,a2 = F . ~'11,(w'22*(F{~}k) 9(G23 x W)) = r. ~,~,~,(~*~*~*(r{~}~)
9(a23 x w ) )
As was done above, we apply (FP) to ~. Since G23 is the graph of a m o r p h i s m from
H ~ ( V ) to V, it follows t h a t : ~,[G23 • W] = [H2(V) • W] = 1 . Next, we apply (FP) to # which is birational. Hence : ~,az
= r.
~ , ~ * ( r { ~ } ~:) = r . pr~p~2,(r{~} k) = j,j*pr~pr2,(F{~} ~')
by l e m m a 2 by (3.7).
Hence :
pr l ,Cal , a2
(V{~:} k) 9 . . , - k ) , by (3.7) and since pr2Pr2*(J*J {c}
= prl,j,j*pr~pr2, = 7
, .J,
prl,j,
= a,
=
.),*
.
Triple formula
38 j*{~}*: = a*c~, = 7,c~. The equalities
But one has (2.7) :
3 pr2pr2*3*7*c~ = f * f , ck p r l , W l , a 2 = 7 *., , 9
follow, since f = pr2 o j o 7 . To summarize, since u~~ = al + a2, we have o b t a i n e d : (3.14)
prl,COl,tY2,o = c ~ : f * f , V + f*f,c~,
c) L e t u s c o n s i d e r t h e o t h e r t e r m s o f t h e s u m (3.12) : We wrote above u~ = ~ i = 0 v~h with (for h > 1) v~h = (--2)hwh, where Wh = W'll*r' ~-2*F- ~3"{~} k-h. ( ( B q * F h - l ) • W ) . A
Let b : B ~
H2(V) • V be the canonical i m b e d d i n g ; t h e n B q * F h-1 can be rewritten
as b.b*q*F h-1 from (FP). Since on B, the restrictions Wll, w21 a n d w31 are equal, one can rewrite Wh as : Wh = Wll*r " wl*V . Wl*{~} k-h. (b,b* q ' F h-1 • W ) .
It follows t h a t in C H ' ( V x W) : ~
OJI,W
But inVxW,
h
F2 . {~}k-h ~ , ( b , b , q , F h - 1 •
~
onehascodimF=m>n=
w l , W h = 0. T h e r e f o r e
o n e h a s r~
dimF. HenceF 2=0. 2t h = 0 for h > 1.
It follows t h a t
To conclude, we have o b t a i n e d the result (see (3.14)): + f * f , ck .
prl,wl,U ' = cJ*f,V
3.2.6 Notation
Computation
of prl.~,u2,
17 : Let 5 : F ~
second
(3.15)
part
H 2 ( V ) be the canonical i m b e d d i n g a n d let 5' be equal
A
toSxidv:FxV'--+H2(V)
a) From n o t a t i o n 15, one has =
x V.
v2 = v~ - v~/ •
cW}
where TM
9
•
cW}
TM
.
Recall t h a t in chapter 1 we introduced the convention of o m i t t i n g or n o t o m i t t i n g the n o t a t i o n i. if i is the canonical i m b e d d i n g of a subscheme, d e p e n d i n g on the case one considers.
The triple formula In
this
paragraph,
39 we
prl,O2,1~,122tt,
calculate
From proposition 1, one has in C H ~ u~I =
w-'FP" { q * s
x V x W) :
x c W } ' . ~ 3 * { ~ } k. (U x W)
k
+
~--~(--2)h~*V
9{q*3,s(F) x c W } " . U33"{5} k - h . (Bq*F h-1 • W) .(3.16)
h=l
One can rewrite u~I as : k
,,
ilh
/"2 ~" E
/J2
(3.17)
"
h=0
b) L e t u s c o n s i d e r t h e f i r s t t e r m o f t h e s u m (3.17) : ~2,lO =
cWy".~5*{e} k.
~l*r. {q%s(F) •
Notice first (see notation 17) that ready noticed (lemma 4.(ii)) that
(O x W ) .
q*6,s(F) = 6~,(s(F) x V).
5
:
U2=lGi3 ,
Also we have al-
Since in H2(~-~) x V x W one
has codim(G,3 x W ) = n, it follows that : 2
{q*3,s(F) x c W } m . (U x W )
=
E{~',(~(F) • v ) • c W F .
(a,~ • w )
i=1 2
=
~ ] { ( ,~1 (s(F)
x
v) 9c~a) x c w y ~+',
by (']~1).
i=1
From (FP), one has 5~,(s(F) x V ) . Gi3 = 5',((s(F) x V) . 51"Gi3). The computation performed in chapter 4 yields 51*Gia=B
for
i=1,2.
(3.18)
Hence :
{q*~,,(F) x c w y " . (O • w ) = 2{(6',(,(F) • V ) . B) x c w y '+" . It follows t h a t : u~I~ = 2 ~ * F . s
k. {(6',(s(F) x V ) . B) x c W } "+" .
(3.19)
As we have seen previously, on B, the restrictions Wll, w21 and w31 are equal. Therefore, one has :
.;10 = 2 ~ * r . ~1"{~} ~ 9{(~:(s(F) • V ) . B) • cW) m+" . Therefore, since c o d i m ( ~ ) = - 2 n and m - n = k, (FP) and (7"42) y i e l d :
~ , 4 '~ = 2 r . {~}~. {~1,(~',((4F) • v ) . B)) • c W } ~ . Let us consider the commutative diagram
40
Triple formula
F•
lee F
with
r 1 =
~'
9 H~(IV) • v
q
c
T 1 o q
V Diagram
5
where ql is the first projection (restriction of' q) and p is the restriction of 7rl H2(V) --+ V. T h e n one gets the equalities :
~xJ',((s(F) • V). B) = p , q l , ( q ~ s ( F ) . B ) = p , ( s ( F ) . q l , B ) , =
p,(s(F). 1),
=
c(V) - 1 ,
by(FP)
because B is a graph of F in V
because F can be identified with the exceptional
divisor of V ' ~ V . Finally, since ~
=
c(V) -1
X
c W , we are left with :
~l,V~'~ = 2 F . (~}k. {~}k = 2j,/,({~}k)2
by (3.7) .
Hence, as usual : p r l , w l , u 2- 0 = 2 p r l , 3 ,9 j .* ( {-c }k)
2
= 27*j*({~}k:) 2 car p r a , j ,
:
or,
:_,.),*
.
In view of (2.8), we have thus proved the result : prx.~.,~
(3.20)
'~ : 2c~ .
c) L e t us l o o k at t h e o t h e r t e r m s o f t h e s u m ( 3 . 1 7 ) : Recall t h a t v~ = "~: 2-./,=O u2,h where, for h > 1, one has u; 'h = (--2)h~'i*r 9{q*5,s(F) x c W } " . ~3"{~} k - h . ( B q ' F h-1 x W ) . But, as a l r e a d y seen in b), the restrictions w11/3 and s u b s t i t u t e w3* by &'i* in the above equation.
0331B a r e
equal. One can therefore
Furthermore, since c o d i m ( B q * F h-1 x
W ) = n + h, one has from (7~1) : {q*5,s(F) x c W } " .
( B q * F h-1 x W )
=
{ ( q * 5 , s ( F ) . B q * F h-l) x cW} " + ' + h
=
{(q*(c~,s(F). F h - 1 ) 9B) • cW} m+'+h
The triple formula Hence
41
:
u'z'h = (--2)h~'(F{~}~'-h) 9{(q*(5,s(F)' F h - ' ) 9B) x c W } "+"*h Since codim(w'i) = - 2 n and k = m - n, we obtain from (FP) and (T42) the equality in C H ' ( V x W) : ~" lib COl,t/2 = (-2)"r{e}
k-'
9{ ~ , ( q ' ( 5 , s ( F ) .
Fh-1) 9B) x c W } k+h
tf one looks at the commutative diagram 5, since B is a graph from F to V, one see from (FP) that : q , ( q * ( 5 , s ( F ) . Fh-1) . B) = 5 , s ( F ) .
F h-l.q,B=5,s(F).F
h,
since
q,B= F
But (see [FU2], p. 70), one h a s : F 5,s(F) -
A
e
1 + f
CH'(H2(V))
(3.21)
Therefore one has : (--1)hq.(q*(5.s(F) 9Fh-1) 9B)
= (--1)h3,s(F). F h Fh+l
: ( - 1 ) h i ~- F = ] _ ~ + F ( - F + ( - F ) 2 + ... + ( - F ) h) Since COl, = 7h,q,, it yields : F Wl,((--1)h(q*(5,s(F) 9Fh-1) 9B)) = 7rl,(-f--~
h
"Jr- E(--F)
i)
i=1 A
But one has ~h,F i = 0 for 1 < i < h, since in H2(V), dim(F / ) = 2 n - i _ ; 2 2 n - h _ > 2 n - k > n (Recall that k <__n / 2 from hypothesis 3). We have also seen (3.21) that : F 5.s(F) - 1 +
F
F
hence
1.(-~-~)
=
c(V) -1
To summarize, one has : 0 2 1 , / 1 2,h
= 2hF {~}k-h. {c(V)-I • cW}k+h = 2hj,j*({-d}~-h. {~}k+h)
by (3.7).
Hence, as usual : 9 "* -- k - - h prl,cOl,V 2I I h =- 2 h prl,3,(3 {c} . j,{~}k+h) = 2h7,j*{~}k-h. 7*j,{~}k+h
because p r l , j , ---- a, = 7*- (2.8) yields the result : prl,021,V
till
2
---- 2 h c k _ h C k + h
(3.22)
In view of (3.20), we have finally found that : k
prl.U-i. ;' = 24 + Z h= 1
(3.23)
42
Triple formula
3.2.7
Conclusion
We had from
(3.6) : ~ = prl,Wl,U and also, v - - / 2 1 - / ] 2 : /'IX -- (D'~ -- /]~/) a c c o r d i n g to notation 13 and 15. Hence we have the equality m3 = prl,wl,ul - prl,oal,U2 + prl,Wl,U~'. Then, according to (3.8), (3.15) and (3.23), one has the equality: k
m-'~ = m 2 f * f , V - c,,f*f,V - f * f , ck q- 2c~. -I- ~
2hc,:-hC,,+h
(3.24)
h=l
Using the known expression for m2 (see theorem 2), one can give for ~33 the more familiar expression (cf. Kleiman [KL1], th. 5.9) : k
m"~ = f * f , m2 -- 2ckrn2 + ~ 2hc~-hCk+h h=l
To conclude, one has shown the following theorem :
T h e o r e m 4 Let f : V ----+ W be an arbitrary morphism of proper, smooth varieties with d i m W = k + d i m V and 0 < k < d i m V /2. Let ~33 E CH2k(V) be the "triple class", direct image of the cycle A
A
M3 = [g3(r)]. [H3(V) • W]
e
A
CI-F(H3(V• W))
where F is the graph of f . Then one has the 'triple formula" k
m"--~= f* f, m2 - 2ckm2 + ~ 2hCk-hCt:+h
(3.25)
h=l
where c~ is the i th Chern class of the virtual normal bundle f * T W - T V and where m2 denotes the double class in CI-I~(V).
R e m a r k 5 : In chapter 5, we will explain (in a particular case) how to interpret the existence of excess components in the cycle/I//3 in the case where the morphism f has Sz-singularities. For example, let f : p2 > F 3 be defined by f[x : y : t] = [x 2 : z y : yZ : t2]. Let H (resp. h) be the hyperplane generator of CHI(~ 3) (resp. CHX(pZ)). Then f(p2) is a quadratic cone C and f,[p2] = 2[C] = 4H. One h a s : f*H = 2h
and
c(/*TF 3 - T ? 2) = f*(1 + H) 4. (1 + h) -3 = 1 + 5h + 6h 6
The equality m2 = 3h follows (note that the double locus of f is degenerate here). Moreover, f . h = 2H 2 and therefore (theorem 4) one has ~33 = 6h 2, hence deg(~33) = 6. Here, the intersection is reduced to the only extra component 1 (defined in chapter 5, page 55), as is easily seen. This example is confirmed by theorem 5.
The triple formula
43
A
H~(V) • V
,<._ V
4
H~(v)
Jo" ~
V
~(v)xv
S
p 4
H~(v)
Chapter 4 Intermediate computations This chapter provides details of the computations used previously.
4.1 Let X be a non-singular variety of dimension p and do C X a doublet of support a point 0 E X. One can choose local coordinates (x, Y2, Y3,- .., Yv) centered at 0 such that the ideal of do in O x is (x 2, Y2, Y 3 , . . . , Yv)" A chart of H 2 ( X ) = H i l b 2 ( X ) at do is then given (see [Ill) by (a, b, c2, d 2 , . . . , cv, dr), i.e. the coefficients of the neighbouring ideal in O x : (x 2 + ax + b, - Y 2 + c2x + d2 , - Y 3 + c3x + d3 , . . . , - y p + Q,X + dr) (The minus signs are used to simplify the computations). Notice that in this chart, the hypersurface D C H2(X) consisting of doublets of support a single point is given by the equation a 2 - 4b = 0.
N o t a t i o n 18 : One will write ~ = (Y2,.--, Yl,), g = (c2,..., %) and d = (d2,..., dr), so that the ideal of do C O x is (x 2, if) and the ideal of a neighbouring doublet is (x 2 + a z + b, - ~ + x ~ + d~ So a chart of H2(X) at do is given by :
(a, b, e, d3 4.2
Flatness
(4.1)
o f 7rl a n d 7r2
Let do C X be again a doublet of support 0 and let us consider H z ( X ) C H 2 ( X ) • X , the tautological cover of H2(X). We want to give a chart of it at do = (do, 0).
Proof of lemma 4.(iv) and
of ~IlH
0
:
45
Let ((,V2,... ,%) be the coordinates of a point Pl E X in the neighborhood of 0. Using the previous notation, one denotes again its coordinates by (~, ~). One expresses H2(~'X) in the chart (a, b, ~, d~(~, ~) of H2(X) • X by requiring all coordinates of Pl to verify the equations defining the doublet, i.e. : ~ + a~ + b -~+@'+
(/ =
0
(4.2)
0
(4.3)
One sees that H2(~-X) can be expressed locally as a graph ( ~ , ~ , a , ~ ~ (b,d-~. And therefore (~, if, a, ~
(4.4)
constitutes a chart of H2(X) at (d0, 0). Let us then express the morphisms 7q : H2(X) ~
A
X and ~2 : H2(X) ~
X.
For ~1, it is easy, since d = (d, pl) must be sent to Pl. This gives :
For lr2, one must find the coordinates of P2 = Res(pl, d). The abscissa of P2 is the other root o f x 2 + a x + b , i . e . - a - ~ . Asp2 is located on the line defined by the equations ~ = xS"+ ~ its ordinate is - ( a + ~)~'+ d = 77- (a + 2~)~', in view of (4.3). Hence :
~1 : ( ~ , ~ , a , ~
~+ (~,~)
(4.5)
7r2 : (~,ff, a, 4
~
(4.6)
(-a-~,~-(a+2~)c~
T h e s e e x p r e s s i o n s s h o w t h a t ~h a n d 7r2 a r e s u b m e r s i o n s ~ a n d t h e r e f o r e flat morphisms. The divisor F C H2(X) corresponding to the non simple doublets, i.e. px = p2, is then given by the equations :
= --a--~
and
~ = ~ - (a + 2~)~
which reduce to
a + 2~ = o.
4.3
Proof
of lemma
4.(iv)
and
(4.7)
of ~ 1 H : 0
How can the two-sheeted covering 0 : H2(X) > H2(X), which is nothing else that the restriction to H2(t-X} of the natural projection H2(X) • X ~ H2(X), be expressed? From equations (4.2) and (4.3), one has in the charts (4.4) and (4.1) : 0 : (~, ~, a, c~ ~-+ (a, b = _~2 _ a ~ , 5", d = ~ -
~c~ .
(4.8)
46
Intermediate computations
Notice in particular that if the ideal (a 2 - 4b) of D is lifted by 0, one finds the ideal (a + 2~) z, which is the square of the ideal of F C He(X). Therefore, we have proved A
the statement (iv) of lemma 4 :
O*[D]=2[F]
in
CHI(H2(---~)).
(4.9)
The Jacobian DO of 0 (in the charts (4.4) and (4.1))is: 1
0
0
0
0
0
I -~I
0 I
-~ - 2 ~ - a 0 0
0 -~"
where I denotes the identity matrix. DO is invertible except if 2~ + a = 0, which is the equation of F ; therefore we have proved (1.1) (see chapter 1). Moreover, the restriction of 0 to F is not ramified, since in the local coordinates (~, ~, ~ of F, 0 is given by :
(~, ~, c-) ~ (-2~, ~2, ~, ~ _ ~ This shows that : ~FiH1
(4.10)
~- 0 ,
which is used in the demonstration of lemma 5.
4.4
Proof
4.(iii)
of lemma
Let us now prove some other statements of lemma 4 (by denoting again the variety by X and not by V). Let (~', 3) be the coordinates of a point P3 E X. In H2(X) x X, (4.4) provides a chart A
(~, ~, a, c~(~', 3)
(4.11)
A
In this chart, the graph G13 of i.e. :
71"1 : H 2 ( X )
~'--~
and
~
X is given by the condition P3 = Pl, 3=~.
In the same chart, the graph G23 of ~r2 (condition P3 = P2) is given by (see (4.6)) : and
3 = 77- (a + 2~)~'.
Consequently, one has the ideals : I(G13) -- (~' - ~, 3 - ~)
and
z(a23) = (~' + a + ~, 3 - ~ + (a + 2~)c-).
Therefore, the ideal of the scheme-theoretic intersection G13 N G23 is : (~'-~,3-~,~'+a+~,3-~+(a+20c-),
(4.12)
Proof of lemma 4.(ii) and (v)
47
or ({' -- ~, @ - r~, a + 2~). This ideal corresponds to a non-singular subvariety of H2(X) x X. Of course, it is the same ideal as the ideal of the graph B of rq from F to X, since B C Gla N G2a and B is non-singular. This proves the statement (iii) of lemma 4 : B = Gla Cl G2a 9
4.5
Proof
of lemma
4.(ii)
and
(4.13)
(v)
Notice that the expression (4.8) locally gives
o ' = e x idx :
H2(X) x X --+ H2(X) x X
(~,~, a,c-)(~',(~) ~
(a, b = - ~ 2 - a ~ , g ,
i=~-~c-)(~',~)
(4.14)
a) Let us look for the inverse image 0 = O'-'(U) in H2(-'--X) • X. In the chart (a, b, < c0({', ~) of H2(X) • X, the subvariety U is given by the ideal (~,2 + a{' + b, - r f + ~'~*+ c/), which can be obtained by replacing the coordinates ({, if) by ({', r~) in (4.2) and (4.3). By lifting it by 0', one finds the ideal of U :
((~' - ~)(~' + ~ + a), q - ? + (( - { ) ~ . Introduce the new coordinates A" = ~' - {, A = {' + ~ + a, 37 = ~ - r~, keeping the other three {, ~"and ft. Therefore, the ideal of 0 is
i(0) : (xA, - Y + Xc-) Also (see (4.12)), the ideal of G13 U G23 in these new coordinates is :
:(0,3 u c~)
(x, Y) n (x, Y + (.A - x ) ~ .
One does have I(U-) = I(Gla U G23), since the inclusion C is obvious and the other one comes from the following computation : if a X + ft. y = 7A + ~" (2~ + (A - X)c-), where "." is the dot product, it follows that 7 A + A6"- 5 E (X, 2)7). Then 3',4 + A6'. ~' = AX + ft. 3~ and therefore /~ (and fi) is a multiple of A, by factoriality. Hence the condition 7,,4 + A6. ~ E (X,A, - y + Xc-'). Therefore, we have proved the equality of the schemes : 0 = G13 O G2a 9
(4.15)
b) Let us look for the inverse image 0'*[R] where R C U C H2(X) x X has been defined above (notation 12). First, one has the equations of U : ~,2+a~,+b=O
and
-rf+~'~'+c~=O.
48
Intermediate computations
Furthermore, one has the equation of R itself, similar to (4.7) : a + 2~' = 0. Thus the ideal of R is (~,2 + a~' + b, -r~ + ~'~'+ ~ a + 2~'). By lifting it by 0', one finds the ideal in H2(-~) • X: ((~' - ~)(~' + ~ -b a), ~ - r? -t- (~1 _ ~)~., a + 2~'). Let us perform the same change of variables X = ~' - ~, A = ~' + ~ + a, y = ~ - rT,
O'-I(R) is
and keep ~, ~'and fT. In these new coordinates, the ideal of
z = (x.4, - Y +
A + x).
On the other hand, the ideal of B = G13 N G23 is, as was seen in w 4.4 : J=
(~'-~, @-~, a+2~) = (X,y,A-X)
= (X,y,,A) .
In the new coordinates (A, 6', ~, ~y,X, ~), let M be the subvariety of ideal K = (6",~, r~). Then one sees that
J + K is the ideal of the origin, while I + K = (XA, ~, 2(+,.4, ~, ~, ~)
is the ideal of a double point. It follows that :
O'*[R] = 2[/5'].
4.6
(4.16)
P r o o f of l e m m a 1
Now, we consider the situation of w 2.1, i.e. X is a subvariety of the non-singular variety Z. Let II1 and II2 : H2(Z) ~
Z be the two natural morphisms. If F C
H2(Z) is the divisor of the non simple doublets, one writes : '~) = F ['l Y I l l ( X )
= f ~ [I21(2)
.
Let us show that this intersection is transverse. Let do be a doublet of D.
We consider the most degenerate case : the case
with the scheme theoretic inclusion of do in X. If p = dim X and q = dim Z, let
(x, Y2,..., Yp, Y~+I,..., Y'q) be coordinates of Z (centered at 0 = supp(do)), in which the Yp, Y~+I,-.-, Y'q)-
equations of X are Y~,+I . . . . . ylq = 0 and the ideal of d0 is (x e, Ye,..., We use a shortened notation, as in notation 18 : (x, ~7, ~)
chart of Z
where the ideal of do is I0 = (z 2, g, ~). Let us use again the notation of w 4.2 and generalize it. As for (4.1), a chart of He(Z) at do is : (a, b, ~', ~ c~, c~)
where
~', c~e Cp-1
and
c~, ~ ff Cq-p
correspond to the neighbouring ideal I = (x e + ax + b, - ~ + x ~ + ~ - g + xd + ~) .
(4.17)
Flatness of P12 and of P3
49
Let ([, ~, ~) be the coordinates of a point of Z in the neighorhood of 0. Then one has the equations of H2(Z) in H2(Z) x Z : { ~2+a~+b=0 -77+ @'+ d = 0
(4.18)
- ~ + ~ + d~ = 0, giving locally H2(----Z)as a graph (~, ~, ~, a, ~', d) ~-~ (b, ~ ~). And therefore:
(~, ~, 775, a, ~', d)
(4.19)
constitutes a chart of H2(Z) at (d0,0). The morphism expressed as (of. (4.6)):
I]2 : H2(Z)
~
(~, ~,r?~, a, ~', c~) w+ ( - a - ~ , g - ( a + 2 ~ ) ~ ' , ~ - ( a + 2 ( ) d )
Z can be
.
Using the same argument as in w 4.2, one sees that in the chart (4.19), F is given by the equation a + 2~ = 0 ; while II~I(x) can be expressed by @ - (a + 2~)c~ = 0. Therefore, the intersection 7) = F Yl I I ( I ( x ) C H2(Z) is transverse,
(4.20)
which proves lemma 1, page 14.
4.7
F l a t n e s s of P12 and of P3
From now on, we concentrate on the variety H3(X) where X is non-singular (see w 3.1). It consists of the elements [ = (t, d12, d23, d31, Pl,P2, P3). Let
P12:
H3(X)
~
H2(X)
and
P3:
H3(X)
~
X
be the natural morphisms, where dl: = (PI, d12). We show that they are flat. We are led to consider the neighborhood of the most degenerate case, i.e. ~ = ( t o , do, do, do, O,O,O)
where to is an amorphous triplet. Recall the following definition (already given in the introduction - see 0.7-) : A triplet t C X is said to be amorphous if supp(t) is reduced to only one point p and if the ideal oft is the square of the ideal of p in a germ of a smooth surface going through p.
(The other triple points are curvilinear, i.e. subschemes of a non-singular curve). Consequently, in a chart (x, y, Y') of X (where ~"C G~'-2 if p =dim X), the ideal of the
50
Intermediate computations
triplet to is (x 2, xy, y2, E) and the ideal of the doublet do is (x 2, y, Z). From ([LB1], p. 937), a chart of H3(t--X) at to is (s, t, ~', c, c',
C It ,
v, fi, g ) .
(4.21)
Notice that in [LB1] it was assumed that p = dimX = 3, but the generalization can be easily performed. However, a chart of H2(X) at do is (a, b, c, d, ~', fi), which corresponds to the ideal (x 2 + ax + b, - y + cx + d, - Z + gx + f). (It is similar to notation 18, except that not only x but both x and y are needed in the calculation ; this is why E is introduced
and not ~7). A
Let (~,r/, 0 be the coordinates of a point close to 0 ; thus a chart of H2(X)
at
do -- (do,0) is : (~, r/, (, a, c, ~
,
(4.22)
A
which comes from the equations of H2(X) in H2(X) x X :
~+a~+b=o,
-~+c~+d=0,
-(+#~+/=
o
which state that the point must be on the doublet. (The calculation is exactly the same as for (4.2), (4.3) and (4.4)). Then one gives the local expression of P12. First, Pl is the point of coordinates (s, t, ~'). From [LB1], pp 934 and 937, one also h a s : a=--2s--s'=--2s+d'v = --y-- ~c.
The local expression of P12 in the charts (4.21) and (4.22) follows :
i - (s, t, ~, c, c', c", ~, ~, ~) ~ ~
= (s, t, ,~ - 2 s + c'% c, - ~ -
~c)
which proves, in the neighborhood of to : the flatness of P12-
(4.23)
Notice once more that we studied P12 in the neighborhood of the most degenerate case ; in the other cases, the computations are similar. The expression for P12 which was obtained previously enables us to show that :
P~I(F) = El2 + E" C H - 5 ~ )
(scheme theoretically)
(4.24)
Indeed, the divisor F C H2(X) of the non simple doublets is given locally in the chart (4.22) by a + 24 = 0 (same calculation as for (4.7)). If this equation is lifted by P12, one finds - 2 s + c'v + 2s = 0, i.e. d'v = 0. From ([LB1], p. 940), it is indeed the equation of E12 + E ' .
Proof of lemma 4.(i)
51
For P3, it is in fact a submersion, since the coordinates of the point P3 are :
(s+s' +s",
g + # + r 7')
t+t'+t",
(see [LB1], p. 935). Then one sees (again from [LB1], (1), (2), (13), (14) and (E)), that P3 = ( s + . . . , t + . . . , F + . . . ) , where the dots denote terms of degree > 2 in the coefficients of the chart (4.21). This proves the flatness of P3.
4.8
Proof
of lemma
(4.25)
4.(i) A
A
Now, let r -- P12 x P3 : H3(X) --+ H2(X) x X be the morphism defined by r -- (~12,p3) and let 0 c H2(I"X) x X be the inverse image of the tautological scheme U C H2(X) • X by O' = 0 x idx (see notation 12). We are going to show that scheme-the0retically, r __ ~ = El3 + E23 + E*. Once again, we just study the neighborhood of the most degenerate case : t~ = (to, do, do, do, 0, 0, 0), with to amorphous (cf. definition 2). One uses again the previous notation : (a, b, c, d, g, f) is a chart of H2(X) at do ; let (~', ~, ~) be the coordinates of a point of X close to 0, so that one has : (~', r]', ~)
chart or X at 0.
(4.26)
The ideal of U in the corresponding chart of H2(X) • X is
(~,2 + a~' + b, - ~ ' + c ( + d, - ~ + ~ ' + y) Moreover, the local expression of 0' = 0 x idx (similar to (4.14)) is (~, % (, a, c, ~(~', ~', ~) ~-+ (a, _~2 _ a~, c, ~7- ~c, ~', ( -
~'~)(~', ~7', ~) 9
By lifting the ideal of V by 0', one finds (similarly to w4.5) the ideal of U in H2(I-X) x X :
((~' - ~)(~' + ~ + a), (~' - ~)c + ~ - 7', (~' - ~)~+ ( - C) The local expression of r (4.22) x (4.26)
r
: H3(X)
n
~ H2(X) x X in the charts (4.21) and
(s,t,~,c,c',c",v,f, 5) ~ (s,t,~,-2s+c"v,c,-fi-Sc)(s+s'+s",t+t'+t",~*+~+r; 0
is now available (see w 4.7), where s', s", t', t", r-;, r'5 are given by ([LB1], relations (1), (2), (13), (14) and (E)) :
52
Intermediate computations
S/
~-
--CH V
SH
~
--C'V
t'
~
t"
=
CS' s"(e + c' + c")
/,
= ~,,(~+ j + ~')
~"
=
- ~ 7 - 3c
J J'
= =
-3c' -3c"
(4.27)
Then by lifting the ideal of 5 by r one finds the ideal of r
C H~3(X) :
((~' + ~")(J + J' + c%), (~' + ~")c - t' - t", (~' + ~")(~+ ~c) + ; + / 9 , i.e., by using the relations (4.27) :
((c' + c")c'vt (c' + ~")c'v, (c' + c")~'~3) = (r + c")c'~) . m
From [LB 1], pp 939-940, one recognizes the equation of the divisor E = Ez8 + E23-t- E ' . Hence : r
= ~
C H~T'X) ,
(4.28)
which is used in lemma 4.
4.9
Proof of lemma 3
We now consider the situation of w 4.6 : the variety X is now a subvariety (of dimension p) of the non-singular variety Z (of dimension q). The two natural morphisms are denoted once again by P12 and P3 : P12 : H 3 ( Z ) ~
H2(Z)
and
P3 : Ha(Z) ~
Z.
Then, let (x, y, 2", 2) be a chart of Z at a point, so that
(4.29)
z3 = 0 are equations of X in Z. Let t~ = (to, do, do, do, 0, 0, 0) be a complete triple in X, with to amorphous ; it is afortiori in Z. Let ~ = (do, 0) C H2(~-X). Similarly to (4.22), one obtains a chart of H3(~) at ~, simply by changing 2" into (2, z~), therefore by changing ~into (~', r-~), fiinto (~7,/) and 3 into (3, o7'). The chart of Ha(-~) at & follows:
(,, t, ~, r~, c, c', c", v, e, p~, 3, J ' ) . ( W a r n i n g : the variable r~ has nothing to do with ~ used in the previous section !). How is H3(X) expressed in this chart ? Simply by ~ = 0, tP = 0, o7' = 0 and therefore : A
A
H3(X) is a non-singular subvariety of H a ( z ) .
(4.30)
Transversality of/3 and E
53
[t remains to check that P t ~ I ( H 2 ~ - ) ) i n t e r s e c t s transversally the divisors E~3, E23 and E" in
Ha(Z). Of course, one uses the chart of H2(Z) at do, similar to (4.22) : (~, 71, ~, C', a, 0, ~, r
.
In this chart, H2(----X) is given by d = 0, e~ = 0. Then one uses the local expression, similar to (4.23) of P12 :
P~ :
H~(z)
(~,t,r
~
~, ~,J') ~
H~(z)
Thus p121(H-5-~)) has for equations r~ = 0 and E13, E23 and E" are transverse at since their respective equations are d + c" = 0,
4.10
Transversality
~c,-7' - r
(~,t,~',~,-2~ + c ' % c , - ~ -
r + ~3c =
0 in H ~ ) .
One h a s :
P121(H2(X)) in H a ( Z ) ,
(4.31)
c' = 0 and v = 0.
of/3 and
a) Finally, let V and W be two non-singular varieties, of respective dimensions n and T/'t.
Let
F C H2(~-~xW) be the divisor of the non simple doublets. Let us prove t h a t a : H2('-~) • W ~-+ H2(V---xxW) is the imbedding onto the horizontal
a*F = F • W, if
doublets (see w 2.2). Let w 9 W and do = (do,0) 9 H2(V), with do of support 0.
Consider d'o =
a(d~o,w). At (0, w), one has a chart of V • W : (x, ?7,~ ) where (x, ~) is a chart of V at do C O v being (x 2, ~). In exactly the same way as in w 4.6, a chart of H~(V'-~xW) at d~ is (~, ~, 71-;, a, ~', c~), which corresponds 0 and ] is a chart of W at w, the ideal of to the doublets close to (do, w), of ideal
(x~ + ax + b, - g + x~+ ~ - r + ~ + ~) In this chart, F is given, as usual, by a + 2~ = 0. However, d = 0 is the equation of
a(H2(--~) • W), since it must correspond to the horizontal doublets, and then ~ is constant. This proves the statement : The intersection F A (H2(V) x W) is transverse in
b) Now let p : H3(~-V)• W r
H2(V x W).
(4.32)
H3(V-~x W) be the imbedding on the horizontal triplets.
One would like to show t h a t it is transverse to ~ = E13 + ~3 + E'. Once again, we study what happens close to the most degenerate case : to = (to, do, do, do, 0, 0, 0) where to is amorphous. Let w 9 W. Consider t~o = / 3 ( ~ , w). At
54
Intermediate computations
(0, w), one has a chart of V • W : (x, y, 5", z'), where (x, y, 5") is a chart of V at 0 and z~ is a chart of W at w, the ideal of to being
(x 2, xy, y2, Z)
and the ideal of do being
(x 2, y, z). Similarly to (4.21), one obtains a chart of (s, t, ~', r-;, c, c',
C tl
H3(V--"-'~W) at
t~:
v, fi, p-;, ~, a-~) .
In this chart, H3(~-V) • W is expressed by ~ = 0 and o~ = 0, since z-; must be constant. The equations of the divisors IE13, 4 3 and E" are respectively
: c ~ + c" = 0, c~ = 0
and v = 0. It follows t h a t m
A
/3 is transverse to E13, 4 3 and E" and then fl*E = E • W in H3(V) x W.
(4.33)
Chapter 5 A p p l i c a t i o n to the case where V is
a
surface and W a v o l u m e Let V be a surface and W be a volume (i.e. dimV = 2 and d i m W = 3) and let f
: V
~ W be a morphism having a S2-singularity at the point 0 C V which
means t h a t in local coordinates, f is written as :
f ( x , y ) ~- (ql(x,y) -~-''', q2(T,y) -~-''', q3(x,y)'~-''') where the qi are quadratic forms, the dots denoting terms of degree > 2. The intersection
H3(F) n (H3(V) x w )
in H3(V • W) then possesses an excess
component I, which consists of the iF = ( T , D12, D23, D31, P l , P2, P3) with Pl -- P2 -- P3 -- 0. The ideal of T is A/t 2, where A4 is the ideal of 0 in (gv. On the other hand, D12, 023 and D31 are arbitrary and therefore set theoretically, one has a bijection :
i
_u, p1 • p:~ • p1 ~
(5.1)
(o12, D23, O31) ,
by writing p1 for P(ToV), where P denotes the set of lines. I n t h e f o l l o w i n g , w e a s s u m e t h a t I is r e d u c e d , w h i c h i m p l i e s t h a t (5.1) is a n i s o m o r p h i s m .
One has the commutative diagram where the arrows are
canonical imbeddings (the dimensions are shown in parentheses) :
Application to the case where V is a surface and W a volume
56
(9)
Ha(---~) • W r
(3)
I
c
, Ha(v--'~W)
(15)
.
(6)
Ha(r)
Then, the corollary 9.2.3 of [FU2], p. 163 yields : A
A
[Ha(P)] . [Ha(V) • W] = c3(u(w)], - u(v)) + ~r where u denotes the normal bundle and R the "residual class".
5.1
C o m p u t a t i o n of c u ( v )
a) Let 0' = f(0) C W. First, I is contained inside H3(V) (identified with H a ( v ) • {0'}) and even contained inside E ' , so that v can be decomposed in the canonical imbeddings : I ~ E" ~ H3(V) ~-~ H3(V) • W . Thus, in the Grothendieck group K ( I ) of the vector bundles, one has
u(v) = u(I, E ' ) + u ( E ' , H a ( v ) ) I , + trivial bundle. But E" is a fibration on V (by T ~-~ pl) and I is the fiber at 0. Thus, in fact,
u(v) = u ( E ~ Ha(V))EI + trivial b u n d l e , which gives the total Chern class :
c.(~) = c.( E', Ha(V) )li If i : E" ~-+ H a ( v ) is the canonical imbedding, one has therefore
ClZ/(E ~ H3(V)) = i*[E'] . Hence cu(E ~ H3(V)) -- 1 + i*[E~
Abusing the notation, one has therefore:
cu(v) : 1 + [E']I, Thus, it remains to find [E~
Notation
(5.2)
(5.3)
in C H I ( I ) .
19 : In C H I ( p 1 > p1 x p1), one writes A = pr~(*), B -- pr~(*), C -- pr~(*),
so t h a t : CHI(~ 1 • P1 • ~ ) -- Z A |
Computation of c~(v)
57
Notice that A2=B 2=C 2=0
and
deg(ABC)=l.
(5.4)
with
(5.5)
For symmetry reasons, one has [E']I/=a(A+B+C)
aEZ.
Let 5 C I ~- F1 x I?1 x ~1 be the "small" diagonal consisting of complete triples (02, D, D, D, 0, 0, 0) where 02 denotes (improperly) the triplet of ideal Ad 2. Obviously, one has 5 _~ p1
b) A
L e m m a 6 By identifying 5 with ~1, one has the equality C1/](~ , H3(V)) = 1. Proof : Let (x, y) be the coordinates of V centered at 0. If D is a doublet of support 0, one denotes by Axis(D) the line it defines in this coordinate system. One sees that 5 -~ p1 is the glueing of two open sets U0 and U~ (each one is isomorphic to C), where : U0 corresponds to the doublets D of non vertical axis, U~ corresponds to the doublets D of non horizontal axis. Y Axis (D ~ - - - " "
0
v
A
Do A
In [LB1], p. 937 was given a chart of H3(V) at To -- (02 , Do, Do, Do, 0 , 0 , 0) where Do is the doublet of ideal (x 2, y). This chart is
(s, t, c, c', c", v)
(5.6)
with the notation of [LB1]. More precisely :
(i) (s, t) are the coordinates in the chart (x, y) of the point Pl close to 0, (ii) One denotes by r C-~
c+d
Ct
+d I
the slope of Axis(D12), of equation y = cx + d the slope of Axis(D31), of equation y = (c + c')x + d + d' the slope of Axis(D23), of equation y = (c + c' + d)x + d + d' + d".
58
Application to the case where V is a surface and W a volume
(iii) Finally v is a coefficient which arises in the ideal J of a triplet close to 02
:
J = (z 2 + u x + vy + w, x y + u'x + v'y + w', y2 + u"x + v"y + w'')
Then let D ~ be the doublet of ideal (y2, x).
If one considers the complete triple
= (02, Doo, D ~ , D ~ , 0, 0, 0), one sees that a chart of H3(I--V) at ~ (S, T, C, C', C", V)
is given by
where
(5.7)
(i)' S = t, T = s are the coordinates of the point Pl in the chart (y, x) of V (and not ( x , y ) ) ; 1 (ii)' moreover C = - since in the chart (y, x), the equation of the line Axis(D12) is x -
C
y c
d 1 (from (ii)). Also, from (ii) again, C + C ' c c+d
and C + C ' + C "
=
1
c+d+d
I
(iii) ~ Finally, one has u" = V, since the ideal J can be rewritten, by exchanging the
roles of x and y, as j = (y2 + v,,y + u,,x + w,,, y x + v~y + u,x + w,, x 2 + vy + u x + w) .
The first generator of J should indeed be written as y2 + Uy + V x + W . But, from [LB1], relations (E), p. 937, one h a s :
(5.8)
= - e v ( c + c')(c + c' + c") .
In the chart (5.6), 5 is given by the equations s = t = d = c" = v = 0, i.e. 5 is parameterized by the direction c of the line y = cx. On the other hand, in the chart (5.7), 5 is given by the equations S = T = C' = C" = V = 0. Thus, one obtains the normal bundle to 5 in H3(V) as the glueing of C* x C5 and C* • C5 by : (c, ds, dt, dc', dc", dv) ~ (C, dS, dT, dC', dC", dV) .
But C = _1 and also, from (ii)' : dC + dC' - - d c - dc' hence dC' = - d c ' C
C2
one has the equality dC" -
dc"
-
C2
"
~
C2
Then, from (5.8), d V = du" = - c 3 d v .
from (i) ~, one has : d S = dt and d T = ds. One obtains the glueing d a t a : C* --+
GL(5, C) dS
Similarly, '
ds
dt
dc ~
dc"
dv
0
1
0
0
0
dT
1
0
0
0
0
dC'
0
0
-1/c 2
0
0
dC"
0
0
0
-1/c 2
0
dV
0
0
0
0
-c 3
Finally,
C o m p u t a t i o n of cu(w)l I
59
which gives u(5, H a ( v ) ) as bundle of rank 5 on P:. But recall t h a t the g l u e i n g o f C* •
with C* •
by (z,~) ~
(z_1, zfs ' where
n C Z, gives a vector bundle on ~:, whose first Chern class is n : section 1 has a zero of order n at infinity. Here, one sees that the first Chern class of u(8, H3(V)) is 2+2-3=1.
O
c) The inclusions 6 C E ~ C H3(V) yield in the Grothendieck group K(5) :
u(5, E ~ + u(E', H3(V))la = u(6, H3(V)) Considering the first Chern classes and taking into account the previous lemma , it follows t h a t (from (5.2)) : deg ClU(5, E') + deg [E']la = 1 . Moreover, the inclusions 5 C [ C E" yield u(5, E') = u(6, I) + u(I,E')la.
But
u(g, I) ~- T5 | T5 (recall t h a t 5 is the small diagonal of I -~ P: • ~: • p1). Also, as said already, u(I, E ~ is trivial since I is the fiber at 0 of E" - - + V. Since 6 -~ 1~:, it follows t h a t : deg [E~
= 1 - deg c:u(5, E ~ = 1 - 4 = - 3 . But one has (see (5.5))
[E']I• = a ( A + B + C). Also, one sees that [5] = A B + B C + C A in CH2(I). Indeed, one can see t h a t 5 is the intersection of the two diagonals A:2 and A:3 in ~1 • ~: • p1, i.e. [5] = (A + t3)(t3 + C) = A B + B C + CA, from (5.4). It follows that, again from (5.4) : - 3 = deg([E']la ) = deg(([E']lz)l~) = d e g ( a ( A -4- B + C ) ( A B + B C + CA)) = 3a Thus, a = - 1 and one has therefore proved the following lemma : L e m m a 7 In CHI(I), the following equality holds :
[e'llr = - ( A +
B + C)
Equality (5.3) yields immediately the total Chern class in C H I ( I ) :
cu(v)
5.2
Computation
of
=
1
-
(A + B + C)
(5.9)
cu(w)l ~
a) Some preliminary computations are needed. We use again the notation of w 3.1 : let X be a smooth subvariety of a smooth variety Z and let P3:
H3(Z)
-+
Z
and
P:2:
~p3 be the two morphisms, where D = (dl2,pl,p2).
H3(Z)
-+
T
~
H2(Z)
Application to the case where V is a surface and W a volume
6O Proposition on H3(X) : 0
2 Using the above notation, one has an exact sequence of vector bundles
~ P ; u ( X , Z) | O ( - E ) ---+ u(Hn(~-X), H3(---~))
> P;2u(g2(--~), H2(----Z))
>0
where E = E23 + E31 + E ~ and u denotes the normal bundle. In order to show this proposition, we need an intermediate lemma. First, introduce the notation :
N o t a t i o n 20 : If ~ is a smooth variety, its tangent bundle is denoted by T ~ . As H3(----X) C P ~ I ( H - ~ ) ) , restriction a morphism
the differential dP12 : T H e )
dP121TH--~) : T H 3 ( X )
P{2TH2("~) gives by
~ P~2TH2(X)
A "normM derivative" morphism of vector bundles d'P12 : u(H3(X), H3(Z)) ----+ P;2u(H2(-~), H2(---'Z)) follows. Similarly, the differential dP3 : T H 3 ( Z ) > P ~ T Z gives, as above, a "normal derivative" : d'P3 : u ( H a ( X ) , H 3 ( Z ) ) > P;u(X,Z) , A
since H3(X) C P f l ( X ) . b) Then, one has the following lemma :
L e m m a 8 With the above notation, (i) d~P12 is a su~ective morphism of vector bundles on H3(X) ; (ii) if K is its kernel, one has an isomorphism of vector bundles : d'P3[g : K
~> P ~ u ( X , Z )
Proof:
We first give local expressions of dvP12 . One can see easily that it is sufficient to consider the case where X is a surface and Z a volume. We only study the neighborhood of the most degenerate case, i.e. To C H3(X) is given by Too = (To, D12, D23, D31, Pl, P2, P3) = ( 02 , Do, Do, Do, 0, 0 , 0 )
,
Computation of cL,(w)l I
61
where 0 is a point of X, Do is a doublet of support 0 and 02 denotes the triplet of ideal A42, if Ad is the ideal of 0 in Ox (i.e. To is amorphous, cf. definition 2). Let (x, y, z) be the coordinates of the variety Z centered at 0, the equation of X being z = 0, so that one takes the triplet 02 with ideal (x 2, xy, y2 z) and the doublet Do with ideal (x 2, y, z). A chart of H3(----Z)at ~ is given in [LB1], p. 937:
(s, t, r, c, c', c", v, ;, ~ ) .
(5.10)
A
In this chart, the equations of Ha(X) are r = p = a = 0 so that a trivialisation of A
A
r,(Ha(X), Ha(Z)) in the neighborhood of To is given b y : (s, t, c, c', c", v)(dr, dp, da)
,
(5.11)
where dr (resp. dp, dcr) represents a tangent vector in the direction r (resp. p, or). Similarly, a chart of H2(Z) at Px2(T0) = Do is: (s, t, r, a, c, e)
(5.12)
It corresponds to a point Pl of coordinates (s, t, r) and a doublet D of ideal (x 2 + A
ax + b, - y + cx -t- d, - z + ex + f) close to Do = (Do, Pl)- In this chart, H2(X) is given by r = e = 0, since z = 0 is the equation of X in Z. Therefore, a trivialisation of ~(H2(X), H2(Z)) in the neighborhood of Do is: (s, t, 6, c)(d~, d e ) .
(5.13)
Now, in the charts (5.10) and (5.12), the morphism P12 is given by (see the two lines before (4.23)) :
(s, t, ~, c, c', c", ~, ;, ~) ~ (s, t, r, - 2 s + c"~, c, - ~ - ~ c ) . Moreover (see [LB1], p. 937), one has the equality : e = - p - a c .
The local expression
of d" P12
d" P12 : (s, t, c, e, c", ~)(d~, dp, d~)
~
(~, t, c, c', c", ~)(d~, - a p - cd~) (5.14)
follows. We have therefore proved that d~Pz2 is a su_~jection of vector bundles, i.e.
(i) of
lemma 8. Let us now express P3 and d~P3 . In the same chart (5.10) of H3(Z) at To, we have seen (cf. the lines above (4.25)) that P3 can be expressed by :
P3(s,t,r,c,c',d',v,p,~)
= (s+s'+s",t+t'+t",r+r'+r") =
(s - v(c' + e,), t - v ( c e ' + ce + c,2 + e e ' ) , r + ~(r + c")(o + ~c + ~c'))
,
62
Application to the case where V is a surface and W a volume
from (4.27). The local expression of d"P3
d"P3 : u ( H 3 ( X ) , H 3 ( Z ) ) (s, t, c, d, c", v)(dr, dp, dcr)
--+ p ~ u ( X , Z ) ~
(s, t, c, c', c", v)(dr + v(c' + c " ) ( d p + (c + c')dcr))
(5.15) follows. Let us then restrict d~P3 to K = Ker(d'P12).
Expression (5.14) yields :
d~P3rK : (s, t, c, c', c", v)(O, - c d a , dcr) ~ (s, t, c, c', c', v)(v(c' + c")c'&r) . But v ( d + c")d = 0 is the local equation of E = E" + E31 + E23 (see (4.28)). Therefore one sees that d'P3rK is an isomorphism of vector bundles on Ha(X) :
K
oo(-E)
and this proves (ii) of lemma 8.
,
(Of course, one should also construct the local
expressions of d"P12 and d"P3 in the neighborhood of other points of H 3 ( X ) leave it to the reader).
- we []
The last step consists in applying the trivial lemma :
L e m m a 9 Let Y
be a variety, D a Cartier's divisor and E, El, E2 three vector
bundles on Y . Let fl : E A s s u m e that :
~ E1 and f2 : E ----+ E2 be two morphisms of bundles.
(i) f2 : E ---+ E2 is a su__~ective morphism of bundles, (ii) fliK~ri2 is an isomorphism from K e r f2 onto I E I , where I = O y ( - D ) . Then one has the exact sequence of bundles on Y : 0 ----+ IE1 ~t4 E - ~
E2 ----+ 0
,
where u = (fliK~I2) -1. We apply this lemma with fi = d"P3 and f2 = d'P12 9 One gets the desired sequence and proposition 2 is shown.
R e m a r k 6 : This kind of computations recovers in a simpler way the results of [LB2] or [LB3].
c) We can now compute explicitly cu(w)l I = cu(H3(~-T),H3(-~xW))jz
when F C
V • W is the graph of f : V ---+ W. Of course, we will apply the previous results withX=FandZ=VxW.
C o m p u t a t i o n of the contribution of I
63
Let us now consider the sequence of m o r p h i s m s (see the d i a g r a m at the beginning of chapter 5) :
8'
I
r
v • w
w
(5.16)
where fit and j are canonical imbeddings and 1 < i < 3. The composition of the four m o r p h i s m s is then constant. Moreover, the normal bundle N = u ( r , V x W ) can be identified with the bundle (of rank 3) j*pr~TW. Then l*
*
/3 p i N = /*
3 / . .Pi . . .J p r 2 T W is trivial of rank 3,
(5.17)
*
which one writes as fi P~ N = O a. F r o m p r o p o s i t i o n 2, one has the exact sequence of vector bundles on
0 ----+ P j N |
---4 u(w) - - 4 PI*2u(H2(F), H2(V x W ) )
O(-E)
Ha(r) :
>0 .
(5.18)
One has also the similar exact sequence on H2(F) (one can refer for example to [LB2] or ILB3]) :
0 ---9 r ; N | O ( - F ) ----4 u(HZ(F), g 2 ( Y x W ) ) ~ where 7rl, lr2 :
H~(r) ~
7r~N ~
0
(5.19)
r are the n a t u r a l morphisms. Of course, one has 7rl o P12 =
P1 and 7r2 o P12 = P2 ; moreover, P ~ I ( F ) = E12 + E" (see (4.24)). Thus, by lifting the previous exact sequence (5.19), one obtains the exact sequence of vector bundles A
on H3(F) :
0 ---4 P ? N
|
(~(-J~12
-
ET*)
H2(V x W)) ~
) i~
J;:2~N ~
0
(5.20)
A look at (5.18) and (5.20) enables us to give the total Chern class :
cu(w) = c(P~ N | O ( - E ) ) . c(P; N | O(-E~2 - E')) . c(P~N) . By a p p l y i n g fl'* to b o t h m e m b e r s of this equality, one obtains from (5.17) cu(w)l I = c(O a | J3'*O(--E)) . c(O a | J3'*O(-E12 - E ' ) ) .
:
1,
which can also be expressed by
cu(w)l~ = 3'*((1 - E23
-
E31 -
E')3(1
-
~12 -
E')
3)
,
(5.21)
since E = E23 + Eal + E ~
5.3
Computation
of the contribution
of I
F r o m w h a t we have seen at the beginning of chapter 5, the contribution of I in the cycle [H3(F)] - [Ha(V) x W] is C3(b'(w)]I
-
-
b'(V)). From (5.21) and (5.3), one knows
the t o t a l Chern class c(u(w)l~ - u(v)) = / 3 ' * ( ( 1 - / ; 2 3 - Eal - E ' ) a ( 1 I+E"
-
El2
-
e ' ) 3)
(5.22)
64
Application to the case where V is a surface and W a volume A
where ,~' : I ~-~ Ha(F) is the canonical imbedding. Moreover (lemma 7), one has /3'*E" = - ( A + B + C) where A, B, C E CHI(I) -~ z 3 were defined in notation 19. A
Furthermore, the divisors E12, E23, E31 of Ha(F) give, when restricted to I, the three diagonals of ~1 x F1 x pl respectively. E12 is for example defined as the set of
= (T, D12, D23, D31, Pl, P2, P3) such that D23 = Dal. If one denotes by Aab, Abe., Aca the three diagonals of p1 • p1 • pl (defined by Aa~ = { (a, b, c) t a -- b}, e t c . . . ) , one sees t h a t ~'*E12 = Abe, since D23 = D31. Moreover, Abe = B + C in CH 1(~1• p l • F1) and similarly, by circular p e r m u t a t i o n : r
23
= /kac = A + C E CHI(~ '1 • F 1 X: P 1)
fl'*E31 = A,tb = A + B E C H ' ( P 1 x ~1 x ~1) . Then one has the total Chern class in C H ' ( I ) : (1 - A)3(1 + A) a - ,(v)) =
- (X T-g
Since A 2 = B 2 = C 2 = 0 and d e g ( A B C ) = 1, it follows immediately that : deg c3(v(w)lI - / 2 ( v ) ) = 6
(5.23)
Therefore, the following theorem has been proved :
Theorem
5 Let f
: V ~
W be a morphism of a smooth surface in a smooth
volume. The rnorphism f is assumed to have a S2-singularity at the point 0 E V. Let F C V x W be the graph o f f . A A Let I be the excess component, of dimension 3, of the inters e ction H 3 (F)gI(H 3 (V) x W ) consisting of the complete triples T = (02, D12, D23, D31, 0, 0, 0) where supp(Dij) =
{0}. The component I is assumed to be reduced. Then, the contribution of I in the O-cycle A
M3 = [H3(F)] " [H3(V) x W]
is of degree 6. Therefore, the contribution of [ is 6 in the O-cycle m ~-~ ff CH2(V).
Part two Construction of a complete quadruples variety
The present goal is to construct a "good" space of ordered quadruples of a variety V, in order to give a definition and a computation of the quadruple class m4 for an arbitrary morphism
f
:X ~
Y
between non-singular varieties (with dim X < dim Y).
We saw in the introduction (in 0.7) that a "naive" generalization H~a~ve(V 4 ) of the construction of H3(V) is not sufficient : the variety H,,a~,~.(V 4 ) obtained in this way is in fact reducible and singular. Therefore, we had to construct an intermediate variety B(V), which is the closure of the graph of the residual rational map (see definitions 3 and 4) :
nes :
i(v)
dCq
...
-~
Hi~b2(V)
...--+ d'=q \ d
The following chapter is devoted to the study of this auxiliary variety
B(V). The
construction of our complete quadruples variety will be given in chapter 7.
Chapter 6 Construction of the variety 6.1
B(V)
Statement of the theorem
Theorem
6 Let V be a non-singular, irreducible variety of dimension dim V > 3
over C . Let B ( V ) be the closure of the graph of the residual rational map Res : I ( Y )
... -~
(d,q)
.-~
Hilb2(Y) d'=q
\ d
where I ( V ) C Hilb2(V) x Hilb4(V) is the incidence variety. The variety B(V) is irreducible and non-singular of dimension 4 9d i m V .
The irreducibility of the variety B ( V ) will be established in w 6.3. of the non-singularity of the variety B ( V ) will be the subject of w 6.4.
The proof One can
go back to the case where V is a variety of dimension 3 in a systematic manner. When n =
dim V _> 4, one just has to replace z by Zl,...,z,~-2 everywhere in the
computations. I n t h e f o l l o w i n g , V will b e a n o n - s i n g u l a r v a r i e t y o f d i m e n s i o n 3. Let us give some definitions :
6.2
Definitions, drawing conventions
D e f i n i t i o n s 7 : Let q be a quadruplet of support a closed point p of V. According to the description of Briangon [B1], the different quadruplets supported by p are given by the different ideals of Ov (in an appropriate coordinate system (x, y, z) centered at p) :
Construction of the variety B(V)
68
(i) I(q) = (x~, y, z) This quadruplet is called a curvilinear quadruplet.
(ii) I(q) = (x:, y2, z) This quadruple point is called a square quadruplet .
(iii) I(q) = (x a, xy, y2, z) This quadruplet is said to be elongated.
(iv) 1(q) = (x, v, z) 2 This quadruple point is said to be spherical.
Drawing conventions : One will use the following drawing conventions : - The following symbol will represent a curvilinear quadruple point :
- The following symbol will be used to represent a square quadruplet :
H - An elongated quadruple point will be represented by the following drawing :
(
9
)
- Finally, the following symbol will represent a spherical quadruplet :
6.3 6.3.1
Irreducibility General
facts
and
dimension
on Hilbert
schemes
Here, we recall some generalities on Hilbert schemes :
of :
B(V)
Irreducibility and dimension of Property
B(V)
69
1 : U n i v e r s a l p r o p e r t y of t h e H i l b e r t s c h e m e
Hilbd(V) comes equipped with a d-sheeted tautological covering, denoted by 'Hilbd(V), and defined in the product V • Hilbd(V). w denotes the projection from 'Hilbd(V) C V • Hilbd(V) on Hilbd(V); the projection w (called universal family) is flat by definition. From a set-theoretic point of view, 'Hilbd(V) The Hilbert scheme
contains the couples (p, ~) such that the point p is a subscheme of ~. The Hilbert scheme
Hilbd(V) is solution of the following universal problem :
Let S be a scheme. Let y C S • V be a flat d-sheeted ramified cover of S (via the first projection). Giving such a subscheme y of S • V is equivalent to giving a unique morphism
f : S ~ Hilbd(V). The family y is obtained by the pull-back of IHilbd(V) by f.
the universal family
One recalls the flatness criterion for a finite morphism : Let ~ : X --+ T be a finite morphism of schemes, with T integral. Then ~ is flat over T if and only if the length of the fibers ~-l(t) is a constant d independent of
tET. Hilbd(V) is a d-uple union of ~1 and ~2 where ~1 is a Hilbd(V) is locally isomorphic at ~ to the product Hilb d' (V1) • HiIb d-d1 (V2), where V1 is a neighborhood P r o p e r t y 2 : Recall that if ~ E
dl-uple of support Pl and ~2 is a ( d - d l ) - u p l e disjoint from p~, then
of V at ~1 and V2 is a neighborhood of V at ~2Improperly, we will say that the d-uple ~ is deforming to ~ in
Hilbd(V) and we
will denote this deformation by ~ -+ ~ when c goes to 0, if the family (~)~ec defined in this way corresponds to a sub-family of V • C, flat over C, via the second projection. Said differently, this deformation of base C corresponds to a unique morphism from C to Hilbd(V).
6.3.2 Recall that
I(V) is the incidence subvariety of Hilb2(V) x Hilb4(V) consisting of
the elements (d, q) such that d is a subscheme of q. Also recall that 1-Is denotes the projection from I(V) onto gilb4(y). One introduces some new notation: N o t a t i o n 21 : F r o m n o w on, o n e d e n o t e s b y
Hd(V) t h e H i l b e r t s c h e m e
Hilb'l(V) o f t h e d-uples o f V. N o t a t i o n 22 : 9 For d < 4, we denote by
H~(V) the dense open subset IF] of Hd(v) containing
the d-uples formed by d simple points.
70
Construction of the variety
I#(V) q E HI(V ).
9 One denotes by such that
I(V)
containing the elements (d, q)
H2(V)
• H3(V) containing the (d, t)
the open subset of
9 One denotes by H3(V) the subvariety of
B(V)
such that d is a subscheme of t. This subvariety is non-singular [ELB]. The second projection
H (V) (d,t)
H (V) t
is generically a 3-sheeted covering.
The Hilbert scheme H4(V) has a natural stratification consisting of five strata H4(V), H42(V), H41(V), H411(V) and H4111(V).
denoted by
- The stratum
H4(V) is the
closed subvariety of H4(V) containing quadruple points
(i.e. the support is only one point). - The stratum H42(V) is the locally closed subvariety of H4(V) containing quadruplets which are the union of two double points. - The stratum
H~(V)
is the locally closed subvariety of H4(V) containing quadru-
plets which are the union of a triple point and a simple point. - The stratum H41 I(V) is the locally closed subvariety of H4(V) containing quadruplets which are the union of a double point and two simple points. - The stratum H4111(V) is the open subset containing simple quadruplets, previously denoted by H~(V). This natural stratification of H4(V) will induce through the projection II2 : H 4 (V) a stratification on of
I(V)
I(V), denoted by I.(V). I#(V).
I(V) --+
Note that the open subset II111(V)
has already been denoted by
Other drawing conventions : -
We recall the drawing convention used to represent the double point d of support a
pointpofV
:
- We will represent a n-uple curvilinear point (i.e. a subscheme of a non-singular curve) by the symbol :
- Then, for an amorphous triplet t (cf. definition 2), we will use the following convention :
Irreducibility and dimension of B(V)
71
Our goal now is to prove the following proposition :
P r o p o s i t i o n 3 The incidence subvariety I(V) of H2(V) x H4(V) is irreducible of
dimension 12 = 4. dim(V). Proof : We will show that the open subset I#(V) C [(V) is dense in I(V). According to property 2, it is enough to prove that each element (d, q) of the stratum /4(V) (i.e. when the support of the quadruplet q is one point) is the limit of elements of I#(V). In fact, when the support of the quadruplet contains at least two points, the result is already known, as shown below : a) In an element of the s t r a t u m / 3 I ( V ) : With our drawing conventions, the elements of this stratum are of one of the four following forms : - The quadruplet q is the union of a triple point t and a simple point m, the doublet d is simple :
p
Om
J
k/
Om
Figure 6.1: d = p U m a n d q = t U m - The quadruplet q is the union of a triple point t and a simple point m, the doublet d is contained in the triplet t :
P
S
Om
t ~ d
Om
d Figure6.2: d C t
andq=tUm
From property 2, the Hilbert scheme H4(V) is locally isomorphic at q to the product H3(V) x V. If the doublet d is simple (figure 6.1), the variety I(V) is locally isomorphic to the product ~H3(V) • V which is an irreducible variety of dimension 12. If the doublet d is a double point (figure 6.2), the incidence variety is in this case locally
72
Construction of the variety
B(V)
isomorphic to the product H3(V) x V which is a variety of dimension 12. Moreover, each element of this stratum can be obtained as the limit of elements of the open subset
Ir
b) In an element of the stratum h2(V): The quadruplet q is the union of two double points dl and d2 of support Pl and P2 :
~ P l
d d p 2~
Figure 6.3:
q=dl U d2
Still from property 2, the Hilbert scheme H4(V) is locally isomorphic at q to the product
H2(V) •
H2(V). If d is the union of the two simple points Pl and P2, the
incidence variety is locally isomorphic to the product 'H2(V) x 'H2(V). If d is one
I(V) is then locally isomorphic to the product H2(V) • H2(V). In these two cases, I(V) is a variety of dimension 12 and the elements of this stratum can be obtained again as the limits of elements of the open set Ir of the two doublets dl, d2, the variety
c) In an element of the stratum/211(V): The quadruplet q is the union of a double point dl of support Pl and two simple points P2 and P3 :
d d ~1
9 P2
9 P3
Figure 6.4: q = da U P2 U P3 If d = dl or d = P2 U P3, the incidence variety is in this case locally isomorphic to the product Hz(V) • H2(V) which is a variety of dimension 12. Now, if d is one of the two simple doublets Pt U P2, Pl U P3, the variety
I(V) is then
locally isomorphic to the
product V x V x H2(V) which is of dimension 12. The points of this stratum belong to the closure of
I#(V)
in
I(V).
d) It remains to study the elements of the stratum I4(V) :
Irreducibility and dimension of B(V)
73
Let us denote by p the support of an element (d, q) of I4(V). Remember t h a t 172 is the projection from I(V) onto H4(V). (i) If q is a curvilinear quadruple point, there is only one element (d, q) in the fiber II~-l(q), where d is the only doublet contained in q. In an appropriate local coordinate system (x, y, z) centered at p (cf. definition 7.(i)), the quadruplet q is defined by the ideal I(q) = (x 4, y, z) of Ov and the doublet d has for ideal I(d) = (x 2, y, z). Let us consider the ideal Ie = (x 4 - r
y, z) ; it is the ideal of a quadruplet qe of
H~(V). The ideal Je = ( x2 - r
y, z) defines a doublet de of H~(V). Since the inclusion of
ideals Ie C Je is eqnivalent to the scheme-theoretic inclusion de C qe, it follows t h a t for every r different from zero, the element (de, qe) is in I#(V). In addition,/~ = I(qe) obviously goes to I(q) when r goes to 0. Similarly, Je = I(d~) goes to I(d) when r goes to 0. So, when the quadruplet q is curvilinear, the element (d, q) of I(V) is the limit of elements of Ir
(ii) If q is square, from definition 7.(ii), one can assume it to be defined by the ideal I(q) = (x 2, y2 z) of Oy, where (z, y, z) is an appropriate local coordinate system centered at p. As the coordinates x and y play a symmetric role, one can always assume t h a t a doublet d ~ which constitutes an element (d ~, q) of the fiber II~-l(q) is given by the ideal I(d '~) = (x 2, y - ax, z), where a is a fixed scalar. For a ~ 0, the ideal Ie = (x(x - r
y(y - as), z) defines the quadruplet qe which
is the union of the four following simple points : 0 Pie
0 0
P2~ 0 0
0
r
r
P3e
O/g
0
P4e
0
a
(The notation m
b represents the coordinates in C3 of the point m. The point m c
is then defined by the ideal (x - a, y - b, z - c). ) Obviously, I(qe) goes to I(q) when c goes to 0. The ideal (x(x - e), y - ax, z) defines the doublet de which is the union of the two simple points Pie and P3e- The doublet de deforms in d ~ in H2(V) because I(de) goes to I(d '~) when r goes to 0. We represent these different configurations as follows :
Construction of the variety B(V)
74 Y
Y
IJ J
m--
J
X
*
Pie l
X
P2e
(d~,qe)
(d~
If a = 0, the element (d ~ q) of I4(V) is the limit of elements (de, qe) of I#(V), where qe is the quadruplet union of the following four simple points :
Ple 0 0 0
e P2e 0 0
C 0 P3e C P4e 0 0
The quadruplet qe is defined by the ideal [(q,) = (x(:c - e), Y(Y - r
z) and it goes to I(q) when e goes to 0. If dE is the simple doublet Pie tOP2e, de is defined by the ideal I(d~) = (x(x - ~), y, z) which goes to I(d ~ when r goes to 0. Again, we represent these configurations as follows :
Y P4e L p 3 e L
~
.
X
i
(d~
X
(de,qe)
So, if q is a square quadruplet, each element (d, q) in I4(V) is the limit of elements of I#(V).
(iii) Now, if the quadruplet q is elongated, from definition 7.(iii), one can assume q to be defined by the ideal I(q) = (x 3, xy, y2 z). A doublet d ~ which constitutes an element (d% q) of the fiber II~-l(q) is the same as the one given by an ideal I(d '~) = (x 2, y - ax, z). For a ~ O, one considers the quadruplet qe which is the union of the four following simple points : 0 e c -~ Pie
0 0
P2e 0 0
P3e a~ 0
P4e
0 0
This quadruplet is defined by the ideal : I(qe)
--
( ~ ( x 2 - c2), y, z) n (x - ~, y - ~ c , z)
__
( ~ ( x ~ _ ~2), y ( x _ ~), v ( y _ ~ ) ,
z)
Irreducibility and dimension of
B(V)
75
This quadruplet qe goes to q in H4(V) when s goes to 0. If de is the doublet union of the two simple points Pie and P3~, d~ is defined by
I(d") Y ~d
ideal clearly goes to
[(d,) = (x(x -
r
y - ax, z). This
when s goes to 0. Let us draw below these configurations :
Y d~.
(
),X
"P4s
(d~
P3e P2e~x
lPle (de, qe)
If a = 0, the element (d ~ q) of I4(V) is the limit of elements (de, qe) of
I#(V),
where qe is the quadruplet which is the union of the four following simple points : 0
s
Pl~ 0 0
P2e 0 0
0
-~
P3e 6" P4~ 0
0 0
I(q~) = (x(x 2- ~2),yx, y(y-~), the ideal I(d~) = (x(x - r
This quadruplet is defined by the ideal doublet de = Pl~ U P2e is defined by
z ) The simple Here are the
configurations : Y
Y t Pae
d~
P4e
(d~ So, when the quadruplet q is elongated, every element the closure of
I#(V)
in
P2e
(de,qe) (d, q) of I(V) belongs to
Ir
I(V).
(iv) If the quadruplet
q is spherical, that is to say defined by the ideal m 2 where
mp
is the maximal ideal of Or, the fiber H~-l(q) is isomorphic to P(TpV) (the projective space associated to the vector tangent space of V at p). As there is no preferred direction in such a quadruplet, it is sufficient to prove that one of the elements (d, q) of the fiber II~-l(q) is in the closure of
I#(V)
in order to obtain the result for all the
elements of the fiber. For an appropriate local coordinate system (x, y, z) centered at p, the doublet d is defined by the ideal (x 2, y, z). The element (d, q) is the limit of elements (de, qE) of
I#(V),
where q~ is the simple quadruplet q~ = Pie Up2~ Up3e UP4E and
de is the
simple
76
Construction of the variety
B(V)
doublet de : Pl~ U Pze. The coordinates of the points are :
Ple
0
g
0
0 0
P2e 0 0
P3e r 0
P4e
Again, one represents these configurations as : Y
Y
d
~
d~
9
x
pl~
Z
~2e
X
z
(d, q)
(de, qe)
I(qE) = (x(x-e), xy, xz, y(y-s), yz, z(ze)), which goes to I(q) when e goes to 0. The doublet d~ of ideal [(de) = (x(x-e), y, z) goes to d in H2(V) when e goes to 0. So, when q is a spherical quadruplet, the element (d, q) of I(V) is in I#(V). The quadruplet qe is defined by the ideal
I#(V). As it is the I#(V) = I(V) follows. On the other hand, the open subset I#(V) is irreducible because there is a birational morphism from I(V) to the irreducible product H2(V) • H2(V) : We have therefore shown the inclusion of the s t r a t u m I4(V) in
same for the other strata, the equality
The incidence variety
l(V)
...-+
(d, q)
...
-~
H2(V) • (d, d' : q \ d)
I(V) is therefore irreducible of dimension 12 = 2. dim (H2(V)).
So, we have proved proposition 3. On the other hand, the projection II2 :
[(V) -+ H4(V) (d, q)
~
isgenericallya6: ( 4i) -sheetedc~176 s d e n
q
s
e
i
n
2
H4(V). Then proposition 4 follows trivially :
The closure B(V) of the graph of the residual rational map Res is an irreducible subvariety of I(V) • H2(V) of dimension 12.
Proposition 4
B(V)
Non-singularity of the variety
77
Proof : Remember that U C map
Res is regular.
I(V)
is the open subset of
The graph
I(V)
Fn,~.wof Res restricted
where the rational residual
to U is isomorphic to U, which
is a dense open set of dimension 12, from proposition 3. The closure in I(V) x H2(V) is then irreducible of dimension 12.
6.4
B(V)
of
Fae,w []
N o n - s i n g u l a r i t y of B(V)
Remember that II denotes the projection :
n:
R(v)
~ H4(V)
(d,q,d')
~-~
q
B(V). Also remember that ~ denotes an element of the fiber C B(V), (see w 0.7, definition 4 and notation 2). The study of the nonsingularity of B(V) reduces to the cases where the support of the element ~ C B(V) and ~r its restriction to ~r-l(q)
consists of exactly one point. When the support of the quadruplet q consists of at least two points, the variety
B(V)
is in fact locally isomorphic at ~ to a smooth variety
of dimension 12 = 4. dim (V) : - If the quadruplet q is the union of a triple point t of support p and a simple point m, according to property 2 the variety product H3(V) x V, where
B(V)
H3(V) denotes
is locally isomorphic at ~ = t U m to the
the incidence subvariety of H2(V) x H3(V)
(cf. notation 22), - When the quadruplet q is the union of two double points dl and d2 of distinct supports, again from property 2, the variety to the product 'H2(V) x 'H2(V).
B(V)
is locally isomorphic at ~ = dl O d2
In these two cases, the result is a smooth variety (of dimension 12) since it is locally the product of two smooth varieties. The goal of w 6.4.1 is to prove the non-singularity of
B(V)
at every element ~o,
where qo is a locally complete intersection quadruple point (i.e. qo is curvilinear or square). Remember that the complete intersection k-uplets ~ are smooth points of the Hilbert scheme Hk:(V). It results from w 6.4.2 the non-singularity of
B(V)
[HI-n-prop. 8.21.A(e).
We will then prove in
at the points 4o where qo is a non-locally complete
intersection quadruple point (i.e. qo elongated or spherical) . Remember [I2, F] that the Hilbert scheme
H4(V)
is irreducible and singular at the spherical quadruplets q,
that is to say defined by the ideal Ad~2,, where M v is the ideal of a closed point p of V. (Also remember that we have assumed dim (V) = 3.) In this whole section, the support of the quadruple point qo is denoted by p. We denote by (x, V, z) an appropriate local coordinate system centered at p (cf. definitions 7), i.e. a system in which the quadruplet qo is defined by the ideal :
Construction of the variety B ( V )
78 (i) (x 4, y, z), if qo is curvilinear,
(ii) (x 2, y2, z), if qo is square, (iii) (x 3, xy, y2, z), if qo is elongated, (iv) (x, y, z) 2, if qo is spherical. 6.4.0
Preliminaries
a) In the following, we will divide elements of the algebra C{x,y, z} by ideals of C{x, y, z). In [B2], Brianqon gives a generalization of the division theorem of Weierstrass, according to the method of Hironaka (generalization in the sense that we divide a germ of an analytic function by several others, with respect to several coordinates). Let us now give some details on this division theorem. For each multi-index a -~ (al, a2, a3), the notation (xyz) '~ denotes the monomial x"ly~'2z '~3. Recall some definitions : Definitions 8 : 9 A non zero linear form with positive integer coefficients is called direction L. For each element f in C{x, y, z}, let
f = ~
a , . (xyz) (~
(xE~
9 The set
N ( f ) = {a E 513 l a~ ~ O} is called Newton's diagram of f. 9 The integer
dL(f) = i n f { L ( a ) l a,~ • O} is called L-graduation of f. 9 The element of C[x, y, z] :
inL(f) =
~
a,~. (xyz) '~
L(,~)=dz(f)
is called initial form of f with respect to the direction L.
Such a positive linear form L defines an order on 1~3, denoted by < : (~ ---- (Or1, Or2, 0/3) < fl ~- (ill, f12,/33)
if and only if : -
either L(ax, a2, a3) < L(/3l, /32, /33),
Non-singularity of the variety B(V)
-
79
or L(al, a2, ha) = L(/31,/32,/3a) and there exists an index i0 such that a~o
for each index j > i0, aj =/3j. The linear form L then allows an ordering of the monomials of C{x, y, z}. D e f i n i t i o n 9 : The smallest element of N(f) for this order is called a dominant exposant of f with respect to the direction L, and it is denoted by expL(f). For each ideal I and for each direction L, one can associate the set of the dominant exposants EL(I) of I and the set FL(I) = (a~, ..., %,) which is the minimal finite subset of N3 such that EL(I) = LJl
(see [B1], [B2], [Gall). Let AL(I) =
N3 -- EL(I) ; there exists a unique basis {f,,} of I, called standard basis of I with respect to L indexed by FL(I), such that
f (x, y, z) = ( yz) +
e F (s) ileAL (I)
Each element of C{x, y, z} is equivalent modulo I to a unique element of C{x, y, z} of the form :
ai,(xYz) t~ flCAL(I)
(see [B2], [Gal l, thm 1.2.5 p. 124). In particular, if I is of finite colength n, the family {(xyz)/~}/~CAL(S) is a basis of the quotient C{x, y, z} over C and # A L ( I ) = n. I b) Moreover, we will frequently use the division theorem of Galligo by a family with parameters ([Gall, thin 1.2.7 p. 126) in the following particular case : Let I = ( f l , . . . , fs,) be the ideal in C{x, y, z} of a n-uple point ~ of Ha(Y). The
c{x, y, z}
quotient ( f l , . . . , fs,) is a C-vector space of dimension n. One assumes ( f l , . . . , fp) to be the standard basis of I with respect a direction n ([B2], [Gall). The monomial basis of (cf{Tx ,.y. , z:}: ~ , ) is denoted by {( x y z ) '~ }oe~.(s), where " - ' ' denotes the class of an element of C{x, y, z} modulo the ideal I. Let
ai.,,(xyz) <* aCAL(I)
One denotes by Os the C-algebra C{_a}. For each element f in Os{x, y, z}, one can perform the division of f by the family F 1 , . . . , F v and the remainder h is an element of the form :
E
'~EAL (I)
where h<,(a) e Os.
Construction of the variety B(V)
80
D e f i n i t i o n 10 ([Gall, def. 1.4.5 p. 135) : Let ~ : X -+ S be a morphism of germs of analytic spaces. One denotes by 9)* : Os ~ Ox the corresponding local C-algebra morphism. There exists ([Gall, thm.
135) a unique C-algebra Or, which is the
1.4.4 p.
quotient of Os, equipped with a canonical surjeetion X) : Os --4 Or, satisfying the conditions :
(i) Ox|
is a flat Or-module,
(i/) For each morphism of algebras A* : Os --+ OT such that Ox|
is a fiat OT-module, there exists one and only one morphism #* : Op --+ OT such that A* =if*ok:},. One calls flattener of the morphism ~o, the morphism ~op : X •
--+ P obtained by
the change of basis P ~-+ S. P r o p e r t y 3 : One denotes by (A', 0) the germ of an analytic subspace of (Ca • Cap, 0) defined by Fl(x,y,z,a) . . . . .
Fv(x,y,z,a ) = 0. The natural projection from
V ( ( F 1 , . . . , Fv) ) C (Ca • C"v, 0) onto (S, 0) = (C '*', 0) will be denoted by 13. The Hilbert scheme Hn(V) is locally isomorphic at { to the flattener of II (see [Gr], prop. 0.4 p. 20). One denotes by (P, 0) the basis of this flattener and Or the corresponding local algebra. (Op is a quotient of Os.) One recalls the flatness and division theorem of Galligo ([Gall, thm 1.2.8 p. 126) which will enable us to compute explicitly local flatteners.
One has the equivalent
statements : 1. The module Os{x,y,z} is Os-flat,
(F1,... ,Fp)
2. For each f in Os{x,y,z}, the remainder of the division of f by the ideal
(F1,..., Fv) is zero if and only if f C ( & , . . . ,
&).
R e m a r k 7 : If I = I(4) is a complete intersection ideal, the morphism II introduced previously is fiat. Then one has (P, 0) = (S, 0) - the Hilbert scheme H"(V) is smooth at 4, of dimension 3n. In the following, we will use the result : R e s u l t 1 : According to the previous property, the Hilbert scheme Hn(V) is locally isomorphic at ~ to the flattener of basis P. [7r
If (fl, f2, f3) is the ideal (x 2, y, z) of C{x, y, z} of a double point of H2(V),
let :
Fl(x,y,z,A) = x 2 + a x + b P~(x,y,z,.4) = y - c x - d ~3(~,y,~,A)
A
=
z-e.-
f
= (a,b,c,d,e,f)
Non-singularity of the variety B ( V )
81
One denotes by Os the local algebra C{A}. For each element f in O p { x , 9, z}, the remainder h of the division of f by F1, F2, F3 is of the form : h ( A , x , y , z ) = ho(,A) . 1 + hl(AA) " x
where ho(A) and hi(__A) are in (pp. [TC.es2] If (fz, f2, f3) is the ideal (x 3, y, z) of C{x, y, z} of a curvilinear triple point of V, let : < ( ~ , > ~ , a ) = x 3 q- al x2 -4- a2x q- a3
Y~(~, y, ~,_~) = y-k a4x2 + asx -4- a6 F~(x,>z,a) = z + a r x 2 + a s x + a 9 =
a
(al,...,a9)
Let Os be the local algebra C{_a}. For each element f in O s { x , y , z}, the remainder h of the division of f by F1, F2, Fa is of the forin : h=
ho(a)+<(~)~+h~(~)x
~
,
where h0(_a), hi(a) and h2(_a) are in Os. [T~esa] If (fl, f2, fa, f4) is the ideal (X2, xy, y2 z) of C{x, y, z} of an amorphous triplet of V, let : Fz(z,y,z,a)
=
x2+alx+a2y+a3
F2(z,y,z,a)
=
xy+a4x+asy+a6
Fa(x,y,z,a)
=
y2+azx+aay+a9
F4(x,y,z,a)
=
z+alox+any+a12
=
(al,...,al2)
For each element f in Op{x, y, z}, the remainder h of the division of f by F1, F2, Fa, F4 is of the form :
h = ho(a) + < ( ~ ) x + h~(a)y
,
where ho(_a), hl(_a) and h2(a) are in Op. [74es4] If (]'1, f2, fa) is the ideal (x 4, y, z) of C{x, y, z} of a curvilinear quadruple point of V, let : Fl(X,y,z,a)
-
z4
Y2(x,y,z,a)
-
y+a5x 3+a6x 2+a7x+a8
F3(z,y,z,a)
=
z+a9 xa+alo z2+anx+al2
a
=
(~1,...,a12)
For each element f in is of the form :
Op{z,
y, z } ,
+ al xa + a2x 2 + adz + a4
the remainder h of the division of f by F1, F2, Fa
h= ho(a)+hl(~)x+h2(~)x2+h3(a)x 3
where ho(a),hl(a),h2(~) and ha(G) are in Op.
,
82
Construction of the variety B ( V )
[T~ess] if (fl, f2, f3) is the ideal (x 2, y2 z) of C{x, y, z} of a square quadruplet of V, let : Fl(X,y,z,a) = x 2+alxy+a2x+a3y+a4 F2(x,y,z,a) = y2+abxy+a6x+a7y+as F3(x,y,z,a)
=
z+agxy+alox+ally+a12
=
(al,...,a12)
For each element f in Op, the remainder h of the division of f by F1, F~, F3 is in this case of the form : h : h0( )
,
where h0(__a),hl(_a), h2(_a) and ha(a) are in Op. [T~es6] If (fl, f2, f3, f4) is the ideal (x 3, xy, y2, z) of an elongated quadruplet of V, let : Fl(x,y,z,a_) = xa + a l x 2 +a2x + a 3 y + a 4 F 2 ( x , y , z , a ) = x y + a 5 z2 + a s x + a y y + a s F3(x,y,z,a)
=
y2 + agx2 + alOX + allY + a12
F4(X , y, z,_a)
=
Z -~- al3x 2 -}- a14x -~- alsy + a16
a
=
(al, . . 9,a16)
For each element f in O p { x , y, z}, the remainder h of the division of f by/'1, F2, F3, F4 is of the form : h = h0(a ) -~- h l ( a ) x ~- h 2 ( a ) x 2 -~ h3(a)y
,
where h0(_a), hi(a), h2(a) and h3(a) are in Or. [7~es~] If ( f l , . . . , f6) is the ideal (x 2, xy, xz, y2, yz, z 2) of a spherical quadruplet of V, let : F l ( x , y , z , a ) = x2 + a l x + a 2 y + a 3 z +a4 F,2(x,y,z,a)
=
xy+abx +aGy+aTz +as
F3(x,y,z,a_)
=
xz +agx + a l o y + a l l z +al2
F4(x, y, z, a_) = y2 _~ alax + al4Y + albz + a16 Fb(X,y,z,a) = yz +alTx + a l s y + a l g z +a20 F6(x,y, z,a)
=
z ~ ~- a21x + a22Y + a23z -t- a24
a
=
( a l , . . . ,a24 )
For each element f in O p { x , y, z}, the remainder h of the division of f by F 1 , . . . , F6 is of the form : h = h0(a) +
x(a)x +
+ h3(a)z
,
where h0(_a.), hi(a), h2(a) and h3(a) are in Op. In each case, the module O p { x , y , z }
is Op-tlat by definition.
Consequently, an
element f in O v { z , y, z} belongs to (F~) if and only if its remainder is zero.
Non-singularity of the variety B(V)
83
R e m a r k 8 : The above result is an application of the division theorem of a family with parameters of Galligo ([Gal] 1.2.7 p. 126) in the following cases :
. a (i) (fl, f2, fa) is a standard basis of the ideal I and the quotient Op{x, (zL y,y,z)z} is C-vector space of basis {1, 5}.
(ii) (fl, f2, 5 ) is a standard basis of the ideal I and the quotient
Op{x,y,~} (xa, Y, z)
C-vector space of basis {g, ~-, x2}.
(iii) (fl, f2, f3, f4) is a standard basis of the ideal I and the quotient
is a
Op{x,y,z} (x 2, xy, y2, z)
is a C-vector space of basis {1, 5, ~}.
(iv) (f~, h, fa) is a standard basis of the ideal I and the quotient
Op{x, y, z} (x4, Y, z) is a
C-vector space of basis {1-,~-, x 2, xa}. (v) (fl, f2, f3) is a standard basis of the ideal I and the quotient
o p { ~ , y, z} . (x2 y2,z) is a
C-vector space of basis {]-, 5, ~yy,~}.
Op{~,y,z}
(vi) (fi, f~, fa, f4) is a standard basis of the ideal I and the quotient (x3 xy, y2, z) is a C-vector space of basis {T, 5, x 2, ~}.
(vii) ( f l , . . . , f6) is a standard basis of the ideal I and the quotient
o~{~,y,~} (x, y, z) 2
is
a C-vector space of basis {]-, 5, y, ~}. c) Finally, note that in our study divisions must be performed by ideals in a polynomials ring. There exists a division algorithm for this purpose (see [E] for example), which was implemented in the algebraic geometry package M a c a u l a y . This package was used to perform most of these divisions. R e m a r k 9 : Let Ff be the graph of a morphism f : X ~ Y of varieties.
The
variety r I C X x Y is a subvariety of X x Y, isomorphic to X via the first projection. If (Xx,..., z,,, Y l , . . . , y,~) denotes a local chart of the product C~ x C" at (0, 0), we will say improperly that the parameters Yl,..., Ym are obtained locally as graphs in
terms of the other coordinates Xl,..., x,,, if Yl,...,Ym are elements of the algebra C{Xl,..., x,,}. In this case, let Yl = fi(Xl,..., x,,) ; the subvariety of C' x C~ defined by {yl - f i ( x l , . . . , x,,) = 0}1
Non-singularity plete
intersection
of B(V)
a t ~o w h e r e
quadruple
qo is a l o c a l l y
com-
point
Recall that the rational application residual Res: I ( V ) . . . -4 H2(V) maps (d, q) on to d' = q \ d and that B(V) denotes the closure of its graph. Recall also that the projection from B(V) onto H4(V) is denoted by 7c.
84
Construction of the variety B ( V )
T h e case o f t h e c u r v i l i n e a r q u a d r u p l e p o i n t We intend to prove the following proposition : P r o p o s i t i o n 5 The v a r i e t y B ( V )
is s m o o t h at ~o where qo E H4(V) is a c u r v i l i n e a r
quadruple p o i n t .
Proof : We are going to provide a chart of I ( V ) in the neighborhood of (do, qo) and to express the residual map R e s in these coordinates. The curvilinear quadruplet qo of support p is defined by the ideal I(qo) = (x 4, y, z) of Ov. A quadruplet q close to qo in H4(V) is given by the ideal I(q)
=
(x 4 + aim 3 + a2x 2 + a3x + a4, y + a5 x3 -~- a6 x2 ~ aTx -4- a8, Z -t- a9 x3 + al0 x2 4- a l l x ~- a12)
,
so that a = ( a x , . . . , a12) constitutes a chart of H4(V) at qo. Such a curvilinear quadruple point, i.e. subscheme of a non-singular curve C, can contain only one doublet : it is the doublet with the same support the point p, subscheme of C. Therefore, the doublet do is defined by the ideal (x 2, y, z). A doublet d in a neighborhood of do in H2(V) is given by the ideal I ( d ) = (z 2 + a x + b , - y + c x + d , - z + e x + f )
,
so that A = (a, b, e, d, e, f) constitutes a chart of H2(V) at do. (The minus signs are just for convenience.) The inclusion of schemes d C q in the neighborhood of (do, qo) is equivalent to the inclusion of ideals I ( q ) C I ( d ) of C{a, Jt}{x, y, z}, which can be formulated by equations in C{_a,.A} that we are going to determine. For each of the three generators of the ideal I(q), the division by the ideal I ( d ) is performed. The remainder of the division of the i th generator of I ( q ) is denoted by r~. According to result [TCesl], w 6.4.0, the remainders ri are of the form : r~ ~ T~,0 ~- r i , l z
where rl,0 and ri.~ are elements of C{_a,__A}. From [7r
again, the ideal I ( q ) is contained in the ideal I ( d ) if and only if the three
remainders rm, r2, r3 are identically zero, which can be expressed by the six equations in C[a,A] : a l a 2 - alb - a2a + a 3 - a 3 ~- 2ab = 0 alab - a2b + a4 - a2b Jr b2 = 0 asa 2 - a s b -
a~a + aT + c = 0
asab - a6b + as + d = O a9 a2 - a9b - aloa + a n + e = 0 a9ab - alob + a12 + f = 0
(6.1) (6.2) (6.3) (6.4) (6.5) (6.6)
Non-singularity of the variety B ( V )
85
These six equations form the generators of the ideal of c { a , A } which defines I ( Y ) in Hz(V) x H4(V) locally at (do, qo). From these six equations, one can see that the variety I ( V ) is obtained locally as the graph of a morphism : C 12
C_ = (al, a2, as, a6, a9, alo, a, b, c, d, e, f)
-_~
C6
~
(a3, a4, aT, a8, a n , a12)
So __Cforms local coordinates of I ( V ) at (do, qo).
L e t us see n o w h o w t h e r e s i d u a l m a p Res c a n b e e x p r e s s e d in t h e s e c o o r dinates. As the quadruplet qo contains only one doublet, the residual doublet d'o = qo \ do is defined by the ideal I(d'o) = (x 2, y, z) of O r . A doublet d' close to d'o in H2(V) is given by the ideal I(d') = (x 2 + a'x + b ' , - y + c'x + d ' , - z + e'x + f') Thus, the parameters .4-- = (a', b', c', d', e', f') form a chart of H 2 ( V ) at d'o. The scheme-theoretic inclusion d' C q of the residual doublet is equivalent to the inclusion of ideals I(q) C [(d') and it is expressed by the six equations (6.1)'-(6.6)' of C[a,.A ~] (where a , . . . , f have been replaced by a ' , . . . , f' in the equations (6.1)-(6.6)). From (6.3)' and the expression of a7 (given by (6.3)), it follows that c' = ab(a 2 - a '2) + ab(b' - b) + a6(a' - a) + c
(6.7)
From (6.4)' and the expression of as (given by (6.4)), it follows that d' = ab(ab - a'b') + a6(b' - b) + d
(6.8)
Equation (6.5)' and the expression of a n (given by (6.5)) yield: e' = ao(a 2 - a '2) + ag(b' - b) + alo(a' - a) + e
(6.9)
From (6.6)' and the expression of a12 (given by (6.6)), we get the equality: f' = ag(ab - a'b') + alo(b' - b) + f
(6.10)
Let us now find the ideal of C{_a,A, .A~} which expresses the inclusion of ideals I(d) 9 I(d') C I(q) in the neighborhood of (do, qo, d'o). For each of the nine generators of the product I ( d ) . I(d'), one performs the division by the ideal I(q). If R~ denotes the remainder of the division of the ith generator of I ( d ) . I(d'), one has from [7r : R~ = R~,0 + R~,lx + Ri,2x 2 +RI,3 x3
86
Construction of the variety B ( V )
where Ri,o, Ri,1, Ri,2 and Ri. 3 are elements of C{a, A , A'}. A g a i n from [7~es4], the ideal I ( d ) . I(d') is contained in I(q) if and only if the nine r e m a i n d e r s R~ are identically zero, which yields the 9 x 4 = 36 equations {R,j = O} . . . . . 0<_j_<3
If the r e m a i n d e r of the division of the element (x 2 + ax + b)(x 2 + a'x + b') is d e n o t e d by R1, one has in p a r t i c u l a r the expressions : /~1,3
:
a + a' -- a 1
R1,2
:
b + b'
+
aa'
-
a2
The expression of the coordinate a' of the doublet d' in the chart C_ of I ( V ) is o b t a i n e d by the equation R1,3 = 0 : a' = al - a
(6.11)
F r o m the expression of a', the equation/~1,2 = 0 can be rewritten :
(6.12)
b' = - b + a(a - al) + a2
a' and b' are then replaced by their expressions in terms of a, b, ax, a2 in the equations (6.7), (6.8), (6.9) and (6.10). New equations are o b t a i n e d : c'
=
a5(a 2 + aid -- a~ -- 2b + a2) + a6(al - 2a) + c
d'
=
as(a 3 - 2a2 al + aa 2 + a2 + alb - ala2)
(6.13)
+ a 6 ( - 2 b + a(a - al) + a2) + d
(6.14)
e'
=
ag(a 2 + ala -- a21 -- 2b + a2) + al0(al - 2a) + e
f'
=
a9(a 3 - 2a2al + aa21 + a2 + alb - ala2)
(6.15)
+ a l 0 ( - 2 b + a(a - al) + a2) + f
(6.16)
F r o m these six equations, the coordinates of the residual doublet d' can be expressed in the chart C_ of [ ( V ) .
T h e n we check with M a c a u l a y
t h a t when a ' , . . . , f '
are
replaced by their expressions in terms of (al, a2, a5, a6, a9, al0, a, b, c, d, e, f ) , the equations (6.1)', (6.2)', {R~j = 0},<,<9 are satisfied.
Thus, the variety B ( V )
is locally
0
o b t a i n e d as the g r a p h of a m o r p h i s m : C 12 ...+ C 12 C_ = (al, a2, as, a6, a9, alo, a, b, c, d, e, f ) ~ (a3, a4, aT, as, a l l , a12, a', b', c', d ~, e', f ' )
(6.17) So, the variety B ( V ) is s m o o t h at ~o, since it is locally isomorphic to the variety C xz. This proves p r o p o s i t i o n 5.
Non-singularity of the variety B(V)
87
T h e case o f t h e s q u a r e q u a d r u p l e t Now, we are going to prove the non-singularity of the variety B(V) at ~ where q E
H4(V) is a square quadruplet, which is represented by the following symbol :
II First, we state a lemma which gives the form of the elements ~ in the fiber over such a quadruplet : L e m m a 10 Let qo be a square quadruplet of support the point p of V. The elements
of the fiber lr-l(qo) C B(V) over the quadruplet qo are of the form (d~,qo,d_~)~c c where d~ is the doublet contained in the quadruplet qo of direction a. Proof : The quadruplet qo is defined by the ideal (x 2, y2 z) of Ov (cf. definition 7.(ii)). A doublet in the quadruplet qo is fixed. Up to a change of coordinates from x to y, one can always assume that this doublet is defined by the ideal (x ~, y + ax, z), where a is a scalar. The figure below represents the element (d~, qo) of I(V) : Y m
~
X
The residual doublet is necessarily contained in the open set of the non-vertical doublets. Indeed, the ideal correponding to the vertical doublet is I = (y2 x, z) and the inclusion I(d~) 9I C I(qo) is not satisfied because the element (y + ax)x does not belong to the ideal I(qo) = (x 2, y2, z). Thus, the residual doublet d' is defined by an ideal of the form (x 2, y + fix, z). The inclusion I ( d , ) . I(d') C I(qo) will be verified if and only if the element (y + ax)(y + fix) is in I(qo) = (x 2, y2 z), which is equivalent to the condition a +/3 = 0.
[::]
Let us now prove the following proposition :
P r o p o s i t i o n 6 The variety B(V) is non-singular at each element (d, qo, d') C B(V)
where qo E H4(V) is a square quadruplet. Proof : In order to prove the non-singularity of the variety B(V) at each element (d, qo, d') where qo is a square quadruplet, we will just prove the non-singularity at the most
Construction of the variety B ( V )
88
degenerate element, i.e. when the two doublets d and d' are identical and horizontal (cf. lemma 10). The different elements of the fiber 7r l(qo) C B ( V ) are represented below : Y
Y X d~
d'=d_~
dI ~do
Extreme case Computations performed for the extreme case will yield the non-singularity of the variety B ( V ) at each element (d, qo, d'). As the square quadruplet qo is defined by the ideal (x 2, y2, z), a quadruplet q close to qo in H4(V) is given by the ideal I(q) =
(x 2 -1- a l x y -1- a2z q- day + a4, y2 + a5xy + a6x + aTy + as, z + aoxy + alox + a n y + a12) ,
so that _a = ( a l , . . . , a12) forms a chart of H4(V) at qo. At the most degenerate point go = (do, qo, d'o) of the fiber 7r-l(qo) C B ( V ) , the two doublets are identical and defined by the same ideal (x 2, y, z) (cf. lemma 10). As before, a doublet d in the neighborhood of do in H2(V) is given by the ideal I(d) = (x 2 + ax + b , - y + cx + d , - z
+ ex + f )
The chart of H2(V) at do is denoted by A = (a, b, c,d, e, f).
Similarly, the ideal
(x 2 + a'x + b ' , - y + c'x + d ' , - z + e'x + f') defines a doublet d' close to d'o. And the chart of H2(V) at d'o is denoted by A ' = (a', b', c', d', e', f ' ) . L e t us d e t e r m i n e t h e local e q u a t i o n s o f t h e s u b v a r i e t y I ( V ) o f H 2 ( V ) • H4(V) at (do, qo). We have to determine the ideal of C{_a,A} which expresses the inclusion d C q in the scheme-theoretic sense. To do so, one performs the division of each generator of I(q) by the ideal I(d). If Ii denotes the remainder of the division of the ith generator of I(q), one has from [T/esl] : l~ = l~,o + li,lx where l~,0 and 1~,1are elements of C{a, A}. From ['~e81] again, the inclusion I(q) C I(d) will be satisfied if and only if the remainders l~ are non zero, which can be expressed by the six equations {l~j = 0} i = 1,2,3 in C[a,A]. More explicitly : j =0,1 11,1 =
- a l a c + ald + a2 + a 3 c - a
(6.19)
Non-singularity of the variety B ( V )
89
11,o =
- a l b c + a3d + a4 - b
(6.20)
/2,i
- a h a c + ahd + a 6 -}- ate - ac 2 + 2cd
(6.21)
12,0 =
- a h b c + aTd + as - bc2 + d 2
(6.22)
13a =
-agac+a9d+a!o+anc+e
(6.23)
13,0 =
- a g b c + a n d + hi2 + f
(6.24)
=
Therefore these six equations define the subvariety I ( V ) of H2(V) x H4(V) locally at (do,qo).
These six equations (6.19)-(6.24), taken in this order, enable one to
express a2, a4, a6, a8, hi0 and a12 as functions of the other coordinates.
Thus, C =
(al, a3, ah, aT, ag, a n , a, b, c, d, e, f) forms a chart of I ( Y ) at (do, qo). L e t us n o w d e t e r m i n e t h e r e s i d u a l a p p l i c a t i o n R e s in t h e s e c o o r d i n a t e s . The inclusion of the residual doublet d I in the quadruplet q in the neighborhood of (qo, d~o) is expressed by the six equations {l'i,y = 0}~b~53 in C[a, .4'] where l~,j is obtained from li,j by replacing a , . . . , f by a~,..., f .
In particular, we get from the inclusion
of ideals I(d) 9[(d') C I(q) that the element ( - y + cx + d)(y - cSx - d') is in the ideal I(q). From [~esh], the remainder of the division of this element by I(q) is of the form ro + r l x + r2xy + ray, where r o , r l , r 2 and ra are in C{a, Ar, A}, and the condition ( - y + cz + d)(y - c'x - d') E I(q) can be expressed by the four equations {ri = 0}~=0,...,a. In particular, we get the expressions : r2
=
c + cI + a l c d + a5
r3
=
d + d' + aT + a3cd
The equation r2 = 0, which can be rewritten as c'(1 + ale) = - a s - c, enables one to express the parameter c' locally as a graph because the element (1 + ale) is invertible in the ring C{C_}. Then one obtains the expression : c'= --(ah+C)(l+alc)
-1
(6.25)
The equation r3 = 0 and the expression of c' yield the following expression for the parameter d' in C{C} : d' = - d
-
a 7 q- a 3 c ( a 5 q-
c)(1 q- ale) -1
(6.26)
From the expression of a2 (given by (6.19)) and the equation l~,1 = 0, one gets the equality : a'(1 + a l c ' ) = a(1 + ale) + al(d' -- d) + a3(c' - c) From the expression of a4 (given by (6.20)) and the equation l'1,0 = 0, one gets the equMity : b'(1 + a l c ' ) = b(1 + ale) + a3(d' - d)
Construction of the variety B ( V )
90
As the element 1 + alc' (= 1 - al(as + c)(1 + alc) 1, by (6.25)) is invertible in the ring C{C_}, one rewrites the two following equalities :
a' = [ a ( l + a l c ) + a l ( d ' - d ) + a 3 ( c ' - c ) ] ( l + a l c ' ) b' =
-1
(6.27)
[b(1 + a l c ) + a3(d'- d)](1 + a l c ' ) -1
(6.28)
d and d' are then replaced by their expressions in terms of al, a3, a5, aT, c and d in the equations (6.27) and (6.28), which enables us to express the coordinates a' and b' of the residual doublet in the chart _C_.From the expression of al0 (given by (6.23)) and the equation l~,1 = 0, one gets :
e'=e-a11(c'-c)+ao(d-d')+a9a'c'-a9ac
(6.29)
The expression of o,12 (given by (6.24)) and the equation l'3.o = 0 yield :
f' = f + a9(b'e' - bc) + all(d - d') It was shown above that a',b',c' and d' are elements of the algebra
(6.30)
C{C}.
When
a', b', d and d' are replaced by their expressions in terms of the coordinates C_ in the equations (6.29) and (6.30), one gets the expressions of the last two coordinates e' and f ' of the residual doublet in the chart C C_. The coordinates a',..., f' of the residual doublet have thus been obtained locally as graphs in the variety C12 of coordinates C_. Then one checks that when a ' , . . . , f ' are replaced by their expressions in terms of the coordinates C_in the equations/~.0 = 0 , l~,1 = 0 and in the equations given by the generators of the ideal of C{_a,4 , . 4 ' } which express the inclusion I(d). I(d') C I(q), one gets indeed 0 = 0. Therefore, the variety t?(V) is locally isomorphic at ~o to the graph of a morphism : C 12
(al,a3, a5, a7, a9, a l l , a , b , c , d , e , f )
-.~
C 12
~
(a2,a4, a6,aa,alo, al2, a',b',d,d',e',f')
The variety B ( V ) is smooth at this point and (al, aa, a5, az, a9, a11, a, b, c, d, e, f) forms a chart. R e m a r k 10 One can also choose (al, a3, a9, air, a, b, c, d, e, f, d, d') as chart of B ( V ) at qo. In this case, the variety/?(V) is locally isomorphic to the graph of a morphism
Non-singularity of the variety B ( V ) from
C 12
91
to C12 which is expressed by the following equations :
+alc) - -
a4 =
aidb(1 + a l c ) - a3d
a~ =
- ( c + c' + alcc')
a6 = a7 --
--ace'(1 § ale) + dc'(1 § - ( d + d' + a a c c ' )
a2 =
as = alo
=
a12 = a'(1 + ale' ) = b'(l+alc') =
a(1
a3c
+ c(d' + aacc')
-bcc'(1 + ale) + d(d' + a3cc') -allc-e-a9d+a9ac agbc- all d - f a(1 + a l e ) § al(d' - d) § a3(c' - c) b(l+alc)+a3(d'-d)
(e'-e)(l§
=
(c' - c ) [ - a l l ( l § aid) § aga § a3a9d] § ag(d' - d)
( f ' - f)(1 + a l c ' )
= agb(c'- c) + ( d ' - d)[a3age'- an(1 + ale')]
These charts will be used for the construction of the variety H4(V) in the product [H3(V)] 4 x [B(V)] ~ (cf. w 7.1.2). From lemma 10, the other elements of the fiber 7r ~(qo) C B ( V ) over the square quadruplet qo are of the form (d,~,qo,d_(~)(,cc,
where the doublet d,~ is defined by
the ideal (x 2, y + ax, z). The computations performed in the neighborhood of (]o = (do, %, do) in order to establish a chart of B ( V ) are still valid for (d~, qo, d_,~). It is sufficient to replace c by C - a and d by C' + a in the equations, where C and C' are local parameters. This ends the proof of proposition 6.
Propositions 5 and 6 lead to the non-singularity of the variety B ( V ) at each element where q C H4(V) is a locally complete intersection quadruple point. We now proceed in proving the same result when q is no longer a complete intersection. 6.4.2
Non-singularity complete
of B(V)
intersection
at
qo w h e r e
quadruple
qo is a n o n - l o c a l l y
point
Recall (cf. definitions 7) that the non-locally complete intersection quadruple points are the elongated and spherical quadruplets. Recall also that R(V) denotes the subvariety of H2(V) • H4(V) • H2(V), the elements (d, q, d') of which satisfy the inclusions :
I(d') c r(q) c I @ n I(d') The variety R ( V ) contains the graph ['n,~.~of the rational residual map Res : I ( V ) 9 9--+ H2(V) since elements (d, q, d' = q \ d) satisfy the previous inclusions. Therefore, the variety R ( V ) contains B ( V ) , the closure of the graph FR~,, in [(V) x H2(V).
Construction of the variety B ( V )
92 T h e case o f t h e e l o n g a t e d q u a d r u p l e t
Recall that if qo E H4(V) is the elongated quadruplet of support the point p of V, in an appropriate local coordinate system centered at p, it is defined by the ideal (x 3 , x y , y 2 z) (of. definition 7-(iii)). The following lemma provides a description of the elements (d, qo, d') of B ( V ) : L e m m a 11 One of the two doublets d, d' which constitutes the element (d, qo, d') of
B ( V ) defines the same line as the one defined by the elongated quadruplet qo. Proof : The two doublets d and d' have the same support as the quadruplet qo. Therefore, the question is to prove that one of these two doublets is a subscheme of the curve of local equations y = 0 = z. In this system of local coordinates (x, y, z), the doublets d and d' are necessarily defined by ideals of the form :
I(d)
=
(x 2 , y + ~ x , z )
I(d')
=
(x 2,y +
z)
,
where /k and A~ are elements in C. Then, the scheme-theoretic inclusions d, d' C qo follow. Then one checks that the inclusion [(d) 9[(d') C I(qo) is equivalent to the condition ~ ' x 2 9 I(qo) = (x 3, xy, y2, z) This condition is equivalent to ~,k' = 0.
[]
According to our drawing conventions, the elements (d, %, d') in the fiber 7r-~(qo) C
B ( V ) are of the form :
w
d
~
qo elongated
with as extreme case the element ~o = (do,qo,d~) where the doublets do et d~ are identical and define the same line as the quadruplet :
A
d~. do qo elongated
)
L e t us n o w p r o v e t h e n o n - s i n g u l a r i t y o f B ( V ) at this e l e m e n t e/o. The two doublets do and d~ are defined by the ideal (x2,y,z). As before (see (6.18)),
Non-singularity of the variety B ( V )
93
A = (a, b, c, d, e, f) denotes a chart of H2(V) at do and Jt ~ = (a', b', d, d ~, e', f ' ) a chart of H2(V) at dro. A quadruplet q close to qo in H4(V) is defined by the ideal : I(q) = (
x 3 +alx 2 +a2x +a3y+a4,xy+a5
x2 + a 6 x + a T y + a s ,
y2 + agX2 + alOX + a l l y + a12, z + a l a x 2 + a14x + a15y + a16)
where a4, as, alo and a12 are explicit functions of al, a2, a3, a5, a6, aT, a9 and a n (cf. Appendix B.1), so that _a = (al, a2, a3, as, a6, aT, a9, a n , a13, at4, a15, a16) constitutes a chart of H4(V) at qo. The scheme-theoretic inclusion d C q in the neighborhood of (do, qo) is equivalent to and it is expressed by an ideal of C[_a,~] the inclusion of ideals (.) I ( q ) C I ( d ) which we are going to determine now. For this purpose, one divides each generator of I ( q ) by the ideal I ( d ) = (x 2 + a x + b , - y + c x + d , - z + e x + f ) . From [ n e s l ] , the remainder ri of the division of the ith generator is of the form : ri = ri,o + ri,lX
where r,,0 and r,,1 are elements of C{_a, A}. The inclusion (*) can be expressed by the eight equations {r~,j = 0}~<~<~ in c{a,A}. More explicitly, one has the expressions : j=0.1
(6.31)
rl,1
=
-ala
+ a2 + a3c -~- a 2 - b
rl.o
=-
2a3asa7 - a l a 2 -]- a~ - a3aG + a2a~ + a 3 a n
(6.32)
- a l b + a3d + ab
(6.33)
r2,1 =
-asa
r2,0
=
- a a a ~ - asa~ + aGaT - a3ag - asb + a T d -
r3,1
=
a l a 2 + a~a7 - 2aSa6 + alag - aTa9 + a S a u - aga + a l l c
+ aG + a T c -
ac + d bc
(6.35)
- a c 2 + 2cd ?'3,0
=
(6.34)
2 a3a~ -- d l a 2 a 7 + a2a 5 Jr- 2a5a6a7 + a3asa9 -- alaTa9 + a~a9 -- a s a T a n
--a~ + a2a9 + a 6 a u -- a9b + a n d - bc 2 + d 2
(6.36)
r4,1
=
- a l 3 a + a14 + a15c + e
(6.37)
r4,0
=
- a 1 3 b + a l s d + a16 + f
(6.38)
Similarly, the incidence relation d' C q can be expressed in the neighborhood of (dro, qo) by the eight equations {r[j = 0}1_<~_<4in C{a,~_~4~} , where r [ j is obtained from j=O,1
ri,j by replacing a , . . . ,
f by a ' , . . . , f ' .
which expresses the ideal inclusion
Let us now exhibit the ideal of C{_a,A,~_~} (**) in the neighborhood of
I(d). I(d') C I(q)
(do, qo, d~o), Let :
I(d)
=
(x2+ax+b,-y+cx+d,-z+ex+f)
I(d')
= =
(fl, f2, f3) (x 2 + a ' x + b ' , - y + c ' z + d ' , - z + e ' x + f f )
=
(f~,f~,f~)
94
Construction of the variety B ( V )
One adds to the nine generators of the ideal I(d) 9I(d') combinations of elements of
[(q), so t h a t one obtains nine new generators P 1 , " " , P9 of the form : Pi = P,,o + P i , l Z + Pi,2z 2 + Pi,3y
The inclusion (**) is equivalent to the new inclusion ( P 1 , ' " , Pc) C I(q).
This last
inclusion can be expressed, from [gesG], by the 9 x 4 = 36 equations {P~.j = 0} ,_<~_<~ l<_j_<3
in C[a,__M,~4']. If P1 (resp. -P2, P4 and Ps) denotes the remainder of the division by [(q) of f l f ~ (resp. f ] f l , flY2 and f2f~), one has in p a r t i c u l a r : a~ nt- a 3 a 5 - - a i ( a -~ a') -- ct2 + act' -~- b -~ b'
(6.39)
P2,3 =
-a3as-a~-a3c'+aTa-b
(6.40)
P4,3
=
--a3a5
Ps,2
=
as( e + c') - ao + ec'
(6.42)
P5,3
=
a7(c ~- c') -- a l l - - d - d '
(6.43)
P1,2
z
(6.41)
-- a2 -- a3c ~- a7 a! -- b'
The equations r2.1 = 0, r~, 1 = 0,7"4.1 = 0 and r~. 1 = 0 enable one to express the parameters d, d', e and e' as Nnctions of the other coordinates. The equations P2,a = 0, P4,a = 0 and P5,2 = 0 enable one to express b, b' and a0 in terms of aa, as, aT, a, c, a' and d.
From the equation P1,2 = 0 and the expressions of b and b', one gets the
following expression for a2 :
a2
•
--a3(as + c + c') + al - 2a~ + aa' + (a + a)(aT - al )
(6.44)
The equation P5.3 - 0 and the expressions of d and d' yield the equality : all
=
2a6 ~- aT(c ~- c') - a(a5 -~ c) - a'(a5 -~ c')
(6.45)
From the expressions of b and d and the equation r4, 0 = 0, one gets the following expression for f :
f
=
ala[aT(a-az)-a3(a~+c')]-a16-als[-a6+a(as+c)-a~c]
(6.46)
Similarly, from the expressions of b' and d' and the equation r~, 0 = 0, one gets : f'
:
-
-
+
e)] -
-
+
+
e')
-
(6.47)
Thus, the equations (6.33), (6.33)', (6.37), (6.37)', (6.38), (6.38)', (6.39) (6.43) enable one to express the eleven parameters a2, ag, all, b, d, e, f, Dr, d', e', f ' as functions of the other thirteen coordinates. Let _C_= (at, a3, as, a6, aT, a13, a14~ a15, a16, a, c, a', c/). These eleven parameters are then replaced by their expressions in terms of the coordinates C in the other forty-one generators of the variety R(V) (of course, we use M a c a u l a y ) . Only four equations are satisfied. We are left with thirty-seven generators with a zero linear part. The ideal of C[C] generated by these thirty-seven generators is denoted by J. Therefore the variety/~(V) is locally isomorphic at (do, qo, d'o) to the subvariety V ( J ) of C13, of coordinates C.
Non-singularity of the variety B(V)
95
12 The subvariety V(J) of C13 is reducible. This subvariety possesses two smooth irreducible components, one of dimension 8 and the other of dimension 12.
Lemma
Proof : One shows (still by using M a c a u l a y ) that the ideal J can be rewritten as a product J =
(h).L,where
h=
C', al -- a7 -- a, al -- a7 -- a') of
al+aT-a-a'
C[C_].
and L is the ideal (a3, a 5 + c , a s +
The hyperplane V((h)) and the linear subspace
V(L) of dimension 8 intersect transversally.
[]
Then the variety R(V) is the union of two smooth irreducible components, one of which is of dimension 12. As recalled at the beginning of w 6.4.0, this variety R(V) contains the variety B(V) which is irreducible of dimension 12, from proposition 4. Then one concludes that the variety B(V) is locally isomorphic to the hyperplane
V((h)). Therefore the variety B(V) is non-singular at the element ~o, the most degenerate of the fiber 7r-l(qo) C B(V). In the neighborhood of this point, the variety
B(V) is locally isomorphic to the graph of a morphism : C12 _+ C12
(a1,a3, a5, a6,aT, a13, a,4, a15, a16, a,c,e')
~
(a2, ag, all,b,d,e,f,a',b',d',e',f')
and (al , a3, a5, a6, aT, a13, a14, als, a16, a, c, cI) constitutes a local chart. The non-singularity of B(V) at any element of the fiber follows immediately. Indeed, from lemma 11, one can always assume the doublet d to be defined by the ideal (x 2, y + c~x, z) where c~ is a fixed non zero scalar and the doublet d' to be defined by the ideal (x 2, y, z) : Y
q elongated A doublet close to d is then defined by the ideal (x 2 + ax + b, - y + (C - c~)x + d, - z +
ex + f).
c is replaced by (C - a) everywhere in the computations, where C is a
local parameter. In this case, the variety It(V) is smooth, irreducible of dimension 12 because the subvariety V(L) of Cla is reduced to the empty set - The ideal L of C{al, a3, a5, a6, at, a13, a14, a15, a16, a, C, a', c'} contains the element (a 5 + C - a) which is invertible. From proposition 4, the inclusion B(V) C R(V) is in fact an equality. This ends the proof of the following proposition : Proposition
7 The variety B(V) is non-singular at every element (d, q, d') E B(V)
where q C H4(V) is an elongated quadruplet.
Construction of the variety B ( V )
96
To complete the proof of the non-singularity of the variety B ( V ) , it remains to study the case of the spherical quadruplet. T h e case o f t h e s p h e r i c a l q u a d r u p l e t If qo E H4(V) is the spherical quadruplet of support the point p of V, it is defined by the ideal (x,y, z) z (cf. definition 7.(iv)). In this case, the doublets d and d ~ which constitute the element (d, qo, d') of B ( V ) have the same support the point p and any directions. The extreme configuration is obtained when the doublets are identical. With our drawing conventions, the elements ~]o are represented as :
d' -
~
d=d'
As usual, we prove first the non-singularity of B ( V ) in the extreme case, i.e. when the two doublets do and d'o have the same direction. One can always assume that these doublets do and d'o are defined by the ideal (x 2, y, z). As usual, ~ = (a, b, c, d, e, f) will denote a local chart of H2(V) at do and ,4' = (a', b', c', d', e', f ' ) a local chart of H2(V) at d'o. A quadruplet q close to qo in H4(V) is defined by the ideal : I(q) = (
x 2 + alz + a2y + aaz + a4, xy + a5z + a6y + aTz + as, xz + agx + aloy + allz + a12, y2 + al3x + a14Y -t- a15z + a16, yz + alTx + alSy + a19z + a20,
z2
-t- a21x + a2~y + a23z
+
a24)
where a4, a8, a12, a16, a20 and a24 are explicit functions of the other eighteen parameters. These eighteen parameters must also satisfy fifteen linearly independent quadratic equations {Hi = 0}1_
again, the remainder rz of the division of the ith
generator is of the form : ~'i = Ti,0 -~ ~i,1 $
where rl,o and ri,1 are elements of
C{a, __A} (H~)I<_~"
The previous inclusion of ideals is
then equivalent to the inclusion (r~,..., rG) C I(d). However, the ideal I(d) can not
Non-singularity of the variety B ( V )
97
contain such non zero affine linear forms r i = ri, 0 -b ri,lX. The previous inclusion is therefore expressed by the 6 x 2 = 12 equations {ri,j = 0}~<,<~. In particular, one has j-0,1 the expressions :
rl,1
=
al + a2c + a3e - a
(6.48)
r2,1
=
as + aGc + ate - ac + d
(6.49)
r3,1 =
a9 + alOC + a l l e - ae + f
(6.50)
rs,1 =
azz+alsc+az9e-ace+de+cf
(6.51)
Similarly, the scheme-theoretic inclusion d' C q in the neighborhood of (d'o, qo) is expressed by the twelve equations {r~,j = 0}1<_,_<~where r~,j is obtained from ri,j by j=0,1
replacing a , . . . , f by a ' , . . . , f ' . Let us now determine the equations in C[a, A, al'] which express the inclusion of ideals I ( d ) . I(d') C I(q) in the neighborhood of (do, qo, d'o). One adds to each of the nine generators of the product I(d) 9I(d') elements of I(q), in order to obtain nine new generators R1, . . . , R9 of the form : Ri ~- Ri,o -~- ~i,1 x Ay Ri,2y -t- ~i,3 z
where R i j are elements in '
c[_~,x, x'] --
,
The inclusion of ideals I ( d ) . I(d') C I(q)
(H~)~<~
is equivalent to the inclusion ( R 1 , . . . , R9) C I(q). From [7~esT], the elements Ri are necessarily zero in order to be able to satisfy the previous inclusion, which is expressed by the 9 • 4 = 36 equations {R~,2 = 0} ~<~<~. The remainder of the division by the 0<j<_3
ideal I(q) of the element ( - y + cx + d)(x 2 + a'x + b') (resp. (x ~ + ax + b ) ( - y + d x + d'),(-y+cx
+d)(-y+c'x
and ( - z + ex + f ) ( - z
+d'), ( - z + e x + f ) ( - y + c l x
+d), (-y+cx
+d)(-z
+e'x + f')
+ e'x + f ' ) ) is denoted by R2 (resp. R4, Rs, R6, R8 and R9). In
particular, one gets the expressions : R2,2 z
ala2c - a2a5 ~- a2a6c - a~ -I- a3aloc - aTalO - a2ca ~
(6.52)
- a 2 d + aGa' - b'
R4,2 z
ala2 c~ -- a2a5 -I- a2a6c I -- a 2 -t- a3alo cl -- a7alo -- a2cla - a 2 d ' + a6a - b
(6.53)
R5,1
=
- a l c c ' + a5(c + c') - a13 + cd' + de'
(6.54)
R~,2
:
-a2cc'
(6.55)
R5,3
=
--a3cc' -1- aT(c -1- e') -- a15
(6.56)
R6,2
=
--a2ec I + age + axoC' -- a18 -- f
(6.57)
R8,2
=
--a2ce' + age' + alo c -
diS -- f '
(6.58)
R8,3
=
--a3ce' + ate' + a11c-- a19 -- d
(6.59)
+ aG(c + c') -- a14 -- d - d'
C o n s t r u c t i o n of the variety B ( V )
98
R9,I
=
--alee' Jr- a9(e Jr- e') -- a21 -~- eft -~ f e '
(6.60)
/~9,2
=
- - a 2 e e ' + al0(e A- e') -- a22
(6.61)
- - a 3 e e ' + a u ( e + e') -- a23 -- f -- f '
(6.62)
F~9,3 =
C l a i m : All these e q u a t i o n s enable us to express the seventeen p a r a m e t e r s a, a', b, b', d,
d', f , if, a13, a14, a15, alT, a18, a19, a21, a22 and a23 in t e r m s of the t h i r t e e n o t h e r coordinates al~ a2: a3~ a5, a6~ aT~ a9, alo: all~ c, e~ c1~ e'. Let us prove this claim. Let C = (al, a2, a3, as, aG, a7, ag, alo, a n , c, e, c', e'). T h e equations rl,1 = 0 and r~, 1 = 0 enable one to express a and a' in t e r m s of the c o o r d i n a t e s C_C_.T h e e q u a t i o n r2,1 = 0 and the expression of a yield the expression for d :
d
=
- a s + c(al + a2e + a3e - a6) - ate
(6.63)
Similarly, one o b t a i n s the expression for d' :
d'
=
- a s + c'(al + a2c' + a3e' - ae) - ate I
(6.64)
F r o m the e q u a t i o n r 3 j = 0 and the expression of a, one gets the equality :
f
=
- a 9 + e(al + a2c + a 3 e -
a n ) - aloc
(6.65)
T h e expression for f ' :
f'
=
--a 9 -Jr- e'(al -]- a2c"-~- a3 e! -- a11) -- aloC"
(6.66)
is o b t a i n e d s i m i l a r l y f r o m the e q u a t i o n r~, 1 = 0 and the expression of a'. T h e e q u a t i o n s R5,3 = 0 and R9. 2 = 0 enable one to o b t a i n a15 and a22 in t e r m s of the c o o r d i n a t e s C. W h e n d and d' are replaced by their expressions in the e q u a t i o n
R5.2 = 0, one gets the expression for a14 : a14
=
2a5 + (c + e')(2a6 -- al) -- a2(e 2 + e '2 + ec') -- a3(ec + e'c')
+aT(e + e')
(6.67)
F r o m the e q u a t i o n s Rs,1 = 0 , (6.63) and (6.64), one gets : a13
=
cc'[al -4- a2(c -4- c') d- a3(e d- e') -- 2a6] - aT(ec' + e'c)
(6.68)
T h e e q u a t i o n s Rs,3 = 0 and (6.63) lead to the expression for a19 :
a19
=
a5 - c[al -~- a2c + a3(e + e') - a6 - an] + a~(e + e')
(6.69)
T h e e q u a t i o n s ~/:~6,2 = 0 and (6.65) yield the expression for a18 : als
----- a9 -- e[al + a2(e + c') + a3e -- a6 -- an] + ai0(c + c')
(6.70)
Non-singularity of the variety B ( V )
99
F r o m the equations r5,1 = 0, (6.63), (6.65) and from the expressions of a18 and of a19, one o b t a i n s the equality : alz
=
ec[al - a6 - alx + a2(c + c') + aa(e + e')] - alocc' - a~ee'
(6.71)
F r o m the equations R2.2 = 0, (6.63) and the expression of a', one obtains : b'
=
a6[al + a 2 ( 2 c + c') + a3e' - a6] - a2c[al + a2(c + c') + aa(e + e')]
(6.72)
+a2aze + aaalOC - aTalo
Similarly, the equations R4,2 = 0, (6.64) and the expression for a yield the expression b =
a6[al + a2(c + 2c') + a a e - a61 - a2c'[al + a 2 ( c + c') + a3(e + e')]
(6.73)
+a2aTe' + a3aloc' - aTalo
F r o m the equation Rg,1 + era,2 + e'R6,2 = 0 and the expression for als, one gets : a21
=
--al0(ec'
-I- e'c) -~ ee'(a 6 + aae - a,1)
+e2[ai + a2(c + c') + a3e - a6 - all]
(6.74)
Finally, the equation Rg. 3 - Rs. 2 - R6.2 = 0 and the expression for als yield the equality : a23
=
2a9 -~ a6(e - e') ~- a l l ( 3 e -~- e') -~- al0(C -t- c') +a2(e'c - 2ec - ec') - 2ale - 2aae 2 - aaee'
(6.75)
The equations r1.1 = r[.1 = R5.3 = R9.2 = 0, and (6.63)-(6.75) then prove the claim. Replacing these seventeen p a r a m e t e r s by their expressions in terms of the coordinates C_ in the r e m a i n i n g equations leads to an ideal J in the ring C[C__]. Therefore, the variety R ( V ) is locally isomorphic at (do, qo, d'o) to the subvariety V ( J ) of C ~a. This variety R ( V ) is locally reducible. The ideal J can indeed be rewritten as a p r o d u c t J = (h) 9L where h = al - a6 - a l l
+ a2(c + c') + aa(e + e') and J is the ideal
(c - c', e - e', aT - a3c, al0 - a2e, a6 - a l l
- a2c + aae) of C[C_] and the two subvarieties
V ( ( h ) ) and V ( L ) are obviously transverse. The variety R ( V ) is the union of two irre-
ducible components, one of which is s m o o t h of dimension 12 because it is isomorphic to V ( ( h ) ) . The other one is s m o o t h of dimension 8 because it is isomorphic to V ( L ) . As the subvariety B ( V ) of R ( V ) is irreducible of dimension 12 (according to proposition 4), B ( V ) is isomorphic to the s m o o t h hypersurface V ( ( h ) ) .
The variety B ( V )
is therefore s m o o t h at 4~ The equation h = 0 enables us to o b t a i n the c o o r d i n a t e al in t e r m s of the other twelve coordinates. We replace al by its expression in the equations (6.48), (6.48)', (6.63)-(6.75). The variety B ( V ) is then locally isomorphic to the g r a p h of a m o r p h i s m : C 12 __~ C ~s (a2,aa, a5, a6, a7, a9,alo, a11,c,e,e~,e I)
~
(a1,a13, a14, a15, a17, a18, a19, a21,a22, a23, a, b, d, f , a I, b', d', f ' )
100
Construction of the variety
and (a2, aa, as, a6, aT, ag, al0, axl,
B(V)
c, e, c', e') constitutes a local chart.
If now the two doublets d and d' contained in the spherical quadruplet qo do not define the same direction anymore, one can always assume the doublet d to be defined by the ideal (x 2, y + a x , z) where a E C* and the doublet
d' is defined by the ideal
(x 2, y, z). The following figure represents this configuration : Y d : x dI Z
(x2+ax+b, -y+(C-a)x+d, -z+ex+f). In this case, (a, b, C, d, e, f ) constitutes a local chart of H2(V) at d. One replaces c by (C - a) everywhere in the previous computations. The variety R(V) is then locally irreducible, smooth of dimension 12 because the subvariety V(L) of Cxa is in this case reduced to the empty set - The ideal L of C{al, a2, aa, a5, a6, at, do, al0, a l l , C, e, d, e'} contains the invertible element (C - a - c'). The inclusion B(V) C R(V) is in fact an equality, by proposition 4. This proves the non-singularity of B(V) at each element A doublet close to d is then given by the ideal
(d, %, dr). One has proved the following proposition : 8 The variety B(V) is non-singular at every element (d, q, d') E B(V) where q E Hilb4(V) is a spherical quadruplet.
Proposition
The proof of theorem 6 is just a consequence of propositions 3, 4, 5 to 8. Let us just
Res(d, q), or Res(d', q) is well defined, it would have been enough to simply compute the dimension of the tangent space of I(V) at (d, q) or at (d', q), in order to prove the non-singularity of B(V). But we were also note t h a t in the case where either
interested in getting local equations which would be useful for the construction of this variety H4(V) of quadruplets of V. Beforehand we are going to give a description of the elements of the excess component of the variety
R(V).
Description of the elements of the excess component The subvariety
of the variety
R(V)
R(V) of H2(V) • H4(V) • H2(V) is the union of two irreducible
components, only one of which dominates H4(V) by the projection II. This dominant
B(V). Let M(V) be the excess component of R(V). The elements of this variety M(V) are elements ~ = (d, q, d ~) in R(V) such
irreducible component is the variety that :
- The quadruplet q is a non complete intersection, - T h e doublets d and
d' are identical and have the same support that the quadruplet
q. W i t h our drawing conventions, one represents the elements of the variety
M(V) as :
B(V)
Non-singularity of the variety
101
T~
with the extreme configurations :
A
~
~d=
q elongated The variety
M(V)
q spherical
is irreducible, smooth of dimension 8. When the support of the
quadruplet q is only one point, the variety subv~riety
dI
V(L) of d
M(V)
is locally isomorphic at ~ to the
3 (cf. w 6.4.2, p. 95 and 100).
To summarize, we proved in chapter 6 that the closure of the graph of the residual rational map
Res defined
by :
ne~:
I(v) ...-+ (d,q) ...-+
HUv) d'=q \ d
is an irreducible, non-singular variety of dimension 4. dim V. In the next chapter, we will use this auxiliary variety to construct the variety H4(V) as a subvariety of the smooth and irreducible product [H3(V)] 4 • [B(V)] 6. Recall that the variety H3(V) denotes the variety of complete triplets of V ([LB1]).
Chapter 7 A
Construction of the variety H4(V) A
Let us now give an explicit construction of the variety H4(V) of ordered quadruples of V, as a subvariety of the product [Ha(V)] 4 x [B(V)] G where Ha(V) denotes the variety of complete triples of V ([LB1]). Let us briefly recall (cf. introduction 0.8) how the element 0 of H4(V) can be constructed from a generic quadruplet q. Let pX,p2,p3 and p4 be the four simple points which constitute this quadruplet. -If tl denotes the triplet contained in the quadruplet q, disjoint from the simple point pi, the complete triple in H3(---'~) corresponding to the point (pX,... ,pi,... ,p4) in V a is denoted by [i. -The doublet, which is the union of the two simple points pi and pa, is denoted by d~j, the residual doublet of d~j in q by d(. and the element (dij, q, d'~j) of B(V) by q-~j. The element ~ C H4(V) constructed from the point (plp2,pa,p4) of V 4 consists of the data (/1,/2,ta,t4, (c]ij)l_
p4
pl
~(2
d13~ p 3_'G.
Thus, an element q of H4(~-V) is an element ([1, [2, [3, [4, q12, q13, (]14,q23, q24, q34) in the
103 p r o d u c t [Ha(V)] 4 • [B(V)] 6 where (P2, P3, P4, d23, d34, d24, tl)
~ij
=
(P1, P3,/94, D13,034,/)14, t2)
:
(7)1,7)2, 7)4, ~ 1 2 , ~)24, ~)14, t3)
:
(Pl, P2, P3, d12, d23, d13, t4) (5#, q~j, 5~j) for 1 _< i < j < 4
=
satisfy the conditions : 1. T h e q u a d r u p l e t s
qij
are all equal to a same q u a d r u p l e t q,
5~j =
2. One has the e q u a l i t i e s :
5~:i
{i,j,k,l}
for
= {1,2,3,4}
3. T h e simple points satisfy the equalities :
(3.i) (3.ii) (3.iii) (3.iv)
p l = P1 = P l
P2 =
7)2 =
P2
P3 =
P3 -- P3
7)4 = P4 = P 4 4. T h e doublets satisfy the equalities :
d12
=
~[~12 :
(~12
d23 = d23 = 523 d13 = D13
=
513
D24 = d24 = 524 "~14 = D14 = (~14 D34 = d34 = (~34 5. T h e four triplets t~ are subschemes of the q u a d r u p l e t q. The points satisfy the conditions : Remark
H4(V).
pj = Res(tj, q)
11 : Let
{pl,p2p3p4}
for j = 1, 2, 3 and P4 =
Res(t4, q).
denote the s u p p o r t of a generic q u a d r u p l e t q of
C o n d i t i o n 3 can be rewritten as : pl
pl
(3.1)
P2 = 7)2 = P2 = p2
= P1 = P l
(3.2)
P3 = P3 = P3 = p3
(3.3)
7)4 :
(3.4)
P4 = P4 = p4
A
D e f i n i t i o n 11 : L e t elements
HI(V )
-
A
be the open subset of H4(V),
~ having a four-point support.
containing
the
A
104
Construction of the variety H4(V) A
The study of the variety H4(V) can be restricted to the case where the support of the quadruplet reduces to one point, since (see property 2, page 69) this variety is already known when the support of the quadruplet contains more than one point : when the support of the quadruplet q consists of two points, the variety H4(V) is locally isomorphic to one of the products V x H a ( v ) or 'H2(V) x 'H2(V) : -If q is the union of a triplet t located at the point p and a simple point m, the data of an element
E H4(V) is equivalent to the data of an element (m, ~) in the
product V x Ha(V). -If q is the union of two doublets dl and d2 of respective support px and P2, the data of an element ~ C H4(V) is in this case equivalent to the data of an element (J1, d2) in the product 'H2(V) x 'H2(V). In both cases, one deals with a locally smooth (of dimension 12) variety, since it is locally the product of two smooth varieties. Moreover, such elements ~ E H4(V) are limits of elements of H~(V). In section 7.1, it will be shown that the variety H4(V) is smooth at each element where q is a locally complete intersection quadruple point and that in the neighborhood of such elements, the open subset H~(V) is dense in H4(V). The next section will be devoted to the study of the neighborhood of the elements ~ where q is a non locally complete intersection quadruplet. A
7.1
Non-singularity of H4(V) at 0 where q is a locally complete intersection quadruplet
7.1.1
Case
of the
curvilinear
quadruplet
Let us see what the elements which can be constructed in H4(V) from an arbitrary curvilinear quadruple point are : Let qo be a curvilinear quadruplet located at the point p. This quadruplet qo, subscheme of a non-singular curve C, can only contain one triplet : it is the (curvilinear) triplet t of support p, subscheme of C. This quadruplet qo also contains only one doublet. We are going to show that in the neighborhood of such an element 0o, the variety H4(V) is non-singular of dimension 4 9 dim V and that the open subset H~(V) is dense in H4(V).The doublets and the triplets contained in the quadruplet qo are represented in the following figure :
A Non-singularity of H4(V) at c) where q is a 1.c.i. quadruplet.
J
105
tl = t2 = t3 = t4 curvilinear
dis equal, 1 < i < j _ < 4 Let (x,y, z) be a suitable system of local coordinates, centered at p, in which the ideal of qo is I(qo) = (x 4,y, z) (see definition 7.(i)). The triplet t contained in this quadruplet is defined by the ideal (x 3, y, z). The ideal (x 2, y, z) defines the doublet d contained in %. R e m a r k 12 Since y and z play exactly the same role, we restrict ourselves to the case where dim V = 2. In this case, let (x, y) denote a local coordinate system centered at p.
A quadruplet q~J in the neighborhood of qo is defined by the ideal f(q~j)
=
(Z4 _~ a[~]x3 _~ a{~]x 2 .4_ a[3]x ij ij "" ij ij + a[4], Y ~A_ a ij[5]x 3 + ai~]x2 + a[r]x + Sis])
The doublets 51j ~ and 5ij ~ which constitute the element qoZj = (Sij , qo, 5~j ~ of B ( V ) are defined by the ideal (x 2, y). The doublets 5ij and 5~j, in the neighborhood of 5ij ~ and 5ij ~ are given by : I(Sij) 6!
I(ij)
=
(x 2 + aijx + b#,
=
I
( x2 + aij
X
!
+ b#,
- y + cijx + dij) I - y + cijx + dijI )
ij a[5], ij a[6], q a#, b#, clj, diA constitute a From (6.17) , the coordinates (C#) = (ai{], a[2], local chart of B ( V ) at %#. Using again the notation introduced in [LB1], one defines a triplet ti in the neighborhood of t by the ideal : Z(t,) = (x 3 + ~ix 2 + ~ix + 7i, Y + Xix 2 + ,iY + a ) Let (__T/) (si, t~, sl, s" ci, Ai) denote the chart of H3(V) at [i - (Pj , Pk , Pz , djk , dkl , with {i, j, k, l} = {1, 2, 3,4} (see [LB1]). The variety H a ( V ) i s locally isomordjz~ phic at [~ to the graph of a morphism from Ca to Cis (see its expression p.130). ~-
t
__
o
o
o
o
o
L e t us e x p r e s s H 4 ( V ) in t h e p r o d u c t [H3(V)] 4 x [B(V)] s or m o r e p r e c i s e l y in t h e C72 o f c o o r d i n a t e s ('T'l, 7-2, ~---,3,T 4 , ~--12, ~--13, ~--14, C--23, ~--24, ~---34). We are going to show that at qo the variety H4(V) is isomorphic to the graph of a morphism 11 sl, . c4, a[5 12], a[6]). 12~ from Ca to Ca4 where Ca has coordinates (C__)= (s4, t4, s~, s4, 9 First, let us show that the sixty four coordinates can be expressed in terms of the other eight.
A
106
Construction of the variety H4(V)
Condition (3-i), which defines a complete quadruplet, gives the equalities of the coordinates of the points : 84
S2
83
t4
t2
t3
Condition (3-ii) can be rewritten as : +
84 -}- S~t
s3
s~
s]
t4 4- t;
t 3 + t~
tI
And the expression of t~ in the chart (~E4) is t~ = c4f 4. Similarly, conditions (3-iii) and (3-iv) can be rewritten as : S 4 -~ S 4. -~- .S 4
.
.
t4 -b tl4 + t~
S2 + S 2
S l + s~
t2 + tI2
tl + tll
s3 + s'a + s "3
s2 + s2' q- S~
I! '31 ~- $11 -~- 81
t 3 +t~ +t~
t 2 +t~ +t~
tl + t ' 1 + t i'
' s a" and Thus, conditions (3-i)-(3-iv) enable us to express s2, s3, t2, t3, s~, Sl, tl, s~, Sz, s 2" in terms of coordinates _C. So, one has the expressions S '3
~
8 '4
(7.1)
S '1
=
84"
(7.2)
S 2'
=
S 4' Jr- 8 4"
(7.3)
$ .2 . =. .
81
(7.4)
83"
=
S "1 --}- 84"
(7.5)
tl
=
t4 ~- C484'
(7.6)
From remark 11, let x[i] be the abscissa of the point pi, contained in the support of the quadruplet q ; one has the change of variables
x[1]
=
s4
I:[2]
=
84 -t- 8~
x[3]
=
s4+s~+s4
x[4]
=
I! It s 4 -]- 8~ -~- s 4 -]- s 1
I!
W e will also d e n o t e b y (C) t h e set o f c o o r d i n a t e s (x[1], x[2], x[3], x[4], ~4, * c4,~[s1,~1~ 12), in o r d e r t o s i m p l i f y s o m e o f t h e e x p r e s s i o n s . a[61 Condition 1 enables us to obtain in particular the following equalities for 1 < i < j < 4 ij
=
a~]
ij
=
a~
a[q a[2] alt]
=
~j a[6] =
12 a[51 12 a[6]
(7.7)
(7.8)
A
Non-singularity of H4(V) at 0 where q is a l.c.i, quadruplet.
107
In particular, let a[bl = a1215]and a[61 = a~6~. Conditions 4 lead to the equalities : a0
=
-x[i]-x[j]
b#
=
x[i]z[j]
(7.9) (7.10)
For {i, j, k,l} = {1, 2, 3,4}, the equalities a~j = akl and b~j = bkl are consequences of condition 2. Moreover, in the chart (~ij), the expression of a(.,j is a'ij = allij ] - aij (equation (6.11), page 86). Using the previous equalities, this equation becomes : ij
- x [ k ] - x[l] = aE1 + (x[i] +
This proves that all the allij ] are equal to : a[1] =
-(x[1] + x[2] + x[3] + x[4])
(7.11)
Similarly, the expression of b~j in the chart (C_~j) (equation (6.12)) leads to the equality : a[2 ]ij =
x[1]x[2] + x[1]x[3] + x[1]x[4] + x[2]x[3] + x[2]x[4] + x[3]x[4]
(7.12)
Let a[2] = a~2~. Moreover, from condition 5, which can be expressed by the inclusions of ideals
:(pi). :(td c :(q) c I(p i) n I(t,)
,
one gets in particular the condition : (*)
(x - x[i])(Y + ~ : 2 + .iY + "4)
9
:(q)
One adds to this element the element a[5](x 4 + a[1]x 3 + a[2]x 2 + a[a]x + a[4]) + (x[i] x ) ( y + a[5]x 3 + a[6]x 2 + a[7]x + a[8]) of I(q), in order to obtain the new element Ri : Ri
=
( - x [ i ] u / + a[4]a[5] + a[s]x[i]) + (ui + ,ix[i] - a[s] + a[3]a[5] + a[7]x[i])x +(#i - x[i])~i - a[7] + a[z]a[5] + a[6]x[i])x 2 + (,~i - a[6] -t- a[1]a[5] + a[s]x[i])x 3
Thus condition (*) is equivalent to the condition Ri 9 I(q). From result [nes4], w 6.4.0, this condition is equivalent to the condition Ri = 0, which implies in particular that the coefficient of x a is zero. This yields four equations : hi = a[6] -- a[5](a[1] + x[i]) which can be rewritten using the expression of a[1] (given by (7.11)) as :
(ed
for
{i,j,k,0={1,2,3,4}
From condition 4, one gets in particular the equalities of the doublets 512
=
~12
=
d12, 523 = d23 = dz3, 513 = dla. These equalities give in particular the equations :
c3
=
c4
cl
=
c4--}-c/4-]-ca~
c2
=
c4-1-d4
d12 =
d4
d23 =
d4 + d~ + d~
d13 =
d4 + d~
A
108
Construction of the variety H4(V)
W a r n i n g : In these equations, dij denotes the sixth coordinate of the doublet 5ij. From the expressions in the chart ( ~ ) of the right hand sides of these equalities and from equation (e4), one gets c12, e3, d12, c23, cl, d23, cl3, c2 and d13 in terms of (C_). The equalities of the doublets 514 = D14, 524 = d24 and 534 = d34 give in particular the equations :
dz4=d2+d~
d23=dz+d~
d34=dl+d~+d'{
(Here too, the symbol d~j denotes the sixth coordinate of the doublet ~j.) Similarly, one replaces the right hand sides of these equalities by their expressions in the chart (_C). Thus, the sixty four coordinates have been obtained as expressions of the other eight coordinates s4, t4, s !4, s /4,I s /1/ , c4, a(bb hi6]. * Then one checks (using M a c a u l a y again) that replacing these sixty four expressions in the remaining equations leads to 0 = 0. In other words, the sixty four conditions which are necessary to define locally H4(V) in the product [H3(V)] 4 • [B(V)] 6 are sufficient. These sixty four expressions enable one to exhibit locally the variety H4(V) as the graph of a morphism from Cs to C~4 where C"~ has coordinates (C). Then the variety H4(V) is non-singular of dimension 4 9dimV in the neighborhood of c]o and (_C_) = (s4, t4, s~, s~, s~~, c4, a[5], a[6]) constitutes a local chart. Moreover, the expression of this chart proves that in the neighborhood of 4~ the open subset H ~ ( V ) is dense in H4(V). 7.1.2
Case
of the
square
quadruplet
In the next section, we prove the non-singularity of this variety at each element where q is a square quadruplet.
We also prove that such elements ~ are limits of
elements of the open subset H I ( V ) . Starting with such a quadruplet q, let us see what the various elements which can be contructed in H4(V) are. Such a quadruplet q can only contain one triplet, which
is amorphous.
From lemma 10, only three of these six doublets contained in the
quadruplet can be arbitrary. Recall that if d~, is a doublet of direction a contained in q, the complementary doublet of d, in q is the doublet of direction - a . Of course, A
in order to prove the non-singularity of H4(V) at such elements, we will restrict the computations to the most degenerate point q, i.e. when the six doublets are identical. Let qo be a square quadruplet located at a point p of V. Consider a convenient local coordinate system (x, y, z) centered at p in which the quadruplet qo is defined by the ideal (x 2, y2, z) of O y (cf. definition 7.(ii)) and the six doublets are defined by the same ideal (x 2, y, z). The only triplet t contained in qo is then defined by the ideal (x 2, xy, y2, z).
A
Non-singularity of H4(V) at ~ where q is a 1.c.i. quadruplet.
109
Let (Aij) = (ai:i,b~:i,ci:i,di:i,ei:i,flj) be a chart of H 2 ( V ) at 5ij ~ and (.A~i:i) = I ! s l / (aij/ , bij, ci:i, i:i a[3], i:i a[9], ,~ a[u], ,~ ,j, ei:i, f~j) be a chart at 5i:iot 9 One denotes by (C_o) = ,[a[1], "~
o
o!
ai:i, hi:i, ci:i, dij, ei i, fi:i, c~j, d~j) a chart of B ( V ) at qoir = (Sij , qo, 5~:i ) (cf. remark 10). If t ~ is the amorphous triplet of ideal (x 2, xy, y2, z), following the notation introduced in [LB1], one defines a triplet ti in the neighborhood of t~ by the ideal : (x 2 + u.ix + viy + wi,
x y + uix t + v , yt + wi, y2 + u~"x + v~'y + w~", z + p~z + aiy + 0~)
II It where w~, w i' and wi" are explicit functions of ui, vi, u~,I vi,l u~, v i ([Ill , [ELB]). (7s
' Vi~~ u"i , Vi" ~ pi~ O_i~ 0~) Vi~ Tti~
So
constitutes a chart of H3(V) at t i.o Let us then denote by
('I-i) = (s~, ti, ri, c~, c~, d{, vl, pi, a~) the chart of H3(-'~) at t~ = (p~,p~,p~, d;~, d]~z, d;z , t;) where {i, j, k, l} = {1,2, 3, 4}, determined by Le Barz in [LB1] (see A.2). A
Now we are going to express
locally H 4 ( V )
in t h e Cl~
of coordinates
(T1, T2, 7"3,~4,C12,C13, C14,C23,C24,C34). We will show that at ~o the variety H4("~) is locally isomorphic to the graph of a morphism from C~2 to C96 where C ~2
12 ~[31, _12 a[91, 12 a[u], 12 P4)has coordinates (C_) = (s4, t4, r4, q2, q3, ~14, ~24, a[~], 9 First let us show that the ninety six coordinates can be expressed as elements of the _
_
algebra C{C_}. The equality of all these quadruplets (condition 1) gives in particular the equalities of the coordinates : ij
~-
a~]
all) i:i
a[3] = ij
a[9] = ij
a[u]
=
12
a[3] 12
a[9] 12
%1].
From now on, the element a[z12] will be denoted by all). Condition 2 yields in particular the equalities of the c o o r d i n a t e s :
/ c~j = d'~j =
(
ckt , for { i , j , k , l } d~t
=
{1,2,3,4}.
The
a[~] equality of all the coordmates " ~ (whose expressions are given in remark 10) provides the equations : a[1]c12c34 -t- c12 -J- c34
--~
a[1]c13c24 -J- c13 -I- c24
a[11c12c34 -J- C12 -j- C34
z
a[1]c14c23 -J- c14 -t- c23
These two equalities can be rewritten as : c34(1 + afllc12)
=
atxjc,3c24 + c13 + c2, - c12
(7.13)
C23(1 -~ a[1]C14 )
=
a[1]C13C24 "-~ C13 -I- C24 -- C14
(7.14)
Since the elements (1 + a[1]c12) and (1 + a[1]c14) are invertible in the algebra C{C_}, these two equalities enable us to obtain the coordinates c34 and c23 as elements of
A
110
Construction of the variety H4(V)
C{C_}. When one expresses the equalities of condition 4 on the third coordinate of the doublets, one gets the equations :
{c1=c231c2=c131c3=c121c4=c1 1 C1
CIll
~
~
C24 - - C23
C2
C34 - - C24
I! C2
~
~
C14 - - C13
C3
C34 - - C l 4
II C3
i
C I 4 - - C12
C4
~---
C13 - - C12
C24 - - C14
It C4
z
C23 - - C13
From equations (7.13) and (7.14), the right hand sides of these equalities are elements of C{C_}. The equality of the points gives in particular :
(Eq)
{
/
Sl
~---
84 ~
tl
=
t4 + t~
S4
r 1
~-
Tr4-~-7't4
{
S2
:
83
t2
=
t3 =
:
t4
r2
=
7"3
7"4
=
S4
When one writes the scheme-theoretic inclusions ti C q, which are equivalent to the inclusions of ideals I(q) C I(t~), one obtains in particular the conditions :
x 2 + a[1]xy + a~22]x + a[3]y + a ~
E
I(ti)
(7.15)
z + a[9]xy + a~o]X + a[ll]y + a~122] C I(ti)
(7.16)
Then one divides each of these elements by the ideal I(t~) (the form of the remainder is given by [~es3]). Condition (7.15) yields in particular the equalities :
(eqi)
a[3] - v~ - a[1]v~ = 0
(This term corresponds to the coefficient of y in the expression of the remainder.) Condition (7.16) gives in particular the equations : ar - Pi - aguli = 0 a[11] - ai - agv~ = 0
(Ei) ( Fi)
(These two expressions correspond to the coefficients of x and y of the remainder of the division.) From equation (eqi) and the expression of v~ in the chart (7-/), one obtains the new equations :
(eq4)'
v4(1 + a[1]c23) = a[a] + all]S4
(eq3)'
v3(1 + a[1]c24) = a[3] + all]S4
(eq2)'
v2(1 -~- a[1]c34) = a[3] q- a[1]s4
(eql)'
Vl(1 q- a[1]c34) -- a[3] q- a[1](s4 - v4(e23 - c13))
Equation (eq3)' enables one to express the coordinate v3 in the ring C{C__},since the element (1 + all]e24 ) is invertible.
From equations (7.13) and (7.14), the elements
(1+a[,]c34) and (1+a[1]c23) ofC{C} are invertible. Equations (eq4)' and (eq2)' enable us to express the coordinates v4 and v2 in C{C__}. Thus, equation (eql)' enables us to obtain the coordinate Vl as an element of C{C}. From equations (Fi) and the expressions
The variety H4(V) at ~ where q is a non 1.c.i quadruplet
111
of v~ in the chart (C), one obtains expressions for the coordinates a~. Then, the expressions of c4, c~, v4, a4 in <:{C} and equations (Eq) enable one to obtain Sl, tl and rl as elements of C{C_}. Equation (E~) (i = 1,2, 3) is replaced by (E,)' = (Ei) - (E4), which can be rewritten as : (el)'
pl ---- ;94 "~- a[9][~}le23c24 -- V4CI2C23]
(E2)'
P2 = ;94 J- a[9][v2e13e14
- V4C12C13]
(e3)'
;93 : ;94 -}- a[9][v3c12c14
- v4c12c13]
These three equations and the expressions of v{ in C{C_} enable us to obtain expressions for ;91,;92 and ;ga in the ring C{C_}. One writes explicitly the equalities 512 = d12, 51s = d13, 523 = d23, 514 = D14, 524 = ~24 and 5a4 = D34 on the coordinates of the doublets. Thus one obtains expressions in C{C_} for the coordinates aij, b~j, dij, eij, fij of the doublets 6ij since the right hand sides of these equalities are elements of C{C__}. Thus, the ninety six coordinates have been obtained as elements of the algebra C{C_}. 9 Then one checks that when these ninety six expressions are plugged into the remaining equations, one does obtain 0 = 0. Consequently, the variety H4(V) is locally isomorphic to the graph of a morphism from C12 to C96 defined in this way. Therefore, the variety H4(V) is smooth at this point and (C_) = (s4, t4, r4, c12, c13, c14, c24, a[1], a[3], a[9], ~[11], P4) constitutes a chart. Moreover, the above calculations show that this element ~o is in the closure of the open subset H ~ ( V ) . If x[i] denotes the first coordinate of the point p{ contained in the support of an element (} in the neighborhood of qo, one has the expressions : x[1]-x[2]
s4
--
s4 - (c23 - c13)(a[a] + a[1]s4)[1 + a[1]c23] - 1
x[3] = x[4] =
s4 - (c23 - c12)(a[a] + all]s4)[1 + a[1]c23]-1 s4 - (e23 - e12)(ar~l + at~ls4)[1 + aE1]c23]-1 - v1(c2~ - c23) A
7.2
The
variety
complete 7.2.1
Case
H4(V) a t intersection
of the elongated
0 where
q is a non
locally
quadruplet quadruplet
Consider an elongated quadruplet qo E H4(V) located at a point p of V. According to the definition 7.(iii), the quadruplet qo is defined by the ideal (x 3, xy, y2, z) of O v for a suitable local coordinate system centered at p. For such a quadruplet, let us see what the different elements which can be constructed in H4(V) are : Recall (see lemma 11) that from such a quadruplet qo, the elements ~o of B ( V ) all are of the form ~o = (d, qo, d') where either d or d' is defined by the ideal (x 2, y, z).
112
Construction of the variety H4(V)
X
A k
Thus, at an element qo of H4(V), only three of the six doublets
{i,j,
d~j,djt,dli,
where
l} C {1, 2, 3, 4}, can be arbitrary. Then the other three doublets are identical and
defined by the ideal (x 2, y, z). The quadruplet qo can contain the amorphous triplet t of ideal (x 2,xy,
y2, z), as
well as the curvilinear triplets t~ defined by the ideals
(x a, y + ax 2, z) where a E C. The doublets contained in the amorphous triplet t can have an arbitrary direction, while, the curvilinear triplet t~ contains only the doublet of ideal (x 2, y, z). We can now give the form of the different elements qo E H4(V) : (i) The four triplets tl are equal and amorphous, only three of the six doublets contained in the quadruplet are arbitrary. The other three doublets are identical and well determined :
d,i3
. j . ~ ajl
dik = djk =
7 " / dd
tl = t2 = t3 = t4 amorphous
(ii) The quadruplet qo contains three amorphous triplets t and a curvilinear triplet t~. The three doublets contained in the triplet t~ are identical and well defined : they all have as ideal (x 2, y, z). The three other doublets contained in the quadruplet qo can have arbitrary directions : tl curvilinear
dJt
7 "
/ dkt
tz = t i = tk amorphous
(iii)
The quadruplet qo contains two amorphous triplets equal to t and two curvilinear
triplets. Five of the six doublets (those contained in the curvilinear triplets) are identical and well determined. The sixth doublet contained in qo has an arbitrary direction :
A
The variety H4(V) at q where q is a non 1.c.i quadruplet
113
dkl ) t~: curvilinear
(
=
~
t~ = tj a m o r p h o u s / / ~
s
)
dil=di~=djl=djk=diJ
h curvilinear
(iv) The quadruplet qo contains an amorphous triplet and three curvilinear triplets. Then the six doublets are identical and well determined : tj curvilinearl ~ tk curvilinear
(
tik.amorphous / ~ tf c u r v l h n e a r
~
"
)
d~j identical
I
(v) Finally, when the four triplets are curvilinear, the six doublets are then identical and well defined :
(
tj curvlhnear) ) tk curvilinear
tz c u r i ~ c u I ! i f f l J a [
dentical
Now, we are going to give local equations of H4(~-V) in the product [Ha(t"V)]4 • [B(V)] 6 at the most degenerate element ~o, i.e. when the four triplets are amorphous of ideal (x 2, xy, y2, z), the six doublets are identical and defined by the ideal (x 2,y, z). We will see (proposition 9) that at this point the variety H4(V) is singular of dimension 12, and that the open subset HI(V ) is dense in H4(V). In the next section, we will give a geometric description of the singular locus of H4(V) in the neighborhood of 0oA quadruplet q~y in the neighborhood of qo is defined by the ideal : ij 2 • ij ij ij a ij x 2 ij ij ij [(qii) = (x 3 + a[1]x . a[2]x + a[a]y + a[4], Xy A_ " [5] q- a[6]xq- a[7]Yq- a[8] q + a[u]y " + a[12], q z+ai~31x2+ a[14]x ~J + a[15]y ij + a[16]) q y2 + alJg]x2+ a[lo]Z ij ij ij ij where a[4], a[s], a[lo] and a[12] are explicit fimctions of a[1], 'J a[2], " a[3], ~J a[51, ~J a[G], 'J a[z], ~J a[9], ~J a[n] ~ (see appendix B.1). As usual, let (_~j) = (aq,bq, cq,dq,eq, fq) - resp. (A'ij) = I ! I / (a~:y, b~y, 4j, d~.d, e~d, f~Id) - denote a chart of H2(V) at 5~yo -resp. at 5~do ! - where 5~jo and 5q ~! are defined by the ideal (x 2, y, z). From section 6.4.2, (•ij) = (all], a[31 ,,5 a[5] ,~) a[6] ,~j ij ij ij ij ij ! -" o o! a[7], a[13], a[14], a[15], a[16], a~:i,c~j, Qj) constitutes a chart of B(V) at %" = (6~j , qo, 6ij ). If t ~ is the amorphous triplet of ideal (x 2, xy, y2 z), and if we use again the notation
114
Construction of the variety H4(V)
introduced in [LB1], we define a triplet ti in the neighborhood of t~ by the ideal : (x 2 + u l x + viy + w~,
t x y + u~x + v~y + wl,t
y2 + ui, x + vi" y + wi",
z + pix +cr~y+O~)
, , i,, (see [I1], [ELB]). So where wi, w i and w i" are explicit functions o f u i , vl,ui,i v~,u~,v
(u~, vi, u~, v~',u"~, v~", Pi, ~i, 0~) constitutes a chart of H3(V) at t~.~ Let us then denote by (7-i) = (s~, tl, r~, ci, c'i, c7, vl, p~, ~r,) the chart of H3(V) at t~ = (p;, p~, p~, d;k , d~l , d;l , t~)
where {i, j, k, l} = {1, 2, 3, 4} (see A.2). Thus we are going to determine the equations A
of H 4 (V) in the Cl~ of coordinates (~1, ~2, ~ , Y-4,-C12, ~13, C~14,C23, •24,
C~34)"
x[i] Remark
13 From remark 11, we denote by
the coordinates of the point pi.
y[i]
4i] We perform the change of variables : X[I]
I
=
S4
y[1] =
t4
z[1]
7'4
:
We will denote the set of seventeen coordinates by either one of the following notation :
~
( v1~v2~va~v4~c12,C13'c14~C23,C24,C34~S4,t4~7.4,~12~[5],
=
(Vl, v2, vs, v4, c12, c13, c14, c23, c24, c34, x[1], y[1], z[1], a[5],12a12113],a[14],12a[151)12
u[13],~12 a[14],12 c~[15] )-12 x
We propose to show the following assertion : Assertion
: the 108 - 17 = 91 other coordinates are obtained as explicit fonctions
of these seventeen coordinates C. Moreover, these seventeen coordinates are linked together by ten linearly independent quadratic equations. Condition (3.1), which defines a complete quadruplet, gives the equations : x[1]
=
$4 :
83
~- $2
y[1] = t 4 -- t 3 = t2 z[1]
=
7"4 :
7"3 =
7"2
Condition 1 in particular gives the equations : a ~J N = all12]
for
1 <_i < j < 4 _ and l C { 1 , 3 , 5 , 6 , 7 , 1 3 , 1 4 , 1 5 } .
Prom now on, we will denote by a N the element a ~ in order to simplify the expressions. Condition 2 (5~j = 5~..l) enables us to obtain the expressions : (*)
c'ij = ckl
,
for
{i,j,k,l}
= {1,2,3,4}
A
T h e variety H4(V) at ~ where q is a non 1.c.i quadruplet
115
{c1=c23{c2=c13{c3=c1 {c4:c12,
From condition 4, one obtains the equations :
('k-k)
C1 l/
C1
C24 - - C23
g2
C34 - - C23
C2
l/
=
C14 - - C13
4
~
C34
C3
II
C14
=
C14 - - C12
C~
~-
C24 - - C14
C4
II
~---
C13 - - C12
~-
C23 - - C13
In the chart (_T_a) of H3(--"~) at t~, the coordinate x[21 is given by the expression: x[21
=
x[l] - v4(c23
-
-
-
-
C13)
(7.17)
C14)
(7.18)
In the chart (_Ta), one has the expression for x[2]: x[2]
=
x[1] - 733(c24
Similarly, in the different charts CT~4), (7-2) and (7-1) taken in this order, the coordinate x[3] is given by the expression : x[3]
:
x[1] - v4(c23 - c12)
(7.19)
x[3]
:
x[1] - v2(c34 - c14)
(7.20)
x[3]
=
x[2] - v1(c34 - c24)
(7.21)
Finally, in the different charts (T_3), (7-2) and (7-1) taken in this order, the coordinate x[4] has for expression : x[4]
=
x[1] - "03(c24
C12)
(7.22)
X[4] =
X[1] -- V2(C34
C13)
(7.23)
X[4] =
X[2] -- vl(Ca4 -- c23)
(7.24)
-
-
-
-
T h e n one keeps the generators (7.17), (7.19) and (7.22). One replaces the generator (7.18) by the generator ( 7 . 1 8 ) ' = (7.18) ~4(c23 - c13)
(7.17) =
:
v3(c24 - c14)
One replaces the generator (7.20) by ( 7 . 2 0 ) ' = (7.20) ~4(c2~ - c12)
=
(7.19)
:
v2(c34 - Cl~)
The generator (7.21) is replaced by (7.21)' = (7.21) - (7.19) + (7.1r) v4(c1~ - c12) The
generator
(7.23)
is
=
replaced by (7.23)' ~(c24
- c1~)
~1(c~4 - c24) = (7.23)
=
~(c~4
- (7.22): - c,~)
:
A
116
Construction of the variety H4(V)
Finally, one replaces the generator (7.24) by (7.24)' = (7.24) - (7.22) + (7.18) :
v~(c14 -
Cl2)
=
~1(c34 - c~)
The five quadratic equations (7.18)', (7.20)', (7.21)', (7.23)' and (7.24)' are necessary conditions to express the equalities between the points (condition 3). The scheme-theoretic inclusions 6~j C t~, for { i , j , k} C {1, 2, 3, 4} enable us to obtain in particular the expressions for the coordinates aij : a,) = -x[i] - x[j]. From the inclusions I ( t i ) . I(p ~) C I(q) (condition 5), it follows in particular that the generator (z 2 + ulx + viy + wi)(x - z[i]) belongs to the ideal I(q). One adds to this generator an element of I(q) in order to obtain the new element R, : x2(-a[1] - a[5]vi - x[i] + ui) + x ( - u i x [ i ] - wi - a[2] - a[6]vi)
Ri =
y(-x[i]~,
- aE~I - aETl~ ) - (x[i]w~ + dE41 + acsl~)
Therefore, the condition (x 2 + uix + viy + wi)(x - x[/]) E I(q) is equivalent to the condition R~ E I(q). From result [Ties6] one obtains in particular the four equations :
We keep equation (el), which one rewrites by using the expression of Ul : a[1] = --3x[1] + y1(c34 - c24 - c23 -- a[5]) + 21)4(c23-- c13) One replaces equation (e2) by (e2) - (el), which one rewrites by using the expressions of Ul,U2,X[2] and the quadratic equations (7.18)', (7.23)' and (7.24)': ~(ai~ 1 + c~)
=
~(~E~3 + cl~)
(7.25)
One replaces equation (e3) by (ea) - (el), which one rewrites by using the expressions of ul, u3, x[3] and the quadratic equations (7.18)' and (7.21)' : ~l(~E~l + c:3)
=
.~(a~] + c1:)
(7.26)
Similarly, one replaces equation (e4) by (e4) - (el), which one rewrites by using the expressions of ul, u4, x[4] and the quadratic equations (7.18)' and (7.24)' : ~gl(a[5 ] -[- C24 )
~-- v4(a[51 -~ C12 )
(7.27)
On the other hand, we have seen (section 6.4.2) that the coordinate a~j of the doublet 5'ij is given in the chart C_ij by the expression : a'ij = a[11 + a[7] -- aij The equalities ~ a~i
(
a,j
= =
au
-x[i]
- ~[j]
lead to the equality
a[7] = -(a[l] + x[1] + x[2] + x[3] + x[4])
A
The variety H4(V) at ~ where q is a non 1.c.i q u a d r u p l e t W h e n a[1], x[2], x[3] and
x[4] are
117
replaced by their expressions in t e r m s of the coordi-
nates C, this equality becomes : dE71 = vl(a[5 j + c23) + v3(c24 - c1~) - x[1] We have seen (equation (6.42)) t h a t the expression in the chart (C~j) of the p a r a m e t e r s ij
a[91 is : /
Using the conditions c(,j = ct:l, these equalities can be rewritten as : a~
=
a~9~ =
a[5](c12 -I- c34) -t- c12c34
a~
=
a~
=
a[5](c13 + c24)-1-c13c24
a~
=
14 a[9 ] =
a[5](e14 -t- c23) -t- c14c23 ij
T h e n the equality of the coordinates a[9] of the q u a d r u p l e t s (consequence of condition 1) gives b o t h q u a d r a t i c equations : a[5](e12 ~- e34 ) + e12e34
aE~l(Cl: +
e~4) + e , : ~ 4
=
a[5](c13 -}- e24 ) + c13c24
(7.28)
e,4c2~
(7.29)
aE~l(c14 + ~:~) +
The scheme-theoretic inclusions ti C q, which are equivalent to the inclusions of ideals I ( q ) C I(t~), yield in p a r t i c u l a r the conditions :
(*)
z + a[13]x 2 + a[14]x + a[15]Y + a[16]
e
I(tl)
One adds to this generator an element of I(t~) in order to o b t a i n the new element R i : R~ = ~(~E~l - P~ - aI~31~,) + Y(aI151 - ~' - aI131~) + ( a i d
Therefore the conditions (*) are equivalent to the conditions
- O, - ~,ai131)
R i C I ( t i ) . F r o m [Nes3],
these conditions can be t r a n s l a t e d into the equations :
all5] -- cri -- a[13]vi = 0
eli) (e2/)
a[16] -- Oi -- wia[13] = 0
(e3d
all4] -- Pi -- a[lalul = 0
For i = 1 , . . . , 4, equation (e2i) enables us to express al in terms of the coordinates C. Similarly, equation (ell) and the expression of ui give Pi in t e r m s of the c o o r d i n a t e s C. T h e scheme-theoretic inclusion t4 C q gives in p a r t i c u l a r the two conditions : X3 -1- a[1]x 2 ~- a[2]x + a[a]y -/- a[4]
C
I(t4)
x y + a[5]x 2 + a[6]x + aF]y + a[8]
C
/(t4)
The first condition yields in p a r t i c u l a r the equality : a[31 = v4(--v14 -t- a[l] -- u4)
,
,
A
118
Construction of the variety H4(V)
while the second condition gives
a[6] = u~ + a[5]u4. The expressions which were
already obtained for u4, u~, v~ and all] enable us to express a[3] and a[6] in terms of the coordinates C. Finally, the condition pX C q gives in particular the equation : z[1] + a[13]x[1]2 + a[14]x[1] + a[15]y[1] + a[16] = 0 which enables us to obtain a[1Gj in terms of the coordinates C. Therefore, the ninety one parameters have been obtained as explicit functions of the seventeen coordinates g. Then one verifies that the necessary conditions imposed by the ten quadratic equations (7.18)', (7.20)', (7.21)', (7.23)', (7.24)', (7.25)-(7.29), in order to define the variety H4(V) locally at qo are sufficient. It suffices to replace all the expressions obtained above in terms of the coordinates g in the remaining equations and to verify that one obtains either 0 = 0, either combinations of these ten quadratic equations. One checks that these ten quadratic equations are linearly independent. Consequently, the variety H4(V) is locally isomorphic at qo to the subvariety of C17, of coordinates C, defined by these ten quadratic equations. Moreover, the computations show that this element qo E H4(V) is the limit of elements of H~(V). Indeed, if 0 is a complete quadruplet in the neighborhood of 0o, one obtains the expressions of the first coordinate of the different points of the support of the quadruplet :
x[2] =x[l]--v4(c23--c13) x[3] = x[1] --v4(c23 --c12) x[41 : x[1] --Y3(c24--c12 ) and one checks that the differences x[2] - x[3], x[2] - x[4] and x[3] - x[4] are not combinations of the ten previous equations. Next, we prove the proposition : A
P r o p o s i t i o n 9 The variety H4(V) is singular of dimension 12 at 0~ Proof 9 We saw that the ten quadratic equations (7.18)', (7.20)', (7.21)', (7.23)', (7.24)', (7.25)-(7.29) constitute local equations of Hi(V) at qo in the ring C[g]. We perform the change of coordinates : X/
z
vi
)(5
=
a[5]+c12
)(-6
=
a[5] + c13
X7
=
a[5]+ c14
Xs
=
a[5]+ c23
X9
=
a[5]+c24
X10 =
a[5]+c34
,fori--1,...,4
The variety H4(V) at ~ where q is a non 1.c.i quadruplet
119
The ideal I generated by the ten quadratic equations can be rewritten in the ring
C [ X I , . . . , Xlo, a[5], a[13j, a[14], a[15], x[1], y[ll, z[1]] as: XnX 5
=
X2X. 7 ;
X4Xs
~- X2Xlo ;
X4X 5
~-
XlX9 ;
X2X6
~-
X3X 5 ]
X4X6
=
XaX7;
XlX8
=
X2X6 ;
X4X~
=
X,X,o
X~X,o
=
X6X9 ;
X4X 8
~-
XaX9
XsXlO
=
XrX 8 ;
;
;
These generators are independent of the seven coordinates a[s}, a[13j, a[14j, a[15], x[1], y[1] and z[1]. Therefore the subvariety V ( I ) of C17 is a product H x C7 where H is the subvariety of C1~ of affine coordinate ring C[X1,..., Xlo], defined by the ideal I, and the coordinates of C7 are a[5], a[13], all41, a[~51, x[1], y[1] and z[l]. Therefore the variety H4(V) is locally isomorphic to the product H • C7. In order to prove the proposition, one has to prove that the subvariety H C C1~ is singular, of dimension 12 - 7 = 5. As the ideal I is homogeneous, one must just study the projective subvariety F H C ?C l~ associated to the subvariety H C C1~ From now on, the projective space associated to the vector space C1~ is denoted by ?9. The affine open subset of F 9 consisting of points of homogeneous coordinates IX1 : ' " : X10] such that Xi # 0 is denoted by Ui. This affine open U,, of coordinate ring C[X1,..., X l o , Xi-1]0 is isomorphic to the variety Cv, of coordinate ring C[Xl,..., x i - l , z i + l , . . . , Xl0]. (The isomorphism is given by Xi ' for j • i.) We now show that the variety ? H is singular, of dimension 4, and irreducible. Through the isomorphism Ui --% r 9, the ideal I N r . . . . , X10, Xi-1]0, xj = Xj
which defines F H Cl Ui, is isomorphic to the ideal Ii. 9 S t u d y o f F H on the open s e t U1 :
The variety P H N U1 is isomorphic to the subvariety V ( I 1 ) of C9, where I1 is the ideal of C[x2, x3,.. 9 xl0] generated by the ten following generators :
X4X 5
:
X22g7
(7.30)
X4X5 --~ X9
(7.31)
X4X 6
:
X3g37
(7.32)
X4X6
=
XlO
(7.33)
X4XS
=
xax9
(7.34)
x4x8
=
x2xlo
(7.35)
x2x6
=
x3x5
(7.36)
x8
=
x2x6
(7.37)
xsxlo
=
x6x9
(7.38)
xszlo
=
xrxs
(7.39)
A
120
Construction of the variety H4(V)
From equations (7.31), (7.33), (7.37), the coordinates x9,xlo and xs are obtained as explicit functions of the other coordinates. When one substitutes x8,x9 and xl0 by these expressions into the seven other generators of the ideal I1, the generator (7.34) becomes proportional to (7.36), the generator (7.39) becomes proportionnal to (7.30) and the equations (7.35) and (7.38) are satisfied. The subvariety V(Ia) of C9 is isomorphic to the subvariety of ~ , of coordinate ring C[x2, x3, x4, x5, x6, xT], generated by the three generators (7.30), (7.32) and (7.36). It is nothing else than the determinantal variety defined by : E M2x3(C) s.t. rankM < 1}
= X5
X6
X7
which is irreducible of dimension 4, as it is shown in the following lemma : L e m m a 13 The determinantal variety M1 defined by =
=
9
Z4 Z5 Z6
M2•
s.t. rankM < 1}
is irreducible, of dimension 4, and singular at the origin. Proof : The condition rankM < 1 can be rewritten as : Z1Z s
=
Z2Z 4
Z1Z 6
=
Z3Z 4
Z2 Z6
=
Z 3 Z5
One easily checks that the projective subvariety PM1 C I?5 associated to the variety M1 is irreducible, smooth, of dimension 3.
[]
On the open set U1, the variety 1?H is isomorphic to the variety M1 and is therefore irreducible of dimension 4 and singular at the origin. The origin of this affine open subset U1 corresponds to the point A1 E p9 of homogeneous coordinates [1 : 0 : ... :0]. 9 Similarly, one checks that on the nine other open sets Ui of the covering, the variety I?H is isomorphic to the determinantal variety M1 and that the singular point of M1 corresponds to the point Ai of p9 of homogenous coordinates [0 : 9.. : 0 : 1 : 0 : ... : 0] (the 1 is the i th component). Therefore, one has proved that the variety P H is irreducible, of dimension 4, and that its singular locus is the union of the ten points Ai of p9. Consequently the subvariety H x C C Ct~ x C7 is irreducible of dimension 12 = 5 + 7, and its singular locus is the union of the ten subvector spaces ~ of Clz where V / = V ( X 1 , . . . , Xi-1, Xi+l,..., Xlo). As in the neighborhood of ~o the variety H4(V) is locally isomorphic to the product H x C7, it follows that this variety is singular of dimension 12 at ~o. Therefore, proposition 9 is proved.
A
The variety H4(V) at 2 where q is a non l.c.i quadruplet
121
In the neighborhood of qo , the singular locus sing(H4(V)) of Ha(V) is the union of the ten subvarieties ]2~ C H4(V). These ten subvarieties are smooth, of dimension 8. The subvariety )2i is locally isomorphic to the vector space V/. Geometric description of the variety sing(H4(V))
at qo :
In this section, we give a geometric description of the singular locus sing(H4(V)) of the variety H4(V) in the neighborhood of 2~ Recall that V/is the subvector space of C1~, of dimension 8, defined by the equations: X1 . . . . .
Xi-1 = X~+t . . . . .
Xlo = 0. At
an element 2 of sing(H4(V)) in the neighborhood of 20, the quadruplet q is elongated and its support p has coordinates (x[1], y[1], z[1]). Then one performs the change of origin : x
=
x-x[1]
y
=
x - y[1]
z
=
z-z[1]
In the local coordinate system (X, y , Z) centered at p, the quadruplet q is defined by the ideal :
I(q) = (X 3, X ( y + a[5]X), ( y + a[5]X) 2, Z + a[13]X2 + (a[14] -{- 2a[13]x[1])X + a[15]Y) This ideal does define an elongated quadruplet (cf. definition 7.(iii)). 9 Geometric description of an element 2 of V~, for i = 1 , . . . , 4 : At such an element, all the doublets and all the triplets have the same support, the point p of V. An element of V1 U ])2 U V3 U ];4 satisfies the conditions : X5 . . . . . Xlo = 0, which can be rewritten as clj = -a[5]. (The coordinate c~i corresponds to the direction defined by the doublet d~j C q.) These six equations yield the equality of the six doublets.
One checks that these six doublets are defined by the ideal
(X 2, y + a[512(, Z + (a[14] + 2a[13]x[1]),-Y+ a[15]Y). For j = 1 , . . . , 4, the condition Xj = 0, which can be written as vj = 0, is equivalent to the condition : the triplet tj is amorphous. Thus, for i = 1 , . . . , 4, the triplet ti is generically curvilinear at an element 2 E ];i (since X~ = v~ is generically non zero on V~). This triplet is defined by the ideal : I(t,) = ( X 2 + v~(y + a[5]X), XN - a[5]vz(Y + a[s]X), y2 + a[5]2vz(y + a[5]X), Z + a[13]X2 + (a[14] + 2a[13]x[1])X + a[15lY) which can be rewritten as follows, if v~ is non zero :
I(t,) -- (X 2 + v i ( y + a[5]X), X 3, Z + a[13]X2 + (a[la] + 2a[13]x[1])X + a[ts]Y) The other three triplets are amorphous and defined by the ideal :
(X 2, X y , y2, Z + (a[141+ 2ai~31x[1])X + ai,51y )
122
Construction of the variety H4(V)
Thus, for i = 1,... ,4, at a generic element ~ of Pi, the six doublets are identical, the triplet t~ is generically curvilinear and the other three triplets are (identical and) amorphous. On can represent a generic element ~ E ]2~ as :
I t~
curvilinear
tj = t~ = ( a m o r p h o u s / / ~
)
d~j identical
9 Geometric description of an element ~ of Vi, for i = 5 , . . . , 10 : For i = 5 , . . . , 10, the elements ~ of ]2i verify the equalities X1 = X2 = X3 = )(4 = 0. These equalities, which can be rewritten as Vl = v2 = v3 = v4 = 0, mean that the four triplets are amorphous. These four triplets are defined by the ideal : I(t~) = (X 2, ,3c'y, y2, Z, + (a[14) + 2a[13]x[1]),'Y + a[15]Y) At an element ~ of ]21, five of the six doublets are identical and defined by the ideal : (X 2, Y + a[5]X, Z + (a[14] + 2a[13}x[1])X + a[15]Y) The sixth doublet dkl contained in q (the doublet for which Xi = a[5]+ ckt is generically non zero) has an arbitrary direction. This doublet dkt is defined by the ideal :
I(dkz) = (X'2, Y + cklX, Z + (a[14] + 2a[13]x[ll)X + a[15]Y) Thus, for i = 5 , . . . , 10, at a generic element ~ of ]2~, five of the six doublets are identical, the sixth doublet contained in q has an arbitrary direction and the four triplets are amorphous (and identical). On ]25 (resp. ]2G,]27, ]28, ]29 and ]210) it is the doublet d12 (resp. d13, d14, d23, d24 and d34) which has an arbitrary direction. Consequently, in the neighborhood of 4~ the singular locus of H4(V) consists of the following elements : - The quadruplet q is elongated, the six doublets contained in q are identical, only one of the four triplets contained in the quadruplet is curvilinear. The other three triplets are amorphous : l only one curvilinear triplet
(
// //~
"- )
dij identical
three amorphous triplets
- The quadruplet q is elongated, only one of the six doublets contained in q has an arbitrary direction. The other five doublets all define the same direction as that
A
The variety H4(V) at ~ where q is a non 1.c.i quadruplet
123
defined by the quadruplet q. The four triplets are amorphous : only one arbitrary doub]
(
five of the six doublets identical
the four triplets are amorphous 7.2.2
Case
of the
spherical
quadruplet
Now, let qo denote the spherical quadruplet of support the point p of V. Such a quadruplet can only contain amorphous triplets. We are going to determine local equations of H4(V) in the product [H3(V)] 4 • [B(V)] 6 at the most degenerate element ~o, i.e. when the four triplets are identical (and amorphous) and the six doublets are identical. In an appropriate system (x,y, z) of local coordinates centered at p, the triplets are defined by the ideal (x 2, xy, y2, z) and the doublets have for ideal (x 2, y, z). Let ~o = (t~ , ~§176 2 , §176 ~ 3 , ~o ~ 4 , ~12 ~o , q13, q14, q23, q24, q34). Let (Ti) = (si, ti, ri, ci, 4, d~', vi, pi, ai) denote again a chart of H3(V) at t:~ (cf. A.2). A quadruplet q~J in the neighborhood of qo is defined by the ideal : ij
ij
ij
ij
ij
ij
ij
ij
I(q~ i) = ( x 2 + afl]x + a[2]y + a[a]z + a[4] , xy + a[5]x + a[6]y + a[7]z + dis] , ij ij z3 ij y2 zl ~1 ij ij xz + a[9]x + a[lo]y + a[u]z + a[121 , + a[la]X + a[14]Y -t- a[15]z -t- a[16] , ij ij ij ij 2 ij ij ij ij yz + a[17]x + a[ls]Y + a[19]z + a[20] , z + a[2a)x + a[22]y + a[23]z -t- a[24]) ij
ij
ij
ij
ij
ij
where a[4], a[s], a[12], a[16], a[20] and a[24] are explicit functions of the other eighteen parameters, which are denoted by a'J. These eighteen parameters a" are linked by fifteen quadratic equations, whose expressions are given in appendix B.2. (These fifteen quadratic equations constitute local equations of H4(V) in the ring C[a#].) As previously, (Aij) -- (aij, bij, clj,dlj, eij, fij) denotes a chart of g 2 ( y ) at 5ij ~
, ' eli; d'ij, e'ij, fij)' a chart at 5ij ~ Let (C_q) =~a[2],'1 a[3] ,,J a[5],'~a[@'3a[7] ,,3 and (A~q) = (aq; bij ij ij ij t t a[9], a[10] , a[u], cq, e~j, c~j, e~j) denote a chart of B ( V ) at qoq = (~j o , qo, ~qo,) (cf. p. 99). We are going to express H4(V) in the Cl~ Let(C)
:
of coordinate ring C[Zl, 22,-T-3, ~Y-~,C_q].
( s 4 , t 4 , 7,4, /94, 0"1,0.2, 0.3, o'4, a[2], 12 a[3], 12 a[7], 12 c12, c13, ~ c 23, c 24, c 341~ d e n o t e (~14,
the
set
of these seventeen coordinates. Remark
14 From remark 11, if
y[i]
denotes the coordinates of the point pi con-
z[i] tained in the support of the quadruplet q, one has the change of variables : =
y[1] = z[1] =
t4 7"4
A
Construction of the variety H4(V)
124 The set of the seventeen coordinates :
12 ' C12~ C13, C14, C23, C24, C34) 12 aft], 12 a[7] (x[1], y[1], z[1], P4, 01, 02, (73, 0.4, a[2],
will be also denoted by (C). Let us show the following assertion : A s s e r t i o n : the 108 - 17 = 91 other coordinates are explicit functions of the seventeen coordinates C. Moreover, these seventeen coordinates are linked by ten linearly independent equations. The computations performed in 7.2.1 to express the equalities of the coordinates of the points of V (condition 3) are still valid (equations (7.17)-(7.24)). Therefore, one has again the five quadratic equations (7.18)', (7.20)', (7.21)', (7.23)' and (7.24)'. Condition 2 (5~1 = 5kl) enables us to obtain again the equalities as well as the equalities e'q = ekl, for {i,j,k,l}
= {1,2,3,4}.
(*) (see page 114), The equations (**)
which were established in 7.2.1, page 115 are again satisfied. Condition 1 gives in particular the equalities :
q ----a}l~ for a[z]
l E {2,3,7}
Then, let aft] = a[~, in order to simplify the notation. The scheme-theoretic inclusions tz C q (condition 5), which are equivalent to the inclusion of ideals I(q) C I(t~), yields in particular the condition :
x 2 + a~]x + a[2]y + a[3]z + a~4l]
E
I(t~)
If Ri denotes the remainder of the division of this element by the ideal I(ti), the previous condition becomes Ri = 0. This condition gives in particular the equations :
vl -- a[2]-a[310.i
,
the element a[2] - v~ - a[310,i being the coefficient of y in the remainder R~. One substitutes vl, v2, v3 and v4 by their expressions in terms of the coordinates C into the equations (7.18)', (7.20)', (7.21)', (7.23)' and (7.24)'. One obtains the new expressions : (a[2] -- a[310.4)(c23 -- c13) ---- (a[2] -- a[3]o3)(c24 (aE2]- aE3m)(c~
- c12)
=
-- 04)
(7.40)
(a[2] -- ai310.2)(c34 -- c14)
(7.41)
(a[2] - a[310.1)(c34 -- c24)
(7.42)
---- (a[2] -- a[310.2)(c34 -- c13)
(7.43)
(a[:] - a E ~ m ) ( c . - c1:)
(a[2]- a[310.3)(c14 - c12) :
(a[21 - a[3]al)(c34
- c23)
(7.44)
A
The variety H4(V) at ~ where q is a non 1.c.i q u a d r u p l e t W h e n one expresses condition 4
on
125
the fifth coordinate of the doublets, one obtains
the equalities : e4 I!
e4 + e~ 4- e 4 e4 4- e~
~--
e3
=-
el2
~
eI
~
e23
~-
el3
----
e 3 4- e~ 4 - e ~
e2
-=
e3 4 - e ~
=
e2 + 6 + e~
=
e14-e~
---- e24
e24-e~
=
e14
e I 4- e I
e34
F r o m the six equalities on the left h a n d side, and from the expressions of e~, e~ and e~~ in the charts (7-i), one obtains the three expressions :
0-1)c23
m
=
p4 + (0-4 -
f12
:
f14 4- (0"4 - - 0"2)C13
P3
~-
fl44-(O"4--0-3)C12
,
as well as the three q u a d r a t i c equations : (0-2 - 0"3)(c14 - el3)
=
(~3 - 0"4)(c13 - c12)
(0"1 - ~2)(c34
- c13)
=
(0"1 - 0"~)(c~3 - c13)
(0"1 - 0"3)(c2~ - c~3)
=
(0"3 - 0",)(c23 - e12)
(7.45) (7.46) (7.47)
Then, the six equalities on the right h a n d side enable us to express the c o o r d i n a t e s el2, el3, e14, e23, e24 and e34 in terms of the coordinates C. Moreover, the inclusion of ideals I(q) C I(ti) (condition 5) gives in p a r t i c u l a r the conditions : 12 12 xy + a[5]x + a[6]y + a[7]z + a~s2]
E
I(t~).
Let rl denote the r e m a i n d e r of the division of this element by the ideal I(t~). The previous conditions become ri = 0. In particular, the coefficients of x and of y of the r e m a i n d e r ri must be equal to zero, which leads to the equations : 12
a[5] = 12 a[6] =
i
u4 + a[z]p4 v~ + a[730"~
The first equation can be rewritten as : a~5~ = - t 4 -
(E,)
c12c13(a[2]- a[310"4)+ a[7]P4. Using
the expression of v~, the equation (E4) can be rewritten as : 12
a[6] = a[7]a4 + (a[23 - a[310-4)c23 - s4 For j = 1, 2, 3, one replaces the equation (Ej) by (Ej)' = (Ej) - (E4), which one rewrites as : a[7](0-1 - - 0 4 ) 3r-
a[2](c34
-
c13) + a[3](0"4c13 - - 0"1c34)
a[7](0"2 -- ~r4) + a[23(c34 -- c23) + a[3](a4c23 -- 0-2c34)
~-
0
(7.48)
:
0
(7.49)
0
(7.50)
a[T](0"3 - 0"4) + a[2](c24 - c23) 4- a[3](0"4c23 -- 0"3c24) =
A
126
Construction of the variety H 4 ( V )
The inclusions of ideals I(p~) . I(t~) C I(q) give in particular the conditions : (~ - ~[i])(z
+ p~
+ 0",y + 0~)
e
I(q).
One divides each element by the ideal I(q). The form of the r e m a i n d e r of the division is given by the result [T~esT]. Since this r e m a i n d e r must be equal to zero, one has in p a r t i c u l a r the four equations :
(eq~)
a~211] = - x[i] -- a[710"i - a[3]pi
(This expression corresponds to the coefficient of z in the remainder.) E q u a t i o n (eql) and the expression of fll in the chart C enable one to o b t a i n an expression for the c o o r d i n a t e a~21] in terms of the coordinates C. For j = 2, 3, 4, one replaces equation
(eqj) by (eqj)' = (eqj) - (eql), which one rewrites a s : (0"2 -- 0"1)(a[7] -- a[3]c34)
(0"3 - 0"1)(aE71 -
aE31c24)
(0"4 -- 0"1)(a[7] - a[3]c23)
=
(a[2] -- a[310"4)(c23 -- c1 3 )
(7.51)
=
(at21 - aE310"4)(c:3 - c12)
(7.52)
=
(a[2] - a[310"2)(c34 -- c13)
(7.53)
The condition xz + a ~ x + a~o]y + a~121]z+ a~2] C I(t4), which is a consequence of the inclusion I(q) C / ( t 4 ) ,
enables us to o b t a i n similar expressions : 12
12
I
a[9]
=
a[11]P4--~4u4--u4P4+04
12 a[10]
=
--Y4f14
-- 0"4~)~ -t-
,
12 a[1110"4
One replaces the right h a n d side terms by their expressions in terms of the c o o r d i n a t e s C, which enables one to o b t a i n the expressions for a~9~ and a[10].12 F r o m the equation (6.56), p. 97 and the equalities c<,j = ckl (*), the expressions /j all5] = --a[3]cijckl + a[7] (Cij 71- Ckl) follow. T h e equality of all the coordinates a[15] ij (condition 1) gives the two equations : a[7](c12 + c34) - a[3]c12c34 aET](cl~ + c34) -
ac3]c1~c34
=
a[7](c13 + c24) - a[3]c13c24
=
ac7](c14 + ~23) -
aE31~14c~3
(7.54)
(7.55)
The ninety one c o o r d i n a t e s have been o b t a i n e d explicitly in terms of the other seventeen C. One checks t h a t one obtains either 0 = 0, or c o m b i n a t i o n s of the sixteen equations (7.40) - (7.55) in the ring C[C], when these ninety one expressions are substit u t e d in the r e m a i n i n g equations. T h e n one verifies t h a t the ideal I of C[C], g e n e r a t e d by these sixteen equations, can in fact be generated by the ten linearly i n d e p e n d e n t
A
Irreducibility of H4(V).
127
equations : (0-3 - 0-4)c12 + (-0-2 + 0-4)c13 + (0-2 - 0-3)c14 = 0 (~2 - ~4)~13 + ( - ~ 1 + 0-4)c23 + (~1 - ~2)~34 = 0 (0-2 - ~ 3 ) ~ . + ( - ~ 1 + ~3)~2~ + ( ~ - ~2)c3~ = 0 (~3 - ~
- ~23 + c2~)aE21 + ( - 0 - ~ 1 3 + 0-3c14 + 0-4~23 - ~3~24)aE3] = 0
(C12 -- C14 -- C23 -J'- C34)a[2 ] -~ (--O-4C12 -~- O'4C23 "4- 0"1C24 -- G3C24 -- G1c34)a[3 ] ~- 0
(c1~ + c23 - c24 - c34)aE2] + (-~3c14 - ~ c 2 3 + ~3c24 + ~3~)aE3] + a E ~ ] ( - ~ + ~4) = 0 (c23 - c34)~E21 + (-~4c23 + ~2c34)~E3j + (-0-2 + ~,)aE~J = 0
(c24 -- c34)a[2] (C13C24
--
(c14c23 --
+ (--0-3c24 + 0"2c34)a[3] -[- (--02 At-0-3)a[7] = 0
c12c34)a[3 ] "~ (C12
c12c34)a[3]+
C13
--
--
-[- c34)a[7 ] ~--- 0
C24
(c12 -- c14 -- c23 + c34)a[7] : 0
We have shown the assertion. A
The variety H 4 ( V ) is locally isomorphic at ~o to the subvariety of C 17 of c o o r d i n a t e s (C), defined by these ten equations. Since the generators of I are i n d e p e n d e n t of the c o o r d i n a t e s x[1], y[1], z[1] and P4, consequently the variety H4(V) is locally isomorphic to the p r o d u c t
V(I)
x C4 C C 13 • C4, where I is in this case considered as an ideal
of the ring C[0-1,0-2, ~r3, 0-4, a[2], a[3], a[r], c12, c13, c14, c23, c24, c34]. One checks t h a t the subvariety
V(I)
of C 13 is irreducible, singular, of dimension 8. As a consequence,
the variety H 4 ( V ) is singular of dimension 12(= 8 + 4) at the point qo. The local equations of H4(V) being in this case slightly more complicated, we have not been able so far to o b t a i n a geometric description of the singular locus
sing(H4(y))
in
the n e i g h b o r h o o d of this element 2~ However, the c o m p u t a t i o n s enable us to prove t h a t in the n e i g h b o r h o o d of this element ~o, the open set coordinate
x[i] of
H~(V)
is dense. T h e first
the point pi contained in the s u p p o r t of an element close to qo has
for expression : x[2]
=
x[1] - (a[2] - a[310-4)(c23 - c13)
x[3]
--
x[l] - (a[2] - a[310-4)(c23 - c12)
z[4]
=
x[1] - (a[2] - a[310-2)(c34 - c13)
and one checks t h a t the differences (x[2] - z[3]), (x[2] - x[4]), (x[3] - x[4]) are not elements of the ideal I. A
7.3
Irreducibility
of
H4(V)
We saw in the i n t r o d u c t i o n of this chapter t h a t when the s u p p o r t of the element 2 E H 4 ( V ) contains at least two points, the variety H4(V) is locally isomorphic to
x' H2(V), Y x H3(V). Therefore such open subset H~(V). The c o m p u t a t i o n s
one of the two p r o d u c t s ' H 2 ( V )
an element
is the limit of elements of the
p e r f o r m e d in
sections 7.1 and 7.2 show t h a t when q is a quadruple point, the element 2 E H4(V)
128
Construction of the variety
H4(V)
A
is again in the closure of the open set H~(V). (See sections 7.1 and 7.2 for explicit deformations of a quadruple point into a simple quadruplet.) Consequently, the open set HI(V ) is dense in H4(V). The variety constructed in this way is irreducible, of dimension 12(= 4.dim V). Furthermore, the projection : H4(V)
--+ H4(V)
q ~
q
is generically a 4!-sheeted cover, because the open set
[I2]).
HI(V ) is dense in g 4 ( y )
(IF],
Appendix A In this appendix, one recalls the local equations of the variety of complete triples of V, derived in [LB1]. A
A.1 triple
Local
chart
of/-/3(V)
at
[,
where
t is a curvilinear
point
Assume that the variety V is of dimension 2. Let us denote by t~ a curvilinear triplet of support the point p of V. Such a triplet can only contain one doublet d. In the neighborhood of the complete triplet t~ = (p,p,p,d,d, d,t~ the variety Ha('--'~) is isomorphic to the graph of a morphism from C6 to fiTM. We recall here its expression. In an appropriate local coordinate system (x,y) centered at p, the triplet t~ is defined by the ideal (x 3, y). Therefore the doublet d is defined by the ideal (x 2, y). A triplet t4 close to t~ is defined by the ideal :
I(t4) = (x 3 + ~4x 2 + 9~x + ~ , y + ~x2 + ,4x + -4) One defines the doublets d12, d23 and d13 close to d respectively by the ideals : I(d12) = (x 2 + a4x + b4, - y + c4x + d4) 1(d13) = (x 2 § (a4 + a'4)x § b4 + bid, --y + (c4 + d4)x + d4 + d~4) I(d23) = (x 2 + (a4 + a'4 + a~)x + b4 + b'4 + b~, - y + (c4 + c'4 + d~)x + d4 + d'4 + d~) The points Pl,P2 and P3 in the neighborhood of p have coordinates
s4 pl
t4
s4+s~ p2
t4+t~
p3 s 4 + s ~ + s ~ t4+t~+t~
In the neighborhood of the complete triplet t'], the variety H3(--'~) is isomorphic to the graph of a morphism ([LBll pp 941-944) : C6
~
CTM
l II l II (s4,t4, s~, S4,it C4' ~4) ~+ (t4,t4,a4,b4, d4, a~4,b4, d4,d4,I a4, b4,d~," ' d4,a4,/34,'f4,#4,
130
Appendix A
defined by :
t~
=
c4~
~4
t~ =
-s~(r
a 4 ~-
--~s 4 -- S '4
54
S4(84-+-$~)
-~
-st
b~ = 4(s4+s~+s~) -~4s~
C" 4 d~
:
~4S~(S4 -~-S~-~-S~)
d4 = t4-c484
a4 = -(3s4+2s~+s~)
b~ =
~4~
~
d~ =
~4848~
//4 -- ~4s4(s4+s~)-t4-Fc4s4
=
-s4(s4+s;)(~+si+s~)
R e m a r k 15 : We have assumed that the variety V is of dimension 2. For V of dimension n, it suffices to replace the coordinate system (x, y) of V by (x, ~7), where ~7 = (Yl,..., Y~-~) (recall notation 18, p. 44). In this case, the ideal of d is (x 2, ~) and the triplet t~ has for ideal (x 3, ~7). Then a triplet t4 in the neighborhood of t~ is given by the ideal :
Similarly, the doublets d12, d23 and d13 in the neighborhood of d are defined by the ideals
I(d12) = (x~ + ~4~ + b4, - Y + c~x + d~) I(d13) = (x 2 + (a4 + a'4)x + 54 -]- Dr4, -y--I- (~4 + 4 ) x, -~- ~ "Jr-~ ) I(d23) = (x 2 + (a4 + a'4 + a~)x + b4 + b'4 + b~, -~7 + (C4 + ~4 + c~)x + ~ + ~ + ~ ) A local chart of H3(~'--V)at t~ is in this case given by (s4, t4, s~, s~, c~, s A
A.2
Local
chart
of H3(V)
a t [, w h e r e
t is amorphous
One recalls here the local equations of the variety Hs(V) in the neighborhood of a complete triplet t'~ = (p~,p~,p~,d~2,d~n,d~3, t~) where t~ is an amorphous triplet of support the point p of V and the three doublets are identical. In an appropriate local coordinate system (x, y, z) centered at p, the triplet t~ is defined by the ideal (x 2, xy, y2, z) and the doublets are defined by the same ideal (x 2, y, z). Then a triplet t4 close to t~ is given by the ideal
(x 2 + u4x + v4y + ~4,
xy + ~ z + v~y + ~ , y2 + u~x + V4" Y + w~, z + p4x + a4y + 04)
A
Local chart of Ha(V) at t', where t is amorphous
131
where w4, w~ and w~' verify the relationships :
{
~4
=
~4~; + ~4v2 - ~
W14 l/
W4
I
I
?s
- v4
II
-- V4% 4
I/
I
2
l/
U4U 4 + U4V 4 __ U~4 -- 7/,4V ~! 4!
--
Using the notation introduced in [LB1], one defines the doublets d 0 close to di~ by the different ideals :
f(dl2 )
=
( z 2 Jr- a 4 x + b 4 , - y - } -
/(d13)
=
(x 2 +
c4x zr- d 4 , - z q -
e4xZr- f4)
(c4-+c14)xq- d4+d14,
(a4 + a 1 4 ) x - + b 4 q - b 1 4 , - y +
--Z -~- (e 4 q- e ; ) Z q- f4 q- ./"4)
I(d~)
=
( ~ + (~4 + < + a~)~ + b4 + b; + b ~ , - y + (c4 + c~ + c~)~ + d4 + d; + d~, - z + (e4 + e~ + e~)x + f4 + ]; + f;')
The coordinates of the points Pl, Pu, Pa in the neighborhood of p are : I
84 P~
t4
I
84 Jr- S 4 P2
r4
t4 + t'4
II
84 ~- 84 ~- 84 Pa
t4 + t~4 + t~ I!
?'4 + rl4
r4 + rl4 + r4
In the neighborhood of a complete triplet t~, the variety Ha("-'-V) is isomorphic to the graph of a morphism from C9 to C27 (c.f. [LB1] pp 934-938) : C9 (s4, t 4 , r 4 , c4, c ~ , c ~ , v 4 , f l 4 , ~ 4 )
--+
C 27
~-~
(s4,t4,r4,s4,t4,r4,a4,
b4, d4, e4, f 4 , a 4 , b4, '
' if4,
a4, b4, d4, e4, f~ , U4, U4, ~)4, U4, V4,'04) . . . . . .
"
"
'
.
.
.
.
Its expression is : 814 ~'~ --C~V4
r
=
8'g = t~ =
-c4c~v4 -c'4v4 -c;~4(c~ + c; +
~)
T4
H
=
a4
=
--2s4 +
b4
=
84(s4 - c~v4)
d4
=
t4 -- C484
e4
=
--P4 -- 04c4
f4
=
r4 ~- 84(P4 -~- 0"4C4)
a4 b~
=
Cl4V4
:
-s4c~v 4
dl4
:
--Cl4S4
l
!
CI4V4(P4 -}- O'4(C4 q- C4 -I- C~))
c~v4
e 4I
__0.4CI4
f'~ =
s4~c'~
a~
=
cI4v4
b4I
=
--C~V4(S 4 -- V4(C ~ -[- C/~))
d~ = e:~ =
c~(-84 + v4(c; + -~4~'~
~))
fg
--
0.4c~(84 - v4(c~ +
r
u4
=
- 2 s 4 -~- u4 (c]~ - C4)
~
=
- c ~ 4 ( c 4 + c~)(c4 + 4 + 4 )
v 4"
=
l II --2t4 + 'v4(c4e~ ~- (c4 ~- cl4)(c4 ~- c 4 ~- 54) )
04
=
- r 4 - p484 - 0.4t4
Appendix B According to [BGS], [Gr], [I1], it is always possible to determine explicitly the neighborhood of H"(V) at a n-uple point .~, which one identifies with the flattener of a germ of an application. (We have at our disposal the division theorem with parameters of Galligo ([Gall) to compute explicitly the flattener of a germ of an application.) In this section, we give local equations of the Hilbert s c h e m e H4(V) in the neighborhood of non-locally complete intersections quadruplets. Recall that we have assumed dim V = 3. Then, let { be a point of the Hilbert scheme H*'(V), of support the point p of V. The ideal of Ov which defines this n-uplet is denoted by I({). One denotes by (x, y, z) a local coordinate system centered at p. Briangon, Granger and Speder (cf. [BGS], [Gr], [I1]) have found a method to obtain explicitly local equations of Hn(V) at ~ : One denotes by ( f ~ , . . . , s
the standard basis of the ideal I({) (once the direction
has been chosen). The quotient
Ov (fl, .., s
is a C-vector space of dimension n. One
denotes by A = {gi-1,. 9 e77,,}a basis of this C-vector space where e l , . . . , en are elements of Ov. Let Fi = fi + ~j'~=laijej, for i = 1 , . . . , n , where all the parameters aij are elements of C. The set of these pn parameters is denoted by (_a). Let Os = C{_a}. The family ( F x , . . . , Fv) defines a morphism at the origin of Ca x O " parametred by (x,y, z), (_a). In the neighborhood of the origin, one considers the subvariety X of Ca x ~
defined by the cancellation of the polynomials F~ (x, y, z, a_),..., Fp(x, y, z, a).
This subvariety X = V ( ( F ~ , . . . , Fv) ) possesses a projection onto O "~, which is the restriction to X of the projection from the product Ca x G"~ onto the second factor : (X,0)
C
(ca •
~',0)
Sn (c,~,0) Brian~on, Granger and Speder ([BGS], [Gr], [I1]) have shown that the Hilbert scheme Hn(V) is locally isomorphic at ~ to the flattener of II. The equations of this flattener can be obtained in the following manner, according to [Gal] - proposition 1.4.7 : One constructs a basis {(Ej) = ( E l i , . . . , Evj)} of relations between the fi. There-
Local cha.rt of H4(V) at an elongated quadruplet
133
fore, for each j, one has the equality : P
~_, E~j fi = 0 i=1
One creates in O s { x , y, z} the products : P
~tj = Z E~jF~ i=1
Then one constructs the remainders of the division of the Mj by the family ( F 1 , . . . , F~,) (cf. [Gal] theorem 1.2.7). One h a s
Hi = ~ hj~(a)~, i=1
Os{z, y, ~}
(This remainder is not zero if (F~, 7 7 7 ~ is not a Sat C%-module.) The equations {s
= 0} are the equations in O s which define H ~ ( V ) locally
at. (. So, when ~ is a locally complete intersection n-uplet, ( is defined by the ideal I(() = (fl, f2, fa) of O v . For j = 1, 2, 3, let Fj = f j + ~ ' = ~ %{e{, where the parameters aj{ are elements o f t . Then the ideal (F1, F2, Fa) defines a n-uplet in the neighborhood of ( in H'~(V) and (aji) constitutes a local chart of H~'(V) at ~.
B.1 L o c a l chart of H4(V) at an e l o n g a t e d q u a d r u p l e t Let qo be an elongated quadruplet of support a point p of V. In an appropriate local coordinate system (x,y, z) centered at p, the quadruplet qo is defined by the ideal I(qo) = (z a, zy, y2, z) of O v (cf. definition 7.(iii), p. 67). Let I(qo) = ( f l , fa, fa, f4).
(-gv In this case, the quotient ~ is a C-vector space of basis {7, 5, a:2, Y} over C. Let F1
--
x 3 + a l x 2 + a2x + aay + a4
-1:2 --
my + asx 2 + a6x + aTy + as
F3
=
y2 + aox2 + alox + a l l y + a12
F4
=
z + alax 2 + al4x + a l l y + al6
The ideal I(qo) possesses five syzygies. The coefficients of these syzygies are given by the columns of the matrix E :
E
0
0
0
-y
-z
-y
0
-z
z2
0
z
-z
0
0
0
0
y2
my
0
xa
134
Appendix B
Then, each column Ej represents a relation between the f i . Using M a c a u l a y and following the previous method, one establishes the equations in C [ a l , . . . , a16], which define H 4 ( V ) locally at %. The equations are : a4
=
2aaasa7 -- ata72 -b a7 a -- a3a6 q- a2a7 + a3a11
as
=
--aaa52 -- a5a72 q- a6a7 -- aaa9
alo
=
ala52 q- a52a7 -- 2asa 6 + a l a 9 -- aTa 9 .-}- a5al~
a12 =
a3a53 -- ala52a7 q- a2a52 + 2a5a6a7 + a3a5a9 -- a l a 7 a 9 qaT2a9 -- a s a r a u -- a62 + a2a9 + aGall
The parameters a4~ as, alo and a12 are explicit functions of the parameters ai, a2, aa, a5, a6, at, a9 and a n . Therefore, (a) = (al, a2, a3, as, a6, aT, ag, a n , a13, a14, a15, al~) constitutes a chart of H4(V) at %.
t3.2 Local equations of
H4(V) at
a spherical quadruplet
If now qo denotes the spherical quadruplet of support the point p of V, qo is defined by the ideal (x 2, x y , xz, y2, y z , z ~) where (x, y, z) is a local system of coordinates centered COv at p. Let qo = ( f l , - - . , fG). The quotient ~ is in this case a C-vector space of basis {I, ~, Y, 7}. One considers a deformation F 1 , . . . , F6 of the generators fx,.. 9 f6 in the cobasis : F1
=
x 2 + a l x + a2y + aaz + a4
F2
=
x y + a s z + a6y + a z z + as
['3
=
x z + a g x + a l o y + a n z + a12
F4
=
y2 + a13x + a i 4 y + a l s z + a16
F5
=
y z + a l T x + a l s y + a l 9 z + a20
F6
=
z 2 + a21x + a22y + a23z + a24
where the parameters ai are elements of C. The ideal I ( q o ) has eight linearly independent syzygies. The coefficients of these syzygies are given by the columns of the matrix E :
E =
0
0
0
0
0
-z
0
-y
0
0
0
0
-y
0
-z
z
--z
0
-y
0
0
x
y
0
0
0
0
-z
x
0
0
0
0
-z
x
y
0
0
0
0
x
y
0
0
0
0
0
0
Using M a c a u l a y again, one establishes the following equations of the flattener :
--a4 -- a3a9 -- a7alo + a l a l l -- all 2 -1- a2a19 Jr- a3a23
~
0
L o c a l e q u a t i o n s of H 4 ( V ) a t a s p h e r i c a l q u a d r u p l e t
135
a2a13 - aloal5 + a T a l 8
=
0
- a 1 2 + a 6 a 9 - ahalO + aloa14 - a6a18 + a l l a l 8 - aTa22
--
0
- a 1 6 + a n a l 3 - a7a17 - alhal8 + a14a19 - a192 + a15a23
=-
0
- a 2 0 + a l l a l 7 + a18a19 - a7a21 - a15a22
--
0
- a 2 4 - a92 - aloal7 + ala21 - alia21 + a5a22 + a9a23
--
0
aloal3 - a6a17 + a l l a l 7 - aTa21
--
0
a9a17 + alTa18 - aha21 + a19a21 - a13a22 - a17a23
--
0
a9a13 - aha17 + ai4a17 - a13a18 - a17a19 + alha21
=
0
a9alo + aloal8 - a2a21 - a6a22 + alia22 - aloa23
~-
0
=
0
=
0
-a 8 + aha 6 -
a92 - a]82 - ala21 + a6a21 + a l i a 2 1 - aha22 + a14a22 -a19a22
a9a23 + alsa23
- -
a T a 9 + aloal5 -- a3a17 -- a7al 8 a 6 a 9 - - a 9 a l l + aloa14 -- a2a17 -- a6a18 + a l i a l 8 -- 2 a t o a 1 9
+a3a21 + a7a21 ahal0
- -
a2a17
al0al9
- -
+
aTa22
---- 0 ----
0
a h a 9 -- a l a l 7 + a6a17 + a l l a l 7 -- ahal8 -- a9a19 + alsa19
--a15a22
---- 0
a h a 7 - - a3a13 -- aTa14 + aGal5 -- a l l a i 5 + a7a19
---- 0
a h a 6 -- a h a l l -- a2a13 -- aloal5 + a3a17 + 2aTa18 -- a6a19
+ a l l a l 9 -- aTa23
=
0
- - a l h a l S + a14a19 -- a192 + a l h a 2 3
=
0
a 3 a 5 - - a l a 7 + a h a 7 + a T a l l -- a2al5 + a3a19
=
0
=
0
=
0
a52 -- a l a l 3 + a6a13 + a l l a l 3 -- aha14 -- aga15
a2a9
--
alalO + a6al0 + a l 0 a l l
- -
a2a18
- -
a3a22
a 2 a 5 -- a l a 6 + a62 -- a 3 a 9 + a l a l l -- a l l 2 -- a2a14 -- a3a18
+a2a19 + a3a23 L e t (a) = ( a l , . . . , d 4 , . . . , d s , . . . , a ] 2 , . . . , a ] 6 t h a t a~ is o m i t t e d ) .
.... ,a~o,...,a~4)
( t h e n o t a t i o n dl m e a n s
F r o m t h e first six e q u a t i o n s , t h e p a r a m e t e r s a4, a s , al2, a16, a20
a n d a24 a r e e x p l i c i t f u n c t i o n s of t h e o t h e r e i g h t e e n p a r a m e t e r s _a. T h e s e e i g h t e e n o t h e r p a r a m e t e r s _a a r e l i n k e d b y t h e l a s t fifteen q u a d r a t i c e q u a t i o n s . O n e c h e c k s t h a t t h e s e fifteen e q u a t i o n s a r e l i n e a r l y i n d e p e n d e n t .
T h e r e f o r e , t h e s e fifteen q u a d r a t i c
e q u a t i o n s of C[a] c o n s t i t u t e t h e local e q u a t i o n s o f H 4 ( V )
at qo. ( R e c a l l ([I2], [El) t h a t
t h e s p h e r i c a l q u a d r u p l e t s a r e t h e s i n g u l a r p o i n t s of t h e H i l b e r t s c h e m e H 4 ( V ) . )
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[B2]
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[BGS]
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[Col
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IEI
EISENBUD, D. "Commutative algebra with a view toward algebraic geometry", Graduate texts in Mathematics 150, Springer-Verlag (1995).
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ELENCWAJG, G. & LE BARZ, P. Ezplicit computations in Hilb3(p2), in "Algebraic Geometry - Sundance 1986," Holme, A. , Speiser, R. Eds. Lecture Notes in Math. 1311, Springer, 1988, 76 100.
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FOGARTY, J. Algebraic families on an algebraic surface. Amer. J. Math. 90 (1967), 511-521.
[FUll
FULTON, W. Personal letter to P. LE BARZ (01/06/1987).
[FU2I
FULTON, W. "Intersection Theory," Ergebnisse der Mathematik, SpringerVerlag (1984).
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FULTON, W. & LAKSOV, D. Residual intersections and the double point formula, in "Real and Complex Singularities," Proc. Conf., Oslo 1976, Holm, P., Ed., Sijthoff & Noordhoff, 1977, 171-178.
[FMPI
FULTON, W. & MAC PHERSON, R. A compactification of configuration spaces, Ann. Math. 139 (1994), 183-225.
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GAFFNEY, T. Punctual Hilbert schemes and resolution of multiple-point singularities.
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[Gall
GALLIGO, A. Thdor&me de division et stabilitd en gdomdtrie analytique locale. Ann. Inst. Fourier, Grenoble, 29, 2 (1979), 107-184.
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GRANGER, M. Gdomdtrie des schgmas de Hilbert ponctuels, Mfim. Soc. Math. France 712 (1982-83).
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GROTHENDIECK, A. Les schdmas de Hilbert, S~minaire Bourbaki n~ IHP Paris (1961).
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HARTSHORNE, R. "Algebraic Geometry," Graduate texts in Mathematics, Springer-Verlag (1977).
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IARROBINO, A. Hilbert scheme of points: overview of last ten years, in "Algebraic Geometry Bowdoin 1985. Part 2," Proc. Sympos. Pure Math. 46 (1987), 297-320.
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KATZ, S. Iteration of multiple point formulas and applications to conics, in "Algebraic Geometry - Sundance 1986," Holme, A., Speiser, R. Eds. Lecture Notes in Math. 1311, Springer Verlag, 1988, 147 155.
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KLEIMAN, S. Multiple-point formulas I: Iteration, Acta Math. 147 (1981), 13-49.
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Index Amorphous triplet 5, 22, 49, 52, 109, 112 113, 121-123, 131 Auxiliary variety 6, 67, 76, 83, 87, 92 irreducibility 76 local chart 86, 90-91, 95, 99-100 smoothness 84, 87, 95, 100 Blowing-up 1, 12 Cartier's divisor 22, 23, 28 Chern class 1, 2, 42, 56 Chern polynomials 33 Complete quadruples variety 3, 6-8, 102-104,105-108, 112-113 birational morphism 4 irreducibility 127-128 local chart 108, 111 local equations 114 117, 127 order-forgetting morphism 4, 128 singular locus 118, 121-123, 127 Complete triples variety 2, 22, 49-50, 52 local chart 129-130, 130-131 Division theorem 79 Double class 1, 16, 21 Double formula 1, 21
Hilbert scheme local chart 44, 109, 131, 133 134 local equations 134-135 universal property 69 "Horizontal" doublets 16, 53 "Horizontal" triplets 26, 53 Incidence variety I(V) of H2(V) • H4(V) 5, 69, 71-76 complementary doublet in 5 birational morphism 76 local chart 85, 89 local equations 93 Incidence subvariety of H2(V) x Ha(V) 70, 72 Induced stratification on I(V) 70, 7176 Intermediate variety - see auxiliary variety Natural stratification of H4(V)
70
Normal bundle 14, 56, 59
Exceptional divisor 12, 18, 19, 30, 45, 53-54 Excess component 13, 23, 55, 64, 95, 99, 100-101 Flattener of a morphism 132-133
Fundamental class 3, 11, 15, 25, 28
80, 81-82,
Quadruple point 67-68, 78, 133-135 curvilinear- 68, 73, 84, 104-105 square - - 68, 73, 87, 108-109 elongated-- 68, 74, 92, 111 s p h e r i c a l - 68, 75, 96, 123 Residual application 5, 67, 83, 91,101 graph 6, 91
140 local equations
86, 89-90
Residual doublet - see complementary doublet Segre class
15, 19, 30, 33
Total Chern class 14, 32, 33, 56, 63 Triple class 3, 26, 42 Triple formula 2, 42 Universal family of H2(V) - see universal two-sheeted cover of the Hilbert scheme Universal two-sheeted cover of the Hilbert scheme 2, 12, 13, 29, 44-46, 49, 72 Virtual normal bundle 1, 2, 21, 32, 43, 56
Index of notation CH'(X) ?7/,k:
~(f) VxV Cd
'mlb~(V) H~(V) Res(n,~) 4 H;.m,,.. (V) z~ Ov R(v) n
i(v) He Res
B(v) T5
A
H4(V)
CH.(X)
[x], 1 0 F
c.(X, z) s(F), s(F, H2(Z)) OL
Chow ring, graded by codimension, 1 h-uple locus of f, ] k-uple class of f, 1 virtual normal bundle, 1 blowing-up along the diagonal, 1 dth Chern class, 1 the universal two-sheeted cover of HilbZ(V), 2 variety of complete triples, 2 complete triple, 2 residual closed point, 3 the na'ive complete quadruples variety, 4 ideal sheaf, 4 sheaf of rings, 4 the subvariety of Hilb2(V) x Hilb4(V) x Hilb2(V), 4 projection from R(V) onto Hilb4(V), 4 the incidence subvariety of Hilb2(V) x Hilb4(V), 5 second projection from I(V) onto Hilb4(V), 5 residual rational application from I(V) to Hilb2(V), 5 closure of the graph of Res, 6 restriction of II to B(V), 6 element of B(V), 6 variety of complete quadruples, 7 complete quadruple, 7 Chow ring, graded by dimension, 11 fundamental class of X, 11 the universal two-sheeted cover of HilbZ(X), 12 exceptional divisor, 12 total Chern class of the normal bundle at X in Z, 14 Segre class of the exceptional divisor, 15 canonical imbedding on the "horizontal" doublets, 16
142
cW F
total Chern class of the tangent bundle exceptional divisor, 18
Av K(V)
diagonal of V x V, 19
TW to W, 18
Grothendieck's group, 21
Hi(Z), Hilb'(Z)
Hilbert scheme of subschemes of length i, 22
E12, E23, E31, E"
Cartier's divisors on Ha(Z), 22
E2a, &~, E"
divisors related to I, 23 canonical imbedding on the "horizontal" triplets, 26
Ha(v x W), 26 H2(V), 29
~a, Ea~, g"
divisors related to
e
universal two-sheeted covering of
R
ramification locus, 29
G13, G23 H
graphs in
H2(Z) x V of natural morphisms, 29
f~lH
Hilbert scheme of subschemes of length 2, 32 sheaf of relative differentials, 32
p-*
conormal bundle, 33
C,('-'*)
Chern polynomial, 33
,5
canonical imbedding of F in H2(Z), 38
DO I
Jacobian of 0, 46
R
residual class, 56
A,,1,, Abe., A,:,,.
the "small" diagonal of I, 57 the three diagonals of ~1 x >1 x ~1 64
excess component, 55
'Hilbd(V)
Hilb'*(V), 69 Hd(V) of the simple d-uples, 69 open subset of I(V), 70 d-sheeted tautological covering of
H (V)
open subset of
&(v) m,(v)
incidence variety of subschemes of length n - 1
HS(V), Hr
i.(v) h,(v) [=(v) ,',(v)
and n, 70 strata of H4(V), 70
H~11 I(V) induced stratification on stratum of
I(V), 70
I(V), 71
stratum of I(V), 72 stratum of I(V), 72 stratum of I(V), 72
a
local chart of H4(V), 84
A Ai
H2(V), 84 H2(V), 85 local chart of B(V), 86
M(v)
excess component of R(V), 100
local chart of
local chart of
143 A
HI(v) (~) sing(H4(V)) v~
A
open subset of H4(V) containing elements having a four-point support, 103 local chart of H3(V), 105 singular locus of H4(V), 121 components of sing(H4(V)), 121 A
A
