Configuration Spaces over Hilbert Schemes and Applications

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen

1647

Springer

Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

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]

Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Dtas, Danielle: Configuration spaces over Hilbert schemes and applications / Danielle Dias ; Patrick LeBarz. - Berlin ; Heidelberg ; New 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 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 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 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