Algebraic Geometry Sundance 1986

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 1311 A. Holme R. Speiser (Eds.) Algebraic Geometry Sunda...

Author: Holme R. Speiser (Eds.)

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1311 A. Holme R. Speiser (Eds.)

Algebraic Geometry Sundance 1986 Proceedings of a Conference held at Sundance, Utah, August 12-19, 1986

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Editors Audun Holme Department of Mathematics, University of Bergen AIIdgaten 55, 5007 Bergen, Norway Robert Speiser Mathematics Department, Brigham Young University Provo, Utah 84602, USA

Mathematics Subject Classification (1980): 14-06 ISBN 3-540-19236-0 Springer-Vertag Berlin Heidelberg N e w York ISBN 0-387-19236-0 Springer-Vertag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations falI under the prosecution act of the German Copyright Law. © Springer-Vertag Berlin HeideIberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

PREFACE

This book presents some of the proceedings of the conference o n Algebraic Geometry held at Sundance, in July 1986, in the mountains near Provo, Utah. Financial support, from the National Science Foundation (grant 86 - 01409) and from Brigham Young University, is gratefully acknowledged. Normally, a proceedings volume collects writups of lecures given at the conference, based on work done earlier, and the present volume, indeed, includes a number of these. We are very pleased, however, that the bulk of this volume presents research begun or carried out right at Sundance. Some of this new work may not have been done at all, had the conference not brought together the individuals involved. Beautiful surroundings, ample and contiguous spaces for lectures and discussions, meals served right beside the working areas: all contributed to an atmosphere unusually conducive to new work. But the major responsibility for the success of the conference lay, we feel, with the participants. Their enthusiasm, their interests, their eagerness, are reflected in the papers which follow. It is a pleasure to thank them here.

Audun Holme

Robert Speiser

TABLE OF CONTENTS

1

P a o l o Aluffi: T h e c h a r a c t e r i s t i c n u m b e r s o f s m o o t h p l a n e cublcs

9

S u s a n J a n e Col]ey: M u l t i p l e - p o i n t f o r m u l a s a n d line c o m p l e x e s

23

S t e v e n D i a z a n d J o e H a r r i s : G e o m e t r y o f S e v e r i v a r i e t i e s , II: I n d e p e n d e n c e o f d i v i s o r classes a n d e x a m p l e s

51

Lawrence Ein, David Eisenbud, and Sheldon Katz: Varieties cut out by q u a d r i c s : S c h e m e - t h e o r e t i c v e r s u s h o m o g e n e o u s g e n e r a t i o n o f ideals

71

L a w r e n c e Ein: V a n i s h i n g t h e o r e m s f o r v a r i e t i e s o f low c o d i m e n s i o n

76

Georges Elencwaig and

Patrlce

Le B a r z :

Explicit computations

in

Hilb 3p2

101

Brian Harbourne: Iterated blow - ups and moduli for rational surfaces

118

A u d u n H o l m e a n d Joel Roberts: On the e m b e d d i n g s of projective varieties

147

Sheldon Katz: Iteration of multiple point formulas and applications to conics

156

S t e v e n L. K l e i m a n a n d R o b e r t S p e i s e r : E n u m e r a t i v e g e o m e t r y o f n o d a l p l a n e cubics

197

J o e l R o b e r t s : Old a n d n e w r e s u l t s a b o u t t h e t r i a n g l e v a r i e t i e s

220

F. R o s s e l l 6

235

Robert Speiser: Transversality theorems for families of maps

253

A n d e r s T h o r u p a n d S t e v e n L. K l e i m a n : C o m p l e t e b i l i n e a r f o r m s

a n d S. X a m b 6

Descamps: Computing Chow groups

LIST OF P A R T I C I P A N T S

• Paolo Aluffi. Department of Mathematics, The University of Chicago, 5734 University Avenue,Chicago, I1 60637, USA. • Patrick Le Barz. Laboratoire de Mathematiques, IMSP Parc Valrose 06034, Nice, France. • Susan Jane Colley. Deptartment of Mathematics, Oberlin College, Oberlin, Ohio 44074, USA. • Steven Diaz. Department of Mathematics, University of Pennsylvania and Piladelphia, PA 19104, USA. • Lawrence Ein. Department of Mathematics University of Illinois at Chicago, Box 4348, Chicago, IL 60680, USA. • David Eisenbud. Department of Mathematics, Brandeis University, Waltham, MA 02154, USA. • Georges Elencwajg. Laboratoire de Mathematiques, IMSP Parc Valrose 06034, Nice, Frange. • Brian Harbourne. Department of Mathematics and Statistics, University of NebraskaLincoln, Lincoln, NE 68588-0323, USA. • Raymond T. Hoobler. Department of Mathematics, City College (CUNY), New York, NY 10031, U.S.A. • Audun Holme. Department of Mathematics, University of Bergen, Bergen, Norway. • Sheldon Katz. Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA. • Steven L. Kleiman. Department of Mathematics, 2 - 278 MIT, Cambridge, MA 02139, USA. • William E. Lang. School of Mathematics, University of Minnesota, Vincent Hall 206 Church St. SE, Minneapolis, MN 55455, USA. • Ulf Persson. Department of Mathematics, University of Uppsala, Uppsala, Sweden. • Joel Roberts. School of Mathematics, University of Minnesota, Vincent Hall 206 Church St. SE, Minneapolis, MN 55455, USA. • F. Rossello Llompart. Facultat de Matematiques, Universitat de Barcelon, Gran Via 585 08007 Barcelona, Spain. • Robert Speiser. Department of Mathematics, Brigham Young University, Provo, UT 84602, USA. • S. Xambo Descamps. Facultat de Matematiques Universitat de Barcelon, Gran Via 585 08007 Barcelona, Spain.

T h e c h a r a c t e r i s t i c n u m b e r s of s m o o t h plane cubics PAOLO ALUFFI Brown University March 1987 Abstract. The characteristic numbers for the family of smooth plane cubics are computed, verifying results of Maillard and Zeuthen

§1 I n t r o d u c t i o n . The last few years have witnessed a revived interest in the search for the 'characteristic numbers' of families, i.e. the numbers of elements in a family which are tangent to assortments of linear subspaces in general position in the ambient projective space. By the'contact Theorem' of Fulton-Kleiman-MacPherson, these numbers determine the numbers of varieties in the family that satisfy tangency conditions to arbitrary configurations of projective varieties: this justifies the central role of the computation of the characteristic numbers in the field of enumerative geometry. The problem received much attention in the last century, when in fact it contributed significantly to the development of algebraic geometry. Schubert's "Kalkiil der a r N h l ~ Geometric" ([S]), published in 1879, is a compendium of the results obtained in a span of some decades by Schubert himself, Chasles, Halphen, Zeuthen and others. The validity of these achievements was soon questioned: in requesting rigorous foundations for algebraic geometry, Hilbert's 15th problem (1900) explicitly asked for a justification of the results in Schubert's book. Algebraic geometry found its foundations in the fifties; the challenge of justifying enumerative geometry had to wait somewhat longer to be accepted. By now, most of the results in the "Kalkiil der ab:zfib&er:defl Geometric" have been verified or corrected, but the enterprise is not yet completed. While rich satisfactory theories are now available for quadrics (Van der Waerden, Vaiusencher, Demazure, De Concini-Procesi, Laksov, Thorup-Kleiman, TyreI1, etc.) and triangles (Collino-Fulton, Roberts-Speiser), and much is known about twisted cubies (Kleiman-Stromme-Xambd), the families of plane curves still offer results which were 'known' in the last century and cannot be claimed such now. The achievements of the classic school are here quite impressive. By 1864 Chasles (and others) had settled conics; already in 1871 a student of his, M.S. Maillard, computed in his thesis ([M]) the characteristic numbers for many families of plane cubic curves, including cuspidal, nodal, and smooth ones. One year later H.G. Zeuthen published a series of three amazingly short papers ([Zl]) again computing the numbers for cuspidal, nodal and smooth cubits; his results agree with Maillard's. Zeuthen finally published in 1873 a long analysis for plane curves of any degree ([Z2]), giving as an application the computation of the characteristic numbers for families of plane quarries. Apparently, noone ever tried to explicitly work out higher degree cases. The problem for cubics or higher degree curves remained untouched - and therefore eventually unsettled- for at least one century. Then Sacchiero (1984) and 1,21eiman-Speiser (1985) verified Zeuthen and Maillard's results for cuspidaI and nodal plane cubits. Kleiman and Speiser's approach replicates and advances Zeuthen and Maillard's, so it is expected to lead eventually to the verification of the numbers for the fanfily of smooth cubits; but that program is not completed

yet. Also, Sterz (1983) constructed a variety of 'complete cubics', by a sequence of 5 blow-ups over the IP9 of plane eubics, giving some intersection relations ([St]). Later, I independently constructed the same variety~ by the same sequence of blow-ups. My approach was in a sense more 'geometric' than Sterz's, and I was able to use this variety to actually compute the characteristic numbers for the family of smooth plane cubics. The result once more agrees with Zeuthen and Maillard's. There is an important difference between this approach and the classical one. Maillard and Zeuthen were computing the numbers by relating them to characteristic numbers of other more special families (e.g. cuspidal and nodal cubics); here, one aims directly to solving the specific problem for smooth cubics, and other families don't enter into play. This makes the problem more accessible in a sense, but it may on the other hand sacrifice the 'general picture' to the specific result. In this note I describe the blow-up construction and tile computation of the numbers. Full details appear, together with partial results for curves of higher degree, in my Ph.D. thesis ([A]), written at Brown under the supervision of W. Fulton. A k n o w l e d g e m e n t s . I wish to thank A. Collino and W. Fulton for suggesting the problem, and for constant guidance and encouragement. §2 T h e p r o b l e m a n d the approach. Let np, n t be integers, with np + ne = 9. The question to be answered is: How m a n y smooth plane cubics contain up points and are tangent to n t lines in general positionf

The set of smooth plane cubics is given a structure of variety by identifying it with an open subvariety U of the lP9 parametrizing all plane cubits. The conditions 'containing a point' and 'tangent to a line' determine divisors in U; call them 'point-conditions'and 'line-conditions'respectively. The question then translates into one of cardinality of intersection of np point-conditions and ne line-conditions in U. One verifies that for general choice of points and lines the conditions intersect transversally in U, so that actually the cardinality of the intersection can be computed as intersection number of the divisors. The first impulse is of course to work in the lP" that eompactifies U: closing the conditions to divisors of lP9 (one obtains hyperplanes from point-conditions, hypersurfaces of degree 4 from line-conditions), and using B~zout's Theorem to compute the intersection numbers. This works if np _> 5: in this case the intersection of the divisors in IP9 is in fact contained in U, and the result given by B~zout's Theorem is correct. If np < 4, non-reduced cubits appear in the intersection of the divisors in p 9 since a curve containing a multiple component is 'tangent' to any line and clearly one can always find non-reduced cubits containing any 4 or less given points. The conclu~sion is that IP9 is not the 'right' cornpactification of the variety U of smooth cubics for this problem, bezause all line-conditions in IP9 contain the locus of non-reduced cubics. The intersection of all line-conditions is in fact a subscheme of IP9 supported over the locus of non-reduced cubits. If we could blow-up IP9 along this subseheme, this would provide us with a compaetification of U in which the proper transforms of the point- and line-conditions don't intersect outside U, and taking their intersection product would answer the original question. But performing such a task requires much non-trivial information about the subscheme, and we are not able to proceed directly.

~,Vhat we can perform without losing control of the situation is the blow-up of ]p9 along a certain smooth subvariety of the locus of non-reduced cubics. The blow-up creates another compactification of U, in which one can again find the support of the intersection of the 'line-conditions'(i.e., of the closure of the line-conditions of U). Again, a smooth subvariety -in fact, a component- of this locus can be chosen as a center of a new blow-up, creating a new compactification. The process can be repeated, under the heuristic principle that at each step, blowing-up the 'largest' possible non-singular subvariety/component of the intersection of all line-conditions shortld somehow simplify the situation. In fact, 5 blow-ups do the job in this case: a non-singular compactification of U is produced in which 9 conditions intersect only inside U. The knowledge of the Chern classes of the normal bundles of the centers of the blow-ups is then the essential ingredient needed to compute the intersections and obtain the characteristic numbers. An intersection formula (see §4) that explicitly relates intersections under blow-ups can be used to reach the result. Apparently, this step (the computation of the Chern classes of the normal bundles and their utilization to get the characteristic numbers) is the only one missing in Sterz's work. Alternatively, one can use the same information to compute the Segre class of the schemetheoretic intersection of all line-conditions in IP9, and apply Fulton's intersection formula (IF, Proposition 9.1.1]). This Segre class has interesting symmetries, which may shed some light on the internal structure of this scheme. §3 T h e blow-ups. In this section I will briefly describe the varieties obtained via the 5 blow-ups. Details are provided in [A, Chapter 2]. The diagram

~':v5

l v4

,

I V3

B4 =

e(c)

1 ,

t

B3 = S3

'

Bg~P 2 x 1P2

1 B,a# 2 x IP 2

B2

,

V~

,

S2

B1

i

V1

I

$1

,

, pg=Vo

,

S=So

,

B*~lP 2

x ~2

1 v3(lP 2 ) = B o

contains most of the notations that will be explained in this section. So is the locus of non-reduced cubics, Bo = v3(tP2) ~-* IP9 is the Veronese of triple lines. Bi will be the centers of the blow-ups, Vi will be the blow-up B~B~_IVi_I of V/-1 along Bi-1, Si will be the proper transforms of Si-i under the i-th blow-up. Z; is a certain sub-line bundle of the normal bundle NB3V3 of B3 in 1/3. A is the diagonal in IP 2 x e 2 .

Also, Ei will be the exceptional divisor of the i-th blow-up, anti ~line-conditionsin V~' will be the closure in Vii of the line-conditions of U: i.e., the line-conditions in Vi will be the proper transforms of the line-conditior~s in Vi-1. For each blow-up I will describe the intersection of all line-conditions and indicate the choice of the center of the next blow-up. The basic strategy is to blow-up along the 'largest possible' non-singular subvariety/component of the intersection of all line-conditions. In fact, the first three blow-ups desingularize the support of this intersection~ the last two separate the conditions. §3.0 T h e IP9 of p l a n e eubies. We noticed already that the intersection of all line-conditions in IP9 is supported on the locus So of non-reduced cubits. This locus is the image of a map

sending the pair of lines (A, #) to the cubic consisting of the line A and of a double line supported on #. The map IP2 x IP2 -~¢ So is an isomorphism off the diagonal A in IP2 x lP v 2; therefore So is non-singular off the (smooth) locus Bo = ¢(A) of triple lines. In fact So is singular along Bo. Bo is the center of the first blow-up.

§3.1 T h e

first blow-up. Let 1/1 be the blow-up of IP9 along B0, E1 the exceptional divisor, S1 the proper transform of So. 5'1 is isomorphic to the blow-up BgzxlP2 x 1P2 of 1P2 x IPl along the diagonal (call e the exceptional divisor of this blow-up); in particular, it is non-singular.

The line-conditions in 1/1 intersect along the smooth 4-dimensional .91 and along a smooth 4-dimensional subvariety of El. To see this, notice that tile line-conditionin 1P9 corresponding to a line ~ has multiplicity 2 along B0, and tangent cone at a triple line ~3 supported on the hyperplane of cubits containing A n 4. Thus, the tangent cones at A3 to all line-conditions in ~9 intersect along the 5-dimensional space of cubits containing ~. It follows that the normal cones to B0 in the line-conditions intersect in a rank-3 vector subbundle of NB0 p9, and correspondingly that the line-conditions in 1/1 intersect also along a lP2-bundle over Be contained in El. Call this subwariety B1, and choose it as the center for the next blow-up. B1 intersects $1 B e z ~ 2 x IP2 along the exceptional divisor e.

§3.2 T h e second blow-up. Let 1/2 be the blow-up of V1 along B1, E2 the exceptional divisor, /~1, $2 the proper transforms of El, $1 respectively. $2 is the blow-up of $1 along a divisor, thus it is isomorphic to $1 and hence to BgA~ 2 x lP2. A coordinate computation shows that the line-conditions in 1/1 are generically smooth along B1, and tangent to El. As a consequence, their proper transforms intersect in E2 along E1 n E~, which is a lP3-bundle over B1 contained in E2. Therefore the line-conditions in P~ intersect along the smooth 4-dimensional $2 and along a smooth 7-dimensional subvariety of E2. Choose this subvariety as the new center, call it B2.

~3.3 T h e third blow-up. Let 1/3 be the blow-up of 1/2 along B2, Ea the exceptional divisor, Sa the proper transform of $2.

Again, $3 is isomorphic to B,AlP2 x lP2. E3 is a lPl-bundle over/32. In each fiber of tiffs bundle there are two distinguished distinct points rl, r2: namely the intersections with the proper transforms of/~1 and E2. Now, over any point in B2 away from Ss N E3, one can find line-conditions that hit the fiber precisely at rl or precisely at r2. This implies that over such points the line-conditions in V3 cannot intersect. Thus the line-conditions in Va intersect only along the smooth 4-dimensional $3. This completes the 'desingularization of the support' of the intersection of all line-conditions, and we are ready to choose/?3 = $3 as the next center. §3.4 T h e f o u r t h blow-up. Let 114 be the blow-up of V3 along B3, E4 the exceptional divisor. The fine-conditions in V4 meet along a subvariety of the exceptional divisor E4 = IP(NB3V3). Notice that above Ba - E 3 ~ So - B 0 , E4 restricts to IP(Ns0_B01Pg). Now, the tangent hyperplanes to the line-conditions in IP9 at a non-reduced cubic A#2 E So - B0 intersect in the 5-dimensional space of cubits containing #. It follows that the line-conditions in 1/4 meet above B3 - E3 along the projectivization of a line-subbundleof 1P(Nn3-E~V3). This fact holds on the whole of B3: the line-conditions in V4 intersect along a smooth 4-dimensional subvariety of E4 = IP(NB~Va), which is the projectivization IP(£) of a line-subbundle of NB~V3. Choose IP(£) to be the next center B4. §3.5 T h e fifth blow-up. Let V5 be the blow-up of V4 along B4, E5 the exceptional divisor,/~4 the proper transform of E4. Finally, the intersection of all line-conditions is empty in Vs. The verification of this fact is similar to the one in 3.3. Here, each fiber of E~ over a point of B4 is a 4-dimensional projective space; in this 1I?4 lies a distinguished IP3, namely the intersection of the fiber with E4. Now, one can produce line-conditions whose intersection is disjoint from this lP3, and a line-condition which intersects the fiber precisely along this IP3. Thus the intersection of the line-conditions must be empty. V5 is the compactification of U we were looking for. By slightly refining the arguments, one sees that the intersection of 9 point/line-conditions in general position in V5 must be contained in U. The characteristic mtmbers are then the intersection numbers of the conditions in Vs, and one can proceed with the actual computation. §4 T h e n u m b e r s . The essential ingredients to obtain the characteristic nmnbers from the construction in §3 are the Chern classes of the normal bundles of the centers of the blow-ups. In fact this information would be enough to determine the whole Chow ring of the blow-ups; but we don't need that much. We have 9 divisors in lP9, and we wish to compute the intersection mtmbers of their proper transforms in some blow-up of lP9, once the Chern classes of the normal bundles of the centers are known. This task can be accomplished directly, by repeatedly applying the i PROPOSITION. Let V be a non-singular n-dimertsionM variety, B ~ V a non-singutar dosed subvariety of V, X ~ , . . . ,X~ divisors on V. Let V = B~BV, and . ~ , . . . X~ the proper transforms o£X1,... Xn. Moreover, let ei = eBA~ be the multiplicity of Xi along B. Then

X1 "'" X~ = ;

X1 " . X~ - .In (el[B] + i*[X1]).-- (e~[B] + i*[X~]) e(NBV)

This specializes to well-known formulas when/7 is a point, and is itself a specialization of a inore general relation among Segre classes (see [A, Chapter 1]). An elementary proof of the form stated here can be obtained by expanding

; Jl/l "'' x'~n = ;(['~1]-[-el[El)'"" ([-~r~] "~ en[E]) (E is the exceptional divisor) and recalling that y~4>0[E] i pushes forward to lary 4.2 and Proposition 4.1(a) in IF].

c(NBV) -1

by Corol-

What we need to compute the intersection numbers of the conditions in Vs is then, for each Vi: (1) The Chern classes of Ns, Vi; (2) The mtfltiplieities of the conditions in Vi along (3) The Chow ring of Bi.

Bi;

We will now indicate how this information can be obtained. As for the multiplicities, they are obtained along the construction: the line-conditions in IP9 have multiplicity 2 along the locus/30 of triple lines, while line-conditions in Vii, i > 0, are generically smooth (hence have multiplicity 1) along Bi. Also, point-conditions never contain Bi, so their multiplicities along the centers are always 0. The Chow rings and the normal bundles of the centers can be obtained as follows. B0 is the locus of cubies consisting of ~triple lines', hence it is isomorphic to ]p2; call h the hyperplane class in B0. In fact B0 is the third Veronese imbedding of p2 in ]p9: it follows that c('NB°Ip9) -

(1 + 3h) i° (1 + h) 3

B1 is a ]P2-bundle over/7o, thus its Chow ring is generated by the pull-back h of h from/3o and the class e of the universal line bundle OB, (--1). A closer analysis of the situation (see §3.1) reveals that Bl is actually isomorphic to the projectivization d the normal bundle to the locus of double lines in the IPs of conics. This determines the relations between h and e, and gives substantial information about the imbedding B1 ~ El. Ns~ V1 is an extension of NB~ E1 and N ~ V1, and one obtaines ' e'(l+ah-e)

l°

C(NB~V~) = (1 ± ) (1 + 2h - e) ? ' Be is a ]P3-bundle over B 1: its Chow ring is generated by the pull-backs h, e of h, e from B1 and 1,3" the class q0 of On2(-1). Recall from 3.2 that B~ =/~1 A E~: Le., B2 is the exceptional divisor in the blow-up of E1 along B1, and hence it is isomorphic to IP(NB~ El). This observation gives relation among h, e, ~. Also, Be =/~1 A E2 gives at once

c(Ns2V2) =

(I -k ~p)(l -I- e - ~p).

B3 = S.3 is isomorphic to the blow-up B~alP 2 x IP2 of IP2 x 1P2 along the diagonal. Its Chow ring is then generated by the pull-backs g, rn of the hyperplanes from the factors, and by the exceptional divisor e. One obtaines the relations

;

~2m2 =1, fB e2~2=-1, 3

~3ea'=-3,

3

£aeam=-3,

B e2r/12 = --1, a

B

e4 = --6"

T h e total C h e r n class of NB3Va can be obtained as ~ :

b o t h c ( T V a ) a n d c ( T B 3 ) can be

computed using the formula for C h e r n classes of blow-ups ( T h e o r e m 15.4 in [F]). T h e result is c ( N s ~ V a ) = 1 + 72 + 17m - 16e + 126rn 2 + 9 9 2 m + 21g2 - 315e2 + 105e 2 + 582•rn 2

+ 23722m - 2517e22 + 1611e2~ _ 358e 3 + 102622m 2 + 9174e2g 2 _ 3912e3g + 652e 4. Finally, t74 = IP(£) is also isomorphic to Bg~]P 2 x ~ 2 ; the C h e r n classes of N B , V4 are easily obtained from c l ( £ ) , which can be c o m p u t e d directly as 3 g + 3 m - 4e. One gets c(NB4V4) : 1 - 52 + 5rn + 18m 2 -- 272rn + 322 + 21e2 - 7e 2 -- 302m 2 + 7522rn -

225e~ 2 + 135e22 -- 30e 3 + 75~2rn 2.

-

Once this information is obtained, 5 applications of the proposition for each n u m b e r np of points ap.d n t of lines give the corresponding characteristic number. For example, the reader may now

enjoy checking by h a n d t h a t numbers of s m o o t h cubics t h r o u g h 4 points and tangent to 5 lines = = 45 - 0 - 0 - 0 - 2 4 - 24 = 976, ()r t h a t mlmbers of s m o o t h cubics t h r o u g h 3 points and tangent to 6 lines = -- 45 - O - O - O - 3 9 0 - 282 = 3424. Tile final result is the list 1

np

4

n p = 8, n t = 1

=

9, n t =

0

16

np=7, nt=2

64

np = 6, n t = 3

256

np = 5, n t = 4

976

np=4, ne=5

3424

n p = 3, n~ = 6

9766

np = 2, n t = 7

21004

n p = 1, n t = 8

33616

n p = O, n t = 9

for the n u m b e r of curves containing np points and t a n g e n t to n~ lines, agreeing with Maillard a n d Zeuthen. !i5 C o n c l u d i n g r e m a r k s .

It seems plausible t h a t the same procedure worked out here for cubics

could in principle be executed to get the characteristic numbers for s m o o t h quartics or for higher degree plane curves, b u t the usefulness of such a n endeavor is questionable at this point. Until these 'blow-up constructions' are part of a general theory, the complication of the technical details is b o u n d to keep the work at the level of b r u t e force computation. P a r t of the construction (essentially the last two blow-ups) can in fact be carried out, giving t h e first 'non-trivial' characteristic

number for smooth plane curves of any degree (see [A, Chapter 3]), but this seems to be in some sense a special case. The next 'non-trivial' number can still be computed for quartics (the results agree with Zeuthen's!), but not via a straightforward generalization from the computation for cubics ([A, Chapter 4]). Perhaps Kleiman and Speiser's approach, pointing in the direction of Zeuthen's monumental 'general theory', will strike more deeply into the heart of the problem. REFERENCES

[A] Alutt~, P., The~ia, Brown University (1987). IF] Fulton, W., "Intersection Theory," Springer Verlag, 1984. [M] Maillard, M.S., Recherche des caractgristiques des syst~mea ~l&nentaires de courbea planes d-a troisi~me ordre, Theses prfisentdes it La Facultfi des Sciences de Paris 39 (1871). IS] Schubert, H.C.H., "Kalkiil der abz/ihlenden Geometric (1879)," reprinted with an introduction by S. L. Kleiman, Springer Verlag, 1979. [St] Sterz, U., BeriihungavervoIlatiindigung fftr ebene Kurven drifter Ordnung I, BeitrS~ge zur Algebra und Geometrie 16 (1983), 45-68; II, 17 (1984), 115-150; III, 20 (1985), 161-184. [Z1] Zeuthen, H.G., Ddtermination des caract~ristiquea des syst&nes ~t&nentaires de cu~iquea, Comptes Rendus Des S&ances De l'Acad&mie Des Sciences 74 (1872), 521-526,604-607, 726.-729. [Z2] Zeuthen, H.G., Alrnindelige Egenskaber ved Syaterner af plane Kurver, Kongelige Danske Videnskabernes Selskabs Skrifter - Naturvidenskabelig og Mathematisk 10 (1873), 287-393. Providence, RI 02912

}@JLTIPLE-POIbrr FORMULAS AND LINE O ~ P I J E ~ S

S u s a n Jane C o l l e y Department of Mathematics Oberlin College O b e r l i n , Ohio 44074 U.S.A.

O.

Suppose xI

of

f: X ~ Y

X

= f(Xr).

such

Call

distinct

and

there exist

x 2 ..... x r E X

xI

a

stationary

"infinitely near" each other. geometry

singularities

and

with

f

is a point

f(Xl) = f(x2) . . . .

a strict (or ordinarN) r-fold point if all of the

call

r-fold

coalesced to become ramification points of

algebraic

An r-fo[d point of

is a map of schemes.

that

xI

Introduction

point

£.

the

determination

the development

xi's

the

are

xi's

have

(We shall say that such points lie

See 1.7 below.)

concerns

if any of

Modern multiple-point

theory in

of

of

of enumerative

the

various

formulas

for

loci the

these

intersection

classes of them. We will not attempt instead mention multiple-point double-point Laksov

only a theory

began

A

general

"method

Alternative

of

the Hilbert

scheme,

treatment

stantially

to g i v e ,

by

revived

Kleiman

in

terms

suitable

of

the

maps.

and g e n u s

g

[K1]

us

the [K2].

thereof,

using

by K l e i m a n [K3~ a n d m e t h o d , b e g a n some Ran IRa] h a s t a k e n

which

formulas in

are

valid

under

sub-

However, Ran d o e s n o t A'(X).

It

is important

a satisfactory

general

of maps w h i c h h a v e S 2 - s i n g u l a r i t i e s .

cite for

and

i n t h e smooth c a s e , a t r e a t m e n t o f b o t h

t h a n t h o s e g i v e n by K l e i m a n .

determination

- g

the

and

o f e n u m e r a t i v e p r o b l e m s may be t a c k l e d by c a s t i n g

Let

5 = 1/2(d-1)(d-2)

refined

theory

in

Most r e c e n t l y ,

to n o t e t h a t n o n e o f t h e t e c h n i q u e s m e n t i o n e d a b o v e y i e l d s

A wide v a r i e t y

Contemporary

and

higher-order

principally

formulas

give any mechanical procedures for generating

treatment of multiple-points

ideas.

t h e o r y and a p p l i c a t i o n s

[Ro].

multiple-point

weaker h y p o t h e s e s

ordinary

initiated

multiple-points

stationary

who

and

in the c o n t e x t of the i t e r a t i o n

a new a p p r o a c h t o w a r d s i t e r a t i o n and

ILl,

been developed,

Roberts,

of t h e work on s t a t i o n a r y

ordinary

of

was

to m u l t i p l e - p o i n t

have also

Le B a r z [LB1], [LB2].

Laksov

figures

Further refinements were made by both Fulton and

iteration"

approaches

recent history of the subject, but

the principal

with

formula of Todd.

IF-L].

so-called

to give a complete few of

of

appropriate

some e x a m p l e s .

t h e number

having only nodes for

6

multiple-point One c a n

recover

the problems

singularities Clebsch's

formula

of nodes of a plane curve of degree

singularities

from t h e d o u b l e - p o i n t

of

d

formula

10

applied

to the normalization map of the curve.

The Rierrann-Hurwltz formula is

nothing more than a special case of a general stationary double-polnt formula (see the formula for

n(2 )

in §3).

Finally. other formulas, both classical and new.

for lines having prescribed contact with varieties in

pn

may also be deduced

(see loll. [~1]. [ ~ 2 ] ) . This article

consists

of a

method i n [K2] t o g e n e r a t e o f a new a p p l i c a t i o n formulas for line tailed

proofs

are

i n §1.

and

themselves

Finally,

i n §4 we g i v e

formulas

to

and

lane

§36).

§§1-3

of

in

are

a sketch

coincidence

the major steps:

The

general

de-

set-up,

multiple-point

the

classes

s h o u l d be i n t e r p r e t e d

valid.

~3 c o n s i s t s

and

of

the

of t h e main i n g r e d i e n t s of t h e c o m p u t a t i o n s . multiple-point

complex p r o b l e m .

would

like

to

thank Robert Speiser

for

arranging

a t S u n d a n c e a n d b o t h Audun Holme a n d R o b e r t S p e i s e r

I.

iteration

and also

of t h e a p p l i c a t i o n of s t a t i o n a r y

here received

timely attention.

to Llnda Miller of Oberlin College for her careful

a magnificent

for ensuring

that

T h a n k s s h o u l d a l s o go

preparation

of this manuscript.

Set-Up and Notation (see [K2]. §4 and [C2], §~1-2)

f: X ~ Y

be a separated map of schemes.

inductive construction: new spaces

outline [C'2].

classes

formulas

a description

the mathematics described

Let

formulas,

the stationary

how t h e s e

resulting

an outline

We

appear

the definition

the

formulas

The a u t h o r

in

I n §2 we d e s c r i b e

under what conditions

the

multiple-point

complexes (see [Sch],

notation,

conference

stationary

of Kleiman's

of t h e t h e o r y to t h e p r o b l e m of c o m p u t i n g c e r t a i n

of the results

necessary given

sketch of an adaptation

set

XO:= Y,

XI:= X.

fr : Xr+1 ~ X r

Xr+ 1 and maps

fo:= f

Consider and,

for

the following r ~ 1,

define

from the diagram below.

Jr+l ~Er+l:= p-l(Ar)

Pl Xr ~

I

! XrXXr_iX r

")A r

(l.l) ideal sheaf I

r

fr-I Xr-1 ~

Xr

fr := P2 p-

This construction defines XrXXr_lX r.

Xr+ 1

Note that when

happens, for example, if

f

Ar

as the residual schaae of the diagonal

Ar

in

is regularly embedded in the £1bred product (as

is a smooth morphlsm), then

Xr+ 1

is the same as the

11

blowup o f t h e d i a g o n a l set

Er+ 1

need not.

equals

P(Ir/I~)

in general,

For

r ~ 1

covering

in the fibred and has

be a divisor

define

product

([K2],

OXr+l(1)

in

on

for

p.

ideal

28).

The e x c e p t i o n a l

sheaf.

However,

Er+ 1

Xr+ 1. i r : Xr+ 1 ~ Xr+ 1

the switch involution

of the self-map

2.2,

XrXXr_lXr

that

reverses

to be the natural

coordinates.

Then

ir

has

the following properties:

ir

[ Er+ 1 = i d

i r 0x

(1.2)

( 1 ) = 0x r+l

(I),

(1.3)

r+l

frir = pl p in (1.1), if

f

is proper,

(1.4)

Y

noetherian, then for

r ~ 2,

s ~ i (1.5)

is~(fs+l "'" fr+s-1)~[Xr+s] = (fs+l ""fr+s-1)~[Xr+s]" (Note that f proper implies that f is, too, for s > 1.) s

The s c h e m e fibres order

of

f,

Xr

may b e s e e n t o p a r a m e t r i z e

including

to identify

those points

r-tuples

which lie

of points

ordered

r-tuples

"infinitely

in particular

near"

infinitely

of points

in the

one another.

In

near configurations,

we n e e d t h e f o l l o w i n g .

Definition

1.6.

Let

a = (a 1 ..... ak)

be a nondecreasing

partition

of

r.

k Set

bs =

_~

a~

Ta C Xr

and define

by

~=s

Ta:=_ (fr_lJr)-I . ..(fb2+iJb2+2)-If~(fb2_IJb2)-1.. -['fb3+13b3+2)",-I£-1b3 •

.

•

-I

.

-I

fa k (fak-13ak)

Note that if

a = (I,I ..... I),

Definition/Proposition Na:= flil...fr_lir_l(Ta)

then

1.7

x

-

Ta = fr l ' " f

([CR],

is a point

points of the fibre through

-..(flJ2)-l(x)

x 6 X

2.3).

l(x) = Xr"

A

geometric

such that there exist

point r - 1

and also such that

aI

of the points (including

x

a2

of the remaining

points are infinitely near each other,

r - aI

itself) are infinitely near each other,

of other

12

a3

of the remaining

the remaining

ak

In the propositon should

be

taken

r - a I - a2

points

are

above,

to

mean

subscheme of the fibre

p o i n t s a r e i n f i n i t e l y n e a r each o t h e r ,

infinitely

the phrase

that

"a. points J points determine

the

with geometric

support

Now we d e f i n e the s t a t i o n a r y c l a s s complete i n t e r s e c t i o n morphisms example, [K2], § I ) .

Definition assume that

Let

1.8. fl .....

= j s ~ ( f s _ l J s )~

£

fr-1

are

is

defined)

class our

J

local

a map ( s e e ,

for

Set

of nonnegative

codimension and

and Js

Then define

f~

f~ . . b 2 ( J b 2" " " J b 3 + 2 ) 5 3 • . . f ~a k ( J a k • . J 2 ) [ X ]

that

:= ( f l .

f r _ l.) ~ [ X r ].

in light

Definition

hypotheses

fl ..... 2.2. ( s e. e

fr-1 Note

[K2],

are also

§5)

all

lci's

that

This

(so

that

na

Kleiman defines

class

is

a

t h e same a s

of 1.5.

2.

so

this

Significance

that

under which the multiple-point throughout

maps) and codimension of

F = (flil)~...(fr_lir_l)~

mr = m r ( f )

Assume

length-a.

point.

R e c a l l the b a s i c n o t i o n s of

a

lci's.

of

universally

"curvilinear"

to a single

all

be part

We p r o v i d e

each other"

lci

the assumption

1)'

a

near

be a projective

will

n(1,1 .....

n .

is as in 1.6.

bs

n a_ : = F ( J r - . - J b 2 + 2 )

We r e m a r k t h a t

(lci

equal

infinitely

Y be n o e t h e r l a n , d i v i s o r i a l , and u n i v e r s a l l y c a t e n a r y .

Let

where

near each other.

the

formulas section

catenary and that

£

of n a

class

na

what

we w a n t a n d

o f §3 a r e v a l i d , that

Y

is a projective

I}e£inition 2.1 ([K2], 4.3, p. 39).

represents

f

is

noetherian,

divisorial,

ici of n o n n e g a t i v e

is r-generLc

and

codimension.

of c o d i m e n s i o n

n

if

f s

is a n Ici of c o d i m e n s i o n

Definition

( r ;a_O-gener~c

2.2. if

n

Let

for

a

0 _( s (_ r-1.

be

a

partition

o£

r.

Then

we

say

f

is

13

(i)

there

exists

r-generic (here (ii)

Note in

closed

and

nr(X )

such

=

r-k

subscripts

in

Js

S

of

Y

COdx(f-l(s),x)

that

+ ..,

and

Js

(r-1)n(x) adapting

=

the

proof

that

] x 2 .....

1

for

1 . S o£

+ r-k

such

fix

> nr(X ) + r-k

+ n(Xr)

cod

in Definition

the quantity

By a p p r o p r i a t e l y

X.

subset

:= m a x { n ( x 2 )

cod(Ta,Xr)

that

show the

-

f-I(s)

> (r-1)n(x)

is + r-k

x r E f-If(x)}).

all

s

which

appear

as

n a.

is

the

"expected"

of

Proposition

codimension 5.3

of

of

[K2],

Na

we c a n

following.

Proposition N

a

2.3.

If

f

is

(r;a~-generic,

then

the

support

of

na

is

all

of

a

few

.

a

Unfortunately, restricted the

cases

1

dlmk(xl~f(x ~

(r;~)-generlcity

the

original

map

singularity

) ~ 2.

property f

has

any

is

the

S2(f)

By m o d i f y i n g

the

inequalities

can

only

obtain

in

S2-singularities. set used

of in

x the

E

proof

Recall

that

such

that

X of

Proposition

J

of

[K2],

we h a v e :

Proposition for

if

Thom-Boardman

•

5.2

the

r

> 4,

following

2.4.

and

S2(f)

f

g 9,

is

(r;aJ-generic

then

r,

o£ c o d i m e n s i o n

a = (a 1 .....

ak),

n > O,

and

n

n

must

constant lie

in

the

range:

r = 1,2,

or 3,

r = 4,

r = 5,

r = 6,

r > 7,

A major parametrize

If

defect

~,

n

arbitrary;

r - k = O,

n = 0,1,2,3;

r - k = 1,

n = 0,1,2;

r - k = 2,

n = 0,1;

r - k = 3,

n = O;

r - k = 0,1,

n = 0,1;

r - k = 2,3,

n = O;

r - k = O,

n = 0,1;

r - k = 1,2,3,

n = O;

r - k = 0,1,2,3,

n = O.

with

ordered r-tuples

the

iteration of points

set-up in

the

of fibres

1.1 of

is f.

that

the

schemes

As a r e s u l t ,

when

X

r f

14

has positive (r - 1)!

codimension,

times.

A similar

stationary

classes.

by 3.3

[K4].

Then

of Xr

((x 1 ..... Xr}

m counts each point in its support r o£ " o v e r c o u n t f a c t o r " a r i s e s when c o n s i d e r i n g the

sort

To d e t e r m i n e Suppose

embeds

in

Xr) e X [ r ]

so that,

Kleirr~m's class

f

this

has

X[r]

:=

no S2-singularlties

al

where

x i g xj

if of

a point

of

factor.

Then it

which arise

of such automorphisms Explicitly,

:

(a

is possible

a

a i ( aj

(r-l)!

factor

then.

under

if

i ( j.

for

mr}

coordinate

ak)

.....

(ql-1)!

in

h a := 1

a

the number of nontrivial

permutations fixed

of

ha .

provides

The n u m b e r

the overcount

as

.

.

.

.

.

a

.....

q2

genericity

embeds

xal+...+a~_l+

to c o u n t

a

)

qp

Then the overcount

is

appropriate

codimension.

maps t o t h e p o i n t

the first

a..

suggested

ak

a = (a 1 .....

.....

Ta

al+a2,,

Ta

ql where

fact,

from nontrivial

which hold write

a

and positive

a2

i ~ j. Ta

In

xal+l

a1

automorphisms

we may f o l l o w a n a n a l y s i s

Xxy...XyX.

] x 1 ....

generically,

factor,

factor

for

na

q2!*--qp!.

For example,

assumptions,

the overcount

(analogous

if

to the

a = (3,3,5,5,5),

factor

for

na

is

for

m r that

6 = (2-1)!3! Finally,

we p o i n t

must be modified, r-k

out

but

that

they

c o d f = O.

then the overcount

be modified

not

for

na

if

factors a

is

such

) O.

3.

Multiple-point (r;a)-generic,

formulas

may

now

positive-codimension Proposition

2.4)

and that

the

be

The Multiple-point

for

(r;a)-generic

below,

ck = Ck(Vf), formulas

for

the

m4

We

maps

first

which

the major steps

n = cod f, where

Formulas

classes

computed,

and then outline

In the formulas class), Note

if

need

v£

may

appearing

below

valid

state

the

have

(the

the virtual in

when main

f

is

results

for

S2-singularities

needed for

mr = m r ( f ) denotes

na ,

(see

the computations.

ordinary normal

the cases

multiple-point bundle

where

of

f.

n = 2 or 3

15 differ

slightly

were r e c e n t l y Doubte-po{nt

from t h o s e g i v e n on p. 48 of [K2]. o b t a i n e d by Kleiman u s i n g H i l b e r t

formulas

v e r s i o n s below

(r = 2)

n(1,1 ) = m 2 n(2 }

The ( c o r r e c t e d )

scheme methods.

= f f~m I -

= Cn+Im I

Trtpte-po~nt forsmlas ( r = 3) n-1 2n_Jcjc2n_jm 1

n ( 1 , 1 , 1 ) = m3 = f f~m2 - 2Cnm2 + j--O

n-I

n(1,2)

2n_Jcjc2n_j+lm 1

= f f~Cn+lm 1 - 2CnCn+lm 1 J---O

n 2 = Cn+ 1 m1 +

n{3 )

~

2 n JcjC2n+2_jm 1

j---O

Qundrupte-potnt

formulas ( r = 4)

n = 1

n ( 1 , 1 , 1 , 1 ) = md = f~f~m 3 - 3clm 3 + 6c2m 2 - 6(ClC2+2c3)ml

n = 2

n ( 1 , 1 , 1 , 1 ) = f~f~m 3 - 3c2m 3 + 6(ClC3+2c4)m2 - 6(ClC2C3+2c12c4+10ClC5+3c2c4+12c6+c32)ml

n = 3

n ( 1 , 1 , 1 , 1 ) = f~f~m 3 - 3c3m 3 + 6(c2c4+2ClC5+4c6)m2 - 6(c2c3c4+2c22c5+10ClC2C6+26c2c7 2 2 +3ClC3C5+12Cl c7+60ClC8+9c3c6+72c9+5c4c5+clc4 )m 1

n = 1

n(1,1,2 )

= ( f ~ f e c 2 - 2 C l C 2 - 2 c 3 ) m 2 - f~f~clc2m 1 - c l f ~ f c2ml - 2f~f~c3ml + (4c12c2+4c22+12ClC3+12c4)ml

n = 2

n(1,1,2 }

= ( f ~ £ ~ c 3 - 2 c 2 c 3 - 4 C s - 2 c l c d ) m 2 - 4f~£~c5ml - 2f~f~ClCdml - f~f~c2c3m 1 - c2f~f~c3ml + (12ClC2C4+4c22c3+4ClC32)ml + (12c12c5+20c2c5+14c3c4+52ClC6+56c7)ml

n = 1

n(2,2 )

2

2

= ( c 2 f f~c2-4ClC 2 - 2 c I c3-8c2c3-10ClC4-12c5)m 1

16

n = 1

2

~

= f f ~ c 2 m1 + 2 f f~c4m 1 + f f ~ c l c 3 m 1

n(1,3 )

- 3(ClC~+C12C3+2c2c3+4cle4+4Cs)ml

Quintuple-point f o r m u l a s n = 1

n(i,1,1,1,1

( r = 5) ) = m5 = f f m4 - 4 c l m 4 + 12c2m 3 - 2 4 ( c l c 2 + 2 c 3 ) m 2 + 2 4 ( c 2c +5e c +6c +c 2)m 1

n = 1

n(1,1,1,2

2

1 3

4

2

1

= (f~f~c2f~f~-4f~f~clc2-2ClC2f~f~-2clf~f~c2-6f~f~c3)m2

)

+ (12c12c2+8c22+28ClC3+24c4)m2 + (2clf~f~clc2+2c12f~f~c2+2f~fe12c2+2c2f~f~c2)ml + (4f~fc22+dclf~f~c3+lOf~f~clc3+12f~f~c4)m1 + (-12c13c2-68c12c3-40ClC22-72c2e3-168ele4-144Cs)ml

formula

Sextupte-potnt n = 1

( r = 6)

n(1,1,1,1,1,1

) = m6 = f f~m 5 - 5 e l m 5 + 20e2m 4 - 6 0 ( C l e 2 + 2 c 3 ) m 3 + 1 2 0 ( c 2c +Se c +6c +c 2)m I 2 1 3 4 2 2 - 1 2 0 ( c 3c +9c 2c +26c c +3c c 2+8c c +24c )m 1

In essence, results. and set

the formulas

More s p e c i f i c a l l y , mI = [ X ] .

2

1

above are generated recall

3

1 4

1 2

by a p p l y i n g

the definition

of Kleiman's

T h e n we may make t h e same d e f i n i t i o n s

for

2 3

various class fs

5

1

operational mr = m r ( f )

where

s > 1,

namely

mr(fs)

:= (fs+l "'" fr+s-l)~[Xr+s]

ml(f s)

:= [ X s + l ] .

We h a v e t h e f o l l o w i n g

is~mr(fs)

Note that

results

for

r > 2,

s ) 1:

= mr(fs)

fs ml(fs-1)

fs~mr(fs)

three

for

= ml(fs)'

(3.1)

assuming

fs

is an lci

(3.2) (3.3)

= m r + l ( f s _ 1)

(3.1)

is simply a restatement

of ( 1 . 5 ) .

We a l s o

need the following.

17

Proposition

3.4.

If

f

is

r-generic

of

codimension

n ~ O,

then,

for

1 <s_(r-1,

(a)

(fsls)~fs

= £s-1 fs-l~

(b)

(fsis)~Cl(OXs+l(1))kfs

(c)

(fsis)~Ck(Ufs)

- C n ( V f s _ 1)

~ = -Ck+n(Vfs_l )

= Ck(Vfs_l)(fsls)

~

k-I k-j ~ [ I (n;J)]cj(Vfs_l)(fsis)wCl(OXs+l(1))k-J

+

j=o ~--o Proof•

Formulas

intersection (with

s = 1)

is just

Formula 3.3 for

(a)

theorem and

m 's. r m with r calculated:

is

the

for

and its

(b)

follow

proof

Once t h e d o u b l e - p o i n t

m2(f )

:=

feature

from t h e s e t - u p S

for

formulas results,

s

recursive

Formula

(c)

by 3 . 2 ,

of this

recursive

procedure

direct

na

with

in

r-k

> O.

cod(Es,Xs) = 1

formulas is easily

by 3 . 4 ( a ) .

an[X ]

ever explicitly

is

formulas

determined,

[X]

fl~il~fl

This

when

the residual

formula itself

by 3 . 1 ,

1.1 w i t h o u t

valid

of

45).

method f o r g e n e r a t i n g

The d o u b l e - p o i n t

> 2.

for

in

p.

fl~[X2] = fl~ilMEX2]

= f~f~[X] -

X

version

Lemma 5 . 5 ,

formula is explicitly

may b e d e r i v e d .

=

An i m p o r t a n t

[}(2],

Lemma 5 . 7 o f [}(2].

t h e key t o K l e i m a n ' s

r > 2

from a g e n e r a l

(see

introducing

contrast To b e g i n

(so that

is that

to

the

with, Es

m may b e c o m p u t e d r the iteration schemes

way

in

we n e e d

is a Cartier

w h i c h we compute the

following

divisor

in

-Cl(O X (1)) • [Xs] = [Es]

easy Xs):

(3.5)

S

js~j s a = -c1(0 X (1))

• a

(3.6)

s

where

a 6 A (Xs)

and

Js

is the inclusion

of

Es

in

Xs .

is~Cl(OXs+l(1)) = cI(OXs+I(1)) The

stationary

multiple-point

formulas

(3.7) may

then

be

computed

by

starting

with

18

Definition involving

1.8

of

only

f,

However,

a

intermediate there

na

and u s i n g

X,

and

i s no g e n e r a l

multiple-point for

n

recursive

m . r

point

Xs ,

formula

0 ( s ( r,

method to s h o r t e n

Currently

Indeed,

if

there are

those given above,

In

classes

the case

formulas

of Kleiman's

(see

stationary It

set-up

have o f 1.1

worth

in

the

2.2

hypotheses the

is

C

X

correct

G in

it

use

of

a~

the

At p r e s e n t

procedure f o r g e n e r a t i n g general

expressions

multiple-point

class

is

must be computed explicitly.

Katz has

recently

closer

of

~ = ~(P,H) P E p3 under

the

Generically,

the

necessary

and

Y

greatly

to a general

do n o t

line

C

C

seem

to

~, n

reduced

the

form f o r

the

apply

to

the

~

some o f t h e

Problem.

Enumerate

possible

planar

pp. 269-270 of [ S c h ] .

X

in

of

lines

in

pencil

embedding, consists all n

In

in

§3

X x Y

of to

be

Ran h a s r e f i n e d

the projection

by using

complex of degree

the multiple

pencils.

this

to of

n,

from

Ta

of

the

a variant

(see IRa],

as

the

§6).

H

under

the

in

points

Plficker complete

n. of all

lines

(which contains

line n

a hypersurface

(ideal-theoretic)

consisting

a

is,

Then,

of degree

be d i s t i n c t . points

that

p3.

in a fixed plane

will

definition

formulas

to Line Complexes

the planar

C N ~

points

the

only that

He d o e s

with a hypersurface

Plficker

cumbersome for

are both smooth varieties,

may be d e f i n e d

which lie the

pencil

rather

to requiring

Application

denote

these

for which pencils

that

t h e g r a p h embedding o f

p5,

p5

fixed point

all

then

always

G = G(1,3)

in

corresponds, contains

the

derivation.

no i d e n t i f i a b l e

techniques

codimension.

denote a general

Grassmanniem

intersection

not

essentially

involving

embedding of

Let

S.

Katz's

remarking

in

d.

Let

mr ,

However,

In the case where

the validity X

its

classes.

(r;a)-genericity

to

terms

the calculation.

needed and i s c o n s i d e r a b l y

[Ka]).

is

valid.

requires for

a formula for a particular

n e e d e d a n d i s n o t one o f

amount of c a l c u l a t i o n

a formula containing

we h a v e o n l y p r o v i d e d a n e f f e c t i v e

formulas.

or

a

r-fold

schemes

We e m p h a s i z e t h a t

to generate

Y.

stationary

iteration

3.1-3.7

G.

through a

P).

Thus,

(i.e.,

The q u e s t i o n

lines is

Then unless in

C p3).

to determine

coalesce.

coincidences

of

C n ~

particular,

verify

as

~

Schubert's

varies results

over on

19

More

specifically,

Eili2...im, that

where

among

the

coincide .....

Schubert's

i k > 1 (k = 1 ..... n

points

points

im

since

particular,

of

formulas

m).

This

~,

i1

C n

coincide

The p r o b l e m d e s c r i b e d classes,

coincidence

(see [Sch],

are closely

we n e e d t h e f o l l o w i n g

the

symbol r e p r e s e n t s points

symbol

the condition

coincide,

i2

points

262).

p.

a b o v e may b e s o l v e d u s i n g

these classes

involve

related

the stationary

to Schubert's

multiple-point

E

conditions.

commutative diagram whose entries

In

are explained

below.

C c

The ~ p Re

j : C C---*G

variety

F

~rametrize

is just

point-line-pl~e

fit •

pencils. variety

is defined

~ps

f

and

that

~

is 5-dimensional.

It

is

addition,

g

not f

that

{or

C

m

are

concerning for~las ~e the

difficult

f

for

F C

similar

~d

g

are

J:

~ ~ I

the

to

a n d h a s no for~las

symbol

that

~us,

C

f

~d

~d

to

~.

for

is "generic"

In

a

line

in the sense

satisfying

Schubert's

namely

5 - ~(ik-1 )

end o f at

Note

Then t h e

valid

corresponds

in principle

the

= C n ~(P,H)"

di~nsion,

the

g

are

~

E. 11---i m {see

{complete)

the inclusion,

f

pencils

have expected

whose p o i n t s

S2-sin~larities.

for

provided

M).

is

f-I(p,H)

E F,

p3

the obvious projections.

(P,H)

par~etrizing all

in

6-dimensional

of

M = q l !- " ' ~ !

where

the

restrictions

n,

no p e n c i l s ,

with

is

to

§2 f o r least,

a

the

class

discussion

all

o£ t h e 43

on p p . 2 6 9 - 2 7 0 o f [ S c h ] m y b e c o ~ u t e d . author

computations

determine

that,

of

flag variety I

~ := g - l { c ) ,

Moreover,

factors

~d

of degree

of points

1 f~{im, .... il )

p3

multiple-point

which contains

empty).

point-pl~e

the respective

finite,

conditions

on

see

stationary

the sets

Eil...i

to

of the line complex in the Crass~nni~.

The s p a c e

by

just

hard is

appropriate complex

are

:C

the inclusion

is the 5-dimensional

the planar

The v a r i e t y

J

c{vf) to

see

has already will

verified

appear

~d

this

that

~:

c~

several

elsewhere

(see

of

the

b e c o m p u t e d from

I ~ F

is

a

formulas

[~]).

~l-bundle

~d

Briefly,

c(vj)

~d

so t h a t

the details one

first

It

is not

e{T~). the

of

must

relative

tangent

20

bundle

T~

fits

into

the standard

exact

0 ~ ~I ~ where

~

is

quotient

$3/S 1

Schubert's

bases

Since

•

and

F

F

on

F.

to describe

t h e Chow r i n g s

codimension

such that

bundles

we n e e d

for

0 0 i ( 1 ) ~ T7 ~ 0,

on

o£ u n i v e r s a l

formulas,

of suitable

include

the rank 2 bundle

sequence

of flag

versions

In fact,

Then to finish

the various

(See,

is

the

i n §3.

of

above in terms

for

the multiple-point

of the ones appearing

~

the verification

Chern classes

varieties.

are both 5-dimensional, 0

I = P(~).

example,

formulas

[E].)

we n e e d

These formulas

are:

n(2 ) = clm 1 n ( 3 ) = ( c 1 2 + c2)m 1 n ( 2 , 2 ) = ( C l £ ~ f Cl - 4 c i 2 - 2 c 2 ) m I . The formulas with

above for

n = O.

n(2 )

p. 5 2 ) .

It

is given

was d e r i v e d

i n 4 . 1 7 o f [C23.

However, Schubert's

n(3 )

0

formula

the

13-43 requires

of all

formulas

cases

o f t h e o n e s i n §3

appears

i n [C1]

codimension

formula

for

approach

to

the use of stationary

must

the formulas

is a very

first be derived.

formulas,

we would need codimension

n(2,3)"

n(2,2,2)'

of codimension

special

n(2,2 )

(see

2.4a,

n(2,2 )

the

which

of

calculation

formulas

in addition

to

In light of the remarks made at the end of §3, this means

the verification

n(2,5)' n(3,4)' we a n n o u n c e t h e

just

for

multiple-point-theoretic

formulas

multiple-point

are

from an arbitrary

the three given above. that

and

The c o d i m e n s i o n

n(5)'

0

formulas

n(2,4)'

process,

since new

to verify all

for the following classes:

n(3,3)'

n(2,2,4)' n(2,3,3)' f o l l o w i n g new s t a t i o n a r y

involved

In particular,

n(2,2,3)'

the

n(4 ),

n(2,2,2,2)' n(6)' For a start,

n(2,2,2,2,2)"

n(2,2,2,3)" formula, valid

for a (4;(4))-generic

map

O:

n ( 4 ) = ( c 1 3 + 3ClC 2 + 2 c 3 ) m I • This

formula

may b e

appropriate

Chern

verification

process

Finally, of

the

are,

of

to

course,

not

formula

check

and

Schubert's

intersection

becomes largely

we r e m a r k t h a t

multiple-point

intersection pencil

used

classes

loci valid.

if

the

formulas.

C

contains

wrong dimension

However,

& l a Le B a r z

to the coincidence

13-15. have

Of

been

course,

once

determined,

the

mechanical.

the complex

have

formulas products

[LB3]

it

may b e

a pencil

and

possible

to determine

the

the to

~,

resulting use

contribution

the

t h e n some formulas residual of

such a

21

The author is most grateful to Steven Kleiman for bringing this question to her attention

( s e e a l s o [KS]).

Bibliography

[c1]

S. J , C o l l e y , " L i n e s h a i n g s p e c i f i e d c o n t a c t w i t h p r o j e c t i v e v a r i e t i e s , " Proc. o f the 198~ Vancouver Conf. i n A l g e b r a i c Geometry, J . C a r r e l l , A.V. G e r a m i t a , P. R u s s e l l , e d s . , pp. 47-70, CMS-AMS Conf. P r o c . Vol. 6, Amer. Rath. S o c . , P r o v i d e n c e , 1986.

[c2]

"Enumerating stationary m u l t i p l e - p o l n t s , " Advances in Mathematics.

[c3]

to

appear

in

., " C o i n c i d e n c e f o r m u l a s f o r l i n e c o m p l e x e s , " i n p r e p a r a t i o n .

[E]

C. Ehresmann, "Sur la topologie de certalns espaces homog~nes," Ann. o£ Math. (2) 35 (1934), 396-443.

[F-L]

W. Fulton and D. Laksov, "Residual intersections and the double point formula," Real and Complex Singularities: Os[o, 1976, P. Holm, ed., pp. 171-177, Sijthoff & Noordhoff, Alphen a an den Rijn, 1977.

[K~]

S. Katz, "Iteration of multiple conics," these proceedings.

[KI]

S. L. Kleiman, "The E n u m e r a t i v e t h e o r y o f s i n g u l a r i t i e s , " Real and Complex S i n g u l a r i t i e s : Os[o, 1976, P. Holm e d . , pp. 297-396, S i j t h o f f & Noordhof£, Alphen s a n den R i j n , 1977.

[K2] (1981),

[K3]

"Multiple-point 13-49.

point

formulas

formulas

I:

and

iteration"

applications

Math.

Acta

to

147

, " M u l t i p l e - p o i n t f o r m u l a s f o r maps," E n u m e r a t i v e Geometry and C l a s s i c a l A l g e b r a i c Geometry, P. Le Barz and Y. H e r v i e r , e d s . , pp. 237-252, B i r k h E u s e r , B o s t o n , 1982.

[K4]

, "Plane forms and multiple-point formulas," Leaf.

N o t e s tn

Math. 9~7, pp. 287-310, Springer, Berlin, 1982.

[KS]

"Open problems,"

lecture at

this conference,

August 14,

1986.

[L]

D. Laksov, " R e s i d u a l i n t e r s e c t i o n s and T o d d ' s f o r m u l a l o c u s o f a m o r p h i s m , " A c t s Math. 140 (1978), 75-92.

[LBI]

P. Le Barz, "G~om~trie ~num6rative pour l e s m u l t i s @ c a n t e s , " L e c t . t n Math. 683, pp. 116-167, S p r i n g e r , B e r l i n 1978.

[u~2]

"Formulas multls~ca_ntes pour les courbes gauches q u e l c o n q u e s , " E n u m e r a t i v e Geometry and C l a s s i c a l A l g e b r a i c Geometry, P. Le Barz and Y. H e r v i e r , e d s . , pp. 165-197, B i r k h ~ u s e r , B o s t o n , 1982.

[LB3]

"Contribution des droites d'une surface multis~cantes," Bull. Soc. Math. France 112 (1984), 303-324.

IRa]

Z. Ran, "Curvilinear no. 1-2, 81-101.

for

the

double

&

Notes

ses

enumerative geometry," A c t a Math. 155 (19S5),

22

[Ro]

J . R o b e r t s , "Some p r o p e r t i e s (1980), 61-94.

£Sch]

H. C. H. S c h u b e r t , KaLbgg[ d e r abTZdh[ertden G e o m e t r t e , T e u b n e r , 1879, r e p r i n t e d by S p r i n g e r , B e r l i n , 1979.

of d o u b l e p o i n t

s c h e m e s , " Comp. Math.

41

Leipzig,

Geometry of Severi varieties. II: IndeDendence of divisor classes and examples Steven Diaz* Department of Mathematics University of Pennsylvania Philadelphia PA 1910d Joe Harris ** Department of Mathematics B r o w n University Providence RI 02912

Supported by NSF Postdoctoral Reasearch Fellowship ~ Supported by NSF grant DMS-84-02209

Contents

§l. Introduction and statements §2. Restriction maps and independence of divisor classes ~3. Examples

~I

Introduction and Statements

In this paper we will continue the analysis, begun in [D-HI], of the g e o m e t r y of varieties parametrizing plane curves of a given degree and genus. We begin by recalling some of the basic constructions and results of [D-H1]. We will denote by pN the projective space parametrizing all plane curves of degree d. Initially, we are interested in the g e o m e t r y of the v a r i e t y V = V(d,6) c pN defined to be the closure of the locus of irreducible curves of degree d and geometric genus g = (d-1)(d-2)/2 - 8, or, w h a t is the same thing, the closure of

24

the Severi v a r i e t y V = V(d,$) of irreducible c u r v e s h a v i n g e x a c t l y $ nodes as singularities. The p r o b l e m is, n e i t h e r of the spaces V or V is ideal for our purposes. ~¢ is too big: it contains points corresponding to s o m e v e r y d e g e n e r a t e c u r v e s - we don't know, in fact, e x a c t l y w h a t c u r v e s a r e limits of nodal c u r v e s C e V - a n d these points t e n d to be e x t r e m e l y singular ones for ~¢. V, b y c o n t r a s t , is m u c h b e t t e r b e h a v e d - - for e x a m p l e , one k e y fact is t h a t t h e r e exists a "universal family" ~ : C -* V of c u r v e s of genus g o v e r V, w h o s e fiber o v e r C • V is the n o r m a l i z a t i o n of C. But V is too small: b y t h e results of [D-HI] it is a n affine v a r i e t y , whose divisor t h e o r y is a t least c o n j e c t u r a l l y trivial. As we said in t h e earlier paper, a basic p r i o r i t y for the f u r t h e r s t u d y of Severi v a r i e t i e s is to find a good compactification W o_f V. Here b y "good" w e m e a n essentially t h a t t h e points of W should a c t u a l l y correspond to g e o m e t r i c objects (i.e., W should r e p r e s e n t a g e o m e t r i c a l l y defined functor), a n d a t t h e s a m e t i m e t h e g e o m e t r y of W should be tractable: for e x a m p l e , it's singularities should be describable, a n d not too bad. At p r e s e n t no such compactification has been found. Instead, w e w o r k h e r e w i t h a partial comDactification of V: we will look first at t h e union V of V w i t h all t h e codimension 1 equisingular s t r a t a in t h e closure V; thus, while 'q is not projective, it is a t least the c o m p l e m e n t of a codimension two s u b v a r i e t y in a p r o j e c t i v e v a r i e t y . We can s a y e x a c t l y w h a t points of IPN lie in V: as in [D-H1] we c a n a p p l y the d e f o r m a t i o n theoretic results of [D-H2] to see t h a t V consists of the union of V w i t h t h e locus CU of reduced a n d irreducible c u r v e s of genus g w i t h nodes and one cusp;

$- i

t h e locus TN of reduced and i r r e d u c i b l e c u r v e s of genus g w i t h

6 - 2

nodes and

one tacnode;

the locus TR of reduced and irreducible c u r v e s of genus g w i t h nodes and one o r d i n a r y triple point; a n d t h e locus A of reduced c u r v e s of g e o m e t r i c genus g two irreducible c o m p o n e n t s , w i t h 8 + i nodes.

-

8- 3

I, having at most

(CU, TN a n d TR a r e p r o b a b l y irreducible, while A is k n o w n to h a v e e x a c t l y one irreducible c o m p o n e n t A 0 consisting of irreducible c u r v e s a n d one

25 irreducible c o m p o n e n t Ai, j whose general m e m b e r is the union of a c u r v e of degree j and genus I with a c u r v e of degree d - j and genus g - i for each pair i , j satisfying 0 ~_ i~_ g/2, 8' = 8 - j(d-j) + 1 ~_ 0, and (j-1)(j-fi)/2_~ i_~ (3-1)(3-2)/2- 8'). We also know w h a t

V looks like in a neighborhood of each of these loci: it

is s m o o t h at points of TN and TP., while in a neighborhood of CU it looks like the p r o d u c t of a cuspidal c u r v e and a smooth (N-8-1) - dimensional v a r i e t y :

9

CU and in a neighborhood of a point C of A it is the union of s m o o t h sheets corresponding to nodes of the c u r v e C (if C is irreducible, t h e r e are 8 + I sheets, corresponding to all the nodes of C; if C is the union of c o m p o n e n t s CI and C2 of degrees d I and d 2 t h e r e will be dld 2 sheets, corresponding to the points of intersection of CI and C2):

We see from the above t h a t , while V m a y be singular, its n o r m a l i z a t i o n W = W(d,8) x~nll be smooth, and for m a n y reasons it is m o r e c o n v e n i e n t to w o r k w i t h this normalization. For example, we once m o r e h a v e a universal flat f a m i l y n : ~ - + W of c u r v e s of a r i t h m e t i c genus g over W, whose fiber over C E W is the n o r m a l i z a t i o n of t h e corresponding plane c u r v e at its assigned nodes (an "assigned" node m a y be defined to be a limit of nodes of c u r v e s Cx £ W lying over V c V and tending to C; t h u s for C ¢ A the fiber of ~ will be the n o r m a l i z a t i o n of C, while for C ¢ A it will be the n o r m a l i z a t i o n of C at all the

26 nodes except for the one corresponding to the sheet of W containing C). It is thus the variety W that we w:ll take as our basic object of interest._ (We w:ll

also refer to W as a Severi v a r i e t y ; indeed, w h e r e the t e r m Severi v a r i e t y is used w i t h o u t f u r t h e r specification in this paper, however, we will m e a n

W.)

Having c o n s t r u c t e d W our first goal is to describe the divisors and line bundles on W. There are essentially two types of divisor classes we will look at on W: t h e r e a r e divisors on W given as the loci in W of plane c u r v e s w i t h a given geometric p r o p e r t y , w h i c h we will call extrinsic divisors; a n d t h e r e a r e divisor classes t h a t arise j u s t f r o m the a b s t r a c t f a m i l y of c u r v e s C -~ W and the line bundle /; pulled back f r o m the plane (i.e., /; = ~ O p 2 ( 1 ) , w h e r e ~ ' C - * p2 is the m a p sending e,~ch fiber of ~ to the corresponding plane curve); we call these the intrinsic divisor classes. Among the extrinsic divisors are of course the

boundary compon~nts CU, TN, TR and the Aid; other divisors c h a r a c t e r i z e d b y geometric prop,~rties of the plane c u r v e s t h e m s e l v e s are for example The divisor :tF of c u r v e s with a hyperflex - - t h a t is, the closure of the locus of c u r v e s ha,,ing c o n t a c t of order four or m o r e with the:r t a n g e n t line at a smooth point

The divisor FN of c u r v e s w i t h a flecnode - - t h a t is. a node such t h a t the t a n g e n t line to one of the b r a n c h e s has c o n t a c t order t h r e e or m o r e with t h a t branch The divisor ]TB of c u r v e s with a flex bitangent - - t h a t is, a bitangent line h a v i n g c o n t a c t of o r d e r t h r e e o r m o r e w : t h the c u r v e at one of its points of tangency the divisor NL of c u r v e s with a node located s o m e w h e r e on a fixed line L c ~2 and m a n y others described in [D-HI]. We can also define divisors on C as well: for example, the divisor N of points of C lying over assigned nodes of the corresponding plane curves, and the divisor F of points lying over flexes. To define w h a t we call the intrinsic divisor classes, we look a t the universal f a m i l y C of c u r v e s of genus g and the divisor classes D and co on ~, w h e r e D = ~"(c1(0p2(1))) is the pullback of the class of a line from ~ 2 and co is the first Chern class of the relative dualizing sheaf of C over W. A n a t u r a l thing to do to define divisor classes on W is to take all t h r e e pairwise products of these

27 two classes and push them forward; specifically, w e define classes in Pic(W) A = Tt,(D2)

B

=

C :

~.(D-co)

and

~.(co2);

we will also consider the divisors A0, Ai,j and A = A0+ ZAi, j as intrinsic divisor classes With this said, the principal result of [D-HI] is simply that all the extrinsic divisors that have been defined are linearly equivalent to rational linear combinations of the intrinsic divisor classes. Among other relations, we have (1.1)

CU

+ 3B

3A

+ C -

A

(1.2)

TN

~ (3(d-3)+2g-2)A

(1.5)

TR

~

(d 2 - 6 d + 8

+ (d-9)B - -5C + -SA 2 2

g+l)A

d-6B

2

2

+ 2C3

i--A 3

(1.4)

NL

(1.5)

FN

(6d + 6g - 21)A + (3d-18)B- 5C + 2 A - (d-2)A0,1

(1.6)

HF

6A

(1.7)

N

(d-3)-D

(1.8)

F

3D

~ (2d- 3)A/2 - BI2

+

+

18B

-

3CO

+

co

-

11C

+

-

5A

+

4A0,1

T{*A

Z~0,1

Indeed, based on the results of [D-H1] w e m a y m a k e the

Coniecture. The Picard group of the Severi v a r i e t y W is generated over @ by t h e classes A, B, and C and the classes of the boundary components A 0 and Al,j. Since by the relations (1.1)-(1.3) above A, B and C are themselves rational

28 linear c o m b i n a t i o n s of CU, TN, TR and the b o u n d a r y c o m p o n e n t s this is e q u i v a l e n t to

&0 and

Ai, J,

Conjecture'. The Picard group of the v ~ r i e t y V of nodal q u r v e s is torsion. The purpose of t h e present p a p e r Is twofold, First, in t h e following sectlon, we will p r o v e t h a t w i t h the exception of the cases g = 0 or ! and 8 = 0 , 1 or 2, the divisor classes A, B, C a n d A a r e indeed independent. To do this, we define m a p s b e t w e e n t h e Picard groups of t h e Severi v a r i e t i e s W(d,8) and W(d,8+l) t h a t play essentially the role of restriction m a p s , a n d describe these m a p s explicitly on the span of the classes A, B, C a n d A. This allows us to r e d u c e our independence s t a t e m e n t to the case of small v a l u e s of g, w h e r e we. m a y v e r i f y it b y exhibiting c u r v e s in the Severi varieties and explicitly c o m p u t i n g their intersection n u m b e r s w i t h the divisor classes A, B, C and A. Then, in t h e t h i r d section, we will consider s o m e special e x a m p l e s of t h e relations above, as applied to v a r i o u s o n e - p a r a m e t e r families of plane curves. F i n a l l y , w e m a k e t w o observations. First, w e observe t h a t w h i l e w e are

concerned here with the Severi variety parametrizing irreducible nodal curves, the same constructions m a y be m a d e for any other irreducible component of the variety of plane curves of degree d with 6 nodes; in particular, the definitions and relations (i.I)-(1~8)hold here as well (with the obvious exception of the definition of the divisor HF, which does not m a k e sense on a component whose general m e m b e r contains a line). Secondly, note t h a t while we a r e dealing h e r e w i t h a p a r a m e t e r space for c u r v e s in IP2, for s o m e purposes one m i g h t w a n t to t a k e t h e quotient of W b y the action of PGL3 a n d look a t t h e moduli space for triples (C,~,V), w h e r e C is a c u r v e , /: a line bundle on C, and V c H0(C,/:) a linear s y s t e m m a p p i n g C b i r a t i o n a l l y onto a c u r v e of t h e a p p r o p r i a t e type. Such a quotient exists, a t least w h e n the degree d z 5, since all the c u r v e s in W will be stable, a n d t h e results of this paper, s u i t a b l y r e p h r a s e d , a p p l y in this context. Specifically, for a n y f a m i l y of triples {(Cx,Z:x,Vx))x(Z - - t h a t is, a f a m i l y ~ : ¢ -~ Z of c u r v e s , w i t h a line bundle ~ on C defined up to twists b y pullbacks of line bundles f r o m Z and a subbundle ~ c ~,~5 of r a n k 3 - - w e a l r e a d y h a v e a divisor class co -cl(co¢/z) on ¢, a n d we c a n define a (rational) divisor class D = Cl(~) b y n o r m a l i z i n g ,~ so t h a t cI(%~) = 0 - - t h a t is, b y setting D = c1(/:) - ~*ci(%~)/3. In this w a y , we c a n define rational classes A, B, and C on Z. Of the extrinsic divisors, t h e ones i n v a r i a n t u n d e r PGL3 - - such as t h e b o u n d a r y c o m p o n e n t s , or the divisors HF and FN - - of course define divisors on t h e quotient; the

29 others can be defined in terms of their relations with A, B, and C (for example, the class of the divisor CP of curves pasing through a point can simply be defined to be the divisor class A = ~,(D2)). With this understood, the relations above continue to hold; we will see an example of this in §3.

.~2 R e s t r i c t i o n

maps

a n d i n d e p e n d e n c e of d i v i s o r c l a s s e s

In this section we define for each d and 8 a homomorphism r : Pic(W(d, 8)) -~ Pic(W(d, 8 + 1)). We also compute r explicitly on the span of the classes A, B, C, and A. This allows us to determine when the classes A, B, C, and A are independent. Recall the definition of V(d, 8) in ~1. Let V'(d, 8) be V(d, 8) U V(d, 8+1) Define W'(d, 8) to be the normalization of V'(d, 8) and A'(d, 8) to be the inverse image of ~/(d, 8+1) in W'(d, 8) with its reduced scheme structure. We have the following commutative diagram (2.1):

7V(d,8) i ./-3 A'(d,8) ~

W(d,S)c j

--..W(d,S)'

9(d,8) •

~ V'(d,8)

ig

W(d,8+l)

9(d,8+I)

The morphisms nl, n2, n3, and n 4 are normalizations and the morphism g comes from the universal mapping property of normalization applied to n 3 . It is clear t h a t g is proper. W' (d, 8) is obtained from W (d, 8) by adding codimension two subvarieties. Also, from the deformation theory of [D-H2] (see as well Lemmas (2.3) and (2.4) below) one m a y see t h a t W' (d, 8) and W (d, 8) are both smooth. This allows us to identify Pic(W(d, 6)) and Pic(W'(d, 8)); call this identification j,. From standard intersection theory (See Fulton, [F]) we get a

30 homomorphism g~n 2

j . • Pic(W(d, 8)) ~ Pic(W(d, 8 + i)).

(2.2) Definitipn: r = g, o n 2 oi x o j . . To c o m p u t e t h e h o m o m o r p h i s m r explicitly we need a description of the local s t r u c t u r e of W'(d, 8) n e a r A'(d, 8) . By t h e d e f o r m a t i o n t h e o r y of [D-H2] we see t h a t we m a y obtain this local i n f o r m a t i o n b y looking in the d e f o r m a t i o n spaces of a p p r o p r i a t e singularities. (2.5) L e m m a . In the d e f o r m a t i o n space of a tacnode

y 2 _ y x 2 + tlx2 + t2 x + t3 = 0 the following loci m a y be described as follows. First, the locus of c u r v e s w i t h two nodes m a y be given p a r a m e t r i c a l l y b y

tI = e, t2 = O, t3 = -e 2 or in Cartesian f o r m as t 2 = O, t 2 -- - t 3.

The locus of curves with cusps is given parametrically by t l = ~ d 2,

t2 = 2d 3,

t3 = 3 d 4

or in Cartesian f o r m as

t2 = 3t3,

9t2 = 3 2 t i t 3.

Lastly', the locus of singular curves (i.e., the closure of the locus of curves with one node) is given in Cartesian form as -64t 3 - 128t 2t 2 - 27t4 + 144tlt22t 3 - 64t 4t3+16t~t2 and p a r a m e t r i c a l l y as t I = s,

t 2 = c3 - 2cs,

t 3 = - ~ c 4 + c 2s.

=0

31 N o t e t h a t t h e i n v e r s e i m a g e in t h e ( s , c ) - p l a n e of t h e t w o node locus is g i v e n

= ~ c 2, while the cuspidal locus is given b y s = h a v e intersection multiplicity 2. (2.4) L e m m a .

by s

c2; and t h a t these two c u r v e s

In t h e deformation space of a triple point

x 2 y + x y 2 + tlxY+ t2x+ t3Y+ t4 = 0 t h e f o l l o w i n g loci m a y be described as follows: t h e locus of c u r v e s w i t h t h r e e

nodes m a y be given p a r a m e t r i c a l l y b y t I = c,

t 2 = t 3-- t 4 = 0

or in Cartesian f o r m by the equations t 2 = 0,

t 3 = 0,

t 4 = 0;

t h e locus of c u r v e s w i t h a t a c n o d e has t h r e e b r a n c h e s , g i v e n p a r a m e t r i c a l l y 1.

t 1 =-2d,

2. t 1 = - 2 a ,

3. t l = - ~ d ,

t 2 = 2d 2,

t5 = t4 = 0

t 3 = - a 2,

t2 = t4 = 0

t2=t3=-ld

2.

t4=

and 3

or in Cartesian f o r m b y equations: 1. t 3 = 0,

t 4 = 0,

t 2 = 2t 2

2. t 2 = 0,

t 4 = 0,

t 2 = 4t 3

3. t 2 = t3,

t I t 2 = t4,

t 2 = - 4 t 5.

The locus of c u r v e s w i t h two nodes likewise has t h r e e branches, given either p a r a m e t r i c a l l y as 1. t I = s,

t 2 = r,

t 5 = 0,

t4 = 0

2. t 1 = s,

t 2 = 0,

t 3 = r,

t4 = 0

as

32 3. t 1 = s,

t 2 = r,

t~ = r,

t4 = r s

or in C a r t e s i a n f o r m b y e q u a t i o n s : 1. t 3 = 0,

t4 = 0

2. t 2 = 0,

t4 = 0

3. t 2 = t3,

t I t 2 = t 4.

o b s e r v e t h a t w h e n w e pull t h e s e loci b a c k to t h e ( r , s ) - p l a n e , t h e l o c u s of c u r v e s w i t h t h r e e n o d e s is g i v e n in b r a n c h 1) b y r = 0, t h e locus of c u r v e s w i t h a t a c n o d e b y s 2 = 4r;

and that these have intersection multiplicity

b r a n c h 2) t h e s e t w o loci a r e g i v e n b y t h e e q u a t i o n s respectively, and have intersection number s2 = - 4 r ,

s2 = 4r

2; a n d in b r a n c h 3) b y r -- 0 a n d

again having intersection multiplicity

m u l t i p l i c i t y of t h e s e t w o loci is t h u s

r = 0 and

2; s i m i l a r l y in

2. The t o t a l i n t e r s e c t i o n

6.

Proof: The proof of b o t h l e m m a s a r e s t r a i g h t f o r w a r d c o m p u t a t i o n s a n d a r e left to the reader.

CU should r e a l l y be denoted b y CU(d, 8) to indicate w h i c h Severi v a r i e t y i t is on; however, we w i l l u s u a l l y s i m p l y w r i t e

CU when no confusion seems

likely, and s i m i l a r l y for the other divisor classes.

(2.5) Theorem: With

r: Pic(W(d, 8)) --* Pic(W(d, 8+1)) as above w e have:

r ( C U ) -- 8CU + 2TN r ( T N ) = (8 - 1 ) T N + 6TR r(TR) = (8- 2)TR r ( N L ) = 8NL r ( A ) = (8 + I ) A .

33 Proof: The formula r ( A ) = (8+1)A is an easy consequence of the fact t h a t A is the h y p e r p l a n e class and t h a t the m a p g in (2.2) has degree 8 + 1. The o t h e r four equalities a r e easily seen to be set theoretically true. W h a t r e m a i n s is to v e r i f y the multiplicities.

Let C be a reduced irreducible c u r v e whose singularities a r e e i t h e r 8+1 nodes, 8 nodes and one cusp, 8-1 nodes and one tacnode, or 8-2 nodes a n d one triple point. Let q be t h e point in FN corresponding to C. Label the singular points of C PI . . . . . Pro. Let B i be the base of the etale v e r s a t d e f o r m a t i o n space for t h e s i n g u l a r i t y of C a t Pi. F r o m the d e f o r m a t i o n t h e o r y of [D-H2] w e see t h a t (after etale base change) a neighborhood of q in pN m a p s to t h e product of the spaces 1:5i and n e a r the origin (0 . . . . . 0) the m a p is s u r j e c t i v e w i t h s m o o t h fibers. This, t o g e t h e r w i t h t h e description of t h e d e f o r m a t i o n spaces of t h e tacnode a n d triple point in (2.3) a n d (2.4), finishes t h e proof of t h e f o r m u l a s for r ( C U ) , r ( T N ) a n d r(TR). The c o m p u t a t i o n of r (NL) likewise reduces to an e x a m i n a t i o n of local d e f o r m a t i o n t h e o r y , in this case t h e condition for a first order d e f o r m a t i o n of a c u r v e C h a v i n g a node a t a point p on a line L to p r e s e r v e the node a n d keep it on L. The condition is e a s y to express: if C is given b y f(x,y) = 0, a n d ~' is the equation of a line t h r o u g h p polar to the line L w i t h respect to t h e two b r a n c h e s of C a t p - - t h a t is, t a n g e n t to t h e c u r v e given b y t h e d e r i v a t i v e of f in t h e direction of L - - t h e n t h e condition t h a t a first o r d e r d e f o r m a t i o n f(x,y) + e.g(x,y) keep the node on L is s i m p l y t h a t g ¢ rrl2+(~'). Now, let P1 . . . . ,Ps÷l be the nodes of a c u r v e C e V(d,8+l); let Ill i be t h e m a x i m a l ideal of Pi in ~2, Li a line t h r o u g h Pl a n d ~ the e q u a t i o n of t h e line t h r o u g h Pi polar to Ll w i t h respect to t h e b r a n c h e s of C a t Pi. Then t h e t a n g e n t space to V (d, 8+1) a t q (the point of ~N corresponding to C) is the space of sections

HO(c, (So(d)® r o t ®

... ®ms+i),

while t h e t a n g e n t space to a b r a n c h of N L a t q corresponding to s m o o t h i n g and keeping a node on Lj , i ~ j, is the space of sections HO(c, @c(d) ® m i ®

®fni® . . . . . .

Pl

®(m2+ ~'.)® ®ms+i), j J ...

That these two spaces are distinct follows from fact that the nodes of C impose independent conditions on curves of degree d - 3 and the monotonicity of Hilbert functions; this verifies the m u l t i p l i c i t y given for r(NL) in the s t a t e m e n t of t h e

34 Theorem.

Remarks: As a l r e a d y observed, in Pic(W(d, 8)) ® © we can use the relations (1.1)-(1.4) of § i to express A, B, C, and A as linear combinations of CU, TN, TR and NL. With this (2.5) can be used to c o m p u t e the image u n d e r r of a n y class in the span of A, B, C and A. This includes most of the geometric divisor classes studied in this paper. As an example the relations of 51 imply t h a t A:

_ cu÷ 366

TR-(

726

NL

T h e o r e m (2.5) now allows us to c o m p u t e r(A) in two ways. They both come out to be equal to (8 + I ) A . This provides a partial internal check on the c o m p u t a t i o n s in the proof of (2.5). (2.6) Theorem: Let S(d, 8) c Pic(W(d, 8)) ® Q be the subspace spanned b y A, 2). Then the dimension of S(d, B,C, and A; a s s u m e t h a t 0 2

for d > 2

b). d i m S ( d , i ( d -

1 ) ( d - 2)) z 3

for d> 3

36 c). d i m S ( d , l ( d -

1)(d-

2)) z 4

for

d >4

d). d i m S ( 4 , 1) ~ 2. To do this, we will c o n s t r u c t one dimensional families of plane c u r v e s and t h e n e v a l u a t e their intersection n u m b e r s w i t h the divisor classes A, B, C and A. By showing t h a t these intersection n u m b e r s are independent we get the desired lower bounds on the dimensions of the S (d, 8)'s. Fatuity One. Let C be a nonsingular c u r v e of genus g (g = 0, ! or 2). Think of C x ~1 as a f a m i l y of c u r v e s of genus g with base ~1, Let F be a fixed fiber of this f a m i l y and S a fixed section corresponding to a general point of C. Fix integers a and b with a>_ 1 and b_> I if g = 0, b>_ 3 if g = 1 and b_> 4 if g = 2. Using the linear s y s t e m laF + bSI m a p C x IP1 to projective space, then take a generic projection of the image to ~2. This will give a f a m i l y of plane c u r v e s of degree b and genus g. On C x p1 the divisors D and co introduced in §1 a r e given up to n u m e r i c a l equivalence b y D

= aF

+ bS

and

co = ( 2 g - 2 ) S

Thus on the base IPI of the family we have: deg(A)

:

D 2 = 2ab

deg(B)

= D.co

deg(C)

= oo2 = 0;

= (2g- 2)a

and of course deg(A) = 0. F a m i l y Two. Let C be a nonsingular c u r v e of genus g (g = 0, I or 2); again, think of C × IP1 as a f a m i l y of c u r v e s of genus g with base pl. Blow up a point on C x p l and call the resulting surface X and the exceptional divisor E; X can still be t h o u g h t of as a f a m i l y of c u r v e s of a r i t h m e t i c genus g. Let F be a fixed general fiber of this f a m i l y and S a fixed section corresponding to a general point of C (in p a r t i c u l a r S.E : 0). Fix integers a and b with a>_ 2 and b_> 2 if g = 0, b_> 4 if g = I and b z 5 if g = 2. Using the linear s y s t e m l a F + b S - E i

37 m a p X to projective space, t h e n take a generic projection of the image to p2. This will give a f a m i l y of plane c u r v e s of degree b and genus g, similar to t h e one c o n s t r u c t e d above except t h a t this one will h a v e one point of (transverse) intersection with the c o m p o n e n t A0, i of A. In the Neron-Severi group of C × ~ 1 the divisors D and co a r e D = a F + bS - E co = ( 2 g - 2 ) S

and

+ E.

Thus on the base ~1 of t h e f a m i l y we have: deg(A) -- 2 a b - 1 deg(B)

=

(2g-

deg(C)

=

-I

deg(A)

=

2)a-

i

I.

Family Three. Consider a generic pencil of plane cubics. All except finitely m a n y of its m e m b e r s will be nonsmgular and t h e rest will h a v e one node. Let X be the blow up of p2 at the 9 base points of the pencil, E l , . . . ,E 9 the exceptional divisors and H the pullback of the h y p e r p l a n e class on p2. X is a f a m i l y of elliptic curves. Let i be a subset of {1, 2 . . . . . 9} w i t h e i t h e r 0, 1 or 2 elements; denote b y [ll the n u m b e r of e l e m e n t s of I. Map X to p r o j e c t i v e space b y the linear s y s t e m [D], w h e r e

D

--

m.H - i~iEi

with m_> i and if m - - 1 t h e n I = ~. Now take a generic projection of the image of X to ~2. This will give a f a m i l y of plane c u r v e s of degree 3 m - Ill and genus 1, with divisor D given as above. To c o m p u t e the class co note t h a t the canonical bundle of ~2 blown up at 9 points is - 3 H + E 1 . . . + E 9 . We m u s t s u b t r a c t f r o m this the pullback of the canonical bundle of t h e base of the family, which is m i n u s two fibers; we h a v e co = - 3 H

+ E i + ... E 9 - ( - 2 ( 3 H

- EX - ... -Eg))

38 =

3H

- El

- ...-Eg.

Thus on the base of the f a m i l y we h a v e

deg(A)

: m 2-1II

deg(B) = 3 m - I I I deg(C) = O. To c o m p u t e the degree of A in this f a m i l y recall the following w e l l - k n o w n l e m m a (see for e x a m p l e Diaz [D]). (2.7) L e m m a : Let r~ : S -* C be a flat f a m i l y of c u r v e s w i t h S a nonsingular surface, C a nonsingular c u r v e a n d all fibers either nonsingular or w i t h o r d i n a r y nodes as t h e i r only singularities. Let g be the genus of the fibers, p the genus of C, a n d 8 t h e n u m b e r of singular points of fibers. Then t h e topological Euler c h a r a c t e r i s t i c of S is given b y X(S)

= (2g- 2)(2p- 2)- 8.

Since t h e topological Euler c h a r a c t e r i s t i c of a blow u p of IP2 a t 9 points is 12, we h a v e deg(A)

= 12.

F a m i l y Four. Consider a general pencil of plane q u a r t i c s double a t s o m e fixed point p • ~2. Let X be the blow up of ~2 a t t h e 13 base points of this pencil, H the pullback of the h y p e r p l a n e class on ~ 2 , E0 the exceptional divisor o v e r p and El, . . . ,El2 the o t h e r exceptional divisors; v i e w X as a f a m i l y of c u r v e s of genus 2 o v e r ~pl. F r o m the dimension calculations of [D-HI] we also see t h a t t h e general fiber of this f a m i l y is nonsingular a n d t h e singular fibers e a c h h a v e only one simple node as a singularity. Let I be a subset of {I, 2 . . . . . 12} containing a t m o s t 3 elements; again, denote b y III t h e n u m b e r of e l e m e n t s in I. Set

D

=

m.H - ~E 1 i~l

39

w h e r e m_> 1 and I = ~ if m = 1; as in the previous case m a p X to projective space using the linear s y s t e m IDI and take a generic projection to p2. This gives a f a m i l y of plane c u r v e s of degree 4 m - III and genus 2. In a m a n n e r similar to Family Three we find that: ¢O = 5 H - 5E 0 : E l - . , . -

E12

deE(A) = m 2 - III deE(B)

= Sin-

111

deE(C)

= 46

and

deE(A)

= 20.

Family,Five. Consider a pencil of c u r v e s of t y p e (3,2) on ~ l x p1. One m a y easily check (by writing down equations) t h a t if t h e pencil is as general as possible the general e l e m e n t of this pencil will be nonsingular and the singular elements will each h a v e only one simple node as a singularity. Let X be the blow up of p l x ~1 a t t h e 12 base points of the pencil and E 1 . . . . . El2 the exceptional divisors. Denote b y ql and q2 the pullbacks to X of the classes of the fibers of ~ i x ~ 1 Let I be a subset of {I, 2 12} w l t h a t m o s t 2 elements and denote b y III the n u m b e r of elements in I; view X as a f a m i l y of c u r v e s of genus 2, Set . . . . .

D and map

=

X

a.q I

+ b.q 2 - i ~ I E i

to p r o j e c t i v e space u s i n g t h e l i n e a r s y s t e m

IDI, t h e n t a k e a g e n e r i c

projection of the image to p 2 This will give a f a m i l y of plane c u r v e s of degree 2a+Sb-lll and genus 2. As before we calculate t h a t on this family:

CO

=

4.q I

+ 2.q 2 -

deE(A) = 2 a b - t l I deE(B)

= 4b-

2 a - III

~

E.

i I=

1

40 deg(C) = 28 deg(A)

= 20.

The proof of independence n o w follows i m m e d i a t e l y f r o m the existence of these families. Specifically, to show a) a b o v e use families i and 2; for b) use families 1, 2 and 3; for c) use families 1, 2, 4 and 5 a n d for d) use families i a n d 4.

~ I e s In this section, we will consider a series of special cases, where the results of [D-HI] and of this paper either yield results about families of plane curves, or m a y be verified directly.

Example i: Projections of a space c u r v e Suppose n o w w e have a curve C c p3, smooth of degree d and genus g. W e can describe a one-parameter family of plane curves by taking a general line L c IP3 and considering the curves Cp obtained by projecting C from the points p ¢ L; as long as the line is general and the c u r v e C does not possess infinitely m a n y q u a d r i s e c a n t s (as it might, for example, if it lay on a quadric surface), these c u r v e s will all correspond to points of o u r partial c o m p a c t i f i c a t i o n of t h e Severi v a r i e t y . Of course, we don't a c t u a l l y get a f a m i l y of c u r v e s in a fixed plane ~ 2 since t h e p l a n e of proJection will h a v e to v a r y w i t h p; r a t h e r , w e get a f a m i l y of c u r v e s in t h e p r o j e c t i v i z a t i o n PH L of t h e restriction HL to L of t h e u n i v e r s a l h y p e r p l a n e bundle H on p3. As we h a v e observed in §1, h o w e v e r , o u r f o r m u l a s should still hold, provided we t a k e as t h e divisor class D t h e r e s t r i c t i o n to C × L of a line bundle on PH L whose restriction to a fiber of PH L is 6I(1), n o r m a l i z e d b y adding a rational multiple of t h e fiber so t h a t D3 = 0. (We o b s e r v e in passing t h a t a n analogous p r o c e d u r e will yield c o m p l e t e s u b v a r i e t i e s of t h e Severi v a r i e t y W of a r b i t r a r y dimension: w e j u s t h a v e to p r o j e c t a fixed c u r v e C c pr f r o m a f a m i l y of planes A c ~r corrsponding to a s u b v a r i e t y of G(r-3,r) o v e r w h i c h t h e u n i v e r s a l quotient bundle is a trivial

41 v e c t o r bundle tensored w i t h a line bundle - - for example, the f a m i l y of linear spaces on a Segre v a r i e t y p m x pn c pr.) With this said, it is e a s y to e v a l u a t e the degrees of the intrinsically defined line bundles A, B, C a n d A on t h e f a m i l y of c u r v e s (Cp}. To begin with, the divisor class co on t h e f a m i l y ~ = C x L of n o r m a l i z a t i o n s is j u s t t h e pullback ~I c°C of t h e dualizing sheaf on C, w h i c h is algebraically j u s t 2g-2 t i m e s t h e class c¢ of a fiber of C o v e r C. Next, to describe t h e divisor class D w e c a n s t a r t w i t h t h e bundle Op(1) on PHL (whose Chern class ~ r e s t r i c t s to the class d.c~ on C x L); if we add a multiple ~,.13 of the class of a fiber o v e r L, we have

(~+%.p)3

= -l+3X

since 0¢3 = - I ; w e t h u s w a n t to t a k e D = (oc + I~/3)[c. We t h e n h a v e

deg(A) =

-- 2d/3

((ec+p/3)lC)

deg(B)

= ((oc+~/3)t~,(2g-2)o:)

deg(C)

-- d e g ( g )

=

=

(2g-2)/3

0.

Now, applying o u r f o r m u l a s (1.1)-(1.6) above, w e a r r i v e at f o r m u l a s for the n u m b e r of c u r v e s Cp in our f a m i l y w i t h cusps, tacnodes, etc.; these in t u r n yield n u m e r i c a l i n f o r m a t i o n a b o u t the space c u r v e C. The m o s t e l e m e n t a r y e x a m p l e of this is t h e o b s e r v a t i o n t h a t Cp wilt h a v e a cusp if and o n l y if the point p lies on a t a n g e n t line to C; t h e degree of t h e divisor CU on o u r f a m i l y will t h u s be the n u m b e r of t a n g e n t lines to C m e e t i n g L, or e q u i v a l e n t l y the degree of t h e s u r f a c e TC s w e p t out b y the t a n g e n t lines to C. We h a v e t h e n deg(TC) = deg(CU) = 2d + 2 g - 2, a n u m b e r readily obtained f r o m t h e R i e m a n n - H u r w i t z - P l u c k e r f o r m u l a s a n y w a y . Similarly, Cp will h a v e a triple point if a n d o n l y if p lies on a t r i s e c a n t line to C; so t h e degree of t h e s u r f a c e SC s w e p t out b y t h e t r i s e c a n t lines to C is

deg(TR)

--

..•

(d 2 _ 6d+8

- 2g+2),d

-

1 d - 6)(g - 1) -~(

42

= --i(d5 - 6d 2 - 5dg + 11d + 6g - 6) 3 Likewise,

Cp will h a v e a tacnode w h e n

p lies on the chord to C joining t w o

points whose t a n g e n t lines intersect; the degree of the surface swept out b y such chords is t h u s deg(TN) = 2(d - 3),d + (4g - 4).d/5 + 2(d - 9)(g - 1)/5 = 2d 2 + 2 d g - 8 d - 6 g + 6 . For example, if C is a q u a r t i c elliptic c u r v e this surface is j u s t the union of the four quadric cones containing C, and so has degree 8 as predicted.

Example 2: reducible c u r v e s We will exhibit here some of the techniques for dealing with families of reducible c u r v e s b y considering the f a m i l y of c u r v e s formed b y taking a general c u r v e C c p2 of degree d-1 and genus g h a v i n g $ = ( d - 2 ) ( d - 5 ) / 2 - g nodes and adding a variable line Lx m o v i n g in a pencil. We view this as a f a m i l y of reducible c u r v e s of degree d and geometric genus g-1 (i.e. w i t h $+d-1 nodes) parametrized by X ¢ p1 To begin with, the s i m u l t a n e o u s n o r m a l i z a t i o n C will consist of t w o disjoint components, the product X 1 ~- C × p1 of the normalization C of C w i t h the parameter c u r v e p 2 and the ruled surface X2 ~ ~-1 swept out b y the lines Lt. On X 1 both the divisor classes D and co are pullbacks f r o m C, and so all pairwise products a r e zero. On X2 the Picard group is generated b y the class of a line L = Lx (that is, a fiber over p1) and the exceptional divisor E, with the divisor D equivalent to L*E and the class of the relative dualizing sheaf co - -L-2E. We h a v e thus deg(A) = ( L + E ) 2 = 1 deg(B) = (L * E).(-L - 2E) = -1

and

deg(C) = ( - L - 2 E ) 2 = 0; of course the degree of A is zero. We thus have by (1.1)

43

deg(CU) = 5 + 5(-1) = 0, as is clear a n y w a y , since none of the c u r v e s Cx has a cusp. Similarly, the c u r v e Cx will h a v e a tacnode w h e n e v e r Lx is t a n g e n t to C, and we observe t h a t b y (1.2), deg(TN) = 3(d-3) + 2(g-1) - 2 - (d-9) = 2d + 2 g - 4, which is of course the n u m b e r of tangent lines to C passing through the base point of the pencil {Lx}. Cx has a triple point w h e n e v e r Lx passes t h r o u g h a double point of C; and indeed we h a v e b y (1.3) deg(TR) = (d 2 - 6 d + 8 ) / 2 =

(d 2 -

5d+6)/2

(g-l) + 1 + (d-6)/2 -

g,

w h i c h is the n u m b e r of nodes on C. As a f u r t h e r check, observe t h a t b y (1.5) the n u m b e r of flecnodes occurring in the f a m i l y (Cx} is deg(FN) = 6 d + 6 ( g - ! ) -

2 1 - ( 3 d - 18)

= 6(d-l) + 3(2g-2), w h i c h is the n u m b e r of flexes of C.

Example 3: The case 8

=

0.

Of course, as has been observed, in case $ = 0 the Severi v a r i e t y W is j u s t open subset of pN consisting of smooth c u r v e s and c u r v e s with one node; the Picard group of W is generated b y the class A = Cl((gpN(1)). We could t h e n d e t e r m i n e the classes B, C and A as multiples of A b y applying the relations CU = TN = TR = 0, b u t it is e a s y enough to do this directly. To this end, let Z c W be a general pencil of plane c u r v e s of degree d; C the blow-up of the plane a t the base points Pi of the pencil Z, H the pullback to ~ of the class of a line in the plane, Ei the exceptional divisor over Pi and E the s u m of the Ei. As observed in ~2, the class co of the relative dualizing sheaf of ~ over Z = ~,1 is

44

the class K~ = -3H+E of the canonical bundle of C plus twice the class d H - E of a fiber of C over Z; thus co

=

(2d-3).H

-

E

and since the class D of 21 is j u s t deg(B)

= (H.((2d-3)H-

E))

H, we have = 2d- 3

deg(C) = ((2d-3)H - E) 2 = (2d-3) 2 - d 2 =

3d 2 - 1 2 d + 9

= 3(d-1)(d-3). Finally, the degree of A m a y be determined in m a n y w a y s (for example, as m

§2 b y applying L e m m a (2.7)); we find t h a t deg(A)

3(d-1) 2.

=

We can n o w v e r i f y directly t h a t the classes CU, TN and TR are all zero. We can also use these relations to determine, for example, the n u m b e r of c u r v e s in the pencil possessing a hyperflex; b y (1.6) this is deg(HF)

=

6.1 + 18.(2d-3) + 11.3(d-l)(d-3)

=

1 8 d 2 - 6 6 d + 36.

- 5.3(d-1) 2

(Observe t h a t this is zero w h e n d = 3, as it m u s t be.) We can likewise use our f o r m u l a s to describe the c h a r a c t e r s of the c u r v e ~ c p2 t r a c e d out b y the flexes of the c u r v e s in t h e pencil Z. By (1.8) the class of t h e divisor F on C is F

-- 3 D

+

30o

=

(6d-6)H

-

3E;

is t h u s a c u r v e of degree (H.F) = 6d-6, with a point of multiplicity (F. Ei) -- 3 a t each of the base points of t h e pencil Z. These, together w i t h t h e nodes of ~ at the singular points of the c u r v e s in Z are all the singularities of ~.

45 Indeed, we can c o m p u t e the geometric genus of can apply the genus formula to ~ c [p2 to find g(~) = (6d-7)(6d-8)/2

in two ways.

First,

we

- $.d 2 - 3(d-l) 2

= 12d 2 - 39d + 25. Alternately, we can realize the n o r m a l i z a t i o n of ~ as a b r a n c h e d cover of Z p l with 3d(d-2) sheets, b r a n c h e d simply over the divisor HF and h a v i n g 4 b r a n c h points (two points of ramification index 3) over each point of A (cf. the description of F in [D-H!]); applying the R i e m a n n - H u r w i t z f o r m u l a we h a v e

2g(~)- 2

=

-2.3d(d-2)

and we deduce again t h a t

+

(18d2-66d+36)

+

4.3(d-i) 2

g(~) = 12d 2 - 39d + 25.

ExamPle 4: the case $ = 1. We consider n o w a m o r e interesting case, t h a t of a f a m i l y of c u r v e s with (generically) one node. We will look a t a generic net ~D c pN of plane c u r v e s of degree d, and take as our f a m i l y the inverse image in W(d,l) of ~D c IPN. Equivalently, Z is the n o r m a l i z a t i o n of the locus Z c D ~ IP2 of singular c u r v e s in the net, w h i c h will h a v e a cusp at the points coresponding to c u r v e s in the net h a v i n g a cusp, and a n o r d i n a r y double point a t points corresponding to c u r v e s in the net with two nodes. Before we s t a r t our analysis, observe t h a t we h a v e the potential for a notational n i g h t m a r e here, with t h r e e s e p a r a t e varieties isomorphic to IP2 appearing in this picture: we h a v e of course the plane in w h l c h the c u r v e s of the net live, w h i c h we will denote b y p2; we h a v e the plane ,D p a r a m e t r i z i n g the c u r v e s in the net; a n d we h a v e the dual plane ~ = ~Dv w h i c h a p p e a r s n a t u r a l l y as the t a r g e t space of the m a p %0: p 2 _ , given b y the n e t JD. The c u r v e Z lives in the plane ,D; t h e dual c u r v e W = Z " c ~ is the b r a n c h divisor of the m a p ~0. Finally, we will denote b y R c p2 the ramification divisor of the m a p %0; observe t h a t R is j u s t the c u r v e t r a c e d out

by the nodes of the curves in Z. Of course, all three curves R c p2, W c ~ and

46 c .D are birational, w h i c h only increases the potential for confusion. We s t a r t our analysis b y d e t e r m i n i n g the degrees of these t h r e e curves. First, we h a v e a l r e a d y d e t e r m i n e d the degree of Z c ~; this is j u s t the n u m b e r of singular elements in a general pencil of c u r v e s of degree d as c o m p u t e d in the last example. Observe t h a t this is also the degree of the divisor A on Z; i.e., deg(A)

=

3(d-l) 2.

Secondly, we can d e t e r m i n e the degrees of R and W b y restricting the m a p

%0

t o a l i n e L c E and its inverse image C = %0-I(L) in p2. This i s a m a p expressing a s m o o t h plane c u r v e C of degree d as a deg(%0) = d 2 - sheeted cover of a line L --- pl; b y the R i e m a n n Hurwitz f o r m u l a this will h a v e 3d(d-l) b r a n c h points. W thus m e e t s L in 3d(d-l) points, so the degree of W is 3d(d-l); likewise, since R m e e t s the c u r v e C c IP2 in 3d(d-1) points, we decude t h a t the degree of R is 3(d-l). Since, as we h a v e observed, R is j u s t the locus in ~2 of nodes of c u r v e s in Z, we deduce t h a t the degree of the divisor NL on Z is likewise deg(NL) = 3(d-l). Since b y (1.4) w e

have

2NL

~ (2d-3)A - B, w e

may

use the last t w o relations to

conclude that

deg(B) = (2d-3).3(d-l) 2 - 6(d-l) = $(d-1)(2d 2 - 5d + 1). Next, to d e t e r m i n e the degrees of C and A on Z, we will use the fact t h a t on Z, the divisors TN and TR are zero. Bearing in mind t h a t here g = ( d - l ) ( d - 2 ) / 2 - i, the relations (1.2) and (1.3) t h e n t r a n s l a t e into 0

-- 2(d2-11)'A

0

=

S(-3d+10)'A

+

2(d-9)'B

-

3(d-6)'B

-

5.C

+

+ 4-C

3.A

-

2"A.

We c a n solve these two relations in t u r n for C and 2C

=

-(4d 2 -

27d+46).A

+

(5d-

18).B

and

A; we a r r i v e at

47

and hence dog(C) = 3(d-l)(3d 3 - 1 5 d 2 + l l d + 14); and similarly 2A = -(8d 2 - 4 5 d + 62).A + (Td - 18).B so t h a t dog(A) = 3(d-l)(3d 3 - 9d 2 - 5d + 22). (Recall that the degree of A is twice the number of curves in the net w i t h two nodes.) We can use the values obtained for A, B, C and A to determine, for example, the n u m b e r of cuspidal curves in the net; this works out to be deg(CU)

=

3.deg(A) + 3.deg(B) + deg(C)

=

3(d-l)(3(d-l) + (6d2-15d+3) + (3d3-15d2+lld+14)

-

deg(A)

+ (3d 3 - 9d 2 - 5d + 22))

= 12(d-1)(d-2).

Exammle 5: cublcs.

As our last example, we will consider the varieties parametrizing plane cubic curves with nodes. Of course, two of these, corresponding to smooth cubics and cubics with one node, have been at least partially described in the previous example; we saw, for example, t h a t the v a r i e t y of singular cubics is a hypersurface of degree 12 in the space ~9 of all plane cubics, double along the v a r i e t y of cubics with two nodes (i.e., reducible cubics) and cuspidal along the locus of cuspidal cubics; and t h a t these two varieties h a v e degrees 21 and 24 respectively. We will consider now the varieties of cubics with two or t h r e e nodes, and verify the relations of §1 for these. Consider first the v a r i e t y of cubics w i t h two nodes. This is just the image of the product ~2 x ~5 of the space of lines w i t h the space of conics, embedded in ~17 by the Segre v a r i e t y and projected to p9 from the subspace of ~17

48

corresponding to the linear relations a m o n g quadrics in p2. In p a r t i c u l a r , its degree is the degree of the Segre v a r i e t y ; denoting b y r11 and q2 the h y p e r p l a n e classes in p2 and p5 respectively, this is j u s t ( q l + "q2) 7 = 21

as p r e v i o u s l y d e t e r m i n e d . As usual, we c a n d e t e r m i n e the classes A, B, C a n d A either d i r e c t l y or b y using s o m e of our relations; we will do the l a t t e r here. To begin with, we h a v e a l r e a d y seen t h a t the class A

= ql

+ q2.

Next, it's e a s y to d e t e r m i n e the clas of the divisor NL: if w e fix the line c o m p o n e n t L of a reducible conic C = L U Q, t h e condition t h a t C h a v e a node on a line L0 c IP2 is j u s t the linear condition t h a t Q pass t h r o u g h t h e point L N L0; likewise if we fix Q the condition is j u s t the q u a d r a t i c condition t h a t L pass t h r o u g h either one of the two points of Q n L0. We h a v e t h u s NL

= 2'~1

+ 112,

a n d applying the relation 2 N L ~ 3 A - B, w e deduce that

]3

=

-ql

+

h2.

Of course the divisor A is j u s t the divisor of reducible cubics Q u L w h e r e Q is a singular (i.e., reducible) conic; since (by e x a m p l e $ a b o v e in case d = 2) the locus of singular conics is a cubic h y p e r s u r f a c e in the space of all conics, w e h a v e A = 3TI2. We can n o w use, for e x a m p l e , the relation (1.1) to d e t e r m i n e t h e class C: since t h e divisor CU is zero, we h a v e 0 = 3 ( q l + "q2) + 3 ( - q l + q2) + C - 3rl2 so t h a t C - -3r12.

49 To v e r i f y t h i s , n o t e t h a t t h e f o r m u l a

(1.3) f o r t h e class of t h e d i v i s o r

TR y i e l d s

(bearing in m i n d that g = -1 here) TR

= 312.A

=

+ 3t2-B

+ 213.C

-

lI3.A

0

as it should. We c a n also check the f o r m u l a (1.2) for the class of the divisor TN: fixing L, t h e condition t h a t t h e cubic C = Q u L lie in TN is j u s t the q u a d r a t i c condition t h a t Q be t a n g e n t to L; and likewise, fixing Q the condition t h a t C e TN is the q u a d r a t i c condition t h a t L be t a n g e n t to Q. We see t h a t

TN

+ 2~2;

= 2q i

and indeed, by (1.2) w e have TN

= -4.A

-

= 2ql

6.B

-

5/2.C

+ 3/2.A

+ 2q 2.

Consider finally the variety parametrizing cubics with three nodes, that is, triples of lines. The Severi variety in this case is ,just the third symmetric product (p2)(3) of the plane p2 (or rather the dual projective plane), minus the diagonals; the Picard group is thus Z, and is generated by the class whose pullback to the Cartesian product (p2)3 is the s u m of the pullbacks of the classes of lines from the three factors. The class A is just this class; A is clearly zero, and w e can use the relations (1.1) and (1.2) together with the fact that CU = TN = 0 to see that 0 = 3A + 3B + C 0

= -6A

-

6B-

5/2.C

and hence that C = 0 and B = - A . Finally, observe that the class of the divisor TR of triples of concurrent lines -- which is visibly just the class A - - is given by (1.3) as

50

TR

=

=

5/2.A + 3/2.B A.

References

[D]

S. Diaz, Exceptional Weierstrass points and the divisor on moduli that they define, Memoirs of the A.M.S. 56 (1985)

[D-HI]

S. Diaz and J. Harris, Geometry of the Seven variety, preprint

[D-H2]

S. Diaz and J. Harris, Ideals associated to deformations of singular plane curves, preprint

IF]

W. Fulton, Intersection Theory, Springer-Verlag Berlin 1984

April 1987 Varieties c u t out b y quadrics: ~ch.eme-theoretic versus homogeneous generation of ideals Lawrence FAn, David Elsenbud, and b-~heldon Katz* Conten~ Positive results 1) C u r ~ on rational normal scrolls 2) Curves m Pq and p5 (Counter-) Examples 3) Deccerminantal constructions 't) General sets of points 5) Elliptic octic curves in p5 Summary In this note we consider cases in which a c u r v e in p r which is scheme theoretically the intersection of quadrics necessarily has homogeneous ideal generated by quadrics. The first case in which this does not happen is for a general elliptic octic in p5; we give a woof of this using the surjectivity of the period m a p for K5 surfaces. *The authors are cra~ful to the N S F for p~rtial support, and to the N S F and Brigham Youn~ Uni~sr~ity for h~vin~ 3upporte~lthe oonfemnc~s on Enumerative C~ometry at Sundanos, Utah, whioh provided a pleasant and ooncenial backdrop for work on this project.

Introduction Several i m p o r t a n t results in the t h e o r y of projective curves assert t h a t a given class of curves has homogeneous ideal generated by quadrics. Such for example is the case of a canonically embedded c u r v e (Noether's Theorem) or a curve e m b e d d e d by complete linear series of high degree c o m p a r e d to the genus of

52

t h e c u r v e . Because direct g e o m e t r i c t e c h n i q u e s a r e available, t h e s e results a r e generally easier to p r o v e s c h e m e theoretically - - in algebraic language, it is easier to p r o v e t h e w e a k e r s t a t e m e n t t h a t t h e ideal g e n e r a t e d b y t h e q u a d r a t i c a n d linear f o r m s v a n i s h i n g on t h e c u r v e agrees w i t h t h e ideal of t h e c u r v e u p to a n "irrelevant" c o m p o n e n t . This reflection gives rise to t h e wish t h a t t h e r e should be s o m e principal saying t h a t , u n d e r suitable hypotheses, a c u r v e c u t out s c h e m e t h e o r e t i c a l l y b y q u a d r i c s h a s ideal g e n e r a t e d b y q u a d r a t i c f o r m s (one c a n i m a g i n e m u c h m o r e general s t a t e m e n t s , b u t p e r h a p s it is well not to be too greedy.) This p a p e r is a n exploration of the e x t e n t to which such a principal m a y exist. The positive results a r e roughly as follows: For c u r v e s on 2dimensional r a t i o n a l n o r m a l scrolls, a l w a y s t h e easiest to study, t h e principal is t r u e in a n e x t r e m e l y strong form, w i t h o u t f u r t h e r hypotheses, a n d e v e n stays t r u e if we replace q u a d r i c s b y f o r m s of higher degree (section 1). It r e m a i n s t r u e for all c u r v e s in p r w i t h r ! 4 (section 2), b u t it c a n n o t be extended to f o r m s of higher degree, e v e n in p3 (section 3). It is also t r u e for p r o j e c t i v e l y n o r m a l c u r v e s in p r w h i c h lie on p r o j e c t i v e l y n o r m a l K3 surfaces c u t out b y quadrics; this includes in p a r t i c u l a r all p r o j e c t i v e l y n o r m a l c u r v e s in p5 (section 2). These last results a r e p r o v e d by combining liaison techniques w i t h a sort of general position result, L e m m a 2.7, w h i c h a s s e r t s t h a t t h e canonical m o d u l e of t h e h o m o g e n e o u s coordinate ring of a n irreducible p r o j e c t i v e l y C o h e n - M a c a u l a y c u r v e is g e n e r a t e d in degree 0.

On the other hand, the principal fails already for some non projectively normal curves in p5. The example of smallest degree is the general elliptic octic in pS, which is, as w e show, cut out scheme theoretically by 5 quadric~, though its homogeneous ideal requires two additional cubic generators (section 5). The example is constructed, following the attack of Mori [1984], by exploiting the surjectivity of the period m a p for K3 surfaces to first construct the K3 surface in p5 which will be the intersection of 3 general quadrics containing C. After the fact, we discovered an explicit example as well, which however w e can only verify by computer, using the program Macaulay of Bayer and Stillman

[1986]. We see f r o m t h e e x a m p l e of t h e elliptic octic t h a t s o m e additional h y p o t h e s e s on C will be n e c e s s a r y in general. P e r h a p s t h e m o s t salient p o ~ b i l i t y

53 in this direction, supported b y t h e results in p5 and on K3 surfaces, is t h a t p r o j e c t i v e n o r m a l i t y m i g h t suffice: Problem: Let C c pr be a projectively normal curve which is scheme theoretically cut out by quadrics. Is the homogeneous ideal of C necessarily generated by forms of degree 0. Since S, the image of S' under [HI,is assumed 2-dimensional, w e m a y write ]~ -~ 0pi(c) @ 0pl(d) with 0 ~_ c ~.d. W e write C O ~ H-dF for the effective irreducible divisor which is the section of the natural projection P(~:)-*pl corresponding to the quotient ~-* Opl(c).

56 With this notation established, we c a n s t a t e t h e s h a r p e r version: T h e o r e m 1.1 bis: With notation as above, t h e following a r e equivalent: i) The h o m o g e n e o u s ideal of C is g e n e r a t e d by f o r m s of degree a a n d ( e - a ) c

2b

l~emark: If t h e r e is a h y p e r s u r f a c e of degree e containing C b u t not containing ~0(C0), t h e n condition ii) is satisfied. Proof of T h e o r e m i.1 bis: Condition i) trivially implies condition ii). Suppose t h a t condition ii) is satisfied. It follows t h a t t h e linear series [eH-C~ does not h a v e C0 as a base c o m p o n e n t . I n t e r s e c t i n g w i t h F a n d CO we see t h a t e - a _> 0 a n d b 2, to show t h a t t h e multiplication m a p H0~C/5~e)®HOOs 0. Since condition iii) implies t h e corresponding condition for larger values of e, we m a y r e s t r i c t ourselves to t h e case k : 1. Writing R for t h e "residual" divisor eH-C', a n d using t h e fact t h a t ~ c / ~ e ) = ~ O s ~ e H - C ~ , we m u s t show t h a t t h e multiplication m a p

(i)

HOOsO.

Theorem 2 2 follows easily from Proposition 2.6 and the following result, essentially a consequence of Green's "Kp,1 Theorem" (Theorem ~lb2 of Green [1984], modified to work for singular curves by the technique of Eisenbud, Koh, and Stillman [1986], for example): Theorem 2.7: If C is a reduced and irreducible locally Gorenstein curve of arithmetic genus >0, and L is a line bundle generated by its global sections with hO(L)_~5, then the module ~HO(y_ -" L n)

is generated, over the ring ~H OLn bV elements of degree i O.

Proof of Proix~ition 2.4: Bertmi's Theorem s h o w s t h a t neither ~ nor D have singularities a w a y from C. Let 3lC~ = JC/~C2 be the conormal bundle of C, and let V = H0~C(2). Our hypothesis implies t h a t V generates ~C~(2) on C. ~ince TtC~ is a vector bundle of rank r-1, it follows t h a t r-Z general sections will be everywhere independent on C, and thus the intersection ~ of the corresponding hypersurfaces will have no singularities along C, proving the first statement. To prove the second statement, fix 5 as above. It suffices to show that a general element of V induces a section of the line bundle JIC/S ~ having

60

only simple zeros. But b y o u r hypothesis V g e n e r a t e s JlC/S ~, so again b y a version of Bertini's T h e o r e m a general section v a n i s h e s on a reduced set of points, as desired. Finally, t h e f o r m u l a s for t h e n u m e r i c a l c h a r a c t e r s of C' a n d COC' follow a t caace if w e p u t t o g e t h e r t h e facts t h a t , b e c a u s e D is a c o m p l e t e intersection of r - 1 quadrics, its canonical bundle is t h e restriction of {3pr(r-3), a n d t h e r e s t r i c t i o n of its canonical bundle to C or C' is t h e canonical bundle of t h e s m a l l e r c u r v e twisted b y t h e divisor on t h a t c u r v e w h i c h is t h e s u m of t h e points of CAC'. (These f o r m u l a s a r e also special cases of exc. 9.1.12 of Fulton [1984].) proof of Corollary 2.5: The first r - 1 of t h e h y p e r s u r f a c e s c u t t i n g o u t C m e e t in CUC °, so t h e r t h m u s t m e e t C' in e x a c t l y t h e (r-3)d - 2g + 2 points of intersection. Thus this n u m b e r m u s t be twice t h e degree of C~. The f o r m u l a s of Proposition 2 2 a n d simple a r i t h m e t i c n o w yield t h e de~sir~l r ~ u l t . Proof of Pro]x~ition 2 6 : First, let D : COC' be t h e c o m p l e t e intersection of quadrics. W r i t e ~C for t h e ideal sheaf of C in p r a n d s i m i l a r l y for C' a n d D. Since t h e canonical bundle on D is given b y OaD:OD(r-3), we h a v e b y t h e t h e o r y of liaison t h a t ~C/~D : (~D:~C')/~D : Hom((~C, , OEO = Hom(0c', ~D)(3-r) = ~C~3-r). F u r t h e r , since D is p r o j e c t i v e l y n o r m a l , w e h a v e for e a c h integer k a n e x a c t 0 -* H0 ~D(k) -~ H0 ~c(k) -* H0 w c ~ k + 3 - r ) -* 0 . The s t a t e m e n t s of t h e proposition n o w follow a t once. If D is instead t h e c o m p l e t e intersection of a p r o j e c t i v e l y n o r m a l K5 s u r f a c e S c u t out b y quadricn a n d a q u a d r i c h y p e r s u r f a c e then, since D is again lorojectively Gorenstein, e x a c t l y t h e s a m e a r g u m e n t applies, using n o w WD=(ws@Os(D))[ D = OD(2). proof of T h e o r e m 2.1: We a d o p t t h e n o t a t i o n of Proposition 2.1, a n d let i = (r-3)d 2g + 2, t h e n u m b e r of points of intersection of C a n d C'.

61 We leave t h e cases r~_5 to the reader. Consider first the case w h e r e r=4. A t h e o r e m of Castelnuovo [1895] (or see M a t t u c k [1954]) asserts t h a t if C is a n y c u r v e of degree d and genus g embedded b y a complete series, t h e n the homogeneous ideal of C is g e n e r a t e d b y quadrics if d _, Zg+2, so we m a y ignore these cases. F u r t h e r , if i > 2d ~, t h e n a n y quadric containing C contains a c o m p o n e n t of C' as well, contradicting our assumptions. These r e m a r k s , together w i t h t h e ideas of section 1 suffice. We r u n quickly t h r o u g h the possible degrees d, assuming t h a t C is not contained in a hyperplane, so t h a t d_~,l: d=8: d=7: d=6:

d~_5:

C is a complete intersection. We h a v e d'=l, so C' is a line a n d g'=0, w h e n c e i=3>2d', so all t h e quadrics containing C contain C' as well. Here d':2, so g'=0 or -1. If g'=-i t h e n we see again i>2dm,a n d we a r e done as before. If g'=0, t h e n g=2, d_>Zg+Z, and Castelnuovds t h e o r e m applies to show t h a t t h e homogeneous ideal of C is g e n e r a t e d b y quadrics. C is contained in a rational n o r m a l scroll by Proposition 12, so we a r e done b y T h e o r e m 1.1.

Proof of T h e o r e m Z2: I m m e d i a t e f r o m Proposition Z.6 and T h e o r e m 2..7. ~ [ I

3) ~ i n a n t a l

Constructions

While it seems to be difficult to c o n s t r u c t varieties schemetheoretically b u t not a r i t h m e t c a l l y c u t out b y quadrics, t h e r e is no difficulty in m a k i n g examples if one a d m i t s equations of higher degree. Perhaps t h e simplest example is t h a t of 18 general points in p2; t h e points a r e c u t out schemetheoretically b y 3 quintics, b u t t h e i r homogeneous ideal requires in addition a sextic g e n e r a t o r (this t u r n s out to be t h e e x a m p l e of lowest degree in p2). A general technique produces this and m a n y other examples: Let A be a p×q matrix with p!q, filledwith a pxp block A Z of general quadratic forms and a px(q-p) block A I of hnear forms over a polynomial ring in r+l variables k[xo,xl,...,Xr].

62

't I

A

A2 deg 2

t00,I 1

I

\

Proposition 3~!: If the entries of A 1 generate the ideal (x0,xl,_.,Xr), then the ideal of all pxp minors of A defines the same scheme as the ideal of all pxp minors of A except the determinant of A2. In particular, if p(q-p) .~ r+l .~ q-l~2 (respectively zq-p+3) and A is chosen as generically as possible, then the pxp minors of A cut out a nonsingular (respectively nonsingular and irreducible) scheme of codimension q-p+ 1 which is scheme-theoretically but not arithmetically cut out by equations of degree < 2p. The case of 18 general points in the plane is obtained by taking p=5, q=4; if instead we take p=4, q--5, r = 5 , we get a smooth irreducible curve in p 5 of degree 52 and genus 109, cut out scheme-theoretically by 4 forms of degree 7, whose homogeneous ideal requires an additional generator of degree 8. Proof. W e need only prove the first statement, as the second follows by considering the generic case and applying Bertmi's Theorem. Considering the relations a m o n g the minors given by the rows of p×(p+1) suhmatrices containing A2, w e see however that (x0,xl,...,Xr).det(A2) is contained in the ideal generated by the px p minors of A other than A 2.

4) General sets of noints The ideas of this section were suggested to us by Jos Harris. Theorem 4.1: If F is a general set of d points in pr with d-r-1 coming f r o m t h e linear n o r m a l i t y of C . ~

5) Elliptic octic c u r v e s in p5 In this section w e w o r k o v e r t h e complex n u m b e r s . For t h e n e c e s s a r y b a c k g r o u n d on linear series on K3 s u r f a c e s t h e r e a d e r m a y consult t h e p a p e r of S a i n t - D o n a t [1974]. The p a p e r of Beauville [1985] a n d t h e first 2 sections of t h e p a p e r of Merindol [1985] provide excellent b a c k g r o u n d on Hodge t h e o r y a n d t h e period m o r p h i s m for K3 surfaces, a n d t h e i r relation to t h e Picard group.

65 Theorem 5.1: The general elliptic octic in p5 is scheme theoretically the intersection of five quadric hvpersurfaces, but its homogeneous ideal requires two generators of deeree three. Example: Having dealt with the general situation, it is pleasant, though not particularly enlightenin& to be able to write d o w n an explicit example: Let E be the ellipticcurve defined in p2 by the equation x3+xz2-y2z = 0 , and let ~0.~-4p5 be the m a p defined by the linear serie~ x 3, x2y, xy2, x2z+y2z, yS+xz2, yz 2. Using the computer program Macaulay of Bayer and Stillman [1986] w e have shown that (in characteristic 51991 and several others) the homogeneous ideal of E in p5 is minimally generated by 5 quadrics and 2 cubics. The product of either of the cubic generators with any form of positive degree lies in the subideal generated by the quadrics alone,so E is scheme theoretically the intersection of the 5 quaclrics. The actual equations involve so m a n y terms that they are probably not interesting to anyone without a computer system like Macaulay to manipulate them, and with such a system they can be generated easily from the data just given, so w e will not reproduce t h e m here. To understand our approach to Theorem 5.1, note that by Proposition 22, such a curve as in the Theorem will have to lie on a smooth surface which is the complete intersection of 3 quadrics. Such a surface is a K3 surface, and w e will begin by constructing a candidate for it: Proposition 5 2: There is a K3 surfac~ who~e divisor cla~s group is of r~Lk 2 with intersection form

Let S be a K3 surface as in the Proposition, and let A, E be divisor classes on S with A2=AE:8, E2=0. Dy Riemann-Roch either A or -A and either E or -E are effective, and w e m a y assume that A and E are. Evidently both are numerically effective and primitive in Pic S, so by Theorem 5 of Mori [1984], and the fact that every intersection n u m b e r on S is divisible by 8, [AI is very ample and IE1 is base point free. Again by Riemann-Roch and the results 2 2 and 7 2 of

66 S a i n t - D o n a t [1974] t h e i m a g e of S u n d e r IAI is a c o m p l e t e intersection of 3 quadrics in p 5 By Proposition 2.6 of S a i n t - D o n a t [1974] the general m e m b e r of IEI is a s m o o t h elliptic c u r v e , w h i c h w e m a y as well a s s u m e w a s E to s t a r t with. ( R e m a r k on references: The results used h e r e w e r e p r o v e d in Characteristic 0 b y M a y e r [1972]; t h e cited p a p e r of S a i n t - D o n a t extends t h e m to Characteristic p, while t h e p a p e r of Mori s u m m a r i z e s s o m e of t h e m in a f o r m t h a t is c o n v e n i e n t for us.) ~v'ith this n o t a t i o n we will show: Theorem 5.3: Let S, A, E be a 113 surface and divisors as above. The complete linear series IAI, restricted to E, embeds E as ar~ elliptico~tic in p5 which is scheme theoretically the intersection of five quadric hvpersurfaces, but w h ~ h o m o g e n e o u s ideal requires t w o g e n e r a t o r s of degree three.

Proposition 5 2 follows easily f r o m t h e s u r j e c t i m t y of t h e period m o r p h i s m for K3 surfaces, via Corollary 1.9 of Morrison [19841. V/e sketch t h e r e q u i r e d ideas, which, w i t h T h e o r e m 5.4, c e r t a i n l y belong to t h e folklore: W e write H for the integral lattice with quadratic form represented by the matrix

t h e "hyperbolic plane', a n d E8 for t h e n e g a t i v e definite q u a d r a t i c f o r m w i t h Dynkin d i a g r a m E8, so t h a t for a K3 s u r f a c e S we h a v e H2(S, Z) = 3 H ~ 2 E 8. We w r i t e V for this integral lattice. W e will say that an integral lattice L with quadratic form is a }{3 latti~ if it can be realized as the Picard group of a K3 surface with the intersection form. Of course if L is a KZ lattice than, l~=~use of t h e index theorem, L m u s t satisfy the index condition that I~@L does not contain a 2dimensional positive definite subspace. Also, L m u s t be emheddable in the sense that L can be embedded in V in such a w a y that the underlying abelian group is a direct s u m m a n d (it will be the intersection of V with the 1,1 forms in H2(S, C) . W e will say that L is nondegenerate if the induced bilinear form on L corresponds to an injection of L into its dual lattice.

67 T h e o r e m 5.4: L is a K5 l a t t i c e if a n d o n l y if L c a n b e e m b e d d e d in V in s u c h a w a y t h a t t h e u n d e r l v i n z a b e l i a n z r o u o of L is a d i r e c t s u m m a n d

a n d ~ ® L±

contains a 2-dimensional positive definite form. C o r o U a r v 5 5 ( M o r r i s o n [1984] Cor.l.9,i): If L is n o n d e g e n e r a t e , t h e n L is a K3 l a t t i c e if a n d o n l y if L is e m b e d d a b l e a n d s a t i s f i e s t h e i n d e x c o n d i t i o n . We begin the proofs with the results on surfaces: P r o o f of T h e o r e m ~.4: The Hodge T h e o r e m a n d t h e s u r j e c t i v i t y of t h e p e r i o d m o r p h i s m for K3 s u r f a c e s i m p l y t h a t L is a K3 l a t t i c e iff it c a n b e w r i t t e n a s a s u b l a t t i c e of V in s u c h a w a y t h a t t h e r e e x i s t s a v e c t o r ¢ o c ¢ ® V w i t h ~2=0, ~

> 0,and

L = (¢~o@¢~)~nV

.

I n p a r t i c u l a r , if L is a K3 l a t t i c e , t h e n L is e m b e d d a b l e . F u r t h e r , s i n c e t h e conditions w 2 = 0 and w~

> 0 a r e e q u i v a l e n t to t h e c o n d i t i o n s (Re w ) . ( I m w ) = 0

a n d (Re ~o) 2 = ( I r a ~o) 2 > O, w e see t h a t ~ ® L x c o n t a i n s t h e p o s i t i v e d e f i n i t e s p a c e spanned by Re c~ and I m w. Conversely, suppose that L is embeddable in V in our sense, and so t h a t R ® L± c o n t a i n s a p o s i t i v e d e f i n i t e s p a c e ,

spanned b y

v e c t o r s ~ a n d 8, s a y .

M u l t i p l y i n g b y a r e a l f a c t o r , w e m a y a s s u m e c¢2= 82. L e t ~ ' = c¢+i~ c ¢ ® V , so that (oo52=o, ~o' ~

> O, and

L c (¢¢o'@¢~nV . W e will f i n i s h t h e p r o o f b y p e r t u r b i n g ~ ' in s u c h a w a y a s to p r e ~ r v e

the first

t w o r e l a t i o n s a n d a c h i e v e e q u a l i t y in t h e t h i r d . The second of the three relations is preserved under all small perturbations of ¢o', so w e m a y

ignore it. The first and third, thought of as

conditions on ~o°, define a complex quadric hypersurface Q in C ® L ± . Suppose x c V - L. Because L is a direct s u m m a n d

of V as an abelian group, w e have L =

VC~((¢®L±)a), so the hyperplane (C®x) ± meets • ® L ± properly. B y our hypothesis, • ® L ± contains a positive definite plane D, and the intersection of 0 with C @ D is then the union of 2 distinct lines. Thus Q is not a double plane, so the

68 h y p e r p l a n e (C@x) a m e e t s Q in a p r o p e r s u b v a r i e t y . There are only countahly m a n y x c V - L, so the complement of the union of all the QN(C@x) ~ is dense in Q, and w e m a y approximate co' by an element co in this set, which will have the desired properties. Proof of Corollary 55: In the nondegenerate case, if L is embedded in V, then ~ @ L is an orthogonal direct s u m m a n d of ~@V. But ~ @ V has signature (5,19), so the dimensions of the maximal positive definite subspaces of R @ L and ~ ® L ± add up to 3. ~[(I Proof of Prooosition 52: Note t h a t t h e lattice in Proposition 5 2 is n o n d e g e n e r a t e a n d satisfies t h e index condition (in fact I~®L is a hyperbolic plane), so t h a t b y Corollary 5 5 it is enough to e m b e d it suitably. In f a c t it c a n be e m b e d d e d a l r e a d y in H@H in t h e desired sense: taking a basis el,fl,e2,f2 of H(BH w i t h ( e l f 1) = (e2f2) = I a n d all o t h e r p r o d u c t s 0, e l e m e n t a r y considerations lead to t h e choice of generators E: e l A : el+8fl+e2-4f 2 for a direct s u m m a n d w i t h t h e r e q u i r e d induced q u a d r a t i c form. Proof of T h ~ r e m 5.5: Regard E c S as e m b e d d e d b y [A[ in p 5 To show t h a t E is ~ : h e m e t h e o r e t i c a l l y t h e intersection of quadrics it suffices, since S is a l r e a d y t h e c o m p l e t e i n t e r s e c t i o n of quadric% to show t h a t t h e residual divisor R=ZA-K m o v e s in a linear series w i t h o u t base points. Note t h a t t h e basis {A,R} of Pic S satisfies t h e s a m e n u m e r i c a l conditions as {A,E), a n d - R c a n n o t be effective since (-R)An)

in

F m.

fhen

Sing(LNX)

The

4348,

OF LOW CODIMENSION

be ~m*.

Sing(XnH),

X ~n.

X >n,

the

conormal P2(Cx)=X

where

of X in ~m.

where

of k - a m p l e

Then

dim(p21(h))~(m-1-n).

Cx=m-1 , dim

sheaf

X

is

vector

variety

of

If hEX

H is the

hyperplane

Thus

is

N(-I)

Then

the

dual

N(-I)

is

variety

of

bundle.) X.

There

, then

is

we may

corresponding

(m-l-n)-ample.

a proidentify to h.

Since

72

In [3], fying that dim

we give

the

G(2,5)

It is well

Jn F 9 and

property

simple

the

dim

intersection

1.5.

X*

and

X*

that

the

allow

f: Fm~

S u p p o s e that 1 6 A u t ( F m) -I (X) is smooth.

of

those

X(m+2)/2

and

and

with

S.

Mori.

X is p r o j e c t i v e l y

normal.

lhen

(a)

K x ~ O x ( t O)

(b)

Assume

for

that

projective]y is a c o m p l e t e Proof. integer

(a) to .

some

integer

Hi(Ox(J)):O Gorenstein.

tO .

for

I(2m)/3.

Let

N be

the

normal

sheaf

and

HI(Ix(1))

of

X in

F m . Then 8

Hi(N

(I))

b

HO(o

m(1))

C

Hi(Ox(1))

d

Suppose for

= O

for

OO d e f i n e

first

Vi-l'

factor

of

Y.

a

D e n o t i n g Xi_ 2

Xi o f YXzY a l o n g

we c a l l

i points X_I b e a

inductively

Y is an Xi_2-variety.

which

at

In particular,

The b l o w i n g - u p

composition,

the projection

of a given variety

[K].

the diagonal

the

blowing-up

Thus we now h a v e a

sequence of morphisms: ~" ~i-1 --~1Xi

.

S i n c e we o n l y b l o w up s m o o t h p r o j e c t i v e the

varieties

projective. blowings-up

Xi

are

But

it

Definition. n S-points 1

an ordered

implicit.

~

varieties

projective

from

~-1 ~ X0

X_l

along

and

Proposition

1.2

that

the

s m o o t h , so ~ i i s s m o o t h t o o .

fibres

In fact,

the blowing-up of an S-point blowing-up

of

YO a t

~i:Xi+l

we w i l l

refer

Y abstractly n

are ---*i i

An ordered b[omtng-up o f YO

1

the

vi

are

i+l points.

o f Y.; by e l i s i o n

n S-points,

--~ Xi of

a sequence of S-morphisms of S-schemes bi:Yi+l---*Yi,

When we c h o o s e t o r e g a r d

morphisms b i,

we s e e t h a t

morphisms ~i:Xi+l

f a m i l y o f o r d e r e d b l o w i n g s - u p o f Xo=P2 a t

is is

smooth centers,

the

L e t S b e a scheme a n d d e n o t e SXkX0 by YO"

such that b. as

follows

o£ p2 a n d t h e r e f o r e

is the universal

at

smooth and

~0 ..

specific

we r e f e r

i=O,---,n-1, to Y

n

blowings-up

as an S-scheme,

itself

b i being

disregarding

t o Yn a s s i m p l y a btowtng-up o f YO a t n S - p o i n t s .

e a c h n>O a n d any scheme S d e n o t e by Bn(S ) t h e s e t o£ o r d e r e d b l o w i n g s - u p

the For

o f S×kX0 a t

n S-points. Remark.

The d i f f e r e n c e

between regarding

Y

n

as an ordered blowing-up as opposed to

104

simply a blowing-up i s more than j u s t f o r g e t t i n g the order in which the S - p o i n t s are blown-up.

Regarding YO a s P2S,

there i s concomitant with the ordered blowing-up

s t r u c t u r e on Yn a b i r a t i o n a l S-morphism Yn ~ P2S.

But Yn regarded a s

P2S' unique up to S-automorphisms of

simply a blowing-up of

P2S may have i n f i n i t e l y many

b i r a t i o n a l S-morphisms Yn --~P2S. d i s t i n c t even modulo S-automorphisms of P2S.

See,

for example, [H2]. It

follows

from

lerama I . l

that

a

f i b r e - p r o d u c t pullback

of

an

ordered

blowing-up i s i t s e l f an ordered blowing-up, and t h i s makes B into a functor, the n moduli functor for ordered blowings-up o£ XO.

To s e t up the lemma, l e t S be a

scheme, Y an S-scheme with an S-point s, and v:Y' --* Y the blowing-up of Y a t s. For any S-schemes T and Z we can form the f i b r e product TZ = T×sZ.

In p a r t i c u l a r ,

TY has a T-point t = TXsS and we denote the blowing-up of TY a t t by TV:(TY)'--~TY. By the u n i v e r s a l property of blowings-up there i s a unique morphism (TY)'--*Y' making

d i a g r a m 1.1 c o m m u t a t i v e . T~

Lemma 1.1.

by ~,

Let ~ denote the ideal

~n@~._~n, diagram

the ideal

s h e a f o f s i n Y.

s h e a f o f t i n TY i s ~ .

1.1,

so

it

is

enough

to

1.2.

The i t e r a t e d

The c a n o n i c a l

the structure

s h e a f o f TY

s h e a f homomorphisms

see ~n~__~n

is

isomorphic.

But

this

is

clear

*

b l o w i n g - u p Xn_ 1 r e p r e s e n t s

the moduli functor

B,

the

e l e m e n t o f B n ( X _ I ) b e i n g t h e f a m i l y 7rn_l:Xn--~Xn_l. One f i r s t

n Xn_l-points, constructed.

checks inductively

t h a t Xn i s a n o r d e r e d b l o w i n g - u p o f Xn_IXkX0 a t

which is not hard using

lemma 1.1 i f one t a k e s

By lemma 1.1 i t now f o l l o w s

ordered

blowing-up

o f Z×kX0 a t

ordered

blowings-up

a n d we must show t h a t

all

Denoting

n n m o r p h i s m ( T Y ) ' = P r o j ( @ ~ ~)--$Proj(@ff @~)=T(Y') o f

n~O, i n d u c e t h e c a n o n i c a l

Proposition

Proof.

(I.1)

i s an i s o m o r p h i s m .

s i n c e ffn@t~--ffn@y(~y®S~T)--~n®S~T---~n~.

universal

~ y' 1 v ~ Y

The canonical morphism (TY)'--OT(Y') of (TY) ' to the fibre product T(Y')

i n d u c e d by d i a g r a m ( I . 1 ) Proof.

(TY) ' i TY

ordered blowings-up.

for any X

n Z-points.

But by i n d u c t i v e l y

1-scheme Z that

Thus m o r p h i s m s the correspondence using

i n t o a c c o u n t how X

ZXXn_IXn i s a n

t o Xn_ 1 c o r r e s p o n d extends

the universal

is

bijectively

property

to to

of blowing

up a n d lemma 1 . 1 , g i v e n a scheme S o n e c a n show t h a t a n y o r d e r e d b l o w i n g - u p o f SXkX0

105

at n S-points

is,

the f i b r e product of

f o r a u n i q u e l y d e t e r m i n e d morphism S---*Xn_1,

Xn w i t h S o v e r Xn_ 1. •

II. Isom

Every b a s i c

surface

Y of

b l o w i n g - u p o f p2 a t n p o i n t s near

points

of

p2

that

completely arbitrary, infinitely

n e a r p,

• ~uny b i r a t i o n a l induces at

Picard

since

be

it

blown-up

must

be

distinct

one s t r u c t u r e

information

functorially

obtain

of

regarded as

¥.

comt~atible w i t h

an

This the

ordered

infinitely

ordering

fact

that

is if

not q

is

However, V can have i n f i n i t e l y

even modulo automorphisms o f p 2

of o r d e r e d b l o w i n g - u p on V.

and each

Thus, w h i l e t h e f i b r e s

t h e s e t o f o r d e r e d b l o w i n g s - u p o f Xo=P2 a t n p o i n t s ,

f i b r e s may h e i s o m o r p h i c a s a b s t r a c t

The

to

t h e n p must be blown-up f i r s t .

of 7rn_l:Xn--*Xn_1 a r e p r e c i s e l y distinct

can be

by s p e c i f y i n g an o r d e r i n g o£ t h e p o s s i b l y

must

morphisms to p 2

least

number n+l

which

fibres

o£

rational ~rn._l a r e

surfaces. isomorphic

by t h e f u n c t o r Isown_ 1. d e f i n e d i n t h e u s u a l way:

can

be

organized

f o r any scheme U and

p a i r o f morphisms f,g:U--~Xn_ 1, ISOmn_l(U ) i s t h e s e t of U - i s o m o r p h i s m s UXfXn=~UXgX n, where UxIXn i n d i c a t e s and ~ n - l '

finite-type

The f u n c t o r

See [SB, Exp. 221];

ISOmn_ 1 i s

i s t h a t 7rn_1 I s a f l a t

scheme In_ 1 l o c a l l y

projective

of

morphism, b u t i n f a c t

a l o n g w i t h In_ 1 come morphisms s and t to Xn_ 1.

i o f In_ 1 p a r a m e t r i z e s an isomorphism ~i b e t w e e n two f i b r e s

may t h i n k o f s ( i ) composition

by a

•

As n o t e d i n t h e i n t r o d u c t i o n , Each p o i n t

representable

t h e o n l y h y p o t h e s i s t h a t n e e d s t o be c h e c k e d i n o r d e r to

~rn_1 i s smooth and p r o j e c t i v e .

of

as being the p o i n t p a r a m e t r i z i n g the source of ~ i ' the

Xn_lXXn_ 1 o n t o i t s where t i s

projection

t h e morphisms f

o v e r Xn_lXXn_l .

apply Grothendieck's result

~i'

of U and Xn o v e r Xn_ 1 v i a

and l i k e w i s e f o r Ux X . gn

Theorem I I . 1 .

Proof.

the fibre-product

structural first

factor.

morphism

In_l--*Xn_lXXn_l w i t h

The t a r g e t ~ o r p h i s m t p a r a m e t r i z e s

the composition of the s t r u c t u r a l

o f Xn_lXXn_l o n t o i t s

the

second f a c t o r .

o f Vn-1; we

where s i s the projection

of

the t a r g e t of

morphism In_l--*Xn_lXXn_l w i t h the

106

III.

Versality

To see that a solution to moduli for basic surfaces of Picard number n is given in the sense we described at the beginning of this paper by s

In-I

t ] Xn-l

we must show that the family ~n_l:Xn--*Xn_l behaves well locally. be

checked

is

versality,

First

recall

the

definition

The first thing to

of

an

infinitesimal

of

the k-scheme U

deformation.

Definition. ( o v e r T)

Diagram III.1 if

it

is

called

is a cartesian

an infinitesitmll

v

~ u

T

l

1

Speck such

that

A

is

(Notationally, fibre

Let

simply

p--~V b e a

diagram

cartesian,

will

Artin

k-algebra

as

an

(III.l)

-----~T

index

and for

UT

is

flat

UT w h i c h

is

over

not

T

meant

=

Spec

A.

to connote

a

product.)

Consider

local

local

T occurs

Defintion.

are

a

deformation

diagram of schemes

Artin

III.2:

T---*T'

D is

family

suppose is

k-algebras,

say that

flat

the and

versa[

both

of

schemes,

where v

parallelograms

morphism

on

Spec's

at

if

v

I

comprised

of to

the closed

point

nondotted a

point

of

V.

arrows

surjection

of

of T to v.

We

for any such diagram

.

.~,~ b

~U''-

T ~

a closed

corresponding

t h e m o r p b i s m T---~V t a k e s

Txv~..~

is

i

]

(III.2)

~V

t there

exist

making the at

morphisms third

every point

Theorem III.1. The p r o o f deformation

of

Artin

is

let

making

We w i l i

just

of

the point

k-algebras

Y'

over T'.

is

flat

case

Let

in

the

say

of at

that

the

diagram ~ is

consider

a

of

Spec's

smooth

commutative

versa[

the

that

is

if

it

and is

so

an infinitesimal

the blowing-up

surface.

To s t a t e

corresponding

surface

f':X'---~Y'

be a blowing-up

note

infinitesimal

that

fact

a point

a deforn~ition

the morphism of

and

over T',

a special

of a smooth surface

T'--*T" b e

see

X'

arrows

t h e f a m i l y ~n_l:Xn----~Xn_l i s v e r s a l .

essentially

deformation that

dotted cartesian.

For each n20,

of a blowing-up

precisely, local

the

o f V.

embedded d e f o r m a t i o n fact

for

parallelogram

Y with

o f Y' a t

deformations

to a an

this

surjection

infinitesimal

a T'-point of

of an

y'.

To

smooth affines

107

are to

trivial, a

so locally

trivial

o n Y' we s e e b y u s i n g

deformation

particular,

of

1.1

blowing-up

f:X-=-~Y o f

Y at

deformation

of X over T'.

the

X' i s a n i n f i n i t e s i m a l

is an infinitesimal

Proposition

that

is

isomorphic

y--Spec(k)XT,Y'. Now s u p p o s e

X" o f X exLenfl.Lng X' o v e r T " ;

deformation

X'

i.e.,

that

In

that

there

X'=T'×T,,X".

T h e n we h a v e :

Proposition

III.2.

T" s u c h t h a t Proof.

Since

underlying

the

deformations

~y,,

extending

there

Since affine and

of a

be

f~X"

of

that

give

This

(where the reader

is

T'

o f y£Y,

and

T"

by

and U',

indeed

set

remark 1.4.1

terms

follows

the

affine.

Thus RIf.~V = 0 implies

deformations

are

just

f"~V"'

it

with

the

follows

[Se]. that

trivializations.

blowing-up

of a T"-point

Lemma I I I . 3 .

Let

surface

U.

Proof.

Asterisks

corresponding is

trivial

to

will

S p e c ( v u / m i ).

By t h e

affine.

sequence

implies

Rlf~

that

all

of U

we h a v e

To s e e get

HI(u,f~EV ) = 0

= 0 and f i n a l l y HI(E)

to

E,

the

an

exact

since

U is

= 0 follows infinitesimal since

v]j,, i s

i n d u c e s a m o r p h i s m fxid:V×T"---NJxT" c o m p a t i b l e

is

the blowing-up

o f U×T" a t

yxT",

so f"

is

the

•

completions

u.

U is

V" ~ V×T" a n d U" ~ UxT" a n d ,

the blowing-up

denote

remains

U be an open

by

Therefore

= O, w h e r e E V i s

the point

you to

deformations

sheaves

of

But f×id o f Y".

of

HI(~x)=O he

So l e t

vanishing

f":V"--4J"

f:V--4J be

Then Rlf~V

that H I ( v , ~ )

that

still

It

induced

since

Again,

But

the

the

infinitesimal

the condition

tangent

spectral

below.)

of

just

which I refer

JR, 4 . 1 a ] .

Leray

lemma I I I . 3

low d e g r e e

for

while

and therefore

of

as Rlf~vx=O ).

O='=-)HItu,f~F-,V)---)HI(V,~v)'-')HO(u,RIf~,V).

ui

of

is

f~VX,,

f":X"---)Y"

that

Denote

Denoting

(The first

f v X,

y".

so is f"

V=f-Iu.

to

case,

sequence

from

use

the

take note

etc.

to

infinitesimally

morphism

Y' o v e r

y'.

isomorphic

isomorphic

of a T"-point

U",

H I ( U , E u ) = 0 a n d H I ( V , E v ) = O.

is

is a

should

via Wahl's

X"--~Y" i s a b l o w i n g - u p

neighborhood

Y

~y,

Y" o f Y e x t e n d i n g

y" extending

scheme deformed

f i s a n i s o m o r p h i s m away f r o m y ,

V over

second,

Vy

will

X'-==oY' .

is reinterpreted

t o show t h a t

sheaf

space

deformation

of a T"-point

o n e may e x p e c t to

[W,Theorem 1.4(c)] finds

structure

scheme,

defining

is an infinitesimal

the blowing-up

topological

the original that

There

X" i s

of a closed

the tangent with

L e t Vi d e n o t e

t h e o r e m on f o r m a l

point

sheaf

respect VXuUi a n d functions

u of a smooth affine

o f V.

to

the maximal

let

Ei denote

[Hs,III.11.1],

h o m o m o r p h i s m (RlfwEv)W==-*invlim H I ( v i , E i ) i s a n i s o m o r p h i s m .

ideal

sheaf

m

EV®~u , w h e r e 1 the

natural

108

S i n c e U i s smooth, V 1 i s pI and i t s have an exact ~pl(2)

sequence (~-~1- ~

= O, i20.

But

s e q e n c e by EV, we o b t a i n : on i u s i n g t h e f a c t It

T e n s o r i n g by ~ p l ( i )

mi/m i + l

that

(R f-~V)

so

the exact

tensoring

the

sequence

last

and h e n c e (R 1 f~EV)

is isomorphic to

exact

T a k i n g cohomology and i n d u c t i n g

i>O, we c o n c l u d e t h a t H I ( v i . E i ) - - O ,

i n v l i m H1 ( V i , E i )

1

R l f . ~ v is c o h e r e n t .

~V1 ( i ) ,

0----~_-1(i)--~i+1---~i--43.

t h a t H1 ( V 1 , E I ( i ) ) = O ,

now i s c l e a r

~

vanish.

i21. But s i n c e

(Rlf~V)®(Og)~ [ ~ t . The.

~].

Sinoe

f i s a n i s o m o r p h i s m away from u, Rlf~Ev h a s s u p p o r t o n l y a t u and so R l f ~ V VU,u-module.

Now Rlf~EV v a n i s h e s s i n c e (VU)~ i s f a i t h f u l l y

[Mat, Thm. 5 6 . 5 ] .

over ~J,u

Suppose t h a t we a r e g i v e n a d i a g r a m s u c h a s I I I . 2 ,

Vn_l:Xn--~Xn_l.

Proposition

flat

is an

*

P r o o f of theorem I I I . l : D---*V i s

Just

and t a k i n g cohomology, we see

Next l e t m d e n o t e rn~V and c o n s i d e r

O--~mi/mi+l--~i+l--~V1--43.

Now we

®~V1---~--~O of s h e a v e s on VI=P1 where EV1 i s

and EV®Wi we d e n o t e E i .

that HI(v1,EI(i))

normal b u n d l e N i n V i s ~ p l ( - 1 ) .

1.2.

T×V~--4r

Thus

By P r o p o s i t i o n

III.2

is

an

ordered

blowing-up

of

i n which T×k P2

t h e s e b l o w i n g s - u p e x t e n d o v e r T' to U ' ;

i.e.,

U' i s o b t a i n e d by a s e q u e n c e of b l o w i n g s - u p of a d e f o r m a t i o n P' of p2 o v e r T ' . course,

P' e x t e n d s T×k P2 o v e r T' and i t

trivialization blowing-up

that

is

compatible

the trivialization

T'.

that

i n d u c i n g TXVD--~T v i a p u l l b a c k

t h e d o t t e d a r r o w s of d i a g r a m I I I . 2 that

with

is trivial

It

pulls

back

a

of TXkP2,

represented

trivialization

take any trivialization A=A'×T,T

Now ( n ' ) - l ( A ' )

TXkP2 a s r e q u i r e d .

The n e x t f a c t

Corollary Proof.

the automorphism funetor

over

III.4.

is

a

T.

Autp2 e v a l u a t e d

trivialization

of

on T.

over

T'

existence 1.2.

of

To s e e

we

have

an

This corresponds But AUtp2 i s

so ~ e x t e n d s P'

an ordered

A' o f P' o v e r

Thus

A to TXkP2.

by t h e smooth g r o u p scheme PGL2 JR2, p . 2 0 ] ,

~'EAUtp2(T' ).

is

would t h e n b e a s s u r e d by P r o p o s i t i o n

a u t o m o r p h i s m of TXkP2 o v e r T t a k i n g t h e t r i v i a l i z a t i o n to a n e l e m e n t n of

t h e n U'--*T'

Of

I f P' h a s a

o v e r T--oT', and t h e r e q u i r e d

TXkP2 e x t e n d s t o P ' , to

s i n c e p2 i s r i g i d .

by

to a n e l e m e n t

compatible

with

•

f o l l o w s " f o r m a l l y " from theorem I I I , 1 .

The morphisms s , t : I n _ l - - ~ X n _ l of § I I a r e smooth.

S i n c e t h e i n d e x n-1 does n o t c h a n g e ,

it will be suppressed;

since the proof

109

for

s is no different

smoothness [F~

than

for

IV 1 7 . 1 4 . 2 ] ,

t,

it

C~

k-algebras.

is

the

We must

commutative.

Define

sa:T--*X a n d

similarly

is

an

infinitesimal

~' :T'---~X i n d u c i n g this

show

of

Xn--~Xn_ 1.

This

deformation

of

pullback

of

versality,

etale

surjection

morphism

T'--*I

By

t~sa.

sa. s o by

T--*I whose c o m p o s i t i o n w i t h

the

by of

the I,

there

D/~, i s

diagram

a

gives

a

that ~a'

is

a morphism

isomorphic

t o Da, and

property

t gives

Artin

morphism

Dta---~a, we f i n d

By v e r s a l i t y

the universal

local

making

induced

construction

Now c l e a r l y

of

a',

of

I

there

is

a

making the diagram

•

section

is to see that

locally

for

p2

number

n

smooth

possibly

blowings-up

of

amounts

showing

to

Xn-~Xn_ 1.

of

Picard that

of

p2

is

Since

we

know

we c a n c o n c l u d e i t

closed point

a

blowing-up

/~a'"

extending

~sa--~ta,

a blowing-up

Theorem I I I . 5 .

a

ordered

and

and extending

task in this

family

~ta

is

to

so composing it with the inclusion

morphism T'--~I extending

Our l a s t

corresponding

the

deformation

commute a s r e q u i r e d .

(iiI.3)

there

be

define

Da ,

implies

~X

that

to

isomorphism extends

smooth

a

morphism

~sa

T-isomorphism Dsa--~t a,

smoothness

I t

T' o:T--~T'

Formal

*I

° I which

be omitted.

s o a s s u m e we h a v e a c o m m u t a t i v e d i a g r a m T

in

will

a

locally

in general

an ordered

this

for

the etale comes

t o p o l o g 5 r any

from

the

noninfinitesimal

blowing-up,

infinitesimal

family

and hence a

deformations

by

by A r t i n a p p r o x i m a t i o n :

L e t Z--*V b e a s m o o t h p r o p e r

family

such

o f v o f V i s a b l o w i n g - u p o f p2 o f P i c a r d

that

the fibre

number n .

Z over a v

Then t h e r e

is an

n e i g h b o r h o o d U o f v s u c h t h a t UXvZ i s a n o r d r e d b l o w i n g - u p o f p 2

Proof.

Let

p2

Z __~p2 s i n c e V

v

Z

Vm--Spec(~v/um ) ,

V

be

the

is where

trivial

an

ordered

u

is

Zm=Vm×vZ i s a n i n f i n i t e s i m a l of ordered

blowings-up

family

(viz.

Xn_l-morphism

Now t h e r e

blowing-up

the

maximal

deformation III.1),

m o r p h i s m s Vm--*Xn_ 1 i n d u c i n g Zm. birational

V×P2.

of

ideal o f Zv.

there

is

p2 sheaf

of

is

Picard

to

Replacing

m>O a c o m p a t i b l e

V by a n e t a l e

n.

Set

v.

Then

o f t h e f a m i l y Xn--oXn_ 1

S i n c e Xn--~Xn_ 1 i s a n o r d e r e d b l o w i n g - u p ,

Zm-~P2V .

morphism

number

corresponding

By v e r s a l i t y for all

a birational

sequence of there

neighborhood

is a

of v if

m necessary, is

there

i s by [A2, C o r o l l a r y 2 . 4 ]

the blowing-up morphism Z __~2 v

a V - m o r p h i s m Z--~2V w h i c h o v e r t h e p o i n t

v

110

Now some c o m p o n e n t o f

the proper

closed

of the morphism Z--~2 V contains

fibres

of the first

blowing-up

c o m p o n e n t E. neighborhood section

In the of v,

of

possibly

restricting

the point first

v is

Let

Y be

a blowing-up to an etale

just

the

transform

as an ordered

as above,

is a section

V.

ZV i s m

the proper

considered

v

same f a s h i o n

there

P2 V o v e r

construction,

of Z

subscheme of Z of positive

after

of the exceptional

blowing-up

possibly

blowing-up

o f YV a t m

of

P2 V a l o n g

there

locus

denote

this

V by an etale

o£ E i n t o

n-1Vm-points.

neighborhood,

factorization

of

of p2

replacing

V--~E a n d by i n c l u s i o n

the

dimensional

this

Z an induced section.

As b e f o r e ,

again

By after

i s a V - m o r p h i s m Z--~Y w h i c h o v e r

the ordered

blowing-up

the

Zv---~P2 t h r o u g h

blowing-up. Continuing

an ordered

in this

By c o n s t r u c t i o n , closed

point

there

the

some e t a l e

m a n n e r , we e v e n t u a l l y

blowing-up both

v.

families

By o n e

functor

obtain

o£ P2 V a n d w h i c h o v e r induce

final

same

application

Isom instead

neighborhood

the

a V - m o r p h i s m Z - - * Z ' , w h e r e Z'

the closed

point

infinitesimal [A2,

Corollary

o£ Hom) we f i n d

that

Z a n d Z' a r e

U of v.

2.4]

is

isomorphism.

(but

at

the

now u s i n g

U-isomorphic

over

•

Pic

Now that we have our moduli solution {In.Xn} in hand, something about

an

deformations

of

IV.

understand

v is

the scheme structure,

will use is the relative Picard scheme.

it will be interesting to

especially of I . n

To review briefly,

A tool

we recall

that we

the following

definitions:

Definitions. (a) locally

Let S be a scheme and let

The Picard free

[SB, Exp. (b) f:V--~S i s (c) S-schemes

group

VS-mOdules.

232-01], The

w h e r e VS

relative

the structural The

relative

Proposition IV.1.

these

is

Picard

is

the group

is a canonical the sheaf group

of

isomorphism classes

of

rank one

i s o m o r p h i s m P i c ( S ) - - - - ~ I ( S , v s ~)

of units

Pic(V/S)

of vS .

is

the

group

HO(s,Rlf~(~V~)),

where

morphism. Picard

to groups defined

Concerning

Pic(S)

There

V and Y be any S-schemes.

objects

functor

Picv/S

is

the

contravariant

as PiCv/s(Y)=Pic(VxsY/Y ).

there

are

the following

Let S be a scheme and let

f:V--~

standard

results:

be an S-scheme.

functor

from

111

(a) with

If

f~V=~S ,

the last (b)

finite

there

homomorphism being

If

represented

then

f

is

by a

is

exact

surjective

projective,

separated

an

fiat,

has

(which

integral

we a l s o

S e e [SB, Exp. 2 3 2 . 2 ]

(b).

fibres,

denote

by

then PiCv/s)

Picv/S

is

locally

of

for

( a ) a n d [SB, Exp. 2 3 2 . 3 ]

a n d [SB, Exp. 2 3 6 . 2 . 4 ]

for

*

Corollary

IV.2.

S-points.

Let

S be a

S-scheme locally

Proof.

This

integral

fibres,

follows

of finite

from

let

a disjoint

f V.V=VS

theorem,

union

f:V---~ be a blowing-up is

exact

let

IV.1

since

are

Theorem IV.3.

Let S be a connected

scheme and let

n>O S - p o i n t s .

Then Picv/S

By c o r o l l a r y

IV.2,

S u p p o s e we show t h a t

Picv/S

fibre

denotes

o£ P i C v / S

over

the

of

fibre

V over

is

a separated

by a

smooth,

projective,

has

the

s of

f:V--~

This

Being a blowing-up

that

over

s(i.e.,

an s-point),

Pic(Vs/S)=PiC(Vs).

well-known

that

of

in

a

line

conclusion

that

Picv/S

is

closed

point. IV

Therefore

the

is

just

freely

classes

Pic(s)---O,

of

of

isomorphic

is of

just

p2

finite

of

Conside:r

Pic(Vs/S ),

w h e r e Vs

=p2,

p2 a t

V

certainly

has a

S

n points

generated

over Z by the pullback

the

transforms

of

type.

t o S.

s o we s e e b y P r o p o s i t i o n

a blowing-up

total

o f P2 S a t

IV.l(a) and

of

it

is

the class

the blowings-up.

The

o f t h e t h e o r e m i s now c l e a r .

To s e e that

B u t Vs

Pic(Vs)=Z n+l,

p2 a n d

while

let

S"

locally

S

section

n,

Thus zn S is a

be a blowing-up

S-scheme

S.

integer

integers. Zn .

union of sheets

point

s.

represented

a s a n S - s c h e m e t o Zn + l

is a disjoint

a closed

n>O

is

For any positive

set-theoretically

isomorphic

Picy/S

is

o f S, Z b e i n g

S-scheme whose fibres

Proof.

P2 S a t

•

S be a scheme.

o f Zn c o p i e s

is

f

by [ECA I I I . 7 ] .

finite

-

and Picv/S

of

t y p e o v e r S.

Proposition

and therefore

To s e t u p t h e n e x t Zn S d e n o t e locally

scheme and

Then O---~ic(S)--~tc(V)--~Pic(V/S)

separated

[EC.A

o v e r S.

t y p e o v e r S.

Proof.

a

O--~ic(S)---~ic(V)--~Pic(V/S)

if V has a section

and

S-scheme

sequence

PiCv/S etale

is

over

a disjoint S.

Since a section

1-7.9.3],

two

union

T h e n show t h a t of a separated

S-sections

of

PiCy/S

on the one hand every S-section

on the other

of

sheets Picv/S

etale

isomorphic has

t o S,

an S-section

first

show

through

any

morphism is both open and closed

which

meet

is a connected

must

in

fact

coincide.

component of Pity/S,

hand the complement of the union of the sections

while

cannot have any closed

112

points

(else

i t would h a v e a s e c t i o n )

Now l e t ' s

see that

t:I---oX r e p l a c e d that

there

by t h e

Picy/S

etale

structural

o v e r S.

Consider

morphism PiCv/s--~.

is always a unique morphism a":T'--*Picv/S

the universal the unique

property

existence

of PiCv/S,

Pic(VxsT'/T')--+Pic(VxsT/T ). Vo=VXsk, a

a corresponds

o f a'" c o r r e s p o n d s

Pic(VxsT'/T' ) mapping to L via

of

is

and so must b e e m p t y . but with

t o do i s

show

making t h e d i a g r a m commute.

to an element

By

L o f P i c ( V x s T / T ) and

is a unique

element

L' o f

homomorphism

But VXsT' and VxsT are j u s t i n f i n i t e s i m a l deformations

smooth complete

r e s t r i c t i o n maps

III.3,

What we n e e d

to showing there

the canonical

diagram

rational

Pic(VxsT(')/T('))--~ic(Vo)

surface, are

and

in

this

isomorphisms

situation

by

the

standard

the

argument. Indeed,

l e t VT be an infintesimal deformation of V0 over T.

A a local ring, we have PicT=O[ffi, p.124].

Since T i s SpecA,

Since A i s f i n i t e dimensional over k,

there i s an element t of A such that tA is a l-dimensional ideal, and i t gives r i s e to an exact sequence O - - ~ t A - - ~ t A - - ~ , where i t denote A/tA by A'. exact

sequence

exponential to

l+tx,

O--~O----~VT---~V,---4),

gives an injective

and

is

n o t a t i o n a l l y convenient to

Flatness of VT over T implies that tensoring by ~V gives an

together

with

where

t~V,

is

(SpecA')XAVT.

The

truncated

g r o u p homomorphism WVo--*(uVT)~ by s e n d i n g a s e c t i o n

t h e homomorphism ( ~ V T ) ~ - - ~ V , ) ~

x

i n d u c e d by ~VT---~V,, we

get an exact sequence O~a~Vo---~VT)~--~V,)X ~.

Taking cohomology we get an exact

sequence HI(v,t~Vo)---~iC(VT)---~ic(V')--CH2(V,~VO) .

The underlying topological spaces

VT and V0 are the same so HI(Vo,~Vo) and Hi(V,~Vo) are isomorphic for a l l i .

But V0

is

is

a

rational

surface

so

HI(Vo.~}o)-O, i~l.

Thus PiC(VT)--~ic(V' )

an

isomorphism, and by induction on the dimension dimkA' = -l+dimkA, we see that the restriction

homomorphism P i C ( V T ) - - - ~ i c ( V o ) i s a n i s o m o r p h i s m .

Finally PiCV/S.

we m u s t

Now V i s

see

that

there

the blowing-up

which to V are divisors,

is

an S-section

through

o f P2 S a l o n g n s e c t i o n s ,

as is the total

S,

as

Since

t o Vs g i v e a b a s i s pointed it

out

factors

above; through

of Pic(Vs), i.e.,

the

Pic(V/S)

any

closed

total

point

of

transforms

of

t r a n s f o r m o f L×kS, w h e r e L i s a l i n e

They i n d u c e l i n e b u n d l e s o n V a n d h e n c e e l e m e n t s o f P i c V. bundles

the

f o r Vs a f i b r e

The r e s t r i c t i o n

via

the

map

of these

of V over any closed

homomorphism P i c ( V ) - - ~ i c ( V s ) Pic(V)--~Pic(V/S)

in p2

point

of

is

surjective.

of

Proposition

113

IY.l(a)

using

surjection

the natural

Pic(V/S)--~Pic(Vs).

is etale

and locally

universal

property

surjectivity

Remark.

If

of finite o£

is

Picy/S,

an

basis

class

in order.

p

corresponds

o£ a l i n e ,

called

possibly

blowing-up

of Picy/S

as

free

latter.

the

elements

o£

PGL2-structures with respect

a

the etale

the

fix

the

therefore

at n points

It

by t h e

of the blowings-up

as described

structure the

gives

would

subscript

n,

an ordered o f P2 S,

for

there also there

is

and

interesting if

the

a the

morphisms

for

s,

to denote

The b a s i c

s,

s o we w i l l

and so its

be c o n v e n i e n t by ~:Z--*X.

facts

use as a the family are

that

I w over open subsets

the components I be

of

situation

PGL2-homogeneous s p a c e s

topology,

W.

scheme

o£ t h e c o m p o n e n t s I w. e s p e c i a l l y

consider

be any exceptional subgroup

generated

is given

Structure

also

It will

a

induces

Since a blowing-up

symmetric with

We w i l l

group

basis

*

w

to

of

I being

i n d e x e d by

somehow c l a s s i f y

the classification

of

the

behaves well

t o Iw, I w, a n d I , .

To b e g i n ,

the

in

ordering

transforms

ordered

different

study is

union of principal

trivial

This

of the total

~heme

to t

can be suppressed,

locally

group.

the

configurations

the

~n:Xn+l-~Xn of ordered blowings-up I is a disjoint

then

exceptional

between

for

By

of an element h of Pic(V/S).

indeed,

want

By t h e

PiC(Yq).

configurations;

I

the

X of

many o r d e r e d b l o w i n g - u p s t r u c t u r e s ,

The s i t u a t i o n

only consider

X,

element

P2 S,

exceptional

V.

section

Since Picy/S

many

correspondence

this

of Picv/S.

infinitely

infinitely

ordered blowing-up structures.

subscript

an

confLguratton.

an exceptional

different

--*X . n n

of

of Picv/S

bijective

In

to

abelian

and the classes

of elements

example, can have possibly

t:I

point

have a

o f P i c v / S o v e r S t h r o u g h p, a s we w i s h e d t o show.

ordered

Such a s e t

which is

may b e

p be a closed

we a l s o

t y p e o v e r S, t h e image q o f p i n S i s c l o s e d .

to a section

Y

distinguished relative

So l e t

of Pic(Vs/S ) with Pic(Vs),

of Pic{V/S)--~Pic(Vq), X is the restriction

But h c o r r e s p o n d s

set

identification

by

transformation

of

a b l o w i n g - u p V o f P2 k a t

configuration the group of

permutations defined

e 3 - - - ~ O - e l - e 2, a n d e i ~ i ,

for

V.

invertible of

as: for i>4.

the

n points

Let W (or, n linear elements

and let

suppressing

transformations {el,-..,en}

eo---*2eo-el-e2-e3 , When n ~ m ( X , aoL) Proof By (7) in the proposition we have

g(X, aL) : U ( ( a - a o ) L ) g ( X , L) Since L is ample, f L 2 ~ - ( m + l ) g m ( X , L) > 0 for all m. This gives the claim. : R e m a r k 3.4 We even have strict inequality in the last corollary unless X is projective space, ao = 1 and L is a hyperplane.

125

4

E m b e d d i n g by the smallest very a m p l e twist Let/2 be an ample inverible sheaf on X, which corresponds to the divisor L as usual. Put

ao = ao(L) = m i n { a e Z laL ~,.~m L' very ample} There are several reasons to expect that the smallest possible embedding dimension obtainabi[e by embedding X with a very ample divisor L' ~,~,,~ a L into P ( H ° ( / Y ) ) and then projecting, is obtained exactly for a = a0. Such a result would be important for the understanding of th.e absolute embedding obstruction: The way 3,~ is defined in section 1, we have to compute the relative embedding obstructions at all ponts of V e t ( X ) , which severely limits the usefulness of the theory. One of the key points now is to find a canonically defined minimal subset of V e r ( X ) , which suffice for the c o m p u t a t i o n of the absolute embedding obstructions. Unfortunately, for the time being we are not able to achieve this general goal. However, below we prove a theorem which at least yields the result in a substantial number of interesting cases, and thus enhances the computability of the absolute embedding obstruction while at the same time increasing the evidence for our general conjecture below. For convenience we introduce the following notation:

B m ( X , L ) = rain

{

~m(a[L])

such that a[L] e Ver(X)

/

The claim we would like to prove is the Conjecture

4.1 If L is ample, then

Bin(X, L) : ~ m ( X , ao (L)L) for all m = r , . . . , 2 r , where r = d i m ( X ) . We axe able to prove the following Theorem

(a) (b) (c) (d) (e)

4.1 Conjecture 4.1 is true in the following cases:

For For For For For

ao(L) = 1 m < 2r-1 r=l, 2 r = 3 and ao(L) _(d - 1)(d - 2) ÷ M2(r + 1) + M(r + 3 - 2d) d-1

As in Section 3, let ~f = sition 3.1:

f L r.

We have the following formula, which is shown as in Propo-

b ( X , (a0 + a)L)

(i4) -- T(aL)b(X,o'oL) + ~ {(o0 + o)~L((a0 + a)L) - a~T(aL)L(aoL)} The remaining case to treat being m = 2r, we only need the last component of the formula, which immediately yields the following: ~ 2 r ( X , ( a 0 + c r ) L ) = ~ ( 2 r ~1 i=0

(15)

Lr-ibr+i(X'6OL) +620~2r

where

r

Ol2r ~" ((TO -~ G) 2r -

orO ~ (

2rr --1-i 1

) ar--iaiO

~--

i=0

0 ~:r(e

,(1 +

Here we have put 2r

( :/ ) (t' -

t : , - ' + ' ) = ~,(t)

i=r+l

Now let

~ : r ( o 0 , o) = z~ = ~ : r ( x , (o0 + o)L) - ~2r(X, o0L)

127

Letting t =

a-~a, we find t h a t r--1

Z ( 5:+_:

+

i=0

:

( 5::: i>2

D i s c a r d i n g t h e t e r m s w i t h i > 2 a n d u s i n g (11) yield t h e following e s t i m a t e for A :

A w

(16)

( 2rr_+: .)t r-1 {(a~6 -- l}(a~6 -- 2) + M ( M ( r + I) + r + 3 - 2o~6)}

- r - 1 10ao26(°o26 - 1)t 2 + ~ - ( ° o 2 - 1)(°o2 - 2)t + °o462~02(t)

= °o462t 4 + 40o462t s + {10Oo2$(crg6 - 1) - 40o462} t 2 + Nowd

= °26>r+2

{ l VO(. o50

- 1)(°o2 - 2) -- Oo462

}

-- 4, so

5 I0-o~6(.o~6 - 1) - 4.o'6 = = 6.o'6 = - 1 0 . ~ 6 = 6.o~6(Oo~6 - ~ ) > 0 and

-~1°-°5 1)(oo~ - 2) - o ~ 3" °-

=

{7oo'~ ~

-

30oo~s + 20} > o.

The l a s t i n e q u a l i t y h o l d s since f ( u ) = gut 2 _ - ~ u + 17 is i n c r e a s i n g w i t h u for u >_ 3. T h u s we h a v e s h o w n t h a t A2(ao,er) > 0 for all °o a n d ° in t h e r a n g e of i n t e r e s t , so t h e c l a i m follows for r :2. N e x t , t a k e r = 3. T h e n (18) b e c o m e s 63 .a3 ZXs > 35°o~6(°o~ - l ) t s + 7 ( o - 1)(°~ - 2)t 2 + °o~62 {t 6 - t + 0 ( t 5 - t ~) + 15(t4 - t s ) }

1 2 8

which yields the estimate

t, _> o ~ 2 t 6 + 6 o ~ 2 t 5 + lSoo~2t • (19) +(2o,:,~,~

- 3so~,~)t~ + ¼(39oo~6 ~ - 189,:,~,~ + 126)t~ - o~,,~t

We n o w utilize this e s t i m a t e in t h e following m a n n e r : Since ° o > 2, we g e t d =

e0~5 > 8,

so t h a t

20oo~2 _ 35Oo3~ > l S . 6 2 s ~ 2 (20) 1 6 2 ~(39oo~

-

18903~

+ 126) > 3.84375°662

T h i s gives (21)

A > o652(t6 + 6t s + 15t 4 + 15.625t 3 + 3.84375t 2 -- t) : a ~ 2 t ¢ 3 ( t )

T h e e q u a t i o n ¢ 3 ( t ) = 0 h a s only one p o s i t i v e root, n a m e l y to ~ 0.15188. T a k i n g ° -- 1 , we find t h e c l a i m to be p r o v e d p r o v i d e d eo _< 6. T h u s we m a y a s s u m e t h a t ° o _> 7. We n o w r e p e a t t h e p r o c e d u r e , using t h e c o n d i t i o n cro > 7 i n s t e a d of o0 ~ 2. T h i s gives b e t t e r inequalities t h a n t h o s e in (20), a n d t h u s a s t r o n g e r i n e q u a l i t y t h a n (21):

20o~

- 35o~6 > 1 9 . s g s o ~ ~

(22)

¼ ( 3 9 o ~ ~ - l S 9 o ~ + 126) > 9 . s o s o ~ ~ w h i c h gives

A > a ~ 2 ( t 6 + 6t s + 15t 4 + 19.898t 3 + 9.505t 2 -- t) = a~6zt~3(t)

(23)

w h e r e we n o w h a v e a n e w p o l y n o m i a l ~bs(t), a n d t h e p o s i t i v e r o o t is ~ p r o v i d e d t h a t Oo < 11, as c l a i m e d . F o r r = 4 we c a n a s s u m e t h a t oo > 2 so t h a t o0~ > 16. We o b t a i n

0.8792. It gives A > 0

A > OoS~2 {t s + 8t "~+ 28t 6 + 56t s + 62.125t 4 + 26.6t 3 - 8t 2 - t} = o ~ 2 ¢ 4 ( t ) w h i c h h a s a p o s i t i v e r o o t ~ 0.25690, w h i c h p r o v e s t h e c l a i m in t h e case cro _< 3. T h u s , we can a s s u m e t h a t e o > 4. In t h a t case, we o b t a i n t h e s t r o n g e r e s t i m a t e : A > aoS52 {t s + 8t 7 + 28t 6 + 56t 5 4- 69.5078t 4 + 38.4125t s

-

8t 2

-

t} = a s 5 2 ¢ 4 ( t )

w h e r e we a g a i n h a v e a n e w p o l y n o m i a l ¢ 4 ( t ) , a n d t h e p o s i t i v e r o o t is ~ 0.2213 It gives 2, > 0 p r o v i d e d t h a t ao 0

(1)

Note t h a t strict inequality in (1) is the Nakai Criterion for ampleness. Thus the points which are nef but not ample represent the boundary of Amp(X) in N u m ( X ) - - or strictly speaking, this applies to the corresponding cones in N u m ( X ) Q . By a theorem of S. Kleiman we may take s = 1 in (1), (but not in the Nakai Criterion). See [17}. For two divisors (or points in g u m ( X ) ) we write

L2 < Li,

respectively L~ < L1

if L1 - L2 is ample, respectively nef. Further, we write L2 -< L1 whenever L I - L2 is numerically equivalent to a non zero effective divisor. All these relations carry over in the natural way to N u m ( X ) . In accordance with the usual terminology for functions between partially ordered sets. we shall say t h a t a function

¢ : Num(X) ~

Z

is increasing with respect to the orderings < , < and -_ 2. Let P be a point. We have:

If 8 g ( X ) + 1 is not an odd square, then e(X) = 3 If 8 g ( X ) + 1 is an odd square, then e(X) = 2 if and only if

o o ( P ) = 1 (3 + v / 8 g ( X ) + 1)

A c t u a l l y t h i s c a n be seen directly w i t h o u t using t h e results from section 4, b y m e a n s of t h e usual C a s t e l n u o v o B o u n d . We even get s o m e w h a t more information: For a s m o o t h space curve of degree d a n d genus g we have 51 ( -d1 ) ( d - 2 )

g
g + 3. Thus we find a0 _ < 9 + 3 From the above one concludes that at least when e(X) = 2, there is no reason to expect that (°0 + 1)P should be numerically equivalent to a very ample divisor: In other words, Ver(X) c N u m ( X ) has one or more "isolated points" on the ray through the element ~ E N u r n ( X ) before all multiples of ~ for a > 9 + 3 are in Ver(X). We also note t h a t if X is a smooth curve of degree d and genus g embedded in p 8 for which there are no integers a and b with

d=a+b,g=(a-1)(b-1) then one has the inequality

g < 6 d ( d - 3) + 1 known as Halphen's Bound, see [12]. Thus if o and g are not of the form

o = a + b , g = (°-1)(b-I) and °~ C V e t ( X ) is such that/32(o~) > 0,then

1 o i> - ~ (3 + vr2~g- -

15)

We collect these observations in the 6.2 If cr~ E V e t ( X ) , then either

Proposition

(i)

o = {(3 +

i n w h i c h ease

s Vgi ) = O, or

(ii) 2(1 + x/g) -< a < {(3 + ~ ) in which case there are inetgers a and b such that o = a + b,g = ( a - 1 ) ( b - 1); or else (iiO

+

< o

The first case for g _ 2 where 8g + 1 is an odd square is g = 3. T h e n e(X) = 2 if and only if no(P) = 4, hence we obtain precisely the non singular quartic curves in p 2 . Let a > 4. We wish to examine when a~" C V e r ( X ) . By the theorem of Halphen quoted above, this is always the case for cr :> 6. We claim t h a t 5~ ~ Y e r ( X ) : In fact, 2(1 + vZg) ~ 5.46, so a = 5 is impossible by (ii) in the proposition. For a > 7 all divisors which are numerically equivalent to a P are very ample. But for o = 6 this is not the case: there are divisors on X of degree 6 which are not very ample. Indeed, a smooth degree 4 curve in p 2 is canonical; let L be a bitangent to X with points of contact P and Q. T h e n K = 2(P + Q) is very ample, and it is easily seen - say by [10], Proposition 3.1 in Chapter IV - that 3(P + Q) is not very ample. This illustrates the fact that in general there is no numerical criterion for very - ampleness, a given class in N u m ( X ) may contain very ample as well as not very ample divisors (all being ample, of course). Incidentally, if g = 2, then a divisor is very ample if and only if it is of degree > 5 (an exercise in [10]): Here we immediately get this from (ii) in the proposition since 2(1 + ~ g ) ~ 4.83, and 2g + 1 = 5. For g = 4 the curve can not be planar, so i f a ~ E Y e r ( X ) , then we must have a > 2 ( l + v ~ ) = 6. F o r a >_ 7 = g + 3 w e h a v e a ~ E V e r ( X ) . I f 6 ~ E V e r ( X ) then X can be e m b e d d e d into PS

135

as a curve of degree 6, which has to be a canonical curve, loc. tit. Proposition 6.3. So X is non hyperelliptic in this case. Thus we have shown that no(P) = 6

~

X not hypcrelliptie

In the hyperelliptic case a 0 ( P ) = 7. Similarly for g = 5: We find that a0(P) _> 7, with equality if and only if X does not have a g~, see loc. cit page 353 for the details.

7

The case of Surfaces

We finally turn to the case of surfaces, and start with embeddings into p3. The surfaces for which there exist an embedding into p 3 form a very special class, in many respects analogous to the class of curves for which there exists a planar embedding. In fact the relevant general case to consider here is the class of smooth connected varieties X of dimension r, for which there exists a projective embedding which identifies it with a hypersurface in some projective space. T h a t such surfaces represent a rather special class is borne out by the propositions given below, which summarize some well known facts: P r o p o s i t i o n 7.1 Let X be a smooth connected surface of degree d in p 3 . Then K x = (d - 4)H, where H is the divisor class of a hyperplane section. In particular (K~c) = d(d - 4 )2,Converse,!y, if X is a smooth connected surface such that ( K ~ ) = d(d - 4) 2 and if there exists a very ample divisor H such that (d - 4)H = K x , then X can be embedded into p3. A proof of this can for instance be found in [13]. Here we note the following immediate C o r o l l a r y 7.2 Let X be a smooth connected surface. If ( K 2 ) is not of the form d(d - 4) 2 J'or some d >_ 1, then X can not be embedded into p 3 /] however ( K ~ ) = d(d - 4) 2 for some d >_ I, then X can be embedded into p 3 if and only if there exists ~ E Y e t ( X ) such that (d - 4)~ = [Kx] T h e facts stated in the next two propositions are also well known and easy to prove: Proposition (a) (b) (e) (d)

7.3 Let X be a surface of degree d in p 3 . Then the following hold:

H i ( x , Ox) : If d ~ 3 then If d = 4 then If d >_ 5 then

0 X is rational X is a K8 surface X is of general type, and K x is very ample

In particular, it follows that irrational ruled surfaces, Abelian surfaces, Enriques surfaces, elliptic surfaces and bielliptic surfaces cannot be embedded into p 3 We conclude this summary of embeddings into P~ by observing that most surfaces with exceptional curves of the first kind cannot be embedded into p 3 . P r o p o s i t i o n 7.4 Let X be a smooth connected surface on which there is an exceptional curve of the first kind. Then X cannot be embedded into p 3 unless X is isomorphic to the smooth cubic surface in p 3 . Proof. If X is a surface in P~ of degree d < 2, then clearly it does not contain exceptional curves of the first kind. Further, if d _> 4 then ( K ~ ) > 0 and ( K x . C ) >_ 0 for every curve C on X. If E were an exceptional curve of the first kind on X , then it would be a non singular rational cm~e with ( E 2) = - 1 . But the adjunction formula would then imply t h a t - 2 = ( E 2) + ( E . K x ) > --1, which is absurd. We conclude that X contains no exceptional curves of the first kind.

136

We turn next to the much harder problem of embeddings into p 4 . In fact, the classification of smooth surfaces in p 4 belongs to one of the central problems in projective algebraic geometry; it has i m p o r t a n t connections to other areas such as the theory of rank 2 bundles on projective space p 4 . Recently there has been some progress in this field, mainly through the work of Christian Okonek in a series of interesting papers [21], t22], [23], [24], I25], and others. Further progress has been m a d e by Geir Ellingsrud and Christian Peskine [6], Alf Aure [2], and Sheldon Katz [16], among others. By reviewing some of the many known examples of surfaces in p 4 , we will see that there are many surfaces which can be embedded into p 4 but not into p3. After that, we will discuss some necessary conditions which imply that there are also many types of surfaces which cannot be e m b e d d e d in p4. In Section 8 we will discuss embeddings of ruled surfaces in some detail. E x a m p l e '/.5 ( S u r f a c e s X c p 4 w i t h H i ( X , Ox) # 0) It is well known known that there are abelian surfaces A C p 4 . (Thus h i ( A , 0A) = 2.) Specifically, A is the zero scheme of a section of the Horrocks-Mumford bundle F. This is the only type of abelian surface in p 4 (As it happens, a smooth surface X which is the zero-scheme of a section of F(t) for some t > 1 is of general type, but it satisfies H i ( X , O x ) = 0.) A more classical example is the elliptic quintic scroll S; we have h l ( S , O s ) = 1. By definition, S = P ( E ) , where E is a normalized indecomposable rank 2 bundle on an elliptic curve Y, with deg(E) = 1. (See [10], Section V.2.) In the notation of loc.eit., the very ample divisor which defines the embedding S ~ p 4 is Co + 2f. E x a m p l e 7,6 ( M i n i m a l K 3 s u r f a c e s ) The only minimal K 3 surfaces X C p 4 are complete intersections of type (2,3). Indeed, the vanishing of 84 implies that d = 6. The results of [22] imply t h a t X is a complete intersection. E x a m p l e 7.7 ( S o m e r a t i o n a l s u r f a c e s in p 4 ) Examples (a), (b) and (c) below are well known; see [10]. The others are discussed in Okonek's papers, as well as in [16] and [1]. In cases where X is obtained by blowing up points P1,. • •, P~ C p 2 L denotes the inverse image of the divisor class of a line (under the structural morphism 7r : X ~ p2), while E l , . . . , E,~ are the exceptional curves on X which are bIown down by 7r. (a) The Veronese surface: (X, Ox(1)) : ( p 2 , 0 ( 2 ) ) . (b) Let X be obtained by blowing up one point P C p 2 . T h e n X embeds in p 4 as a cubic scroll; the hyperplane sections are the strict transforms of conics through P , i.e. H = 2L - E. (c) Let X be obtained by blowing up points P1 . . . . ,P5 E p 2 , no three of which are collinear. T h e n X has an embedding in p 4 such that the hyperplane sections are the strict transforms of cubics through P 1 , . . . , P 5 , i.e. H = 3 L - P1 - ... - P5 • This is a Del Pezzo surface, thus K x : - H . It follows that X C p 4 is of degree 4, and the hyperplane sections are elliptic curves. (d) The Castelnuovo surface X is obtained by blowing up points P t , . . . , P s E p 2 which are in sufficiently general position. It is embedded in p 4 , with hyperplane section H = 4L - 2E1 E2 - ... - Es. It is easy to check that X C p 4 is of degree 5 and that ( H . K x ) = - 3 , so that the hyperplane sections are curves of genus 2. (e) The Bordiga surface X is obtained by blowing up points P 1 , - . . , P10 E :p2 in sufficiently general position. It is embedded in p 4 with H = 4L - E1 - . . . - El0 • It is easy to check that X c p 4 is of degree 6 and that ( H . K x ) = - 2 , so that the hyperplane sections are curves of genus 3. (f) Let X be obtained by blowing up points P 1 , . . . , P l l E p 2 in sufficiently general position. T h e n X is embedded in p 4 with H = 6L - 2(E1 + ... + E6) - E7 - . . . - E l l . One finds that X is of degree 7 and that ( H . K x ) = - 1 . (g) Let X be obtained by blowing up points P 1 , . . . , P l l ~ p 2 in sufficiently general position. T h e n X is embedded in p 4 with H = 7L - E1 - 2(E2 + ... + E l l ) . For the existence proof, see [1]. In this case, X is of degree 8, and ( H . K x ) : 0. Comparing this with (f), we see that if the

137 eleven points satisfy a sufficiently strong genericity condition, then X has at least two essentially different embeddings in p 4 (h) Let X be obtained by blowing up points P 1 , . . . , P16 C P2 in suitable, this time special position. T h e n X is e m b e d d e d in p 4 with H : 6L - 2(E1 + . . . + E4) - Es - . . . - E16. Thus, X is of degree 8 and ( H . K x ) : 2. The Riemann-Roch T h e o r e m yields x ( X , 0x(1)) = ~(H 2) - ~ ( H . K x )

+ x(Ox) = 4 - 1 + 1 = 4

It follows that Hx(X, Ox(1)) ¢ 0. (Actually, hi(X, 0 x ( 1 ) ) = 1.) In the previous examples of rational surfaces, x(X, Ox(1)) = 5. Since those surfaces are rational, H2(X, 0 x ( 1 ) ) = 0, it follows t h a t H i ( X , 0 x ( 1 ) ) = 0 for those examples. E x a m p l e 7.8 ( O t h e r s u r f a c e s w i t h e x c e p t i o n a l c u r v e s o f t h e f i r s t k i n d . ) (a) Let X0 be a K3 surface of degree 8 in p s , specifically a complete intersection of type (2,2,2), and let 7r : X --~ X0 be obtained by blowing up a point P C X0. T h e n X is embedded as a surface of degree 7 in p 4 , such that (H.Kx) = 1 , where H is a hyperplane section. (See [1611 or [25] for a proof.) (b) We give another surface constructed by blowing up one point on a minimal K3 surface. In this case, X is e m b e d d e d as a surface of degree 8 in p 4 , with ( H . K x ) = - 2 . This is obtained by starting with a minimal K3 surface Xo c pZ and projecting from a tangent plane of Xo. (c) A surface X obtained by blowing up 5 points on a minimal K3 surface X0 is given as follows. It is embedded as a surface of degree 9 in p 4 , where it arises as the residual intersection to a degree 7 rational surface via a pencil of quartics. See [16], where it is shown that one .can embed X0 as a surface of degree 14 in pS and then obtain X by taking 5 points of X0 which span a p z and projecting from that p 3 . This implies that ( H . K x ) = - 5 , so that X(X, 0 x ( 1 ) ) =: 4. Thus, H i ( X , 0 x ( 1 ) ) # 0. 7 . 9 0 k o n e k and Katz show that there are surfaces in p 4 of Kodaira dimension 1. Clearly, such surfaces are not complete intersections. We refer the reader to [16], [21] and [25] for a discussion of these surfaces.

Remark

We will conclude this section by giving some necessary conditions for the existence of an embedding into p4. We begin with a simple numerical criterion. 7.10 Let X be a nonsingular projective surface. If X can be embedded into p4, then f s2(X) = 2(K~c ) - 12X(0x) is a square modulo 5. In particular, f s2(X) - 0,1, or 4 (rood 5).

Proposition

Proof. Since 0 = ~4 = d 2 - 10d - 5 ( H . K x ) - f s z ( S ) , we have d 2 - f s2(S)(mod5). Since - 1 is a square modulo 5 , the last statement also follows. Proposition 7.10 implies that (minimal) Enriques surfaces cannot be e m b e d d e d into p 4 It also implies t h a t if X is constructed by blowing up m points of p 2 , then X cannot be embedded in P ' if m -- 2 or 4 (rood 5). (Indeed f s2(X) = - 6 + 2m.) This does not rule out the possibility of an embedding if m = 3, but we will see below that an embedding does not exist in t h a t case, at least if the three points being blown up are not collinear. Let X be a nonsingular projective surface, and let £ be a very ample sheaf on X . We will say that ~ is nonspecial, or that the embedding defined by H°(X, L) is nonspecial, if H i ( X , £) = H2(X,/~) = 0. (If H 2 ( X , •) = 0, but H i ( X , f~) • O, we say that /~ is superabundant.) 7.11 Let X C p 4 be a nonsinguIar surface of degree d which spans p4 and is not a projection of the Veronese surface. If the embedding is nonspeeial, then d 2 - 15d + 50 - 2(K~:) +

Proposition

2x( Ox) = o .

138 Proof. T h e Veronese surface is the only nonsingular surface which s p a n s P " w i t h n > 5, which can be projected isomorphically onto its image in p 4 . Therefore, our a s s u m p t i o n s imply t h a t x ( X , Ox(1)) = h ° ( Z , 0 x ( 1 ) ) = 5. Using the R i e m a n n - R o c h t h e o r e m , we see t h a t 5 = ½d - ~ ( H . K x ) -4- X ( O x ) or equivalently ( H . K x ) = d - 10 -4- 2 X ( 0 x ) . Since X C p 4 we have ~4(X, Ox(1)) = d 2 - 10d - 5 ( H . K x ) - 2(K~:) -4- 12X(Ox) = 0, and the claim follows. C o r o l l a r y 7.12 If ~ : X -* p 2 is obtained by blowing up three non-collinear points, then X cannot be embedded in p 4 .

Proof. We know t h a t K x = - 3 L + E1 + E2 -4- E3 , a n d t h a t - K x is very a m p l e [10], C h a p t e r 5, T h e o r e m 4.6. It follows immediately from this t h a t if 0 x ( 1 ) is a very ample sheaf on X , t h e n h2(Ox(1)) = h ° ( w x ( - 1 ) ) = 0. If H is a h y p e r p l a n e section, t h e n - K x -4- H is very ample, so t h a t we can use Serre duality and the K o d a i r a vanishing t h e o r e m to deduce t h a t h l ( O x ( 1 ) ) = h l ( w x ( - 1 ) ) = 0. (Over a field of positive characteristic, R a m a n u j a m ' s version of K o d a i r a vanishing applies since H I ( 0 x ) = 0.) Thus, every e m b e d d i n g of X is nonspecial. If we had an e m b e d d i n g X c p 4 , t h e n Proposition 6.11 would imply t h a t d 2 - 15d + 40 = 0. (Note t h a t (K~:) = 6 and X ( O x ) = 1.) Since this quadratic equation has no integer solutions, we conclude t h a t X c a n n o t be e m b e d d e d in p 4 . C o r o l l a r y 7 . 1 3 Let rn > 0 and let X be obtained by blowing up P I , . . . , P,~ ~ p 2 . If there is a nonspecial embedding X C p 4 , then m 8 in these cases. For 3 < d < 12 t h e following are consistent w i t h the equation: d

3 or 4 or 5 or 6 or

12 11 10 9 7 or 8

m

1 5 8 10 11

Finally we will show t h a t (d, rn) = (12, 1) is impossible. Thus, let r : X --* p 4 be o b t a i n e d by blowing up P C p 2 let E be t h e exceptional divisor on X , and let H be a h y p e r p t a n e section. T h e n H = aL - bE for some a > 0 and b > 0, where L = 7r*(line). Hence a S - b2 = d = 12, while

5=X(OX(1)) = ~ ( a 2 - b 2 ) ÷ ~ ( 3 a - b ) =

i

so t h a t 10 -- a 2 - b 2 + 3 a - b ÷ 2 . Since a 2 - b 2 = 1 2 , this leads to 3a--b.÷4 = 0 , which implies t h a t b2 = 9a 2 -4- 24a -4- 16. Using the identity a 2 - b2 = 12 once again, we see t h a t 8a 2 -4- 24a -4- 28 = 0, which is clearly impossible for a > 0. This completes the proof. ' / . 1 4 The cases (d, m) = (3, 1), (4, 5), (5, 8), (6,9), (7,11), and (8,11) are covered by Example 7.6. In his interesting paper [1], J. Alexander shows that d > 10 is impossible, see his Proposition ~.8. Furthermore, in his Theorem 1 on page 2 he shows that the case (d,m) = (9,10) actually occur. Thus from a numerical point of view, we have a complete picture.

Remark

139

8

E m b e d d i n g s of r u l e d surfaces

In this section, we s t u d y e m b e d d i n g s into p 4 of surfaces of t h e f o r m X = P ( ~ ) , where is a r a n k 2 v e c t o r b u n d l e on a n o n s i n g u l a r curve C. Surfaces of this t y p e o t h e r t h a n the cubic scroll are m i n i m a l ruled surfaces, i.e. they contain no exceptional curves of the first kind. In fact, it is k n o w n t h a t all m i n i m a l ruled surfaces (except for p 2 ) are of this form; see [26], C h a p t e r V, t h e o r e m 1. We will review some k n o w n results a b o u t e m b e d d i n g s of m i n i m a l ruled surfaces in p . t We will also prove some t h e o r e m s which can be p a r a p h r a s e d as saying t h a t the question of w h e t h e r or n o t a given m i n i m a l ruled surface X has an e m b e d d i n g into p a c a n be settled by c o m p u t i n g e m b e d d i n g o b s t r u c t i o n s for finitely m a n y classes of very a m p l e sheaves. Finally, we will apply our t h e o r e m s to o b t a i n fairly precise i n f o r m a t i o n a b o u t t h e possibilities for e m b e d d i n g s of m i n i m a l ruled surfaces of low genus. We will always use t h e n o t a t i o n a n d terminology of [10], C h a p t e r V, Section 2. T h u s , g will always d e n o t e t h e genus of t h e c u r v e C, a n d ~r : X = P ( ~ ' ) --~ C is t h e s t r u c t u r a l m o r p h i s m . (We will often say t h a t X is a ruled surface of genus g.) It will always be a s s u m e d t h a t ~" is normalized; t h u s H ° ( ~ ') ¢ (0), b u t H ° ( ~ ® •) = (0) if ~ is any line b u n d l e of degree < 0 on C. As is customary, we set e = - d e g ( c l ( ~ ) ) . T h e r e is a section a : C -* X of r which corresponds to a n invertible q u o t i e n t ~ = A 2 8 of ~. We set Co = a(C). It follows t h a t Co ~ C a n d t h a t Ox(Co) = 0 p ( ~ ) ( 1 ) . Therefore (C02) = - e . Finally, we set f = r*(point). It is k n o w n t h a t the t h e n u m e r i c a l equivalence classes of Co a n d f form a basis of N u m ( X ) . Let H be a very a m p l e divisor of degree d on a n o n s i n g u l a r surface X ; let K be the canonical class a n d let ~2(X) be t h e second Segre class of t h e t a n g e n t b u n d l e of X . We recall t h a t t h e c o r r e s p o n d i n g e m b e d d i n g o b s t r u c t i o n is/~4 = d 2 - 10d - 5 ( H . g ) - f s 2 ( X ) , w h e r e d = (H2). For a m i n i m a l ruled surface X over a curve C of genus g, this becomes (1)

/34 = d 2 - 1 0 d - 5 ( H . K ) + 4g - 4

In p a r t i c u l a r , it follows t h a t if 84 = 0, t h e n d 2 ~ 4 - 4g -~ g - 1 (mod 5). Therefore, g - 1 m u s t b e a s q u a r e m o d u l o 5, a n d we have: P r o p o s i t i o n 8 . 1 Let X = P ( ~ ) be a ruled surface of genus g. If X can be embedded into p 4 , then g =_ O, 1, or 2 (rood 5). If X is embedded as a surface of degree d, then (i) in the ease g =- 0 (,~od 5), we have d - 2 or 3 (rood 5). 5i) in the cas~ g =- ~ (rood 5), we have d ~ 0 (rood 5). (iii) in the ease g ==_2 (rood 5), we have d =_ 1 or 4 (rood 5). If = denotes n u m e r i c a l equivalence, t h e n H - aCo + bf for uniquely d e t e r m i n e d integers a,b, while K - - 2 C 0 + (2g - 2 - e)f. Since (Co2) = - e , (Co.f) = 1, a n d (f2) = 0, we o b t a i n

d = (H 2) = 2 a b - a 2 e and ( H . K ) = ( 2 g - 2 ) a + a e - 2 b Therefore, we have: (2)

/34 = d 2 - 1 0 d - 10a(g - 1) + 5 ( 2 b - ae) + 4g - 4

Using this formula, we c a n easily prove a general b o u n d e d n e s s result: T h e o r e m 8°2 Let X = P ( ~ ) , where ~ is a rank 2 vector bundle o~ a curve of genus g > 1. If X is embedded into P~ as a surface of degree d, then d < 10g

Proof. If X is e m b e d d e d in p 4 w i t h h y p e r p l a n e section H , t h e n / 3 4 ( X , H ) = 0. Since d = a ( 2 b - a e ) > 0, it follows t h a t 2 b - a e > 0 a n d 0 < a < d. Using e q u a t i o n (2) a n d the fact t h a t g - 1 _> 0, we see t h a t t h e equality/34 = 0 yields the inequality d 2 - 10d - 10d(g - 1) < 0. This is equivalent to the conclusion of the t h e o r e m .

140

We will p o s t p o n e f u r t h e r e x p l o r a t i o n of t h e c o n s e q u e n c e s of T h e o r e m 8.2, in o r d e r to s t u d y e m b e d d i n g s o f s o m e special t y p e s of r u l e d surfaces. We set do = (H.Co) = b - ae, so t h a t do is t h e d e g r e e of a v e r y a m p l e d i v i s o r o n Co ~ C . In t e r m s of this n o t a t i o n , we h a v e d = 2ado + a2e a n d ( H . K ) = (2g -- 2)a - ae - 2 d o . T h e r e f o r e , /34 = (2ado + ace) z - 10(2ado + ace)

5((2g - 2)a - ae - 2d0) + 4g - 4

T h i s c a n b e r e w r i t t e n as: (3)

{4a2d~ - 20ad0 + 10d0 - 10a(g - 1) + 4(g - 1)} + {a4e 2 + 4aadoe - 10a2e + 5ae}

In this e q u a t i o n , t h e e x p r e s s i o n inside t h e first set of b r a c k e t s gives t h e e m b e d d i n g o b s t r u c t i o n for a v e r y a m p l e d i v i s o r of b i d e g r e e (do,a) on Co × p 1 a n d t h e e x p r e s s i o n i n s i d e t h e s e c o n d set of b r a c k e t s is to b e r e g a r d e d as a c o r r e c t i o n t e r m . T h u s , it is clear t h a t t h e first e x p r e s s i o n > 0 . E q u a l i t y h o l d s only in one trivial s i t u a t i o n : Proposition 8 . 3 Let C be a nonsingular curve of genus g. I f g > O, then C x P 1 has no embedding into p a . The only embedding of P 1 x p 1 into p 4 is the quadric surface p t x p 1 C p 3 C p 4 .

Proof. C o n s i d e r a very a m p l e d i v i s o r H o n C x p 1 of b i d e g r e e ( d 0 , a ) . In this s i t u a t i o n , f o r m u l a (3) simplifies to 84 = 4a2d~ - 20ado + 10do (5a 2)(2g 2). Since 2g - 2 _< do2 - 3do it follows t h a t 84 ~ (4a 2 - 5a + 2)do2 - (ha - 4)do If a = 1, t h e n / 3 4 _> do~ - do . If a > 2, t h e n 4a 2 - 5a + 2 > 5a - 4 so t h a t 84 > (5a - 4)(d~ - do) T h e r e f o r e , 84 = 0 only in t h e case w h e r e a = do - 1. T h i s finishes t h e proof. Proposition 8 . 4 Let X = P ( $ ) be a minimal ruled surface with invariant e > O. I f X is not isomorphic to p 1 x p 1 or the rational cubic scroll in p 4 , then X has no embedding into p 4 .

Proof. We will b e g i n by d e t e r m i n i n g t h e cases in w h i c h t h e " c o r r e c t i o n t e r m " in (3) is _< 0. We d e n o t e t h i s t e r m by e = ae(a3e + 4a2d0 - 10a + 5). If e = 0, t h e n e = 0 a n d t h e value of/34 is e x a c t l y t h e s a m e as in t h e case X = C x p 1 . If C = p 1 a n d e = 0, t h e n X = p t x p 1 , since all v e c t o r b u n d l e s o n p 1 are d e c o m p o s a b l e . T h u s , we m a y a s s u m e t h a t e > 1. If do = 1, t h e n g = 0 a n d e = a e ( a Z e + 4 a 2 - 10a + 5), so t h a t e > 0 e x c e p t in t h e case e = 1 a n d a = do = 1, i.e. for t h e r a t i o n a l c u b i c scroll (see t h e e x a m p l e b e l o w ) , if do > 2 ( a n d e > 1), t h e n e _> a(a z + S a 2 - 1 0 a + 5 ) so t h a t e > 0. It follows t h a t t h e cubic scroll is t h e o n l y ease w i t h e _> 1 w h e r e X c a n b e e m b e d d e d into p 4 . C o r o l l a r y 8 . 5 If X = P ( ~ ) , where ~ is decomposable but not trivial, then X has no embedding into p 4 unless X is isomorphic to p 1 × p 1 or the rational cubic scroll in p 4 .

Proof. Since we are a s s u m i n g t h a t £" is n o r m a l i z e d , it follows t h a t ~ --- 0 @ /~, w h e r e deg(~,) < 0, see [10] C h a p t e r V, S e c t i o n 2. We c o n c l u d e t h a t e > 0. T h e r e f o r e t h e c l a i m follows by P r o p o s i t i o n 8.4. B y d e f i n i t i o n , a scroll in P n is a r u l e d s u r f a c e X = P ( c c) e m b e d d e d into p n in such a way t h a t all o f t h e fibers of 7r : X - - ~ C are e m b e d d e d as lines in P'~. If H - a C o + b f , t h e n X i s e m b e d d e d as a scroll if a n d only if a = 1.

141

E x a m p l e 8 . 6 ( T h e r a t i o n a l c u b i c a n d t h e e l l i p t i c q u i n t i e s c r o l l s ) First, we take C = p 1 a n d ~" = 0 ~3 0 ( - 1 ) . T h u s , e = 1. T h e r e is a very a m p l e divisor H ~ aCo + b f on X = P ( ~ ) with a = 1 a n d b = 2. (See [10], C h a p t e r V, Section 2, T h e o r e m 2.17.) It follows easily t h a t d = 3. T h i s is t h e s a m e surface as t h e one m e n t i o n e d in E x a m p l e 7.7(b). Next let C be a n elliptic curve a n d ~¢ a vector b u n d l e which fits into a n exact sequem:e 0 --+ O c -* ~ --* O c ( P ) --+ 0, w h e r e P is a p o i n t of C. T h u s , e = - 1 . It is k n o w n t h a t t h e r e is a very a m p l e divisor o n X = P ( 6 ) w i t h a = 1 a n d b = 2. (See [10], C h a p t e r 5, exercise 2.12.) It follows t h a t d = 5. It h a s b e e n s u s p e c t e d for some t i m e t h a t these are t h e only scrolls in p 4 A l l Aure, [2] has recently proved t h a t this is indeed true. T h e basic s t r a t e g y of his proof is b a s e d on the same idea as a n earlier p a p e r of A n t o n i o Lanteri, [19]. Aure also studies t h e dual variety of t h e G r a s s m a n n variety G(1, 4) in order to clarify some p a r t s of the proof. T h u s , we have: Theorem

8 . 7 The rational cubic and the elliptic quintic are the only scrolls in p 4

We will now consider t h e question of w h e t h e r or not ruled surfaces w i t h i n v a r i a n t e < 0 can be e m b e d d e d into p 4 , at least to the e x t e n t of s t u d y i n g the consequences of T h e o r e m 8.2. Since t h e r e is only one scroll in p 4 with e < 0, we can consider very ample divisors H - aCo + bf with a >_ 2. As n o t e d before, we have 2b > ae. It is conceiveable t h a t b could be n e g a t i v e and theft some very a m p l e divisors could c o r r e s p o n d to points (a,b) w h i c h are n o t in t h e first q u a d r a n t . T h i s suggests t h a t it could be interesting to express t h e e m b e d d i n g o b s t r u c t i o n / 3 4 in t e r m s of a a n d t h e p a r a m e t e r y = 2b - ae. T h u s , we can rewrite e q u a t i o n (2) as: ~4 = a Z Y 2 - 1 0 a y -

lOa(g-1)

+ 5y+ 49-4

T h i s can be r e a r r a n g e d i m m e d i a t e l y to: (4)

~4 = a2y 2 - (10a - 5 ) y -

(10a - 4)(g - 1), w h e r e y = 2 b - ae

As a simple a p p l i c a t i o n of this formula, we have: Proposition

8 . 8 Elliptic quintic scrolls are the only m i n i m a l ruled surfaces of genus 1 in p 4 .

Proof. By P r o p o s i t i o n 8.1 a n d T h e o r e m 8.2, we know t h a t d < 10 a n d t h a t d is divisible by 5. Therefore ay = d = 5. T h e equality ~4 = 0 yields d 2 - 10d + 5y = 0. Since d -- 5, we conclude t h a t y = 5 a n d a = 1. T h i s implies the conclusion of the proposition. Since we have d -- ay for an e m b e d d i n g of X = P ( 6 ) into p 4 , T h e o r e m 8.2 implies t h a t one c a n calculate B4(X) b y calculating the e m b e d d i n g o b s t r u c t i o n s of finitely m a n y very ample divisor classes. In fact, we c a n s h a r p e n t h a t result s o m e w h a t : T h e o r e m 8 . 9 Let X = P ( $ ) , where ~ is a rank 2 vector bundle on a curve of genus g >_ 2. I f X is embedded into p 4 with hyperplane section H - aCo + bf, then we have: (5)

a2y 2 - (10a - 5)y - (10a - 4)(g - 1) -- 0

where y = 2b - ae, and (6)

5g+5 2

2 _< a < - -

Thus, if H - aCo + bf is a very ample divisor and (a, b) is outside the finite set of pairs/'or which these conditions hold, then ~4(X, H ) ~ 0.

142

Proof. If X is e m b e d d e d into p 4 w i t h h y p e r p l a n e s e c t i o n H , t h e n f l 4 ( X , H ) = 0. T h u s , e q u a t i o n (5) is i m m e d i a t e . In p r o v i n g (6), we first s h o w t h a t (5) h a s no p o s i t i v e i n t e g r a l s o l u t i o n s w i t h y = 1. T h e n , we refine t h e p r o o f of T h e o r e m 8.2 to s h o w t h a t if (a, y) is a p o s i t i v e integral s o l u t i o n w i t h y > 2, t h e n (6) holds. S u p p o s e t h a t ( a , y ) were a n integral s o l u t i o n o f (5), w i t h y = 1. T h i s w o u l d i m p l y t h a t a 2-

(lOa-5)-

(lOa-4)(g-

1) = 0

or e q u i v a l e n t l y t h a t a 2-10ga+

(4g+1)

=0

T h e r e f o r e , t h e d i s c r i m i n a u t D = 100g 2 - 4(4g + 1) m u s t be t h e s q u a r e of s o m e e v e n i n t e g e r w h i c h is < 10g. It follows f r o m this t h a t 100g 2 - 4(4g + 1) < (10g - 2) 2 = 100g 2 - 40g + 4. T h i s w o u l d i m p l y t h a t - 1 6 g - 4 < - 4 0 g + 4, or e q u i v a l e n t l y t h a t 24g < 8. Since g is a p o s i t i v e integer, this is a c o n t r a d i c t i o n . We c o n c l u d e t h a t (5) h a s no integral s o l u t i o n s w i t h g > 1 a n d y = 1. We n o w c o n s i d e r integral s o l u t i o n s of (5) w i t h y _> 2. For t h e s e s o l u t i o n s we h a v e a _< d2 • T h e r e f o r e , e q u a t i o n (2) yields t h e i n e q u a l i t y d 2 - 10d - 5d(g - 1) < 0, o r d 2 - 5d(g + 1) < 0. T h u s , d < 5g + 5, a n d t h e c o n c l u s i o n follows. C o r o l l a r y 8 . 1 0 Let X = P ( ~ ) , where ~ is a rank 2 vector bundle on a curve of genus 2. I f X is embedded into p 4 with hyperplane section H =_ aCo + bf, then a = 7 and y = 2b - ae = 2. Therefore, we m u s t have e = - 2 , d = 14, and b = - 6 . Proof. B y T h e o r e m 8.7 a n d T h e o r e m 8.9, we m u s t have 2 < a < 7. Since g = 2, e q u a t i o n (5) b e c o m e s a2y 2 - (10a - 5)y - (10a - 4) = 0. A n e c e s s a r y c o n d i t i o n for e x i s t e n c e of an integral s o l u t i o n for y is t h a t t h e d i s c r i m i n a n t D = (10a - 5) 2 + 4a2(10a - 4) b e a s q u a r e in Z. We have t h e following values: a

D

2

152 + 162 : 481

3 4

252 + 3 6 . 2 6 = 1561 352 + 6 4 . 3 6 = 3529

5 6 7

452 + 100-46 = 6625 552 + 144-56 = 11089 652 + 196.66 = 17161 = 1312

T h e values o f D c o r r e s p o n d i n g to a = 2 , . . . ,6 are n o t s q u a r e s .

For a =

7, we have

6s+131 = 2. B y P r o p o s i t i o n 8.4, we m u s t h a v e e < 0; s i n c e a is o d d a n d Y ~ " ( 1 0 ~2-a s 2) + v ~ ~ 98 y = 2b - ae is e v e n , e m u s t be even. It is k n o w n t h a t e > - g = - 2 , [10] C h a p t e r V, E x e r c i s e 2.5. We c o n c l u d e t h a t e = - 2 . Finally, d = ay = 14, a n d 2b = y + ae = - 1 2 , so t h a t b = - 6 . Remark W e do not k n o w whether or not there exists a ruled surface in p 4 with the invariants described in Corollary 8.10. We will n o w verify a n e l e m e n t a r y t e c h n i c a l r e s u l t , w h i c h will b e u s e d in p r o v i n g a s t r o n g e r v e r s i o n o f T h e o r e m 8.9 for r u l e d s u r f a c e s of genus > 2. Lemma 8.11 (i) I f g = 2, then (a, y) = (7, 2) is the only positive integral solution o f equation (5} with y = 2. (ii) I f g > 3, then equation (5} has no positive integral solutions with y = 2. (iii} I f g > 2, then equation (5) has no positive integral solutions with y = 3 , 4 , 5 , o r 6.

143

Proof. We c a n rewrite e q u a t i o n (5) in t h e form (7)

a 2 y s - 1 0 a ( y + g - 1) + 5 y - 4 = 0

If g a n d y are given, t h e n a necessary c o n d i t i o n for existence of a n integer solution for a is t h a t t h e discrirninant D be a s q u a r e in Z. We have: D=

100(y+g-1)

=100(y+g-1)

2-4y2(5y+dg-4)

2-16y2(y+g-1)-dy

3

T h u s , y m u s t be t h e s q u a r e of a n integer of the form 10(g + y - 1) - 2k, w i t h 0 < k _< 5(y + g - 1), so t h a t lO0(y+g--1) or equivalently, (8)

2-16y2(y-t-g-1)-4y

10k(y+g-1)-k

3= lO0(y+g--1)

s-4Ok(y+g-1)-dk

2

s =dy2(y+g-l)-FyS

If (8) holds, t h e n we m u s t have 10k(y + g - 1) > 4y2(y + g - 1), so t h a t ~ < k _< 5(y + g - 1). T h e values for which these inequalities hold will be called feasible values of k. The case y = 2. E q u a t i o n (8) reduces to 1 0 k ( g + l ) - k 2 = 1 6 ( g + 1 ) + 8 . T h e smallest feasible value is k = 2. For a solution of (8) w i t h k = 2, we have 20(g + 1) - 4 -- 16(g + 1) + 8, so t h a t 4(g + 1) -- 12, or g -- 2. If t h e r e were a solution of (8) w i t h 3 3 0 ( g + 1 ) - 9 T h i s would i m p l y t h a t 14(g + 1) _< 17, so t h a t g _< 1. T h i s proves (i) a n d (ii). Proof of (iii). Since e q u a t i o n (5) has no solutions w i t h g -- 3 or 4 a n d n o solutions w i t h g = 2 a n d y > 2, it is e n o u g h to show t h a t t h e r e are no solutions w i t h g >_ 5 a n d 3 _< y _ 5 0 ( g + 2) - 25

This would imply t h a t 14(g + 2) 2 t h e r e are no solutions of e q u a t i o n (5) w i t h y = 3. The case y = 4. In t h i s case, t h e smallest feasible value is k = 7. If t h e r e were a solution of (8) w i t h k -- 7, we would h a v e 70(g + 3) - 49 = 64(g + 3) + 64. T h i s is clearly impossible, i[f t h e r e were a solution of (8) w i t h k = 8, we would have 80(g + 3) - 64 = 64(g + 3) + 64, or g -- 5. However, w h e n g = 5 a n d y = 4, e q u a t i o n (5) becomes 0 -- 16a s - 80a + 36 -- 4(2a - 1)(2a - 9). T h u s , t h e r e is no solution w i t h k = 8. If t h e r e were a solution of (8) w i t h 9 < k < 5(g + 3), t h e n we would have: 64(g+3)+64= 10k(g+3)-k 2 >90(g+3)-81 T h i s would imply t h a t 2 6 ( g + 3 ) < 145, so t h a t g < 2. Therefore, w h e n g _> 5 t h e r e are no solutiorLs of e q u a t i o n (5) w i t h y = 4. The case y = 5. In this case, the smallest feasible value is k = 11. Since y = 5, the right h a n d side of (8) a n d t h e first t e r m on t h e left h a n d side of (8) are divisible by 5. Therefore, there are n o solutions of (8) w i t h y = 11,12,13, or 14. If t h e r e were a solution of (8) w i t h 15 < k < 5(g + 3), t h e n we would have: 100(g + 4) + 125 = 10k(g + 4) - k s _> lS0(g + 4 ) - 225

144

This would imply t h a t 50(g + 4) < 350, so t h a t g < 3. Therefore, w h e n g ___ 5 there are no solutions of equation (5) w i t h y = 5. The case y = 6. In this case, the smallest feasible value is k = 15. Since y = 6, the right h a n d side of (8) and the first t e r m on the left h a n d side of (8) are even. Therefore, we need to consider only even values of k. If there were a solution of (8) w i t h k = 16, we would have 160(g + 5) - 256 = 144(g + 5) + 216, or 16(g ÷ 5) = 472. This is impossible. If t h e r e were a solution of (8) w i t h k = 18, we would have 1 8 0 ( g + 5 ) - 3 2 4 = 1 4 4 ( g + 5 ) + 2 1 6 . Thus, 3 6 ( g + 5 ) = 5 4 0 , so t h a t g = 10. However, w h e n g = 10 and y = 6, equation (7) becomes 0 = 36a 2 - 150a + 66 = 6(3a - 11)(2a - 1). Thus, t h e r e is no solution with k = 18. If t h e r e were a solution of (8) with k = 20, we would have 200(g + 5) - 400 = 144(g ÷ 5) + 216. Thus, 56(g + 5) = 616, so t h a t g = 6. However, w h e n g = y = 6, equation (7) becomes 0 = 36a 2 - l I 0 a + 50 = 2(2a - 5)(9a - 5). Thus, t h e r e is no solution with k = 20. If t h e r e were a solution of (8) w i t h 22 < k < 5(g + 5), t h e n we would have 144(g + 5) + 216 = 10k(g + 5) - k 2 _> 220(g + 5) - 484. This would imply t h a t 76(g + 5) < 700, so t h a t g < 4. Therefore, w h e n g _> 5 t h e r e are no solutions of e q u a t i o n (5) with y = 6. This completes the proof of L e m m a 8.11. We can now s t a t e a n d prove our final main result. T h e o r e m 8 . 1 2 Let X = P ( $ ) , where $ is a rank 2 vector bundle on a curve of genus g > 2. I f X can be embedded into p 4 with hypcrplane section H =- aCo + bf, t h e n equation (5) holds and 10g + 60

(9)

2 < a < -

-

49

R e m a r k B y Proposition 8.1, the hypothesis implies that g >_ 5. Proof. E q u a t i o n (5) holds, by T h e o r e m 8.9. By L e m m a 8.11, we conclude t h a t y > 7, so t h a t we need only consider integral solutions of (5) with y > 7. For these solutions, a ~ ~. Therefore, equation (2) yields d 2 - 1 0 d - ~ d ( g - 1) < 0, so t h a t 7d 2 - (10g + 60)d < 0 Thus, d < ~(10g ÷ 60). The conclusion follows immediately from this. For low values of g, T h e o r e m 8.12 provides r a t h e r strong restrictions on the value of a. Thus, for g = 5,6, or 7 we find t h a t a < 2, while for g = 10, 11, or 12 we find t h a t a < 3, and so forth. It is not h a r d to verify directly t h a t there are no positive integer solutions of (5) for those values. A simple calculation yields the following solutions of (5) in the d o m a i n g _< 227, T h e o r e m 8.12 was used to limit the range of values of a which were checked. As far as we know, no examples of minimal ruled surfaces in p 4 with these invariants have actually b e e n c o n s t r u c t e d .

genus

a

y

degree

genus of hypcrplane s e c t i o n

26 47 47 50 50 56 80 162 210 211 227

3 3 8 2 11 5 7 3 3 15 2

10 13 8 16 7 11 11 23 26 12 32

30 39 64 32 77 55 77 69 78 180 64

86 152 397 107 575 298 587 507 654 3235 469

In a weak sense, at least, the relative scarcity of solutions c o r r o b o r a t e s Okonek's conjecture t h a t ruled surfaces of high degree in p 4 are very scarce [22], page 570. While it is entirely possible

145

that such surfaces are even more scarce than what this list would indicate, it does not appear easy to prove that this is the case.

References [1] J. Alexander. Surfaces rationelles non - speciales dans p4. Prepublications, Universite de Nice, 1986. [2] Alf Aure. Surfaces in p 4 PhD thesis, Department of Mathematics, University of Oslo, 1987. [3] Wolf Barth. Transplanting cohomology classes in complex projective spaces. American Journal of Mathematies~ 92:951 - 970, 1970. [4} Magnar Dale. Terracini's lemma and the secant variety of a curve. Proceedings of the London mathematical Society~ 49:329 - 339, 1984. [51 P. A. Griffiths E. Arbarello, M. Cornalba and J. Harris. Geometry of Algebraic Curves. Volume 276 of Grundlehren der mathematischcn Wissenshaften, Springer - Verlag, Berlin, Heidelberg, New York~ 1985. [6] Geir Ellingsrud and Christian Peskine. Sur les surfaces lisses dans P4. Preprint, University of Oslo 1987. [7] T. Fujita and Joel Roberts. Varieties with small secant varieties: the extremal case. American Journal of Mathematics, 103:953 - 976, 1981. [81 Wiliam Fulton and Johan Hansen. A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings. Annals of Mathematics, 110:15(.} 166, 1979. -

[9] Joe Harris. A bound on the geometric genus of projective varieties. Ann. Scoula Norm. Sup. Pisa, 8:35 - 68, 1981. [10] Robin Hartshorne. Algebraic Geometry. Graduate tezts in Mathematics, Springer - Verlag, Berlin, Heidclberg, New York, 1977. [11] Robin Hartshorne. Ample Subvarieties of Algebraic Varieties. Volume 156 of Springer Lecture Notes in Mathematics, Springer Verlag, Berlin, Heidelberg, New York, 1970. [12] Robin Hartshorne. On the classification of algebraic space curves. In Vector Bundles and DiJ'ferential Equations, Universit~ de Nice, 1979, Progress in Mathematics, Volume 7, Birkh£user Verlag, Boston, Basel, Stuttgart, 1979. -

I13] Audun Holme. Embedding obstruction for smooth, projective varieties I. In G. C. Rota, editor, Studies in Algebraic Topology, pages 39 - 67, Advances in Mathematics Supplementary Series, Volume 4. Addison - Wesley Publishing Company, 1979. Preprint from 1972, University of Bergen Preprint Series in Pure Mathematics. f14] Audun Holme and Joel Roberts. Pinch points and multiple locus for generic projections of singular varieties. Advances in Mathematics, 33:212 - 256, 1979. I15] Kent W. Johnson. Immersion and embedding of projective varieties. Acta Mathematica, 140:49 - 74, 1978. [16] Sheldon Katz. Hodge numbers of linked surfaces. Duke Mathematical Journal, 55:89 - 96, 1987.

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[17] Steven S. Kleiman. Towards a numerical theory for ampleness. Annals of Mathematics, 84:293 344, 1966. -

[18] Dan Laksov. Some enumerative properties of secants to non singular schemes. Mathernatica Scandinavica, 39:171 - 190, 1976. [19] Antonio Lantieri. On the existence of scrolls in p 4

Rend. Accad. Lineei, 69:223 - 227, 1980.

[20] A. Ogus. Local cohomological dimension of algebraic varieties. Annals of Mathematics, 98:327 365, 1973. -

[21] Christian Okonek. Fl~ichen yore Grad 8 in p 4 M a t h e m a t i c a Gottingensis. Schriftenreihe des Sonderforschungsbereich Geometrie und Analysis., 1985. Heft Nr. 8. [22] Christian Okonek. Moduli refiexiver garben und fl£chen yon kleinem grad. Mathematisehe Zeitshrift, 184:549 - 572, 1983. f23] Christian Okonek. On codimension - 2 submanifolds in p 4 and p 5 . University of California, Berkeley., 1986. [24] Christian Okonek. Reflexive garben auf p4. Mathematische Annalen, 260:211 - 237, 1982. [25] Christian Okonek. Uber 2 - codimensionale untermannigfaltigkeiten v o m grad 7 in p 4 und p s . Mathematisehe Zeitshrift, 187:209 - 219, 1984. [26] I. R. Shafarevich. Algebraic Surfaces. Volume 75 of Proceedings of the Steklov Institute of Mathematics, 1965, American Mathematical Society, 1967. English translation. [27[ Robert Speiser. Vanishing criteria and the Picard group for projective varieties of low codimension. Compositio Mathematica, 42:13 - 2I, 1981. [28] J e a n - Claude Vignal. Embedding obstructions for Veronese embeddings. D e p a r t m e n t of Mathematics, University of Bergen, 1976.

Master's thesis,

It is a pleasure to express our profound gratitude to the Brigham Young University and the National Science Foundation, who made the con[erence at Sundance possible and thus provided the authors of the present paper with the opportunity to come together and finish this work. Above all we would like to thank Bob Speiser for the truely superb job he did in organizing this conference, and all the participants for the extraordinary scientific atmosphere from which all benefitted so much. This work was also supported by the NSF under the grant of the second author MCS 8501728.

ITERATION OF M U L T I P L E POINT FORMULAS AND A P P L I C A T I O N S TO CONICS Sheldon Katz Department of M a t h e m a t i c s U n i v e r s i t y of O k l a h o m a Norman, Oklahoma 73019 Introduction. This paper grew out of the author's desire to more easily compute m u l t i p l e point formulas.

It turns out that parts of m u l t i p l e point formulas, as described

show up in all higher m u l t i p l e point formulas some s i m p l i c a t i o n in computations.

in [KI],

This result affords

It also raises other natural questions,

why should there be such a formula? c o m p u t a t i o n only;

(Theorem 1.1).

Unfortunately,

e.g.,

this theorem is the result of

it is the author's opinion that there is a deeper i n t e r p r e t a t i o n

w a i t i n g to be discovered. The refined iteration formulas can be applied to compute the class of locus of space curves w h i c h meet a given curve

k

times,

for

1 ! k ! 8 .

This is done in

§2. As an application, we compute the number of conics on a generic quintic threefold by a d e g e n e r a t i o n method.

The number was already known to be 609,250 [Ka2].

In a d d i t i o n to providing a test of the formulas for the incidence loci of conics found in §2, this d e g e n e r a t i o n m e t h o d has independent

interest because of potential

g e n e r a l i z a t i o n of the m e t h o d to twisted cubics.

i.

Interation of M u l t i p l e Point Formu]as. W e recall the s i t u a t i o n and n o t a t i o n of

smooth, p r o j e c t i v e v a r i e t i e s over

k = k .

[KI].

Let

f: X ~ Y

be a map of

Consider the d i a g r a m R

= P(I(A))

~p Pl X ~........ X : ~ y X f

and

let

X x yX,t denote

define

fl

= P2P, i

the

= Cl(0R(1)) the

analogue

involution

, m 1 = [X] for

ms: = fl'~m" s-i

f]

of

of , vf

any

~

*P2

Y~-

X

R

covering

= f*Ty of

the

-

Tx

above

" this is the locus of

~ X

the ,

"switch"

e k = Ck(Vf)

constructs

for

involution .

We u s e f

.

s-fold points of the map

of a

prime

to

We i n d u c t i v e l y

f .

Let

n = dimY - d i m X , and assume that the derived maps

fl,'--,f s

all have c o d i m e n s i o n n.

T h e o r e m i.i.

of weight

(i+l)n

classes of

There exist rationai p o l y n o m i a l s ~f

(depending on

n)

so that

P. 1

in the Chern

148

s-1

Z

ms+ 1 = f*f,m s -

(-l)i+Is(s-l)---(s-i)Pims_i

i=O Remarks I.

The point

of the Theorem

very desirable 2.

Contained P.

Lemma

1

1.2.

classes

vf

are independent

of

s .

It is

form. for the

rational

=

and Lemma. .... Bs

We recall

(1.3)

Qi,g

of weight

in + 8

in the Chern

and

Bs+l)

some more

i (-l)1(s-l)'"(s-i+l)Qi,gms_J+l i=l

We call the statement

of the Lemma (As+ 1

polynomials

so that

Proof of Theorem

implies

P. l in closed

i n t h e p r o o f o f t h e Theorem i s a c o m p u t a t i o n a l g o r i t h m

fl,i,t~ms

statement

P. i

•

There exist

of

is that the

to have the

We will prove

of the Theorem

AI, BI, then

(A i

A

and

s B.i

, and the for

i ~ s)

.

formulas

from [KI]:

a)

fl,l,fl

b)

k * f!~i, t fl = -Cn+ k

= f*f~ - c

c)

i,m s' = m's

d)

f*. m I = m I

e)

fl,i,c~ = Ckfl,i , + j ! O ~ ! 0 l(n-J)~c.f ~ g ) ~ ] ,i, tk-j

(k > i)

k~lik-j

From the double-point the verification

of

Define polynomials

(1.4)

formula,

B 1 , with a.. I]

f l,i,P~. =

A1

is trivial,

forcing

P0 = c n

(l.3b) yields

Ql,g = Cn+ ~ " of weight

(i+l)n-j

in the

ck

by the formula

(i+l)n ~ aij f l,i,t]

j=0 Note that

aio = P. 1

ms+ 2

We compute

f 1,i,m's+ 1

f],l,

lfl,m s

n s

s l(-l>i+is<s-l)'''(s-i>P~.m's - I i=l

149

s-i [ (-l)i+is(s-l)"'(s-i)Pims_i+l i=l

= f*f,ms+ 1 - Cnms+ 1 - S C n m s + 1 +

s-i (1.5)

+

(-l)i+is(s-l)'''(s-i) i:O

= f*f,ms+ 1 - (s+l)cnms+ 1 +

s-i (-l)i+is(s-l)'''(s-i) i=j Note that for verifying

As+ 1 , the terms involving

(-1)S-as(s-1).-.(a)

Putting

e = 1

s-i ~ (-l)l+is(s-l)'''(s-i)Pims_i+l i=l

(i+l)n s-i ~ a.. ~ (-l)~(s-i-l)'"(s-i-$+l)Qg,jms_i_g+l j=l ~J ~il

i < ~ ~ s , the coefficient of

(1.6)

(i+l)n ~ aijfl*i*tJms i = j=l

m~

ms+ 1

check out.

Now, for

in (1.5) is

I-

(a-1)Ps_a+ 1 +

(s-a) (i+l)n ~ 7 aijQs_i_a+l,j i=O

in (1.6~, we see that if

As+ 1

j=l

l

is to be true, we must put

s-i ( i + l ) n (1.7)

p

Inductively, comes

s

1 s+l

~ i=j

~ j=l

aijQs-i,j

(1.7) is then also true for all subscripts less than

s .

So (1.6) be-

(-l)S-as(s-l)'''~[(a-l)Ps_a+ 1 + (s-~+2)Ps_a+ I] = (-l)S-~(s+l)(s)'''(~)Ps_~+l

as required.

This proves

The proof of

Bs+ 1

As+ 1 . is similar (and simpler).

We merely note the analogue of

(1.7). s-i ( i + l ) n (1.8)

Qs+l,~ =

~ i=0

~ j=0

This completes the proof of Theorem Now (1.3),(1.4),(1.7),(1.8) P.

l

.

(1.9)

aijQs-i,~+j

l.l and Lemma 1.2.

QED

combine to give a computational algorithm for the

We list the first few polynomials

Pi

for

n=l .

P0 = Cl P1 = c2 P2 = ClC2 + 2c3 P3 = c2ic2 + 5ClC 3 + 6c 4 + c~ P4 = e~c 2 + 9c~c 3 + 26elc4 + 3ClC~ + 8c2c3 + 24c5 P5 = c~c 2 + 14c~c3+ 6c~e~ 7 1 c ~ c 4 + 3 7 c l c 2 c 3 + 1 5 4 C l C 5 + 2 c ~ + 4 2 c 2 c 4 + 1 4 c ~ + 1 2 0 c 6

,

150

R@moytk:

2.

P0,-,.,P4

Incidence

which

genus

g

then

of t h i s are

we

We use

some

Sy

say

Y

, for

P(E)

and

may

be o b t a i n e d

Let

M

be

the m o d u l i

where

z = Cl(0M(1))

G = p3*

~ G

.

.

The

The

have

used

following

of a s u b v a r i e t y

and

We n o w

Let

~'

•

L

the n a t u r a l = P(K)

(2.4)

is a n a t u r a l .

There

the

scheme

of d e g r e e

n

vector

fibers

of

d

and

bundle

of

E

projection

Qy

.

on

X

,

Usually,

~: Y ÷ X

is a t a u t o l o g i c a l

bundle

in p l a c e

of

conics.

Recall

A*(gl)

.

Then

rank which

. k

on

Y

are

[F,

Chap.3])

that

.

we

Sy

known.

.

that

Note

(e.g.

g{ = P ( S y m 2 S ~ )

is well

The x

Again

generators

is the

are

x,z

"hyperplane

M

,

class"

that zgx 2 = -4

identification

are w e l l

known.

zSx 3 = I

A8(M)

~- Z

We use

meeting

a fixed

line]

[conics

through

a fixed

point]

a bit

I P

whose

Then

.

in the

quotient

used

z7x = 6

of n o t a t i o n .

We d e n o t e

Consider M'

is a r a n k

[conics

= {(p,P,C)

3 bundle

of

(dimM

= 8)

the n o t a t i o n

.

[V]

for

the

class

.

establish

N = P(Q~3)

nk curve

~ Qy + 0

x = - ~ * c l ( S G) known

formulas

(2.3)

rank

n-k

G = (;(3,4)

ring

the n a t u r a l

V

E

planes

There

of s p a c e

Let

z 8 = -4

we

k

G(I,E)

rank

space

Chow

It is w e l l

(2.2)

that

.

of

is u s u a l l y

as f o l l o w s .

~: M

class space

sequence

0y(-1)

write

general

If

of

0 + Sy ~ ~ * E

k = i , then

where

G(k,E)

instead

(2.1)

of

bundle

tautological

b y an e x a c t

If

the c y c l e

to a fixed

conventions.

is the a s s o c i a t e d

a name,

connected

in [KI].

.

the n o t a t i o n

subbundle

is to c o m p u t e

incident

establish

G(k,E)

we give

section

k-fold

, 1 ! k ~ 8

First,

implicitly

Loci

The goal conics

appeared

the n a t u r a l ~ P

fiber

, C at

"evaluation

Note

that

at

we h a v e

0 + QN ÷

p

Let

N = {(p,P) N + p3 p

C p3

by

p

E C c p}

is the v e c t o r map"

Sym2[

+

exact

sequences

+ p*Op3(2)

~ 0 ÷ 0

0"0p3(2)

× p3*

I P

~ P}

•

Note

. .

space

natural

[ ÷ P * 0 P 3(I)

0 ÷ K -~ S y m 2 /

= zx

map

is a conic, (p,P)

= z + 2x

On of •

N

there

linear Let

K

is a

forms be

on

P

.

its k e r n e l .

151

We use

(P to d e n o t e

the n a t u r a l

map

I~' -~ N

, and

~

to d e n o t e

the n a t u r a l

map

M'+N. The above

c a n be s u m m a r i z e d

constructions

by

the d i a g r a m

= M' ~ M = P(Sym SG)

P(K)

(* P ( ~ p 3 ) = ~4 Cp

O

p3 We

let

that

Let

n = %*Cl(0N(1)) ~*z

= m

Now,

let

M~ =

(po%)-l(D)

striction

to

, m = cl(0M,(1))

, ~*x = n D c p3

D

.

.

More

We a r e

to the

restriction

bundle

"~%'I) .

For

this

irreducible

generally,

going ~D

purpose,

%

(2.1)

and

(2.4)

curve

we u s e

to a p p l y of

D

.

of d e g r e e

as a s u b s c r i p t

the m u l t i p l e

to

g{~ .

we e m p l o y

0 ÷ 0 + p D*Q p*3 with

(po¢)*cl(0p3(1)

It is e a s i l y

seen

. be a smooth,

section

together

, h =

the

point

We m u s t exact

c(T~ll)

.

and

genus

to d e n o t e

formulas

find

g .

re-

of the p r e c e d i n g

the v i r t u a l

normal

sequences

® 0 N D ( I ) + T N D + 0DT D*

to c o m p u t e

d

+ 0

To c o m p u t e

c(TM)

, we employ

the

u

exact

sequence *

0 ÷ 0 + ~ Sym together

with

as c l a s s e s

C(TG)=l+4x+6x2+4x3

in

A*(~')

2*

S G @ 0M(1)

. We

*

÷ TM ÷ ~ TG + 0

can now compute

the

ci=ci(vgD ) ~ A*(M~) ~A*(~')

.

c O = dh 2 cI =

(3d+2g-2)h 3 + dh2(m+n)

c 2 = h3{d(m+2n)

+

c 3 = h3{-7dn 2 +

(2g-2)mn}

c~ = 7 h 3 { 2 d ( m n 2 +

(2g-2)(m+n))

2n 3) +

c 5 = -7h3{3d(m?n2+4mn

W e can The

now

we

c 7 = -7h3{5d(m~n2+8m3n

3) + 4 ( 2 g - 2 ) m 3 n 3} + 3 5 d h 2 m ~ n 3

apply

(1.9)

see

that

we

obtain

2.5.

description),

Theorem

classes

For

Theorem

2 ( 2 g - 2 ) m n 3} + 2 1 d h 2 m 2 n 3

3) + 3 ( 2 g - 2 ) m 2 n 3} - 2 8 d h 2 m 3 n 3

formula. easily

(2g-2)n 3} - 1 4 d h 2 m n 3

e 6 = 7h3{d(4m3n2+24m2n

desired

(2.3),

3) +

+ dh2mn

+ 7dh2n 3

i.i a n d

can n o w

},h 2 = z + 2 x

n 8 , we a l s o

Under

be f o u n d

suitable

use

to

the a b o v e

by p u s h i n g

, ~ , h 3 = zx (2.2).

.

Putting

dimensionality

formulas

down

to

We may all

M:

now

these

hypotheses

to f i n d

apply

(Ak(M~)

[KI]

.

From

the p r o j e c t i o n

calculations

(see

mk

n k = ~mk/k!

on a computer,

for a m o r e

complete

152

n] = d(z+2x) n 2 = (d)z2 + {(d-l)(2d-l)

- g}zx + (2d)x2

n 3 = (3)z3 + {(d-l)2(d-2)

- (d-2)g}z2x + ~{(d-l){(d-2)(2d-l)-

2g}/6!zx2 + (23d)x3

n 4 = (d)z4 + {(d31)(2d-3 ) - (d22)g}zx3 +

{

3(d31)(2d-3)

- (4d2-18d+19)g/2

+ gP/2iz2x2+

{ (2d3-1)(d-3) - (d-l)(2d-3)g}zx 3 n5 = (d32)(d-2)(2d-3)

- (d-4){(2dl)-lOd+13)g + g2}/2}z3x2 +

2(d31)(d-4)(2d-3)

- @-2){(2d2-11d+13)g

+ g2}/2}z2x3

n6 =

8(d42)(2d-3)(2d-5)

- {(6d~-8]d3+399df'-855d+677)g

-3(2d2-15d+26)g 2 + g3)/6

z ~ x

-

n7 = ( 5

(d-B) (2d-5) -

I0(d52) (d-3) (2d-5) - (d-6){ (2d~-29d3+157d2-B79d+B47)g -3(d-4)2g 2 + g3}/6}z~x 3 n 8 = (d53) (23d3-209d2+662d-756)/84

-

(d26){ 3dg-45d3+259d2-683d+710)g

3.

- lO(d2-8d+18)g2

+ i0g3}/60

Conics on quintic threefolds In this section, we apply the results of §2 to verify the number of conics on a

general quintic [Ka2].

threefold

in

P~ , which had previously been found to be 609,250 in

This method has been developed

in the hope that it can also be applied to

find the number of twisted cubics on a general quintic We recall a situation which appeared general pencil of quintic quartic

(G = 0)

equations [Kal,(1.7)]

threefolds,

degenerating

and a general hyperplane

F = G = H = 0 .

Let

C

gives the condition for

in [Kal].

threefold.

Let

F t = tF + GH = 0

to a transverse union of a general

(H = 0) .

Let

D

be a smooth conic contained C

be a

be the curve with in

G

or

H .

Then

to deform with the pencil to first order.

A

153

simple

computation

(entirely

order o b s t r u c t i o n s , The s i t u a t i o n

Lemma

3.1.

CO

m a y be s u m m a r i z e d

CO c G

However,

3.2.

gular, to

attained

If the limit CO

CO

.

t

in

easily

G

or

there are no higher deforms

of conics

C

on

t

, and meets

are limits

with

the pencil.

D

of conics

F is smooth, t 8 times~ ConCt .

Furthermore, []

limit.

0

, L c G

of conics , L' c H .

Ct

on

Ft

, and

L'

is trisecant

Conversely,

are limits of conics

Ct

is sin-

all of the

Furthermore,

CO

is

i . to twisted

H , and

cubics)

and

that

w = x = yz = 0 , H

[Ka2,

CO

is to d e f o r m

proof

is reduced.

f,g,h

are cubics. that

3.1]

that

Thus we may a s s u m e

has the e q u a t i o n

the c o n d i t i o n

the ideal of

of T h e o r e m

y = 0

To d e f o r m

the equation

of

, and

G

CO that

has the

CO

to first order,

F

lies

t

in the

t

tF + y(fw+gx+hz)

the p r o b l e m

(linear).

C

I.

from [Kal]

fw + gx + hz = 0 , w h e r e

C

that

then

L ~ N G = (L' N D) U (L' N L)

(which will adapt

It follows

0

CO c H

approaches

C O = L U L'

t2 = 0 , and w r i t e out

ideal of

where

as

just d e s c r i b e d

the e q u a t i o n s

equation

or

just d e s c r i b e d

as a limit w i t h m u l t i p l i c i t y

be c o n t a i n e d

we let

approaches twice,

can have a singular

CO

that

CO

The m e t h o d

has

t D

is reducible,

conics

C O = L U L' cannot

as

C

conics

D , in such a w a y

reducible

Proof.

CO

shows

are met,

as a l~mit w i t h m u l t i p l i c i t y

smooth

then

to [Kal,§3])

as follows.

and m e e t s

all of the conics

is a t t a i n e d

Lemma

analogous

if these c o n d i t i o n s

If the limit

then either versely,

i.e.

= (fy+tm)(w+tg)

is to solve

Extracting

for

m,n

the c o e f f i c i e n t

+

(gy+tn)(x+tg')

(quarries) of

t , we

, p

+

(h+tp)(yz+tq)

(cubic),

q

(quadratic)

, g,g'

see that

F = fy$ + m w + gyg' + nx + hq + pyz Putting

w = x = y = 0 , we get

common

factor

of

(L' N D) @ (L' N L)

h .

there are no higher

3.3.

Proof.

The conics

of

D

adjunction

to count

A generic

of T h e o r e m

quintic

CO c H

.

L'

trisecant

is

computation

Note

that

(analogous

GiL , = hZIL, to

D

.

The

, while

to [Kal,§3])

L' N G

shows

threefold

We s u b s t i t u t e

CO c G

S = F N H , a quintic

CO .

contains

are those m e e t i n g

intersection,

it is easy D

D

twice.

This

exactly

D

609,250

8 times.

to see that

has d e g r e e

these v a l u e s

are those m e e t i n g

surface,

=

that QED

the conics

2.5 are satisfied.

formula.

The conics

= hqlL,

order o b s t r u c t i o n s .

as a c o m p l e t e

potheses

F!L, that

An o m i t t e d

We are n o w ready

Theorem

says

From the explicit the d i m e n s i o n a l i t y

20 and genus

into T h e o r e m twice.

conics. form hy-

51 by the

2.5, o b t a i n i n g

Equivalently,

CO

187,850.

meets

locus can be found by a d o u b l e

point

154

calculation

in

p4

to 5 h y p e r p l a n e s , may

assume

generates section

so

S

becomes

locus

can be c a l c u l a t e d

P~O--.uP
O. First suppose ec>O. Denote by T the intersection subscheme 1(~-1 ,~,8). Then T is a complete curve, in the copy of I~ in N++ by dimensional transversality, because the points and lines are general and the complement of N has codimension 2. Further, T parametrizes a 1-parameter family {Nt} for which the identities (5.9) hold. (Although 5.9 requires T to be smooth, we can proceed to the general case exactly as in the discussion of the elementary systems at the end of Section 5.) Denote by i the incJusion of T in H++. Then we have

T

N~'

by virtue of Lemma 6.1, because with general points, lines and flags, and because T is a curve, we can avoid the VX(qk,mk ). However, 1~=7/2 by (5.9), where 7 counts the number ot t such that Nt os of type 7This )', the characteristic number of curves of type ), analogous to N~_ i .8,6, will be discussed below.

193 If 13>0, we use (t) instead of (p), and compute Na~,6=p. ' for the analogous lparameter family. Here (5.9) gives iJ.'= (2~+7)/3, where the new invariant ~ counts the number of t such that Nt has type ~. Here ~ (resp. 7) is the characteristic number for curves of type ~ (resp. 7) analogous to N(x,la-1,6, also to be treated below. Curves of type ~f. Here the characteristic numbers for the condition (p, t) will be denoted by )'~x~,6, where oc*13+2~= 7. They are: 75.0,1 . . . . . 70,5,1 = 20, 56, 156, 272, 392, 400; (6.4)

73,0,2 . . . . . 70,3,2 = 16, 40, 80, 112; )'1,0,3,

70,1,3

= 12,24.

We define them via cohomology, then compute them by reducing to cuspidal cubtcs, there employing [KS, 8.3 and 8.5]. The reduction ls made possible by the following Identity:

[pl]lr = [pl]K + [cP]K. Here the left side is the flag class (pl) E A2(N ++) to the locus r in N++of curves of type 7. The terms on the right denote the flag class and the cusp-at-a-point class for cuspidal cubics, transported from K to F. To prove the Identity, consider the conormal scheme CN of the canonical family over N++. The restriction CN_IFsplits into 2 components. To describe them, suppose N is a curve of type 7. Then N is a cuspidal cubic, denoted K, plus a vertex at the cusp, which corresponds to a line, denoted V, in i~2. Write K_(resp. V) for the total space of the family of all K in I)2 (resp. all V in 152). Then these families have conormal schemes, denoted by CK and CV, in I x r . Now F is a G-orbit, and the families, hence the conormal schemes, are equivarlant. Therefore the results of Section 2, which describe the fibers of CNI£ as limits, show that we have [cNIIr = [CK]+ [CY] as cycles on ! ×£. Hence, for any T/F, we have

[CN_]I T : [CK]I T + ICY]IT, as cycles on IxT. To prove the displayed Identity above, first Intersect wlth the slice (p,l)xF, and then push down to the base.

194

Curves of type 1/. Here the characteristic numbers for the condition (p,l) will be denoted by 1/(~,8,6,where o{+13+28=7. They are: 1/5,0,I.....1/o,5,1 = 32, 74, 126, 158, I04, 22; (6.5)

1/3,0,2, -.-,1/0,3.2 = 22, 40, 44, 28; 1/1,0,3. . . . . 1/0,1,3 = 12,12.

We define them via cohornology, then compute them by reducing first to curves of type o, and ultimately to complete conics, using the codimension 2 ldentity (pl) I V = LZ + mn/2 + 2a 2. Here the left side is the flag class (pl) e A2(N++) to the locus q/of curves of type 1/. On the right, by abuse of notation, the summands denote the classes corresponding to the terms indicated by the same letters in the computations for curves of type 1/in Section 5. The middle term is the pullback of the flag class for conics. The proof is by an argument similar to that for the previous displayed identity. The conormat scheme of a curve N of type t/has 3 components, one for the line, one for the conic, and a double one for the double vertex. Hence CNI~ has 3 components, the last one double. These yield our identity.

Remarks. (I) The number No,o,4 for nodal cubics appears neither in 6.4, nor in [Z]. It cannot be found by the method used to prove 6.4. (2) The method used for the last two cycle identities gives an efficient proof for the identity of [KS, p.266] for curves of type (~. We no longer need to restrict to an open subset there. COROLLARY 6.6. (Characteristic Numbers for the Condition (np), Zeuthen [Z, p.607]). We have: N~6, 0 .... , N~o, 6 = 1,4, 16, 52, 142, 256, 304.

Proof. These numbers follow immediately, using 6.2, from those in 5.13 and 6.3. Remark. From the action of G = PGL(2), one shows as before, in each characteristic ntwnber, that every solution curve is a nondegenerate nodal cubic, and that it counts exactly once if the characteristic is O.

195 References

[BK]

E. Brleskorn, H. KnOrrer, Ebene algebralsche Kurven, Blrkh~user ( 1981 ).

[DH1]

S. Diaz, J. Harris, Geometry of Severi varieties, preprint (1987).

[DH2]

S. Dlaz, J. Harris, Geometry of Severl varieties, It, In this volume.

[F]

W. Fulton, Intersection theory, Ergebnisse (3. Foige) 2, Springer (1984).

[FKM]

W. Fulton, S. Ktelman and R. MacPherson, About the enumeration of contacts, in Algebraic geometry - - open problems (C. Ciliberto, F. Ghlone, F. Orecchia, eds.), Springer L. Notes 997 (1983) 156- ! 96.

[G]

D. Grayson, Colncidnce formulas in algebraic geometry, Comm. In AIg. 7(16) (1979) 1685-171 I.

[K]

S. Kleiman, About the conormal scheme, Proc. of Arcireale Conf. 1983, Springer L. Notes. S. Kleiman, The enumerative theory of singularities of mappings, Proc. Oslo Symp. 1976 (P.Holm, ed.), Sijthoff and Noordhoff (1977) 297-396.

[K2]

S. Kleiman, Intersection theory and enumerative geometry, a decade in review (Section 3 written jointly with A. Thorup) Proc. of Bowdoin Syrnp. (I 985), Amer. Math. Soc., to appear.

[K3]

S. Kleiman, Problem 15: rigorous foundation of Schubert's enumerative calculus, in Proc. of Symp. in pure Math. ~ Amer. Math. Soc. (1976) 445-482.

[K4]

S. Klelman, Chasles' enumerative theory of conics: a historical Introduction, In Studies in Alg. Geom. CA. Seidenberg, ed.), Math. Assoc. of Amer. Studies in Math. 2_.0(1980) 117-118.

[KS]

S. Klelman and R. Spelser, Enumerative geometry of cuspidal plane cublcs, Vancouver Proc., Canad. Math. Soc. Conf. Proc. ~ Providence (1986).

[KT]

S. Kleiman and A. Thorup, Section 3 of [K2] above.

[M]

S. Maillard, Recherches des characteristiques des syst~,mes elementaires de courbes planes du 3 me ordre, Cusset, Paris ( 1871 ).

[RS

d. Roberts and R. Speiser, Enumerative geometryof triangles, I, Comm. in Algebra 12( I O) (1984) 1213-1255.

[RS2]

J. Roberts and R. Speiser, Enumerative geometry of triangles, II, Comm. in Algebra 14(I)(1986) 155-191.

[Sa]

Sacchiero, G., Numeri caratterisici delle cubiche plane nodale, preprint (1985).

196

[Sch]

H, Schubert, Kalk(Jl der abz~hlenden Geometrie, Teubner ( t 879) repr. by Springer (1979).

[XM l]

S. Xambo Descamps and J.M. Miret, Fundamental numbers of cuspidal cubics, preprint, University of Barcelona (1987).

IXM2]

S. Xambo Descamps and J.M. Miret, work in progress on nodal cubics.

[z]

H. Zeuthen, Determination des characteristiques des systemes ~lementaires de cubiques, cubiques douses d'un point double, C.R. Acad Sci. Paris 74 (1872) 604-607.

Copenhagen and Djursholm June 24, 1987

Old and New Results About the Triangle Varieties Joel Roberts School of Mathematics, University of Minnesota Vincent Hall, 206 Church St. S.E. Minneapolis, MN 55455

1. Introduction. In this paper I will discuss the triangle varieties which have been studied in [Se], [Ty], [RS1], [RS2], [RS3], and [CF]. I will also partly explain how Schubert's methods of obtaining enumerative formulas for triangles, as in [Sch], are related to the intersection theory of the triangle varieties. The definitions of these varieties are sketched in §2 below, at least in sufficient detail for an introduction to this subject matter. The reader is referred to [RS1] or [Se] for further details. It is reasonably accurate to describe §§2, 3, 4, and 5 as expository - - an introduction to some of the results proved in the newest six papers mentioned above. In the first half of §6, the results of §5 are applied to the solution of certain enumerative problems, and in second half of §6, I discuss Schubert's approach to proving the results mentioned at the beginning of § 6 . This turns out to be related to two other constructions of W*, due to Speiser and others; see [Sp] and [HKS]. The material in §7 and §8 is new. In §7, I discuss the question of how to find natural dual bases for the intersection pairing on a nonsingular variety which is constructed by blowing up a subvariety of a nonsingular variety whose intersection pairing is unimodular. This question had been of some interest when we were doing the research which led to the results reported in §5, but it was not possible to use it effectively in that context. The results about that are actually useful in connection with Speiser's new construction of W*. In §8, I discuss the question of whether two specific triangle varieties - W* and B - are isomorphic. The main result of §8 is that there is no isomorphism which is compatible with the natural action of the symmetric group S 3 on W* and B. Sections 2 through 6 correspond fairly closely to the material covered in my lecture at the Sundance conference. Comments made by Susan Colley, Bill Lang, Bob Speiser and others have ted to several improvements in the content and exposition. An explanation by Bill Fulton of methods used by him and Collino has led to the present version of §4, which is much more accurate than the report on their work presented in my lecture. I would like to thank all of these people for their helpful comments. It is also a pleasure to acknowledge Bob°s many significant contributions to our joint work, and to thank him for organizing this conference which has led to such good progress in enumerative geometry. The research reported in this paper was partially supported by National Science Foundation Grant MCS-8501728.

2. Definitions and basic facts.

The most basic triangle variety is W c (p2)3 x (p2)3

the set of all points ( Xl, x 2, x 3, £'1, ~'2' ~'3) such that x i E #'II for all i t j. The exceptional set X c W consists of all points for which x 1 = x 2 = x 3 and ~'1 = &2 = 6 3

In 1954, Semple [Se]

198 [Se] showed that W is an irreducible variety and that X . , Sing(W). Clearly, dim(W) = 6. We think of the Grassmann variety G(2,5) as parametrizing 2-dimensional families of plane conics. The variety of Schubert tdanales W* is defined to be the closure in (p2)3 x (1~2)3 x G(2,5) of the set of all points (x 1, x 2, x 3, ~1, P-'2'P~3,7") such that x 1 , x 2 , x 3 are distinct, x i e ~

for all i ~ j ,

and Y_. is the set of all conics which contain x 1, x 2, x 3,

There is an obvious morphism qw : W* --, W. We define the exceptional set X* = qw -I(X). Semple proved that W* is a nonsingutar variety, and that X* is a nonsingular subvariety of codimension 2. We define B to be the full diaaonal blowup of (p2)3. Thus we blow up the small diagonal A c (p2)3 to obtain a nonsingutar variety A. Then, we blow up the strict transforms

Aij+c A

of the large diagonals

&ij c: (p2)3

to obtain B. It is not hard to show

that there is a morphism PW: B --> W and that the exceptional set nonsingular subvariety of codimension 2 in B.

X B : = pw-l(X)

is a

Finally, we define ~/ to be the blowup of W along X, and we define the exceptional divisor Y, in the usual way. Semple proved that ~/ is nonsingular. Speiser and I proved in [RS1] that there is a commutative diagram

pJ

\q

B

W*

W

We also proved that p identifies ~/ with the blowup of B along W with the blowup of W* along X'.

X B and that q identifies

3. The Picard arouDs and the intersection rinas. We will use the following notation for some of the divisor classes on W*. For more details, see [RS1] or the appendix at the end of this paper. a i = pullback of c1(~(1)) from the i-th p2 factor. Thus a i is represented by the set of all triangles such that the i - th vertex lies on some fixed line. c~j = pullback of c 1(~(1)) from the j-th 152 factor. Thus ~ is represented by the set of all triangles such that the j - th side contains some fixed point. c = ~ivisor class of concurrent trianeles, defined by x 1 = x2 = x3

and

Z = {conics

with a singularity at x I = x2 = x 3 }. Thus, c = [D*] in the notation of [RS1,§4]. 1' = divisor class of collinear triangles, defined by ZI= Z2= ~3

and

Y_. = {conics

containing the line Z 1 = Z2 = ~'3 }" Thus, 1' = [C*] in the notation of [RS1,§4]. eij : defined as the class of the closure of the set of triangles not in c or t' such that x i = xj and ~ = ~ , Le.

8ij = [Dij*] in the notation of [RS1,§4].

t99

Theorem (3.1'L [Ty, Theorem 1; RS1,(4.8)] {a 1'a2,a3,~ 1,ct2,ma,c} is a Z-module basis.

Pic(W*) is a free Z-module, and

Theorem (3.2L [RS3,(1.8)] The groups Qk(B), Qk(W*), and Z-modules of finite rank for every k. The ranks are given by : k

0

1

2

3

4

5

6

rank(Qk(B))

1

7

17

22

17

7

1

rank(Qk(~/))

1

8

20

26

20

8

1

rank(Qk(W*))

1

7

17

22

17

7

1

£1k(~/)

are free

Indeed, the statements about freeness and the ranks follow immediately from our constructions and standard properties of blowups. Some further properties of blowups can be used to show that the rational equivalence rings are generated by divisor classes. Thus, let X be a regularly embedded closed subscheme of a variety Y, of codimension d and normal bundle Ft. Denote by Y the blowup of Y along X, and let ,X = P(ri.) be the exceptional divisor. We have the Cartesian square:

g$ X

J i

Sf ~Y.

Prooosition (3.3X [RS3, (1.9)] Assume also that X and Y are nonsingular. (1) If O.I(Y) generates Q'(Y), and if i* Ell(Y) generates G'(X), then Ql(q() generates Q'(¢0(2) If QI(¢() generates Q'(~'), and if i* maps the Chern classes of /"l into the subring of Q'(Y) generated by 0.1(Y), then Ql(y) generates O.(y). Corollary_ (3.4). [RS3, (1.10)] We have : (1) the component £11(B) generates Q'(B), (2) the component Q~(~/) generates £1"(~/), and (3) the component QI(W*) generates Q'(W*).

Since C* and D* are disjoint, we have c'y = 0 . Using the identity al + a 2 + a 3 + ~' = ~1 + (72 + °~3+ c from [RS1 ,§4], we obtain a basic relation of integral dependence 2

C = 3ac -e~lC - c~2c - c~c, where a c : = alc = a2c = a3c.

200 Therefore we obtain the following result. Theorem (3.5k Each of the groups Qk(W*) is generated by elements of the type listed in [Sch, §2F], i.e. monomials in which the "degenerate" divisor classes c, 7, and eij occur only to the first power. At this point, it is easy to do direct calculations in order to obtain explicit Z-module bases for CL2(W*) and Et3(W*). The method is to use some basic relations among the monomials of a given degree, say k, to find a "small" set of monomials of degree k which generates Elk(w*). When we have found a set of cardinality = rank(Qk(w*)), then we have a basis. Using similar methods, but with somewhat more work, it is possible to obtain explicit Z-module bases for CL4(W*) and (3.S(w*). For details, see §2 and §3 of [RS3].

4. The results of Collino and Fu!toq. The variety W* has recently been studied by Collino and Fulton [CF], using methods different from those used by Speiser and me. They show that W* has a torus action with only finitely many fixed points and then use results of Bialynicki-Birula [B1] to obtain a cellular decomposition of W* in which the cells are affine spaces. Thus, W* has properties which are similar to certain properties of other varieties such as Grassmannians and flag varieties. As a consequence of this, we have: Corolla~ f4.1L If the ground field is C, then H(W*) _=CV(W*). Corollary (4.2). The pairings

QI(W*) x Q6-i(w, ) ._> Z

are unimodular, i.e. these

pairings induce isomorphisms EL6 i ( w * ) -~ Hom (Qi(W*), Z). Proof. Corollary 1 follows immediately from [F, Example 19.1.12]. To prove Corollary 2, one notes that the results of [B2] provide two cellular decompositions which are dual to one another. This shows that the groups (~i(W*) are free Z-modules; the basis of Qi(W*) provided by one of the cellular decompositions is dual to the basis of 0- 6 "i(w*) provided by the other decomposition. This proves unimodularity.

Collino and Fulton have also determined the multiplicative structure of CL(W), by a method somewhat different from the method described in §3. The idea is to use a few simple facts about the geometry of W* to determine some relations which must be satisfied by the generators of CL'(W*). This provides a ring generated by 7 elements which admits a homomorphism to Q'(W*).

Collino and Fulton use elementary algebraic considerations

and the fact that O.'(W*) is a free Z-module to show that this homomorphism is an isomorphism. The reader is referred to their forthcoming paper [CF] for further details. I will conclude this section by mentioning that Collino and Fulton have corrected an error in Schubert's solution of the problem of enumerating the triangles inscribed in one curve and circumscribed about another curve. Some information about this is explained in [K2]; for further details, see [CF].

201

5. Exolicit descriotion of the oairinos O.i(W*) x CL6 " i ~ . By doing elementary computations, one can actually see directly that these pairings are unimodular. Theorems (5.1), (5.3), and (5.5) stated below are proved in [RS3, §§4,5]. As a consequence of these theorems, and corollaries (5.2) and (5.4), we can obtain solutions to certain types of enumerative problems. The method will be discussed in §6. One can recover the results of this section from those enumerative results and knowledge of the fact that the intersection pairings are unimodular. The enumerative results were known to Schubert, except that he needed an extra hypothesis. Thus, the results of this section are very closely related to results which were known to Schubert. In stating our results, we will use the notation of [RS3, §§3-6]. Thus, ac : = alc = a2c = a3c

and similarly

~7: = cc17 = 0-2~' = °t-3Y.

Since aieij = aj0ij and c~ieij= (zj()ij when i~j, we define a0ij: = aigij = ajSij and

cceii: =

%eij

=

Gj0ij when

i ~ j.

Finally, we set ~ = [X*], and we define a~: = a l v = a2N/ = a3N/ and

c ~ : = ccl~ = 0~2~ = c~.3~".

Theore~ (5.1). [RS3, (4.3)], The pairing QI(w*) x QS(w*) --~ Z is described by the following table. The elements listed at the top of the table form a Z-basis of Q~(W*), and the elements listed along the left edge form a Z-basis of QS(w*).

a22a323 1 10 J0 I I a12a323 011 10

la=a 1

l

24

I 1 10 I

0

O1O111

..........

I I

o

Io 11o I-il

o

,,,!

o

,1__1!

Corollary (5.2). The pairing Q1(W*) x O~5(W*) ~ Z i s described by the identity matrix relative to the bases 151'c QI(W*) and 155cGS(w*), where 155 is the basis of cLS(w *) mentioned in Theorem (5.1) and 151' is obtained from the basis of

O.I(W *) mentioned in

Theorem (5.1) by replacing c with d : = c + c~1+ o~2+c¢3 = y+ al+ a2+ a 3.

Remark (5.3). We can describe the geometric significance of the self-dual class d as follows. Let x, y, and z be 3 noncollinear points of p2, let T_, be the 2-parameter family of conics which contain x, y, and z, and let D = {Schubert triangles which are inscribed in some member of D}. Then d = [D]. See [Sch, pp. 174-5] or [RS3, p. 1953] for details.

202

Theorem (5.3). [RS3, (4.7)]. The pairing CL2(W*) x O4(W*) --~ Z is described by the following table. The elements listed at the top of the table form a Z-basis of C]-2(W*), and the elements listed along the left edge form a Z-basis of CL4(W*).

aio~

aOjk

I

O~Ojk

( ~i.k$ = {1.2.3~ - {i} a22ac. I a12a,.=Io111o

=Iolo11

!%2o~ 21 21

o

10IO 0 110

0

0

0

0

0

0

0

) 0

o o11 I a2o~O,

11010

I a2o~O'

II°o lIo~l l I °

la2~o. a, (z2 a ,(7.2 ' I

0

La

11olo 01110 01o11

0

o I olo

o

,a2c I

~cl

c,2~, a2~!

0

0

0

0

II0

0

o]1 -11-11-1 o o]o 11o o 1 o 1 o -1 -11 -1 o!1 o

0

Corollary (5.4L The pairing Q2(W*) x £L4(W*) --->Z i s described by the identity matrix relative to the bases 132CQ2(W *) and 134'CG4(W*), where 132 is the basis of Q2(W*) mentioned in Theorem (5.3) and 134' is obtained from the basis of (Z4(W *) mentioned in Theorem (5.3) by replacing oc2-~ with a2~ + (0.1 + o~2 + 0~3)a2c.

(z2~+ (a 1 + a2+ a3)o~2y and

a2~ with

203

Theorem (5.~). [RS3, (5,1)]. The pairing Q3(W*) x (~3(W') ---) Z is described by the following table. The elements listed at the top or along the left edge (in the same order) form a Z-basis,

a,%'ai=~, J

I~_L.LJ

a~; ~

aI.2;

Ill0J .]~--0 ~ ,

a,;

~J-

la~ [

a,

Ill0J-

o

J

a:

J

0

0

J nJ~'--'

I a,~

I

a% ,I

I

t

J

I 0

o~= I ~=oik lala2a3 1

0

0

0

I 0

0

o

o

I -I

0-I

0

I

0

o

o

o

o

O J0 J 1

,,

o

o

o

I,~2ei131

o

o

o

ala2a3

0

0

0

I czZYl la2c!

o

o

o

'

o

o

i--~;'1~, ' I l~/Cl

J

o

-lJ-lJoO 1101 OO

'

'1

'

I

o

o

~a2t)Ia2el 'J

l

I

o

Io

t

rio Io I 1-i o-tlo i olo 1-11-~loOIlo o ~I

II

°1'1 oliio

J I

I

o

0

I

0

o

J I

0

I o

o

o o 1 O

~-I 0 J .............. I

Corolla _ry (5.6). [RS3, (5.5)] As a Z-module with a symmetric bilinear form, CL3(W*) is diagonalizable, with diagonal entries +I and -I each occurring 11 times. While t don't know of a basis o f CL3(W*) which diagonalizes the intersection pairing and is convenient for enumerative calculations, it is still interesting to describe a basis which diagonalizes this pairing. To do this, consider the following submodules of O.3(W*). 3

M=Z

Z.c~ i2ocj ;

3

N=ZZ.a~y~ZZ.a2e,,

I~J

i=1

P = Z-ala2a 3 •

"

1~1= Z

i<j

i=1

Z-cz~c • Z Z-oc2e,j " t<j

Z.ocloc2(z.3 • Z.0c27 ~ Z.a2c.

Clearly, M, N, I~1, and P are mutually orthogonal, and Q3(W*) = M ~ N @ N • P. We can describe these submodules as direct sums of the following Z- modules:

204 I+ = Z, with bilinear form xy;

I_ = Z, with bilinear form -xy;

U=Z~ Z, with basis {e 1,e2} such that e l e ~ = e 2 e 2 = 0 and el.e 2= 1. This module is called a ; ~ . . e . . [ ~ y . . . ~ . Corollary (5.7~. The submodules M, N, N, and P are mutually orthogonal, and Qa(W*)=M~N(~N@P.

Moreover,

Pr9Qf. It is clear that

M-=3U;

N.~3U;

1~I_=3U; and

P_=2I+~2I_.

M _~3U. To prove N ~ 3U, we note that N = Nle)N2e)N 3, where N 1 = Z.(a2y+ a2012 ) ~) Z.a2823 N 2 = Z-(a22y+ a202 3 ) ~Z-a2013 2 N 3 = Z.(a2y+ a2813 ) ~)Z.a 812.

Clearly, N~, N2, and N 3 are mutually orthogonal and N i_=U for each i. Similarly To prove that P =__2 ]+ @2]

we consider the elements:

u 1 = ala2a 3 - a2c -

u 2 = (/.1-2- 3 - .2..i,

Vl = .2], + a2c. al a2a3.

v2 = .2.y + a2c _ " 1 " 2 " 3 •

It is easy to check that u 1, u2, v 1, v 2 are mutually orthogonal and that while

J'vl.v 1 = Jv2.v 2 = -1.

N=3U.

.l'u~.ul = Ju2.u 2 -- 1,

Thus {U 1, U2, v 1, v2} is a basis of P, so that P _=_2 ][+@ 2 ! .

Rem6rk. The classes u 1 and u 2 are represented by effective cycles. To see this, let H~, H 2, and H 3 be distinct but concurrent lines. Then ala2a 3 is represented by the set of all triangles (xl, x 2, X3,~.1,~.2,.~.3,T) s u c h t h a t xiEH i for i = 1,2,3, If {P} = HlnH2c~H3 , then one component of this is the set of triangles of the form (P,P,P,~.I,~.2,&3,.~,) such that Pe'~i and T_.= {conics with a singularity at P}. This represents a2c, so that ala2a3- a2c is effective. We now describe the basis of Q3(W*) dual to the basis given in Theorem (5.5). Corollary (5.81. If M, N, N, and P are as above, then 2 2 2 2 2 2 (i) M has dual bases { a l . 2, a2.1 , a l . 3 , a3.1 , a 2 . 3 , a3.2} 2 2 2 2 2 2 2 {a2" 1 , ala- 2 , a3-1 , alcf3 , a3- 2 , a3cc2 , a2- 3 },

and

2 2 (ii) N has dual bases {a~y, a2Y, a3Y, a2023, a20t3 ' a2012} and 2 2 2 2 {a2023 ' a2813 ' a2012 ' a0' + a2e~2 + a2el 3' a2")'+ a2012 + a2823 ' a3Y + a e~3 + a2023 }. (iii) I{I has dual bases {.~c , .22c , .32c , .282 3' 0~2013, .2e12 }

and

2 {O.2e23 , "2013, "2012 ,"~C + "2012 + "2~) 1 3' (~.~C+ C(2~)12+ "2~)23, "3 C + "2013 + "2023 } . (iv) P has dual bases {u 1, u 2, v 1, v2} and {u 1, u 2, -v 1, -v2}, with u 1, u 2, v 1, v 2 as above.

205 6. The solution Qf certain enumerative oroblems. If S and S' are two families of triangles such that dim(S) + dim(S') = 6, and if S and S' are in general position, then the triangles common to S and S' are counted, with suitable multiplicities, by .f[s].[S']. Using the explicit descriptions of the intersection pairings Qi(W*) x Q6-i(W* ) __>Z from §5, we immediately obtain an explicit formula for this intersection number, as in Proposition (6.1) below. This type of result is a modern formulation of the classical ~ethocl of characteristics. We will state the general form and one specific instance. Let 1 ~ ---)"0. It is well known that Pic(X) is generated by a and oh the pullbacks of the hyperplane classes from p2 and p2

209 respectively, and o~ = ~ = c1(£ ), where £ = O(1) is the tautological invertible sheaf on X = P(~-). Therefore, acz = a2+ (z2. If "8" is defined by the exact sequence (*), then cl(~'")=~-a. Thus, Cll(X) has first basis {a,cc} and secondbasis {a,c~-a} while C12(X) has first basis {a 2, ac~} and second basis {a(oc- a) = cz2, a2}. (Cf. [RS2, Proposition (2.3)].)

,F~J£i3£&Et#....(~. Let W' be the variety which parametrizes triangles with one side omitted. Thus, W ' c (p2)3 x (152)2 consists of all points (x 1, x 2 , x 3 , &2, #-3 ) such that x i e &j when i ~ j. Clearly, we can construct W' by iterating the Pl-bundle construction of the point-line incidence correspondence, starting with the first p2 factor, adding the two 152 factors next, and finally adding the last two p2 factors. Thus, for any p > 0, CI.P(W') has a first basis consisting of monomials ali(z2JcL3ka2rna3n, where 0 < i < 2, 0 < j, k, m, n < 1, and i + j + k + m + n = p. Here, a~ and c~i are the pullbacks of the hyperplane class from the i-th p2 factor and j-th

p2 factor respectively.

We obtain a second basis, dual to the first, by replacing al i o~2Jo~3ka2 m a3 n with a12-i ((z2.al)l -j ((z3_a1)l - k(a2.cc3)l - m(a3_or.2)l- n and then reversing the order of the elements in the resulting list. For instance, the first basis of QI(W') is (a 1, a 2, a 3, or.2, (z3). After doing some calculation, we find that the elements of the second basis of QS(W') are: ala2a3~2~ 3 - a12a2a3~ 2 - a12a2a3~3 , a12a20~2~3,

a12a20~3(a3-cc2),

a12a3~2~ 3, and

a12a30~2(a2-0~3).

Now we will study blowups. Thus, let X be a regularly embedded closed subscheme of a projective variety Y, of codimension d and normal blowup of Y along Cartesian square:

X, and let

X = P(~)

bundle

"1~.. Denote by ~' the

be the exceptional divisor.

We have the

J

X

i

~ Y.

Since f is birational, Qk(~') = f* Qk(Y) e~ Ker(f*)k for every k. If fact, we can use standard facts about cycles modulo rational equivalence on a blowup (see [F, Proposition 6.7]) to show that O-k(q0 = f* Qk(Y) B)j*(Ker(g*)k); in fact Ker(f,) = j,(Ker(g,)). Let ; be the locally free sheaf on X defined by the exact sequence: 0 ---, ~" ~ g*(lq.) ---> O"(1) ~ 0. Using the facts mentioned at the beginning of this section, we can show that Ker(g,) is a free C;.'(X)-module with basis {1, ~ ..... ~r-2}, where ~=c1(~(1)). Similarly, we check that

210 {1,c1(~"v) ..... Cr.2(;")} is also an £t'(X)-module basis of Ker(g.). Theorem ~7.5'~. Let the assumptions and notation be as above. Assume further that X and Y are nonsingular, and let n = dim(Y). The intersection pairing on ~' is unimodular if and only if the intersection pairings on Y and X are unimodular. Specifically: (1) (2)

f*(G(y)) is orthogonal to j.(Ker(g.)) under the intersection pairing on Y . j"

~, (3)

f*(a)-f*(b) =

j"

a.b

i n-i for a e G (Y), be Q (Y).

Y

j*(g*(Q'(X)).c i (~-v)) is orthogonal to j.(g*(G'(X)).~ r" J 2) if 0 ~ i,j < r- 2 and j ~ i.

• . (a)ci ( ~-v)).j.(g.(b)F=r-i-2) = - j ' a b (4) j" j.(g

~,

if

a,be Q'(X) are homogeneous

x

elements of complementary codimensions and 0_< i,j (xel, xo2, xa3, ~al, ~ , &a3 )- The structural map B--> W identifies B with the blowup of W along ~,p = { (x 1 , x 2 , x 3 , ~ 1 , ½ ,

~ ) • w I x~ = x 2 = x 3 }.

Since Ap is invariant under the action of S 3, we obtain an action of S 3 on B. On the other hand, the structural map W* ~ W identifies W* with the blowup of W along any one of the subschemes Aij = { (Xl, x2, X3, ~'1, '~2, ~J ) • W I xi = Xj and ~ = ~.j }. Since the action of S3 permutes the Aij, we obtain an action of S 3 on W*. ProDosition (8.1L There does not exist an isomorphism of W* and compatible with the actions of S 3 on W* and B.

B which is

Remark. It is tempting to conjecture that the automorphism group of either of the varieties W* and B is generated by the finite group discussed above, together with a

215 continuous part coming from the action of the projective linear group. If this is correct, then one could describe S 3 as QLL$(B) / {elements which act trivially on Q'(B)} and similarly as QtJ~t.(W*) / {elements which act trivially on Q'(W')}.

Proposition (8.1) and this conjecture

would imply that W* and B are not isomorphic. Proof of Prooosition (8.1X compatible with the action of

If there were an isomorphism

CL1(W *) =G.I(B),

which were

S 3, then the group of invariant elements of QI(W*)

be mapped isomorphically onto the group of invariant elements of that

W* --) B Q~(B).

would

It is known

and it is easily checked that the group of invariant divisor

classes is generated by a 1 + a 2 + a 3 , o~1 + o~2 + (z3 , and c, or equivalently by a 1 + a 2 + a 3 , (z1 + o~2 + 0{3, and 7. We also recall the identity of divisor classes: a 1 + a 2 + a 3 + ~,= (z1 + (z2 + (z3+ c. In G~(W *) there are two invariant divisor classes whose product is zero, viz.

c~= O.

Thus, we can complete the proof by showing that this does not happen in O.I(B). In fact, we claim that the six distinct products of the invariant divisor classes a~ + a 2 + a 3 , o~1 + o~2 + o~3 , and c are linearly independent in Q2(B).

In verifying this claim we begin by noting that

O.2(B) has a Z-module basis consisting of the monomials al 2, a22, a32, a l a 2 a l a 3, a2a 3, ~12, 0[.22, 0~32, o~10~2 ot.1~3, ot.2°t-3, al(z I , a2(z2 , a3(z3 , ac, (zy. (The proof is based on Theorem (3.2) and Corollary (3.4) above, along with calculations similar to those in the proof of [RS3, Proposition (2.2a)]). Now, it is easy to calculate, using relations which hold in both Q'(W*) and Q'(B): (1)

(a, + a 2 +

a3)2

(z3)2

2 = a~ + a~ + a 3 + 2(ala 2 + a~a 3 + a2a3) " 2 2 2 = (:z1 + o~2 + o~3 + 2(0~cc2 + 0~10~3 + ~2(:z3) -

(2)

((z1 + e.2 +

(3)

2 2 2 2 2 2 ( a 1 +a2+a3)((:Zl +(:z2+c(3) = 2 ( a 1 + a 2 + a 3 ) + 2 ( c ( 1 +e.2+c~ 3) + al(z 1 + a20~2 + a30~3 .

(Proof: Use the incidence relation: aio~j = ai2+ C(j2 if i ~ j.)

(4) (5)

( a 1 + a 2 + a 3 )c = 3ac ; 2 2 2 2 2 2 ((z 1 +e.2+c(3)c = 2 ( a 1 + a 2 + a 3 ) + 2 ( o ~ +0~2+c{ 3) - 2( 0~10~2 + Or.lO~3 + 0~20~3 ) + alc~ 1 + a2(z2 + a30~3 + 3c~y.

216

To prove (5), write (oc1 + e,2 + o~3 )c = (a 1 + a 2 + a 3 )(oc1 + oc2 + cc3 ) - (81 + o~z + oqj )2 + 3c~y. T h e n u s e (2) a n d (3).) The calculation of c 2 in Q ' ( B ) is s o m e w h a t different from the c o r r e s p o n d i n g calculation in Q'(W*). To do the calculation in Q.'(B) w e recall that D12 ~ D13 = O in B, w h e r e the Oij are the strict transforms of the d o u b l e d i a g o n a l s -&ij c (p2)3. This implies that 0120t3 = 0. N o w 812 = a 1 + a 2 - e..3 - c a n d 813 = a 1 + a 3 - cc2 - c so that 0 = 812813 = c 2- 4ac + (co2 + oc3 )c + al 2 + a l a 2 + a l a 3 + a2a 3 - at(oc2 + cc3 ) - a2~ 2 - a3oc3 + cc20c3 . Equivalently: c 2 = 4 a c - (~2 + cc3 )(a+ + a 2 + a 3 ) + (oc2 + cc3 )(0~1 + oc2 + o~3 ) - 2cc~, - a12- ( a l a 2 + a l a 3 + a2a 3 ) + al((Z 2 + or.3 ) + a2(z2 + a3oc3 - 0¢2o~3 . Therefore, w e obtain: (6)

c 2 = 4ac - 2cc7 - (a12 + a22 + a32 ) - ( a l a 2 + a l a 3 + a2a 3 ) + ((7"1~2 + °~1°~3 + °~2°~3 ) •

Relative to the linearly i n d e p e n d e n t invariant e l e m e n t s .~ai2, %aia j , ,~,cci2, .~,c~iczj , T.aio~j , ac, oc),, of Q'(B), w e have the following matrix of coefficients for the e x p r e s s i o n s on the right hand sides of e q u a t i o n s (1) ..... (6): 1 0 2 0 2 -1

2 0 0 0 0 -1

0 1 2 0 1 0

0 2 0 0 -2 1

0 0 1 0 1 0

0 0 0 3 0 4

0 0 0 0 3 -2

If w e a d d suitable multiples of the s e c o n d and third rows to the fifth row, w e obtain: 1 0 2 0 0 -1

2 0 0 0 0 -1

0 1 2 0 0 0

0 2 0 0 0 1

0 0 1 0 0 0

0 0 0 3 0 4

0 0 0 0 3 -2

Clearly, the third row can't occur in any nontrivial relation; this implies that the s e c o n d row a n d hence the sixth r o w can't occur either. W e c o n c l u d e easily that the rows of either matrix are linearly i n d e p e n d e n t .

This s h o w s that the q u a d r a t i c m o n o m i a l s in o u r t h r e e invariant

divisor c l a s s e s are linearly i n d e p e n d e n t , so that the proof of Proposition (8.1) is c o m p l e t e .

217

Appendix DIvisor classes on W* In each case the diagram shows a typical element (x 1,X2,X3,&l,~..2,~.3,F--) of the divisor which represents the indicated divisor class, The symbol Z always denotes a 2-parameter family of conics.

\

L//'-- fixed line (

~

~

/ ~=0. We only need to take care of the proof of T h e o r e m 1 (Section 3) and the L e m m a (Section 2) because these are the elements used in the proof of T h e o r e m 2. As far as the lemma goes, it is enough to consider, instead of the diagram in the proof of the Lemma, the diagram 0 --) H2k+I(E) ---) H2k(P) ~

r~k(~) --~ o

H2k(P)

$ct~ 0 ---) A k(P)®Z/ ~ A k(P)®Z/ ~ A k(E)®ZI ~ 0 and reason in the same way as there, but using the definition of "el isomorphism" given for the positive characteristiccase. For the proof of Theorem I, notice that step 0 of the induction is stillvalid. If n>0, let K denote the kernel of the map Ak(Xn.l) --~ Ak(Xn), which is free because by the inductive hypothesis the group Ak(Xn.1) is free. Consider the diagram

0-OH2k(Xn_l)

"~clxa_I

--)

H2k(Xn)

$c1~

~

H~(Z.) ~

0

kn "~ClT_,

0 ~ K ® Z I --)'Ak(Xn.I)®Z / ----) Ak(Xn)®Z / -'->Ak(Zn)®Z / ~ 0.

Now the same argument as in the proof of Theorem I shows that the middle vertical arrow is an isomorphism and that 0 case.

K®Z t

is 0. Hence K = 0 and the proof can be continued as in the characteristic

233 List o f notations and conventions (after Elencwajg and Le Barz)

A bold line (resp. point) stands for a fixed line (resp. for a point of the triple). An ordinary line stands for a variable line, and a small circle for a variable point o f t h e triple. A cross denotes a fixed point o f the plane,

H = {n-iples o f p2 with one o f its points on a given line} -

S A = [AI3p2] =

h

=

{triples with a fixed point} =

I

0 •

a = [triples that are colinear with a given point}

0

/

p = {triples with two points on a given line} =

f ° o

o

13= {one point on a fixed line and the other two on another} =

bi(X) = Betti number of X = rank Ai(X)

A c k n o w l e d g e m e n t s . The authors want to thank the referee for his suggestions, which have lead to the improvement o f the manuscript at a number o f points.

234

References

Bialynicki-Birula, A. [1973]. Some theorems on actions of algebraic groups. Ann. Math. 98(1973),480-497. Bialynicki-Bitula, A. [ 19761. Some properties of the decompositions of algebraic varieties determined by actions ofa torus. Bull. Acad. Polon. Sci., Set. Math. Astr. Ph. 24(1976), 667-674. Elencwajg, G., I..¢ Barz, P. [1985a1. Applications Enum~ratives du calcul de Pic(Hilbkp2). Preprint, 1985. Elencwajg, G., Le Bar-z, P. [1985b]. Anneau de Chow de Hilb3p2. CR Acad. Sc. Paris 301 (1985), 635-638. Ellingsrud, G., Strelmrne, S. [1984]. On the homology of the Hilbert scheme of points in the plane. Preprint, 1984. (Inventiones 87 (1987)). Fulton, W. [1984]. Intersection Theory. Ergebnisse 2 (new series), Springer- Verlag, 1984. Fulton, W., MacPherson, R. [1981]. Categorical framework for the study of singular spaces. Mem. Amer. Math. Soc. 243 (1981). Iversen, B. [1986]. Cohomology of sheaves. Universitext. Springer, 1986. Kleiman, S. [1976]. Rigorous foundations of Schubert enumerative calculus. Proc. Sympos. Pure Math. 28, Amer. Math. Soc. (1976), 445-482. Kleiman, S. [1979]. Introduction to the repnnt edition of Schubert [1879]. Laumon, G. [19761. Homologie dtale. Ast~risque 36-37 (1976), 163-188. Le Barz, P [1987]. Quelques calculs dans la variiti des alignements. Advances in Math. 64 (1987). 8%117. Rossell6, F. [1986]. Les groupes de Chow de quelques schdmas qul pararndtrisent des points coplanaires. CR Acad. Sc. Paris 303 (1986), 363-366. Schubert, H. C. H. [1879]. Kalkfil der abzahlenden Geometrie, Springer-Verlag (1979).

TRANSVERSALITY THEOREMS FOR FAMILIES OF M A P S Robert Speiser D e p a r t m e n t of M a t h e m a t i c s , 292 TMCB B r i g h a m Young U n i v e r s i t y Provo, U t a h 84602, USA

Begun b y Bertini, for e x a m p l e , m a n y y e a r s ago, the algebraic s t u d y of t r a n s v e r s a l i t y continues to evolve. While Kleiman's article [K] on t h e t r a n s v e r s a l i t y of the general t r a n s l a t e , based on ideas of Grothendieck, reflects a fully m o d e r n point of view, it h a s now b e c o m e clear t h a t m o r e general results a r e needed. An i m p o r t a n t step in this direction w a s m a d e b y Laksov, w h o found a p e n e t r a t i n g generalization [L, Th. 1] of t h e final m a i n result of [K], a n d applied it to t h e s t u d y of deformations. The w o r k to be described h e r e continues t h e process of generalization a n d recasting, extending Laksov's r e s u l t as well as e v e r y m a i n s t a t e m e n t a b o u t group t r a n s l a t i o n s in Kleiman's article. To introduce t h e n e w work, let's first recall the definition of t r a n s v e r s e m a p s . Suppose t h a t we a r e given m o r p h i s m s f : X ~ Z a n d g:Y--*Z of s m o o t h varieties, a n d set W=X×zY. A point of W will be w r i t t e n (x,y), w h e r e xeX and y e Y h a v e a c o m m o n image, denoted b y z, in Z. (Throughout this p a p e r we shall w o r k in the c a t e g o r y of v a r i e t i e s o v e r a n algebraically closed field.) Set d ~ dim(X) +dim(Y)-dim(Z). We shall s a y t h a t f a n d g a r e t r a n s v e r s e if e i t h e r W is e m p t y , or is s m o o t h of p u r e dimension d. Equivalently, in o u r setting, f a n d g a r e t r a n s v e r s e if, for e a c h (x,y)¢W, t h e t a n g e n t space TzZ is s p a n n e d b y t h e i m a g e s of TxX a n d TyY. If f a n d g a r e not t r a n s v e r s e , w e c a n t r y to m o v e one of t h e m a p s , s a y f, in a suitable f a m i l y , so t h a t f a n d g b e c o m e t r a n s v e r s e . A t r a n s v e r s a l i t y t h e o r e m , for us, will be a criterion for telling w h e n t h e general m e m b e r of a f a m i l y of m a p s f is t r a n s v e r s e to a n y given g in a suitable wide class of m o r p h i s m s . For e x a m p l e , Z m a y c o m e equipped w i t h t h e action of a n algebraic group G, a n d w e can ask for conditions on f such t h a t the general t r a n s l a t e ~f, for ~eG, is t r a n s v e r s e to a n y g. (In c h a r a c t e r i s t i c p, it will be n e c e s s a r y to a s s u m e , as a m i n i m u m , t h a t the d e r i v a t i v e of g does not kill too m a n y t a n g e n t vectors.) This situation w a s a m a i n c o n c e r n of [K], in the special case of a t r a n s i t i v e action. Other n a t u r a l situations do not involve group actions a t all. For e x a m p l e , f m i g h t be a n embedding, m o v i n g in a n a t u r a l f a m i l y of e m b e d d e d deformations. Such a situation led Laksov to t h e t r a n s v e r s a l i t y t h e o r e m of [L].

236 More r e c e n t l y , e n u m e r a t i v e p r o b l e m s w h e r e a g r o u p acts, b u t not t r a n s i t i v e l y (for e x a m p l e , PGL(2), operating on t h e space, say, of nonsingular plane cubics, w h e r e t h e r e a r e infinitely m a n y orbits), h a v e led to a fresh look a t the subject. This p a p e r is a b o u t t r a n s v e r s a l i t y criteria for families of mappings, not j u s t those arising w h e n a g r o u p acts. Our f u n d a m e n t a l outlook is v e r y strongly t h a t of [EGA IV], a n d the g r e a t e r g e n e r a l i t y of our results, c o m p a r e d w i t h their predecessors, s t e m s f r o m a deeper a p p r e c i a t i o n of Grothendieck's vision. There a r e several levels of generality. At the highest level, no group acts: we s i m p l y m o v e f in a family, and investigate t r a n s v e r s a l i t y w i t h a n y g in a given wide class. Here flatness a n d dimension a r g u m e n t s p r e d o m i n a t e . Each section begins w i t h s t a t e m e n t s of this kind (all new, except for Proposition 1.1). Progressing to the n e x t level, we consider a group action, b u t we no longer a s s u m e , as in [K], t h a t t h e action is transitive. We allow infinitely m a n y orbits. The s t a t e m e n t s here, while m o r e powerful, s e e m j u s t as elegant as those of [K] w h i c h t h e y replace. The guiding principle is t h a t if f is t r a n s v e r s e enough to t h e orbits on 7, t h e n a general t r a n s l a t e of f is t r a n s v e r s e to all suitable g. The organization of the exposition is similar to t h a t of [K]. The first section t r e a t s t r a n s v e r s a l i t y a r g u m e n t s in c h a r a c t e r i s t i c 0, while providing the technical foundation (criteria for proper intersections) for the l a t e r sections. Section 2 applies dimension a r g u m e n t s to the t a n g e n t bundles of X, Y a n d Z, to obtain results which a r e valid o v e r a n y base field. The last two sections describe joint r e s e a r c h w i t h Laksov, w h i c h generalizes the t r a n s v e r s a l i t y t h e o r e m of [L]. This work, too, is independent of the c h a r a c t e r i s t i c . A still m o r e general version, o v e r a r b i t r a r y base schemes, will a p p e a r in [IS]. Section 3, w h i c h introduces d e t e r m i n a n t a l pairs of m a p s , provides the technical f r a m e w o r k . Then in Section 4, we obtain first the t r a n s v e r s a l i t y t h e o r e m of [L], a n d t h e n a n e w generalization, for group actions w h i c h need not be t r a n s i t i v e , of t h e final m a i n result of [K]. Along t h e w a y , applications a n d e x a m p l e s illustrate the discussion. These focus on Bertini's t h e o r e m for the singularities of linear s y s t e m s of divisors, w h e r e t h e s u b j e c t began. Although the results of this p a p e r give criteria for generic t r a n s v e r s a l i t y to hold, I think it is still of i n t e r e s t to investigate w h e n this condition can fail. In a n e n u m e r a t i v e calculation, for example, failure would c o n t r i b u t e , according to the results of Section 2, a n order of i n s e p a r a b i l i t y to t h e m u l t i p l i c i t y of e a c h solution in t h e count. This e x t r a m u l t i p l i c i t y ought not, I think, be viewed as a pathology - - on t h e c o n t r a r y , it expresses s o m e t h i n g i m p o r t a n t a b o u t t h e g e o m e t r y of t h e p r o b l e m w h i c h it helps to solve. We need to l e a r n m o r e a b o u t this.

Conversations w i t h S t e v e Kteiman during the S u n d a n c e conference led to i m p r o v e m e n t s in Section 1. The w o r k in Section 2 was influenced b y l a t e r

237

discussions with Dan Laksov about the joint work described in Sections 3 and 4. I owe both colleaugues special thanks. Suggestions b y Torsten Ekedahl a b o u t Section 3 w e r e also helpful. This article w a s w r i t t e n a t Mittag-Leffler, based on r e s e a r c h done there, in addition to work done previously at BYU and Sundance. It is a pleasure to t h a n k these organizations for their help, as well as the NSF, w h i c h contributed funds for the conference.

1. First Results

T h r o u g h o u t this paper, we shall work in the c a t e g o r y of f i n i t e - t y p e separated schemes over Spec(k), w h e r e k is a n algebraically closed field of a r b i t r a r y characteristic. By a v a r i e t y we shall m e a n an integral such scheme.

First, suppose given a fiber product diagram W

X

~Y

.....

~Z f

of s m o o t h varieties,

W e shall say that f and g meet properly if either W is empty, or, for each (x,y)(W, over zcZ, w e have dim(x,y)W = d i m x X + d i m y Y - dimzZ. For example, two subvarieties of Z meet properly in the usual sense exactly w h e n their inclusions meet properly in our sense. At the other extreme, a flat morphism X-*Z meets any m a p to Z properly. W e shall say that f and g are transverse if either W is empty, or, for each point (x,y) of the fiber product W, the tangent spaces TxX and TyY span TzZ, where z is the image of both x and y in z. In our situation, by [EGA IV, 17.13.6], the m a p s f and g are transverse if and only if they meet properly and W is either e m p t y or smooth.

238

Two smooth subvarieties of Z, for example, are t r a n s v e r s e exactly w h e n t h e y m e e t properly and their intersection scheme is either e m p t y or smooth. Also, a smooth m o r p h i s m X-*Z is t r a n s v e r s e to a n y m a p of a smooth v a r i e t y into Z. We shall f r e q u e n t l y abuse this language, and say t h a t X and Y m e e t properly (resp. a r e transverse). To emphasize the role of X, we shall also say t h a t X m e e t s Y p r o p e r l y (resp. is t r a n s v e r s e to Y), instead of saying t h a t f and g m e e t properly (resp. are transverse). Next we consider the case w h e r e f is a f a m i l y of maps, p a r a m e t r i z e d b y a base scheme. Hence, we suppose given smooth varieties X,Y,Z and S, with W=XxzY, fitting into a c o m m u t a t i v e diagram with fiber square, as follows. W

-~Y

X

,Z f

7I

S. Denote p:W-~S the composite of the m a p W-~X and TO. For seS, write Xs for the fiber T~-I(s), and Ws for the fiber p-l(s). Then, clearly, we h a v e Ws=Xsxz Y. The next result is [K, i, p.288]. PROPOSITION l.i. In the situation above, (i) a s s u m e t h a t f is flat. Then t h e r e is a dense open set U in S such t h a t Xs m e e t s Y properly, for all sEU. (2) Assume t h a t f is smooth. Then the generic fiber Xcr is t r a n s v e r s e to Y, and, if the c h a r a c t e r i s t i c is 0, t h e r e is a dense open set UI in S such t h a t Xs is transverse to Y, for all sEU 1. Sketch of proof, For (1), f is d o m i n a n t , and W/Y is flat, h e n c e dim(W/Y) =dim(X/Z), all because f is flat. Since W/Y is flat, we find dim(W)=dim(X)+dim(Y)-dim(Z). If W does not d o m i n a t e S, we a r e done, w i t h e m p t y general W s. Otherwise, b y generic flatness, the general W s has dimension dim(W)-dim(S), and (1) follows. (In this p a r t of the proof, no smoothness a s s u m p t i o n is necessary.) For (2), one replaces flatness w i t h smoothness; in the last step, of course, generic smoothness (Sard's L e m m a ) requires c h a r a c t e r i s t i c O. (This a r g u m e n t , specialized to the case of a group action, is spelled out in [H, pp. 273-4].) This completes t h e sketch.

239

In the sequel, w e shall use the following criterion for flatness, essentially [EGA IV2, 6.1.5, p. 136]. It is a partial converse of the well-known result [E6A IV 2, 6.1.2, p. t35] that a flat m a p has fibers of constant, hence expected, dimension. To state it, w e shall denote by X z the fiber of f:X-*Z at zcZ. PROPOSITION 1.2. Let f:X-~Z be a m o r p h i s m of schemes, w h e r e X is CohenM a c a u l a y a n d Z is smooth. If, for z e Z a n d xeX z, w e h a v e dimx(X) = dimx(Xz) + dimz(Z),

then f is flat a t x. Group actions. Suppose given a n action of a n algebraic group 6 on t h e s m o o t h v a r i e t y Z. Let f:X-~Z be a m o r p h i s m f r o m a s m o o t h X. We shall s t u d y t h e natural map F 6×X

~Z

induced b y t h e action. (i) We shall s a y t h a t f (or X) m o v e s p r o p e r l y u n d e r the action of G if X m e e t s e a c h G-orbit of Z properly, w i t h a n o n e m p t y i n t e r - section. If this holds, it follows f r o m 1.2 t h a t F is flat. Indeed, fix zeZ, a n d let D be its orbit. Denote b y Gz the stabilizer of z. Then w e h a v e X×ZD = [(x,~z) [ f(x)=~z } = {(x,l;z) I (~-l,x)¢F-l(z) } -= {(x,~z) I (~,x) e F-i(z) }. Since D -= G/Gz, it follows t h a t d i m ( X x z D ) = d i m ( F - 1 ( z ) ) - d i m ( G z ) . Because X m e e t s D p r o p e r l y , w e find dim(F-l(z)) = dim(XxG) - dim(Z), so F is flat b y 1.2. (2) We shall s a y t h a t f (or X) is t r a n s v e r s e to the action of 6 if X m e e t s e v e r y G-orbit of Z, a n d for e a c h xcX, zcZ a n d I¢~G such t h a t z=lCx, t h e t a n g e n t spaces TxX a n d Tl~6 s p a n TzT.. If this holds, t h e n F is smooth. Now a s s u m e given a n action of a n algebraic group G on a s m o o t h v a r i e t y Z, a n d t w o m a p s , f:X--*Z a n d g:Y--*Z, f r o m s m o o t h varieties. For ICeG, denote b y ICf:~X-~Z t h e t r a n s l a t e of f b y ~, a n d b y W~, the fiber p r o d u c t ~'X ×ZY.

240

THEOREM 1.5. In the situation above, (1) a s s u m e f m o v e s properly u n d e r the action of 6. Then t h e r e is a dense open subset U of G, such that, if I¢¢U, t h e n ~f and g m e e t properly. (2) Assume t h a t f is t r a n s v e r s e to the action of 6. Then for the generic point ~0EG, the t r a n s l a t e ~'0f is t r a n s v e r s e to g. If the characteristic is 0, t h e r e is a dense open subset U 1 of G, such that, if ~'eU1, then ~'f is t r a n s v e r s e to g. Proof. Combine 1.1 and the assertions (1) and (2) preceding the s t a t e m e n t . REMARK. Denote b y e the identity point of G, pick zcZ, and denote b y D t h e orbit of z on Z. In characteristic 0, the d e r i v a t i v e of the n a t u r a l m a p TeG~Tz[) is surjective. It follows t h a t a m a p f:X--*Z is t r a n s v e r s e to the action if and only if it is t r a n s v e r s e to each G-orbit of Z. This condition is often not difficult to check in practice. Homogeneous spaces. If the action of G on Z happens to be transitive, it follows i m m e d i a t e l y , b y translating generic flatness, t h a t the induced m a p F:6×X-,Z is flat. By the r e m a r k above, it also follows, in c h a r a c t e r i s t i c 0, t h a t F is smooth, but, in c h a r a c t e r i s t i c p, this smoothness m a y fail. Hence nothing is said about the t r a n s l a t e b y the generic point of G in assertion (2) of the next s t a t e m e n t , which is [K, 2, p.290]. COROLLARY 1.4. Assumptions as in 1.3, suppose also t h a t 6 acts t r a n s i t i v e l y on Z. (I) There is a dense open subset U of G, such t h a t , if ~¢U, t h e n ~'f and g m e e t properly. (2) Suppose the c h a r a c t e r i s t i c is O. Then t h e r e is a dense open subset U 1 of 6, such that, if ~'EUI, t h e n ~'f is t r a n s v e r s e to g. COROLLARY 1.5. (Bertini's Theorem in Characteristic 0.) Suppose X is a smooth v a r i e t y in characteristic 0, and t h a t {Dt} is a linear s y s t e m of divisors on X, w i t h o u t base points. Then the general Dt is nonsingular. Proof, Let I:X-~P r be the m o r p h i s m defined b y the linear s y s t e m , and let g be the inclusion of a h y p e r p l a n e in pr. Since P6L(r) is t r a n s i t i v e on p r we m a y apply the last result.

2. P r o j e c t i v e Tangent Bundles

Again w e suppose given smooth varieties X,Y,Z and S, fitting into a c o m m u t a t i v e diagram with cartesian square:

241

~Y

W

g

X

......

)Z

S. Now we shall also a s s u m e t h a t v[ is a s m o o t h m a p . For s¢S, we shall again w r i t e Xs for ~-l(s). Our goal f r o m h e r e on will be to investigate conditions, i n d e p e n d e n t of t h e c h a r a c t e r i s t i c , u n d e r w h i c h the general X s will be t r a n s v e r s e to Y.

S o m e motivation. Denote by T(XIS) (resp. TY, TZ) the relative tantent bundle (rep. tangent bundle). The central idea behind the results of [K] for arbitrary characteristics is to study the fiber product T(X/S) xTzTY as a W scheme. So, consider the natural projection ~o TX xTzTY

~ W.

At a point (x,y)£W, over zcZ, the inverse image ~0-i(x,y) is the fiber product, TxX ×TzzTyY, of the tangent spaces. Hence, if tp-1(x,y) has the correct (and minimal) dimension, it will follow from the transversality criterion [EGA, IV.17.1~.6] that f and g are transverse near x and y, so that W is smooth near (x,y). In this way, by studying the tangent bundles, transversality questions can be reduced to pure dimension statements, and these can be handled independently of the characteristic. W e can run into problems, however, if either f or g is ramified, as w e shall see in two examples to be presented later. For f, the difficulty is that flatness of the bundle m a p Tf:T(X/S) -~TZ generally beaks d o w n along the the zero-section, which is the given m a p f:X~Z. This suggests that w e remove the zero-section, but w e can do so safely only if both f and g don't killtoo m a n y tangent vectors. Further, it is more convenient in m a n y applications to consider the projective tangent bundles, and the projective tangent spaces instead. (These are conormal shemes.) Hence, at the outset, w e shall state our results for projective bundles, and mention some alternatives at the end of the section.

242

The p r o j e c t i v e setup. Denote b y PTY (resp. PTZ) the p r o j e c t i v e t a n g e n t bundle of Y (resp. Z), a n d denote b y PT(X/S) the p r o j e c t i v e r e l a t i v e t a n g e n t bundle of X/S. Since v~ is smooth, PT(X/S) is a p r - b u n d l e o v e r X, w i t h r = d i m ( X ) - d i m ( S ) - l . Taking t h e d e r i v a t i v e of f, we obtain an induced bundle m a p Tf:T(X/S)~TZ. We shall s a y t h a t f is n o n r a m i f i e d o v e r S if, for each xcX the induced m a p tf f is nonramified, w e h a v e a n induced morphism

TxF:Tx(X/S)---*Tf(x)Z is injective,

PT(X/S)

PTf ...... , PWZ

of p r o j e c t i v e bundles. More generally, we shall s a y t h a t f is not too ramified if TxF is not the zero m a p , for a dense set of xEX. (This condition fails if either f is a f a m i l y of c o n s t a n t m a p s , so t h a t f factors t h r o u g h the s t r u c t u r e m a p mX-~S, or the c h a r a c t e r i s t i c is p>O, and f factors t h r o u g h the Frobenius m a p . The first possibility is trivial, b u t the second is interesting.) If f is not too ramified, t h e n PTf will be a rational m a p , whose d o m a i n of definition m a p s to a dense open subset of X. Similarly, g will be n o n r a m i f i e d (resp. not too ramified) if Tg is injective (resp. not zero on a dense set of y¢ Y). PROPOSITION 2.1. In t h e situation above, suppose t h a t ~r is smooth, t h a t f and PTf a r e flat surjections, a n d t h a t g is not too ramified. Then t h e r e is a dense open subset U of S, such t h a t W s is generically smooth, for all seU. Proof. We m a y a s s u m e t h a t W d o m i n a t e s S, for o t h e r w i s e t h e r e is nothing to prove. First w e p r o v e t h e proposition u n d e r the additional hypothesis t h a t f a n d g a r e n o n r a m i f i e d o v e r S, so t h a t PTf a n d PTg a r e defined e v e r y w h e r e . Using t h e m o r p h i s m PTg: TY--*PTZ induced b y the n o n r a m i f i e d m a p g, w e c a n f o r m t h e fiber product, denoted W*, of PT(X/S) and PTY o v e r PTZ. We h a v e a n a t u r a l projection, denoted g: W* ~ W , which is surjective. At a point (x,y) of W, o v e r s¢S and zeZ, t h e fiber of g is ~0-1(x,Y) = PTx(X s) x pTzZ PTyY. Because g is n o n r a m i f i e d , w e c a n t r e a t PTvY as a subspace of PTzZ. Set d=dim(X)+dim(Y)-dim(Z). Since f : X ~ Z is flat, we h a v e dim(W)=d. Set e=dim(S). Because PTI is flat, we h a v e d i m ( W " / P T V ) = dim(PT(X/S)/PTZ) 2(dim(X)-dim(Z))- e. =

Hence dim(W *) = 2d - e -1, so d i m ( W * / W ) = d - e - 1.

243

By generic flatness, there is a dense open set V in W, such that ~0 is flat over V. Over V, by flatness, ~-l(x,y) has dimension exactly d i m ( W * / W ) = d-e-i = dim(Xs)+dim(Y)-dim(Z) -I. Counting dimensions, it follows easily that PTxf(PTx(Xs)) and PTyY span PTzZ. Therefore Tx(Xs) and TyY span TzZ and hence, by [EGA, IV.17.15.6],it follows that W s is smooth near (x,y). Since flat m a p s of finite-type noetherian schemes are open, w e see that p(V) is open in S. If scp(V), w e have shown that W s is generically smooth. By generic flatness, there is an open, dense U I c S , so that the composite m a p p : W ~ S is flat over U I. Let U= p(V^p-l(ui)), an open subset o! U I. Since V and p-l(Ul) are dense in W, it'sclear that U is dense in S. Hence the proposition holds if f and g are nonramified. N o w w e consider the general case. The difference here is that PTf can become a family of projections, and g can become a projection. However, w e only need to find a dense set of (x,y)EW such that W s is smooth at (x,y). Since g is not too ramified by hypothesis, and since f is not too ramified, because PTf, assumed flat,is dominant, the whole argument above applies on the fiber product of the domains of PTf and PTg, which is dense (in particular, nonempty) and open on W*, over a dense open set of W. Choose any (x,y) in this open set. Because Tfx and Tgy can be projections, hence undefined on proper linear subspaces of PTxX and PTyY, restriction to the domains of PTf and PTg replaces PTx(X s)x pTzzPTyy by a dense open subscheme. This does no harm, however, because the dimension is unchanged, so the same argument as in the nonramified case completes the proof. Group actions. Again, suppose an algebraic group G acts on Z, and that f:X~Z is a given morphism, from a smooth X. Denote by F : G x X ~ Z the m a p given by the action. Then G x X ~ G is smooth, and the relative tangent bundle T((GxX)IG) identifies with GxTX, in such a w a y that the corresponding m a p TF GxTX

~-TZ

coincides with the m a p obtained from the derivative Tf:TX~TZ, under the n a t u r a l G-action on TZ. In this setting, suppose t h a t f and g are not too ramified. For a n y smooth v a r i e t y V, denote b y PTV the projective tangent bundle. Then, because f and g a r e not too ramified, we h a v e induced rational maps PTf:PTX~PTZ and PTg:PTY--~PTZ. For ~¢G, we shall denote ~'X×zY b y Wl~, and we shall write d = dim(X)+ dim(Y)-dim(Z) for its expected dimension. W e shall say that a rational m a p is flat,or moves properly under (resp. is transverse to) an action on its target, if it is so w h e n restricted to its domain of definition. If PTf:PTX--*PTZ moves properly under the action of G on PTZ, it follows easily that the m a p TF above is flat. Hence w e obtain the following result.

244

THEOREM 2.2. In t h e situation above, w i t h g not too ramified, a s s u m e t h a t PTf:PTX--*PTZ m o v e s p r o p e r l y u n d e r the action of G on PTZ. Then t h e r e is a dense open set U of G, such t h a t W~ is either e m p t y , or generically s m o o t h of p u r e dimension d, for all ~¢ U. Homogeneous spaces. When G is t r a n s i t i v e on TZ, or, m o r e generally, on PTZ, t h e last r e s u l t applies. The n e x t result generalizes [K, 8, p. 292]. COROLLARY 2.4. In the situation above, w i t h f and g not too ramified, suppose t h a t the action of G is t r a n s i t i v e on PTZ. Then t h e r e is a dense open set U of G, s u c h t h a t W~ is either e m p t y , or generically s m o o t h of p u r e dimension d, for all ~¢U. Applying 2.4 in a special case, we obtain a version of Bertini's Theorem. Let Z -- p r , and G=PGL(r). Then PTZ is t h e incidence correspondence of points and h y ~ e r p l a n e s , on w h i c h G is t r a n s i t i v e . Let f be t h e inclusion of a h y p e r p l a n e in p r , a n d let g:X--,Pr be t h e m o r p h i s m defined b y t h e linear s y s t e m {Dt} of divisors on a s m o o t h v a r i e t y X. We shall s a y t h a t t h e linear s y s t e m {Dt} is not too r a m i f i e d if g is not too ramified. If {Dt} is not too ramified, the last result yields t h e following statement. COROLLARY 2.5. (Bertini's T h e o r e m in a n y c h a r a c t e r i s t i c , version 1.) Suppose X is a smooth variety, and that {Dt} is a not too ramified linear system of divisors on X, without base points. Then the general Dt is nonsingular almost everywhere. (In particular, the general Dt has no multiple components.) Hypotheses on f a n d g a r e needed, as the following e x a m p l e s show. E x a m p l e 1. Let Z be p l , w i t h PGL(1) acting, and let Y={P}, for PeZ. A s s u m e t h a t t h e c h a r a c t e r i s t i c is p> 0, t a k e X=P 1, and let f:X-*Z be t h e m o r p h i s m which is t h e i d e n t i t y on t h e u n d e r l y i n g p l , but the p t h - p o w e r m a p on t h e s t r u c t u r e sheaves. Since the c h a r a c t e r i s t i c is p, t h e d e r i v a t i v e Tf is the 0 - m a p on each t a n g e n t space. Hence g is not too ramified, but f isn't. In p a r t i c u l a r , f c a n n o t be t r a n s v e r s e to t h e action. Each W~ is the n o n r e d u c e d divisor p(~'P), for a n y ~ c t ~ L ( I ) . (Because f is given b y a linear s y s t e m , w e also h a v e a c o u n t e r e x a m p l e to the full Bertini Theorem.) E x a m p l e 2.

Exchange X a n d Y. This this t i m e f is not too ramified, b u t g

isn't. Here G t r a n s l a t e s X, a point, a r o u n d Z=P 1. Since Z is a c u r v e , w e h a v e PTZ=Z, a n d t h e action is t r a n s i t i v e , so PTf is t r a n s v e r s e to t h e action. Since the W~, a r e t h e s a m e as in t h e last e x a m p l e , generic s m o o t h n e s s fails again.

245

Other formulations. One could work w i t h t h e t a n g e n t bundles TX, TY and TZ instead of t h e i r projective analogues, except t h a t the m a p TX--*TZ is not usually flat along the zero-section. Removing the zero-sections, one obtains results no stronger t h a t those above, b u t in a n artificial formulation. The idea of working directly on t h e t a n g e n t bundles, w i t h o u t projectivizing, becomes n a t u r a l , however, if we drop the d e m a n d for flatness a t the t a n g e n t bundle level, Instead, w i t h o u t filtering the question t h r o u g h a flatness a r g u m e n t , w e can t r y to bound t h e dimension of t h e fibers of W ~ W directly. This a p p r o a c h will be explained next.

3. D e t e r m i n a n t a l pairs..

We continue in the same setting as in the last section, w i t h s m o o t h varieties X,Y,Z and S, fitting into a c o m m u t a t i v e diagram with c a r t e s i a n square,

W

,Y g

X

' ') Z

S, w h e r e n is a smooth map. F r o m h e r e on, we shall a s s u m e t h a t g is nonramfied. Suppose also, for now, t h a t f and g m e e t properly (e.g. if f is flat). Our goal will be to find criteria, independent of the characteristic, for the general W s to be smooth, not j u s t generically smooth. The results which follow w e r e obtained j o i n t l y with Dan Laksov. Denote b y E and F the pullbacks of T(X/S) and g*TZ/TY, repectively, to W. Because S is smooth and ~ is flat, E is a bundle. Because Y and Z are smooth and the d e r i v a t i v e of g has c o n s t a n t rank, so t h a t g~TZ is a sub-bundle of TY, it follows t h a t F is also a bundle. Our goal now will be to s t u d y the bundle map, denoted b y

E

,F,

246

induced b y the d e r i v a t i v e Tf. Denote b y V the closed s u b s c h e m e of W w h e r e o~ h a s less t h a n m a x i m a l r a n k , and set p

=

Irank(E) - rank(F)l

+I.

Then it is w e l l - k n o w n t h a t e i t h e r V= 9 , or the codimension of V in W is a t m o s t p. Because f a n d g m e e t properly, it is also e a s y to check t h a t t h e i m a g e of V in S is precisely the set of points s¢S w h e r e X s a n d Y a r e not t r a n s v e r s e ! We n o w investigate the i m a g e of V in S. Assume first t h a t W d o m i n a t e s S. Since f and g m e e t properly, we h a v e dim(W) = dim(X) + dim(Y) - dim(Z) _>dim(S). Therefore w e find p = Idim(X)- dim(S) - (dim(Z)- dim(Y))l+l = d i m ( W ) - dim(S) +1. Now suppose t h a t the codimension of V in W is e x a c t l y p. By the calculation above, we h a v e dim(V) = d i m ( S ) - l , so Z m a p s to a closed subset of S whose c o m p l e m e n t , denoted U, is dense. If, h o w e v e r , W doesn't d o m i n a t e S, t h e r e is obviously a n open, dense U on S, such t h a t t h e fiber p r o d u c t W s is e m p t y , for all s¢S. Set d = dim(Xs)+dim(Y)-dim(2). Then we h a v e established the following result. PROPOSITION 3.1. Suppose, in the situation a b o v e , t h a t f is flat, t h a t g is n o n r a m i f i e d , a n d t h a t V h a s codimension e x a c t l y p in W. Then t h e r e is a dense, open subset U of S, s u c h t h a t XsxZY is e i t h e r e m p t y , or is s m o o t h of dimension d, for a n y s¢ U. This result depends, of course, on the m a p g:Y~Z. In the spirit of our earlier results, h o w e v e r , we should look for a s t r o n g e r hypothesis on f, so t h a t the conclusion holds for a n y n o n r a m i f i e d g. To e m p h a s i z e the d e p e n d e n c e on g, we shall n o w w r i t e pg, Vg a n d Wg in place of the p, V a n d W above. The following definition highlights the role of g in 4.1. We shall s a y t h a t t h e pair of m o r p h i s m s (f,~) as above, is d e t e r m i n a n t a l if t h e following condition holds:

247

(~)

For e v e r y nonramified m a p g:Y--)Z, from a smooth v a r i e t y Y, either Vg= ~, or codim(Vg,Wg) = fag.

As before, set d=dim(Xs)+dim(Y)-dim(Z) , independent of g. PROPOSITION 3.2. Suppose in the situation above that f:X-~Z is flat, and that the pair (f,rt)is determinantal. Then, for any nonramified m a p g:Y--*Z, there is a dense, open subset U of S, such that Xs×ZY is either empty, or is smooth of dimension d, for any scU. Proof. By the first assertion of 1.1, which holds in any characteristic, there is a dense open subset of S over which X s × z Y is either empty, or has dimension d. If empty, w e are done. Otherwise, instead of shrinking S, w e m a y assume the fiber product has dimension d. Then 3.1 applies, so 3.2 follows. The condition (*) is obviously local on Z, but it looks a w k w a r d to verify, because it involves all possible nonramified m a p s into Z. Compare, however, of the analogous cases (1) of flat maps, which are characterized by meeting every g properly, and (2) smooth maps, which are characterized by meeting every g transversalIy. One should certainly hope that (~) would follow from a more managable condition on the fibers of f, and a central observation of [LS] is that, indeed, it does. Passage to the fibers. Suppose given f:X-*Z, as above. Pick a point zeZ, and a linear subspace LCTzZ. For any xcf-l(z), differentiation induces a natural m a p (XL, x

Tx(X/S)

, TzZ/L.

W e set 9L = dim(X)+dim(L)-dim(Z)-dim(S)+l, and w e define V L C f-i(z) to be the set of points x where aL, x has m a x i m u m is at most 9L-

rank. Hence the codimension of V L in f-l(z)

Let f:X--*Z,as above, be a morphism, and fix a point zcZ. W e shall say that the pair (f,~) is determinantal at z if the following condition holds:

For e v e r y linear subspace LCTzZ, either VL= ja, or codim(VL, f-i(z)) = PL.

248

THEOREM 3.3. Suppose given a flat m a p f : X ~ Z as above, such t h a t t h e pair (f,~) is d e t e r m i n a n t a l at each z~Z. Then, for a n y n o n r a m i f i e d g:Y~Z, t h e r e is a dense, open UCS, such t h a t Xs×zY is either e m p t y , or is s m o o t h of p u r e dimension d, for a n y s~ U. Proof. The t h e o r e m follows i m m e d i a t e l y f r o m 3.2 a n d t h e n e x t s t a t e m e n t . LEMMA 3.4. Suppose, for a flat, s u r j e c t i v e m o r p h i s m f:X--~Z, t h a t the pair (f,rc) is d e t e r m i n a n t a l a t e a c h zcZ. Then the pair (f,rc) is d e t e r m i n a n t a l . Proof. Suppose given a n o n r a m i f i e d m a p g:Y--~Z. Choose a point (x,y)EW o v e r z£Z, take L=image of TyY in TzZ, and a p p l y (*)z- We find t h a t VL=V,~I-i(z) is e i t h e r ~ or h a s codimension PL in f-l(z). Since dim(L)=dim(Y), we h a v e pL=p, so v a r y z: it follows t h a t V is either ~ or has codimension p in W. This proves the lemma. While the condition (*)2 does not involve auxiliary spaces Y, it stillseems very strong. Still,as w e shall see, it often holds.

4. Laksov's theorem and related results.

W e continue the joint work begun in the last section. Suppose given smooth varieties S, X, Y and Z, and a flat,surjective morphism F:SxX-~Z. W e shall n o w consider a diagram with Cartesian square W

~Y

S×X

~Z

S, where this time n is the firstprojection, which is a smooth map. Laksov's Theorem. To interpret the condition (*)z in this context, w e shall m a k e the following definition. For any x E X and zcZ, denote by T(x,z) the subset

249

{s¢SlF(s,x)=z}. The isomorphism S~S×{x} identifies T(x,z) w i t h (Sx{x})~,F-l(z), so T(x,z) is in a n a t u r a l w a y a closed subscheme of S. Thinking of the special case of a group action, we shall call T(x,z) the transporter of x to z. Denote by p the restriction to F-l(z) of the projection SxX~X. Then, for a n y xcX, we h a v e p-l(x)--T(x,z). Hence, given zcZ, to check (*)z for a given LCTzZ, it suffices to check t h a t VL,,T(x,z) has codimension exactly PL in T(x,z), for each xcF-l(z). Indeed, F-l(z) is the total space of the algebraic family {T(x,Z)Ix~x. Now for each given LCTzZ, we h a v e a n a t u r a l morphism of schemes 8L T(x,z)

, Homk(TxX,TzZ/L),

induced by differentiation. Denote by 7.L the Schubert cell in Hom(TxX,TzZ/L) parametrizing the m a p s of less t h a n m a x i m a l rank. The codimension of E L is exactly PL, so, if T(x,z) meets 7.L properly for e v e r y L, t h e n (")z will hold. Next we t r y to eliminate the dependence on L. In the case L=0, we shall write

8X,2 T(x,z)

-~Homk(TxX,TzZ)

for 50, and compare this with the general 5L. The quotient projection induces a surjection ¢0: Hom(TxX,TzZ)--*Hom(TxX,TzZIL), clearly a flat map. Hence, if we a s s u m e t h a t 8x.z is also flat, the codimension of 7.L will be preserved under the pullback to T(x,z) via 8L=~OSx, z. Conclusion: if Sx,z is flat, then (*)z holds, so the pair (f,~) is d e t e r m i n a n t a l . For a n y s¢S, denote by F s the morphism X ~ Z induced by the restriction FI{s}×X. We obtain the following s t a t e m e n t . T H E O R E M 4.1. (Laksov, [L, Th.l, p.275].) Suppose w e have a flat,surjective m a p F:SxX--~Z as above, and suppose, for all xcX and zcZ, that the morphism 6x,z above is flat. Suppose also that w e are given a nonramified m a p g:Y-~Z. Then there is an open, dense subset UcS, such that F s is transverse to g, for every s£U.

The condition t h a t the 6x, z should be flat is v e r y strong. It implies, for example, that every T(x,z) is nonempty. Hence, w h e n the m a p F is given by a group action, Z must be a homogeneous space. In the case of a group action, however, special features will allow us to improve the last result.

250 Group actions. Suppose a group G acts on Z. We suppose given a flat m a p I:X~Z, such t h a t the induced m a p F:GxX-~Z is flat (e.g. if f m o v e s properly u n d e r the action). For zcZ, we denote b y D its orbit. Note t h a t we h a v e F - I ( D ) : G x f - I ( ~ ) , as schemes. In particular, if f is t r a n s v e r s e to the action, t h e n I-I(D) and F-I(D) are both smooth, so we can consider the n a t u r a l m a p ~X,Z

T(x,2)

,

Homk(Txf-l(13),Tz0).

Then we h a v e t h e following result. PROPOSITION 4.2. In the situation above, suppose t h a t f is t r a n s v e r s e to the action of G on Z, and, for all zEZ and all x c f - l ( D ) , t h a t Sx,z is flat. Then for a n y nonramified m a p g:Y--*Z f r o m a smooth v a r i e t y Y, t h e r e is a dense open subset UCG such t h a t the t r a n s l a t e gf is t r a n s v e r s e to g, for all gcU. Proof. Choose yEY, and let z=g(y)EZ. Denote b y L t h e image of TyY in TzZ. Because f is t r a n s v e r s e to the action, F is smooth. The orbit 0 is smooth. Hence, since Z and F are smooth, so is f-l(D). If D=Z, the proposition follows directly f r o m 4.1. Hence suppose t h a t D is not the only orbit. If the general W~, is e m p t y , t h e n we are done. If not, b y 11(1) t h e r e is an open, dense subset UoCG , such t h a t Wt~ has dimension e x a c t l y d=dim(X)+dim(Y)-dim(Z), for all ~¢U 0. Consider the m a p fl: f-I(D)-*D- By t h e hypothesis on Sx,z, we know t h a t ( . ) z holds for the action m a p F l : G x f - l ( o ) - ~ O . Hence t h e r e is a n open, dense set UyCG such t h a t the n a t u r a l m a p induced b y differentiation, (x

T(g,x)(F-I(O)/G) -~ TzD/(L~TzD), has m a x i m a l rank, for all (~',x)¢F-I(D)=GxI-I(D), such t h a t YEUy. Denote by M the image of T(Lx)(F-I(D)/G) in TzD. Then it follows i m m e d i a t e l y t h a t we h a v e (1)

(L~,TzD) + M : TzD.

Write N for the image of T(~,,x)(GxX/G) t r a n s v e r s e to the action, we find (2)

TzD +N

=

=

(¥Txf)(TxX) in TzZ. Then, since f is

TzZ.

251

Since (L,~TzD) c L and M C N , w e obtain from (I) and (2) the equality (S)

L+N = TzZ.

Suppose now ~¢¢U0, so the intersection o v e r ~ is proper. Then, b y (S), if (~,x)~F-t(z), w i t h ~¢Uo,~Uy , the m a p s f and g a r e t r a n s v e r s e a t the point w--((~,x),y) ¢ W. We conclude t h a t t h e r e is a n open neighborhood VyC W of the fiber Wy above y c Y , whose intersection with the pullback of Uo,~Uy to W is smooth over U0 ~ Uy. Push Vy f o r w a r d u n d e r the flat m a p W-*Y. (The smooth m a p F is flat, so the base change W--*Y is c e r t a i n l y flat.) We obtain an open neighborhood of y. Now v a r y y ¢ Y. Since Y is quasicompact, we can choose Yl ..... Yr c Y so t h a t the resulting neighborhoods cover Y. Hence the Vyi c o v e r W. Define U to be the intersection of U0 and t h e Uyi. Then U is a dense open subset of G, such t h a t the pulback W U is s m o o t h o v e r U. This proves the proposition. For zcZ, denote b y Gz the stabilizer of z in G, and b y ~z the group homomorphism Ez Gz ) GL(TzD),

induced by differentiation. Clearly ~z is flat if and only if it is surjective. Indeed, as a group homomorphism, Sz is flat onto its image, by 1.2, and the image is a closed subgroup. Here is o u r m a i n result.

T H E O R E M 4.& Suppose that i:X-*Z is rlonrami£ied and transverse to the action, and suppose that for each zcZ the m a p cz is surjective. Then for any nonramified m a p g:Y-~Z from a smooth variety Y, there is a dense open subset U c G such that the translate Iffis transverse to g, for all ~¢U. Proof. For x~X and z~Z, denote by Txf:Txf-l(O)-*TzD the derivative of the restriction of f. Since f is nonramified, Txf is injective. The morphism

~X.Z T(x,z)

~ Homk(Txf-l(f}),Tz D)

is t h e composite d t h e translation isomorphism T(x,z)--.*G z, the flat surjection Sz:Gz-~(Tz]D) given by the hypothesis, and the flat surjection

GL(TzD)-*Homk(Txf-I(I}),TzD) induced by Txf. It follows that Ex,z is flat, with image the open subset of injective maps. Hence the theorem follows from 4.2.

252

The original proof of the n e x t result was a stimulus for [L], and hence for m u c h of this paper. COROLLARY 4.4 [K, I0, p. 294]. Suppose the action of G is t r a n s i t i v e on Z, and for all z e Z the m a p gz is surjective. Then for a n y nonramified m a p g:Y~Z from a smooth variety Y, there is a dense open subset UcG such that the translate ~'f is t r a n s v e r s e to g, for all ~'¢ U. Since GL(r+I), acting on p r , is easily seen to satisfy the condition on gz, we obtain the following application [K, 12, p.296] to linear systems.

COROLLARY 4.5. (Bertini's T h e o r e m in a n y characteristic, version 2.) Suppose Suppose X is a smooth v a r i e t y , and t h a t {Dt } is a linear s y s t e m of divisors on X, w i t h o u t base points, which separates t a n g e n t directions on X. Then the general Dt is smooth.

References

[EGA, IV]

A. Grothendieck and J. Dieudonne, Elements de g e o m e t r i e algebrique, Chap. IV, Publ. Math. de I'IHES, 2__0,24, 2__88,5_22 (1964-67).

[H]

R. Hartshorne, Algebraic geometry, Springer, Graduate Texts in Math. (1977).

[K]

S.L. Kleiman, Transversality of the general translate, Compos. Math. 28 (1973), 287-97.

[L]

D. Laksov, Deformation of determinantal schemes, Compos. Math. 30 (1974), 273-292.

[LS]

D. Laksov and R. Speiser, Notes on transversality, in progress.

Djursholm April 6, 1987

In our view, the variety of complete quadrics ranks with Grassmannians and flag manifolds as one of the most important special varieties. --[5], p. 5 COMPLETE

BILINEAR

FORMS

ANDERS THORUP Matematisk Institut, KObenhavns Universitet Universitetsparken 5, DK-2100 KObenhavn 0, Denmark STEVEN KLEIMAN$ Mathematics Department, 2-278 M. I. T. Cambridge, MA 02138, U. S. A.

INTRODUCTION T h e r e is a special compactification Br of the space of bilinear forms of rank at least r, and its T-points are the r-complete bilinear forms on T. Its subspace -R - r sym is a special compactifieation of the space of symmetric bilinear forms. These spaces possess a similar geometric structure, whose richness and b e a u t y are dazzling. These magnificent spaces were discovered and explored little by little during the course of the 19th century, primarily by enumerative geometers, who were treating quadrics, correlations and collineations by the m e t h o d of degeneration. Their work has been secured and advanced during the 20th century b y m a n y geometers, remarkably many. T h e whole history makes fascinating reading; for starters, see [6], [7], [9], [10] and

[11]. In the present work, we treat the basic geometric properties of B~ and B~ ym over an arbitrary ground scheme. We give no applications to enumerative geometry. We do not discuss the different, but beautiful, representation-theoretic approach of Demazure, De Concini, Procesi, Goresky, MacPherson, and Uzava; for that, see [3], [5] and [15]. Here we advance the fundamental work of Tyrrell [14], Vainsencher [16], [17] and Laksov [10], [12] based on multilinear algebra; we clarify it, refine it, extend it, and surpass it. Their main (non-enumerative) results are completely recovered: the structure via blowups, the description of the normal bundle of each center, the embedding in projective space, the identification of some key T-points, the structure of the orbit closures, and Schubert's basis-change relations. Our main new results include the following: the identification of all the T-points, an extensive theory of splitting and joining forms, the first explicit system of equations, the equivalence of the two natural definitions of symmetric r-complete forms, and a m o d e r n t r e a t m e n t of duality. :[:Supported in part by the National Science Foundation of the United States and by the National Science Research Council of Denmark. The author is grateful to the members of the Mathematics Institute of the University of Copenhagen for all their hospitality. 1980 Matherna~cs Subject clazsif~ations: 14M99, 14N99, 15A63, 15A69

Typeset by .~Ms-'rF~

254

The present work benefited greatly from discussions with Laksov and Vainsencher and from the opportunity to study preliminary versions of their works. We are grateful to them. To appreciate the advances made here, recall the gist of what Tyrrell, Vainsencher, and Laksov did. Tyrrell worked over the complex numbers. He considered nonsingular r by r matrices u and formed the set of r-tuples (u, A 2 u , . . . , / ~ r u), which he embedded using Plfic~er coordinates in the product Sr of the appropriate projective spaces. He formed the closure and took it as Br. This procedure is motivated by the following old observation: If u represents a collineation, then A i u represents the induced i-plane-to-i-plane correspondence; if u represents a correlation, then A{u represents the induced i-plan .e--to-(r - / ) - p l a n e correspondence; if u is symmetric and represents a quadric, then A ' u represents the family of tangent /-planes. Hence a boundary point of B r represents a degenerate collineation, correlation, or quadric, completed with its higher-order "aspects". Tyrrell described the orbit structure: the 2 r-1 subsets I of the interval [1, r - 1] index the orbit closures Or [I]; moreover, each Or [I] is smooth of codimension card(I), and Or [I] and Or [J] intersect transversally in Or [I U J]. (Thus, in the words of the representation theorists, Br is a "marvelous", or '~wonderful", compactification of its open orbit, Oriel.) To prove it, Tyrrell used the decomposition of a nonsingular square matrix with (1, 1)-entry 1 into the product L D U , where L is a lower triangular matrix with unit diagonal, D is a diagonal matrix with (1, 1)-entry dl = 1, and U is an upper triangular matrix with unit diagonal. The decomposition yields an open subset of Br, which is isomorphic to the affine space of dimension (r 2 - 1). In it, Or [I] is defined by linear equations q{+t = 0 for i C I. The coordinates of the affine space are the entries of L, those of U, and the q{ for 2 < i < r. The q{ are not the diagonal entries d~ of D; rather, dl -- ql :--

1,

d 2 : - q l q2 ,

• • •,

d r ~- q l q 2 "" • q,-.

Replacing L by P L a n d U by U Q , where P and Q are variable permutation matrices, yields a covering of B r. Tyrrell easily verified Schubert's basis-change relations, Schubert's formulas expressing the rational equivalence classes of a divisorial orbit closure in terms of the Pliicker hyperplane classes. Tyrrell concluded via a degeneration that the hyperplane classes generate the Picard group. Finally, he observed that the preceding theory and t h a t for B~ym are parallel (and in fact he concentrated on the latter). Vainsencher worked over a base scheme S t h a t is normal, Cohen-Macaulay and of finite type over an arbitrary algebraica~y closed field, of characteristic different from 2 in the symmetric case. Instead of matrices, he considered maps u: 8 ---* jr. ® L where £, jr and L are bundles of ranks e, f and 1. In this setup, he generalized Tyrrell's results. In addition, he proved t h a t Br+l is equal to the blowup of Br along a smooth center Vr. In fact, he proved that Vr is the proper transform of a certain subscheme Z~ +1 of Bo, the space of all u; namely, Z~ +1 is the locus of those u of rank at most r. Vainsencher identified the normal bundle of Vr in Br, and he described the exceptional locus and the other orbit closures. He also gave a parallel treatement of

255

Bsym Thus he was able to do intersection theory and to verify many of Schubert's numbers, including all the fundamental numbers of the quadrics in p 3 . Vainsencher's approach is different from Tyrrell's. Instead of covering Br with affine open sets, Vainsencher made some involved b u t intelligent constructions with sheaves, and he used the general characterization of a blowup as the universal a t t r a c t o r rendering the center a divisor. To treat the particular center Vr, he employed the difficult and delicate t h e o r y of saturated and normal ideals; for this reason, he had to assume t h a t the base scheme S is normal and Cohen-Macaulay. Vainsencher characterized the T-points of the orbits when r -- e < f as follows. Let So := 0 < sl < ... < sk < sk+l := r and I := { S l , . . . , s k } . r

•

T h e n a T-point of Or [I] corresponds to (a) a flag of subbundles £i of the pullback CT of corank si for 0 < i < k + 1, and a flag of subbundles ~ of the pullback jrT of corank si for 0 < i < k + 1, and (b) a sequences of maps vi : £i --~ ~* ® )¢i on T for i = 0 , . . . , k, where Xti is a line bundle and )40 = LT, such t h a t (*)

•+1 = Ker(vi) and ~*+1 = Cok(vi) ® N~-1 for i = 0 , . . . , k.

Laksov went further. He worked over a completely a r b i t r a r y base scheme S, assuming 2 invertible in F(S, O s ) in the symmetric case. Like Vainsencher, he considered maps u : £ --* jr* ®/~ where £, 7 and • are bundles of ranks e, f and 1. However, he generalized Tyrrell's construction of a special a f ~ e open covering of Br: he worked locally on S and modified the r by r triangular decomposition L D U by augmenting L with an f - r by f - r identity matrix, by augmenting D with an f - r by e - r m a t r i x whose ( i , j ) - e n t r y d4j is of the form d~,j = qi,jdr where dr is the r t h diagonal entry of D, and by augmenting U with an e - r by e - r identity matrix. Working in these affines, he proved t h a t B r + l is the blowup of Br along the appropriate center V~; he did so easily, without using the theory of saturated and normal ideals. Laksov identified a significantly larger collection of T-points of the principal orbit closures Or [I]. (If r < e < f , there are other orbit closures; for example, Vr.) T h e identification is, in fact, the heart of Laksov's approach. These T-points correspond to a pair of flags {~'i}, { ~ } and a sequence of maps v~ as above but satisfying (**), which is the weakened version of (*) obtained by replacing Cok(v~) with its quotient modulo its torsion subsheaf and by requiring in addition t h a t all the nonzero sheaves of minors of vi be invertible. Fixing such a pair of flags and sequence of maps vi, Laksov constructed an invertible quotient ~ j of A j £ ® A j jr for j = 1 , . . . , r by appropriately combining the sheaves of minors of the v~. Next, fixing a point t of T and ordered bases x l , . . . , x e and Y l , . . . , Yl of £ and jr in a neighborhood S ~ of the image of t in S, he noted t h a t there exist a reordering of the x's, one of the y's and a neighborhood T ~ of t on which the image of (xl A . . - A xj) ® (Yl A . . . A yj) generates ~ y for each j . Then, proceeding by induction on j and using elimination to clear the appropriate columns and rows, he constructed matrices L, D and U as above such t h a t L and U change the given bases into ones (a) t h a t are compatible with the filtrations of jr b y the ~ and of ~" by the £i and (b) in which each vi is given by the lower ( f - s,) by (e - si) s u b m a t r i x of D

256 with its lower ( f - s i + l ) b y (e - s i + 1 ) submatrix replaced b y the zero matrix. Finally, L, D, U and I define a T'-point of an a/fine space W[I], whose coordinates are the entries of L and U and, as before, appropriate factors qk and q{,j of the entries of D, wherel 0. Let w: A ~ P -~ ~ be a bilinear m a p , a n d q: Q - - ~ P a m a p of pairs. Denote by h(w, q) or h ( w , Q) the m a p of pairs i--i

i

267

associated to the following composition: i--1

i

P ® AQ

i

i

AQ'r-,AP

(w,w*')

AP

,

Note that, if i ---- 0, then A i-1 q = (0, 0) and h(w, Q) = (0, 0). If i ---- 1, then the bilinear map w: P --~ ~ will be called regular if K e r h ( w , P ) (0, 0); t h a t is, if h(w, P ) is injective.

--

LEMMA ( 1 . 8 ) .

Let P be a pair, Q a subpair, and p: P --+ ( P / Q ) the canonical s u r j e c t i o n . L e t w : A~P -* ~ be a bilinear m a p w i t h i >_ 1, a n d u: P --+ f. a f o r m . Then:

(i) Q c__K e r h ( w , P )

i f f w i n d u c e s a bilinear m a p , i

we/Q: A(P/Q)

-~ ,.G.

(ii) p - l K e r h ( w p / Q , P / Q ) ----K e r h(w, P). (iii) I f u: P -+ £ is a regular f o r m a n d i f P is a p a i r o f bundles, t h e n these b u n d l e s are of the s a m e rank.

(iv) If P is a p a i r o f b u n d l e s o f t h e s a m e r a n k , t h e n t h e f o r m u: P - ~ £ is regular i f f e i t h e r o n e o f the c o m p o n e n t s of h(u, P ) is i n j e c t i v e . PROOF: Assertions (i) and (ii) axe obvious. Assertion (iv) is easy to derive from (1.5). In (iii), u is regular; so A i h ( u , P ) is injective for all i by (1.4)(i*). However, if the two ranks are not the same, then taking i to be the larger one yields a contradiction, because f is invertible. DEFINITION ( 1 . 9 ) .

Let u: P --~ /" be a form. The induced map (1.8)(i) Ureg : = U P / K e r h ( u , p )

:

P/Kerh(u,P)

-~ £

will be called the regularization of u. (It is regular by (1.8)(ii).) DEFINITION ( 1 . 1 0 ) . A form u: P --+ f will be said to be of rank r if the quotient P / K e r h ( u , P ) is a pair of bundles and both are of rank r. (Both are of rank r if one of t h e m is by (1.8)(ii)(iii).) DEFINITION ( 1 . 1 1 ) . A bilinear map u: P --+ • will be called the direct s u m of bilinear maps u i : P i --+ £ , and u = @iui will be written, if P=~iPi

and

and if each ui is induced by u.

PiC_Kerh(u, Pj)

forall

i#j

268

DEFINITION ( 1 . 1 2 ) .

A form u : P ~ • induces, for each i _> 0, a form i Au:

i A P ---* ~®i,

which will be called the ith exterior power of u; locally A ~u is given by i

A u(el A . . .

A e,; ® f l A . . .

A S,;) = det

[u(ej ® f k ) ] .

PROPOSITION ( 1 . 1 3 ) . L e t u: P --* £ be a form on a pair o f bundles o f t h e same r a n k r. F i x i, 1 < i 0. Call the image of the ith exterior power A i u the ith m i n o r of u and denote it by N~ or Hi(u), i

:= zm((AP)®

z:®,).

Denote the surjective bilinear map (or nowhere vanishing form) induced by A ~u by i ui: A P ~ )Vt~. (Note t h a t u ° = 1, the identity map of the structure sheaf Os. )

269

PROPOSITION ( 1 . 1 7 ) . Let Q be a pair of bundles of the same rank s, emd u: Q --* £ a form. Then u is regular iff J~8(u) is invertible, iff A S u induces an isomophism onto its image,

A PROOF: Being a form on a pair of invertible sheaves, A s u is regular iff it is injective, iff its image ~ , ( u ) is invertible. Now apply (1.13). LEMMA ( 1 . 1 8 ) . L e t s >_ 1 andletu: P --~ £ beaformsuch that ~ ( u ) isinvertible. Let Q be a pair of bundles of rank s, and q: Q ~ P a map of pairs. Consider the form uq: Q ~ £. Assume that ~ , ( u q ) -- Jvts(u). Then: (i) q is injective and left invertible, and uq is regular. (ii) h(u% q) is surjective and right invertible. (iii) K e r h ( u , q ) = K e r h ( u i , q ) = K e r h ( A ~u,q) for any i _< s. (iv) u = (uq) @ ( u l K e r h ( u , q ) ) .

PROOF: Consider the diagram, Q

h((uq)',Q), Hom(A~_X Q

i, h(~°,Q) p

® A 8 Q % A%(uq))

F i_iom(A~-1Q ® A • Qt,, Ms(u))

,

where b is the obvious map. The diagram is plainly commutative. By hypothesis, J~,(uq) = ~ ( u ) . So b is an isomorphism. By hypothesis, Q is a pair of bundles of rank s, and ~ ( u ) is invertible. Hence, by (1.17), uq is regular, and (A 8 Q)® = ~8(uq). It follows that h((uq)% Q) is an isomorphism. Thus, q is left invertible; in particular, it is injective. Thus, (i) holds; moreover, (1.18.1)

P = q Q @ K e r h(u 8, Q).

Furthermore, (ii) holds. Consider the diagram, h( A' ~,Q)

P

l

H o m ( A i - : Q ® A ~Q~r,L®~)

>

h(u,Q)

H o m ( Q tr, £)

g >

Nom(A ~-~ I-Iota(O% L) ® A ~Q% L®i)

where d is the map induced by Ai-Z(h(uq, Q)) and g is the map induced by the following composition of the natural maps: i-1

i

i

i

H o m ( Q t r , £)® A H o m ( Q t r ~)@ A Qtr ___,A H o m ( Q t r , •)@A Qtr ~ (~®i, L®i).

270

The diagram is easily seen to be commutative. Since u q is regular, h ( u q , Q) is injective by definition, (1.7). So, A ~-1 h ( u q , Q) is injective by (1.4)(i*). Therefore, d is injective by (1.5). Now, g is injective because Q is a pair of bundles of rank s _> i. Plainly, K e r h ( A ~u, Q) = K e r h ( u i, Q). Hence, (iii) holds. So (1.18.1) and the definition of direct sum, (1.11), yield (iv). PROPOSITION ( 1 . 1 9 ) . Let u: P ~ £. be a form, and s > O. Then .Ms(u) is invertible iff, locally, there exists a subpair Q of P of bundles of rank s such that, if v := u]Q, then v is regular and .Ms(v) = .Ms(u). Moreover, if .Ms(u) is invertible and if a system of generators for each component of P is given, then locally there exist subsystems generating a suitable such subpaJr Q. PROOF: Locally .Ms(u) is generated by elements of the form,

/ ~ ~(el A... A e~ ® A A-.. A f~). If .Ms(u) is invertible, then locally it is generated by a single element of this form. Working locally, let Q be a pair of free modules of rank s and let q: Q --~ P be the map defined by the e's and f's. Then .Ms(uq) = .Ms(u) by construction, and the assertions hold for Q :-- q Q by (1.18). The converse holds, because .M8(uq) is invertible by (1.17). COROLLARY ( 1 . 2 0 ) . Let u: P ~ ~ be a form, and r > 0. Then u is of rank r iff .Mr(u) is invertible and .Mr+l(U) 0. =

PROOF: Suppose u is of rank r ; t h a t is, P / K e r h ( u , Q) is a pair of bundles of rank r. Consider the regularization ursg: P / K e r h ( u , Q) -* ~,.

Obviously, .Mi(ureg) = .Mi(u) for all i. By (1.17), .M~(ureg) is invertible. Obviously, .Mr+z(ur~g) --0. Thus, half the assertion is proved. The converse is a local statement. So, by (1.19), we may assume that u = v G v ~, where v is regular of rank r. Then, obviously, .Mr(v) .MI(?)/) C .Mr+I(U)

=

0.

Hence, since .Mr(v) is invertible, .Ml(v') = 0; that is, v' = 0. Hence, since v is regular, v = u~eg. Thus u is of rank r.

2.

QUADRATIC RELATIONS

DEFINITION ( 2 . 1 ) . Let £ be a sheaf, and i , j , k >_ O. Denote by Ae or A the (usual) exterior product map, ¢

5

j+¢

:(Ae®Ae) ,Ae,

271 Denote by VE o:" v the (usual) exterior coproduct map, :+i

i

j

V(Ae)----E(--1)IAel®Aeo~'

v:

I

where e :---- ( e l , . . . , ej+i) and A e : = e l A . . . Aej+i, where I denotes an ordered subset of the intervM [] , j + i] with i elements and C I denotes the ordered complement, and where e1 denot, s the i t u p l e of the ek for k 6 I, and ( - 1 ) 1 denotes the sign of the p e r m u t a t i o n ( I , C I ) . Finally, denote by ~ 8 or ~ , j , k or ~ the composition, i

jWk

A

®AE

i

j

k

k

j

i

k

jq-i

I®^,A~®A~

where s w is the isomorphism switching the first and third factors. LEMMA ( 2 . 2 ) .

Let ~ be a bundle, a n d i , j , k > O. Then:

(i) VE. ---- (/' ~)* and < ~ . = ( ~ ) * , where t h e " * " i n d i c a t e s t h e dual. (ii) Suppose t h a t ~ is o f r a n k j + i. I f k < i, t h e n ~ is surjective; i f k = i, t h e n i t is equal ~o the identity. PROOF: (i) To establish the firsl~ equation, consider the following diagram:

A~+~(~ *)

,

I (A j+~ ~)*

A ~~* ® A ~ .~.

i , (A ~~" ® A j ~)*.

T h e top and b o t t o m maps axe the two in question. T h e two vertical maps are the maps associated to t~e appropriate eocLerior powers of the canonical form, E~ ® ~* ---* O s (and they exist whether or not t' is a bundle); they axe the s t a n d a r d !isomorphisms used to identify their sources and targets, g being a bundle. It is easy to see that the diagram is commutative, using Laplace development of the determinant of an (3" + i) × (3" + / ) - m a t r i x along the first i columns. Thus the first equation is valid. It is easy to derive the second equation from the first. (ii) T h e question is local, so we m a y assume that E is free. Let e l , . . . , ej+~ be a basis. Then, sirce k < i , clearly e j + l A . . . A e j + k @ el A . . . A ej+i = ~>(ej+l A . . . A e j + I @ el A . . . . ~ e j + k ).

Permuting the o's yields (ii). LEMMA ( 2 . 3 ) . Let 8 be a sheaf, ~ , ~ / , K s u b s h e a v e s , a n d i , j , k , rn, n >_ O. Cons i d e r t h e following diagram, whase u p p e r a n d l o w e r m a p s are i n d u c e d b y e x t e r i o r multiplication'.

A ~ ~a® A j+k 9 ® A "~~, ® h" K

..~ A ~+'~ C @ A j+(k+~)

1~9 ®i@i Ak~ ® AJ+~ ~ ®Ame ® A"~

, A k+~ £ @ AJ+(~+m) £.

272

Then, it commutes modulo the image ! of the foIIowing map: k+l

n--1

j--1

i

m

k+n

j+i+rn

Ae® A

e

PROOF: The assertion is easily checked on local sections. DEFINITION ( 2 . 4 ) . the pair

Let P = (~, Jr) be a pair, and i , j , k >_ O. Let i,j,k

A

i

j+k

k

Ai'J'kP denote

jq-i

P::cA e ® A e, Ajr® A Jr)

LEMMA (2.5). Let u : P --~ £ be a form, and i, j, k >_ 0. Then the following diagram is commutative:

(1,0) ,

A ~,j,k p

i

A ~P ® A j+k p

IA'

(o,8

'~®A ; + " '~

A kP ® Aj+~ e

A ~u®A" u

,

/2®(J+i+k)

PROOF: Say P = (g, Jr). To check commutativity, we need consider only a finite number of local sections at a time. Take some section, replace S by their common support, and replace ~" and jr by free modules of finite rank covering the submodules the sections generate. Clearly, the diagram in question is commutative iff the following one is:

A ~ ~ ® AJ+ k ~

, , y o m ( A ~ jr

® AJ+ k Jr, L®(J+~+k))

~+ A k ~" ® A j+~ ~

i~om(, ') , )4om(A k 7 ® A j+~ Jr, L®(J+~+k)).

Now, this second diagram is commutative because (>~, =- ((>7)* by (2.2)(i) and because ~ is obviously functorial. PROPOSITION ( 2 . 6 ) . Let u: P -~/~ be a form, and i , j , k >_ O. Then: (i) These inclusJons hold: ~ j + i c_ ~ i A4j C_ Ati £®J C_ f®U+~). (ii) If one of the two components of P is a bundle of rank j + i, and i l k 0 with i < s. Consider the ~

~y+~ C ~t~ ~4j+~.

This inclusion is established in (2.6)(ii) under the hypothesis t h a t one of the two components of P is a bundle of rank j + s, and in (2.8)(iii) under the hypothesis that 3Ms is invertible. Closely related results are already in the literature. A statement nearly equivalent to (2.6)(ii) is proved in Muir [13], ¶148, p. 132. T h e inclusion is established in [4], T h m . 6.1, p. 1540 under the hypotheses (1) t h a t the base is of characteristic 0 and (2) that P is a pair of bundles of a r b i t r a r y rank; in fact, the inclusion is generalized to the case of a product of several ~ ' s . It is also pointed out there that, in characteristic p > 0, this result may be false unless p is suitably large. On the other hand, it is possible under the hypotheses (1) and (2) to show t h a t the bilinear map i,y,s

i

y+8

(1,0): is surjective; in fact, there is an explicit formula for the inverse map, and the denominators are integers whose primes are ~ s + j . Hence, the proof of (2.8)(iii) works as well under these hypotheses. T h e special case where P is a pair of bundles of rank 2s, and i : = s - 1, and j := 1 is closely related to what is called Redei's identity in [1], art. 8, exer. 25, p. AIII.196. DEFINITION ( 2 . 1 0 ) . I ~ t u: P --~ L be a form such t h a t )v[8 is invertible, s > 0. Set J ~ - i := £ - 1 . Then, denote by 2"~(u) o r / 8 the following ideal, see (2.8)(iii):

2", : = ( ~ _ ~ ~t~+,) ® ~,l~ -2 = n~-~ ® 2"m(~,_, ® n,+, ~

n , ® n,).

Moreover, denote by Vs(u) or V~ the subscherne whose ideal is 2"8; in other words, V~ is the scheme of zeros of the map m. LEMMA ( 2 . 1 1). Let s _ 1 and j > 0. Let u: P --+ /2 be a form such that ~8 is invertible. Let Q be a subpair of P of bundles of rank s. Consider v := u[Q. Assume $4~(v) = $18(u). Set Q ' : = K e r h ( u , Q ) . Then:

(i) P = q ~ q'. (ii) When restricted to Vs, the following diagram commutes: A ~+j p

- -

As+J p

u*+'/A A~Q®NQ

'

,

~+j

where the left map is the projection arising from the decomposition in (i).

276

PROOF: Set v' := ulQ'. By (2.7)(i), u = v @ v'. In particular, (i) holds. Moreover, as in the proof of (2.7)(ii),

~+~

~G0[ "-~

J+~

]

Fix 1 1 and j > O. Let ~ be a sheaf and ~' a subsheaf such that ~ / ~ ' is a bundle of r a n k s. Let p: ~ --* (~ / ~ ' ) be the canonical surjection. Then: (i) There ex/sts a unique map Ve,c, maldng this diagram commute:

A ~ ~ ® A j E'

A ,

IA' l

A~+j

iv,.

A~(~/~ ') ® A j ~, _ _

A~(~/~,) ® A s ~,

(ii) Vc,c, is surjective. (iii) I f ~ ---- ~' • ( ~ / ~'), then V~,e, is s i m p l y the corresponding projection. PROOF: Because ~ / ~ ' is a bundle of rank s, the horizontal map A is surjective. Hence, VE,c, is unique if it exists. Therefore, its existence may be checked locally. However, locally, ,¢ --- ~" O (~'/~'), because ~ / £ ' is a bundle of rank s. Moreover, whenever £ "~ £ ' @ ( £ / C ' ) , then r e , e , may be defined as the corresponding projection. PROPOSITION ( 2 . 1 3 ) . L e t s > 1 and j >_0. Let u : P --+ £. be a form such that JM~ is invertible. Set V := V~ and K := K e r h ( u ~ [ V , P [ V ) and R := ( P I V ) / K . Then: (i) R is a pair of bundles of rank s. (ii) u s induces an isomorphism, (u~[V)R: (A ~ R) ® ~ At,IV. (iii) There ex/sts a unique m a p n rnaldng the following diagram commute:

A.R@Ai K (~'rVo)R®l)qs@(A3K) ® Tv~-,v., K A "+j Prv

~~'+' r~',

~s+j Iv

277

Moreover, n is surjective. (iv) Let Q be a subpair of P of bundles of rank s. Consider v := u[Q. Assume Ats(v) = ~ s ( u ) . Set Q' := K e r h ( u , Q ) . Then (a) K = Q'[V and R = Q[V and (b) the following diagram is commutative:

(A s QlV)®

(Aj Q'IV)®

® (Aj K)®

.M~+~IV

~,+llV

Moreover, the m a p at the top, ( u s I V ) R ® 1, is an isomorphism. (v) T h e m a p n induces a surje~tion: ~ 8 - 1 ® K --* ~ s ® (Zs/Z~).

PROOF: The four assertions are clearly local; so by (1.19) we may assume t h a t there exists a Q as described in (iv). Then P = Q • Q' by (1.18)(iv). Clearly, uS: (A 8 Q)® -~ ~8 is an isomorphism. Let p : P{V --* Q[V be the projection, and consider the canonical form, 8

A(qlv) (A(qlv))® associated to Q{V, a pair of bundles of rank 8. Then (2.11) with j := 0 yields K = K e r h ( w p , PIV). Clearly K e r h ( w , QIV) = o. Now, w obviously factors through WQlV0. So, by (1.8)(ii), Q ' = K e r h ( w , P [ V ) . Hence, (iv)(a) holds. Consequently, (i) and (ii) hold. By (iv)(a) and (ii), the map at the top in (iii) is surjective. By (2.13)(ii), the map at the left is surjective. Hence the map n is unique if it exists. By (iv)(a) and (2.12)(iii), the triangle in (2.11) restricts to the lower left half of the diagram in (iii). By (iv)(a), the other half is equal to the triangle in (iv). This triangle may be used to define n, because the map at the top is an isomorphism. Since u*+JA is surjective, so is n. Taking j := 1 and twisting n yields this map: 1 ® (.M: 1 ® n) : 9Ms-1 ® K -~ (J~s--1 @ ~ s + l @ J~-l)lv" Finally, the target maps canonically onto (Ats ® !~)[V by the definition (2.10) of Is. 3.

DIVISORIAL FORMS

DEFINITION ( 3 . 1 ) . Let u: P --~ £ be a form, and r > 0. Then u wilt be called r-divisoriat if its minors, see (1.16), ~ 1 , . . . , J~r are invertible. LEMMA ( 3 . 2 ) . (i) Let u: P -~ £ be a I-divisorial form. Then ~ i ( u 1) = Hi(u) for i > O, and u l : P --* ~ 1 is r-divisorial i f f u is. (ii) Let u: P -* £ be an r-divisoriM form, T / S . Then the pullback uT: P T --* £ T is r-divisorial iff, for 1 < i _ 0. By an r - f l a g / P will be meant a pair :---- ({Qi}, q) where {Qi} is an increasing sequence of pairs (0,0) = Q0 c Q1 c . . -

c Qr

suc~ t h a t each quotient Q i / Q i - 1 is a pair of line bundles and where q is a m a p of pairs q : Q~ -+ P . Let ¢ be an r-flag. A form u: P ~ £ will be called ¢~-split if (i) there exists a direct s u m decomposition (3.3.1)

P = P~ @ " " @ P r @ P r + l

such that q induces an isomorphism (3.3.2)

Qi

~,~ ( P l @ ' " @ P i )

for l < i < r

and such that the restrictions vi := u]P~ furnish a direct sum decomposition U :

V1

(~''"

(~)V r

(~Vr+ 1

and (ii) there exist a chain of "linking" maps (3.3.3) and a '%railing" form (3.3.4)

v': P r + l --~ P r~

such t h a t each restriction vi : P i --~ 1~ is the form vi: P i --~ ~ associated to the composition P ~ - ~ - . - --~ P ~ --, L. LEMMA ( 3 . 4 ) .

Let ¢ be an r - n a g / P , a n d u : P -~ L a ~-sp//t form. Then:

(i) For 1 < i < r, (3.3.2) induces an isomorphism of pairs (Qi/Q~-I)

~'

Pi

and the sheaf P/@ is invertible. (ii) The tensor product of the vi factors through (3.4.1)

vl ® - . - ® vi: P ~ ® . . . ® P ~ --~

A

p

P: __~ f ® i

if 1 < i < r

and its image is equal to Ali. Similarly, there is a factorization

(3.4.2) V l ® . . . ® v r ® A v r + l : p ~ . . . ® p ~ ® ( A p r + and its image is equal to Ati. (iii) The following statements are equivalent:

1

__,

p

--* ~,~ if i > r

279 (a) For 1 < i < r, (3.4.1) is injective. (c) For 1 < i < r, ,~ : P ~ -+ L is i,jective. (e) ~ r is invertible.

(b) For i = r, (3.4.1) is injective. (d) A11 the maps of (3.3.3) are injective. (f ) u is r-divisoriM.

(iv) If u is r-divisorial and if 1 < i < r, then JMi = ~ 4 i - l ® P i@ C_ L ®~. Moreover, then the image of the trailing form v' : P~+I -+ P ~ is equal to Ir ® P ~ where ~fr is the ideal of (2.10). (v) If 1 < i < j (d). Thus (iii) holds. Clearly (iv) follows from(ii) and the implication (f) => (a). Finally, (v) follows from (ii) because of (3.3.2). PROPOSITION ( 3 . 5 ) . Let (I) = ({Qi},Q) be an r-flag/P, and u: P --+ • a G-split form that is r-divisorial. Then: (i) The direct sum decomposition (3.3.1) is uniquely determined; in fact, Pi = Kerh(ui-llqQi,qQi-1)

for 1 < i < r,

P r + l -- K e r h ( u r , q Q r ) -

(ii) The lintdng maps in the chain (3.3.3) and the form (3.3.4) are uniquely determined; in fact, P ~ --+ £ is the restriction of u, and for 1 < i < r, the map P~+I -+ P ~ is the unique map whose tensor product with (3.4.1) is equM to the composition,

(3.5.1)

P ~ ® " " ® P ~ - I ® P % l -~ ( A ' P ) ~ --+ )4i = )4i-1 ® P ~ , where the equality is that of (3.4)(iv).

PROOF: (i) The assertion follows immediately from (2.7)(i) applied with u := vi and Q :-- Q i - 1 because of (3.4)(v). (ii) By (3.4)(i), the tensor product of the linking map with (3.4.1) is equal to (3.5.1); it is the unique such map because of the implication (3.4)(iii)(e)=~(a). DEFINITION ( 3 . 6 ) . Let u: P -+ £ be a form, and ¢ an r-flag/P. Define U(C),u) as the maximal open subscheme of S on which all the following maps are surjective (so isomorphism@ uiAiq: (AiQi) ® -* ~ i for 1 < i < r.

280

LEMMA ( 3 . 7 ) . Let u: P --* /~ be an r-divisorial form. Then: (i) Let • be an r-flag/P. Then U(6P, u) is equaJ to the maximal open subscheme U of S such that u[U is (OIU)-split. (ii) Set Q0 := (0, O) and Qi := Q i - 1 @ ( Os, Os) for 1 < i < r. Then given a point s of S and a system of generators for each component of P , there exists an ordered subset o f r dements (e, f ) of the cartesian product A of the two systems such that if q: Qr --+ P denotes the corresponding map and if • = ({Qi}, q), then the open subscheme U := U ( O, u) contains the given point s. PROOF: (i) If ulU is (OIU)-split, then U C U(O, u) by (3.4)(v). Now, to lighten the notation, replace S by U(O, u). Then it remains to prove that u is O-split. Proceed by induction on r. T h e case r = 0 is trivial. Suppose that r k 1. Set P1 := qQ1 and P ' := K e r h ( u , q ) . Set Vl := u]P1 and v' := uIP'. T h e n (1.18)(i) and (iv) yield that q: Q1 ~ P is injective and left invertible and t h a t u = viGv'. Since u q : Q1 ~ -Ml(u) is surjective, ~ l ( V l ) = Jql(U); hence, Vl: P ~ ~ ~41(u) is an isomorphism. Obviously, ~ l ( v ' ) C JV[l(U). Hence, v': (P')® --+ £ factors through P ~ . Set Q~ := Q i + I / Q 1 for 1 < i < r and 12' : = P ~ . Let u ' : P ' --* /Y be the form induced by v' and let q': Q'r ~ P ' be the map induced by q. It follows from (2.7)(ii) that u i' Ai q: A t Q~., --+ .Mi(u') is surjective for 1 < i < r - 1. So, by induction, u' is ({Q~}, q')-split. It follows that u is O-split. (ii) T h e case r = 0 is trivial. Proceeding by induction on r, suppose t h a t r > 1. Since )vtl is invertible, there exists a pair (e, f ) in A such that u(e ® f ) generates )Vtl at the given point s. T h e n (e, f ) defines a map q l : Q1 --~ P such t h a t if O1 := ({Q0, Q1}, q l ) , then U1 := U(Ol, u) contains s. Replace S by U1. T h e n u is 01-split by (i), and (3.4)(ii) with r := 1 yields (3.7.1)

~l+y(u) = ~1(vl) ~i(v2)

for j > 0.

Hence, v 2 : P 2 --*/~ is (r - 1)-divisorial. Consider the images of the given systems of generators under the projection P --* P 2 associated to the decomposition (3.3.1) with r := 1. By induction, there exists an ordered subset of r - 1 d e m e n t s of A such t h a t if q ' : Q r - 1 --+ P 2 denotes the corresponding map and if O' := ({Qi}i=o, q ' ), then V := U(O', q'), u) contains s. "-' Replace S by V. T h e n (3.6) yields (3.7.2)

M i ( v 2 [ q ' q i ) = Ny(v2)

for 1 _ 1, define a scheme Bi = B~(u) and a m a p bi = b~(u) inductively as follows:

hi: Bi.

--+ B i - 1

is the blowing-up of B i - 1 along the scheme of zeros of PROPOSITION ( 3 . 9 ) .

A uIB _.

Let u: P --* £ be a form, and r ~ 1. Then:

(i) The pullback u[Br is r-divisorial. (ii) For i < r, the formation of N i commutes with pullback along br;

.Mi(uIB, ) = b:.Mi(ulB,_l). (iii) If T is any S-scheme such that u[T is r-divisorial, then there exists a unique S-map t: T --+ Br. Moreover, ~(u[T) = t* ~t~(u[B). PROOF: By Definition (1.16), ~ i ----~ i ( u ] B r ) is the image of A i u t B r . In particular then, ~ ® (/~®(-r)tB,) is the ideal of the exceptional divisor of b~. Hence, ~ is invertible. Proceeding b y induction on r, assume t h a t u l B r _ l is (r - 1)-divisoria]. T h e n the pullback to Br of the inclusion, ) q i ( u l B , _ l ) --+ I:®~IB,_I, is still injective b y the following lemma, (3.10), applied with b~ as b, the complement of the exceptional divisor as C, the inclusion as c. Hence (3.2)(ii) yields (i) and (ii). Assertion (iii) results by induction on r from the following well-known fact about the blowup B of an ideal I on S: if I . OT is invertible, then (a) there exists a unique map T -+ B and (b) t * ( I . OB) = I . OT. (Assertion (a) follows from the characterization of the T-points of a P r o j , and (b) holds as the map t * ( ! . OB) --+ I . OT is surjective, so an isomorphism.) LEMMA ( 3 . 1 0 ) . Let b: B ---+ A be a m a p of schemes, and v: ~ --~ )4 an injective map of quasi-coherent sheaves on A such that fl{ is locally free. Suppose that there exists a map c : C -~ B such that OB --+ c. Oc is injective and such that C / A is flat. Then the pu11-back VB : ~ B --+ )¢s is injective too. PROOF: Consider the following commutative diagram: .~[B

t XIB

, c.(NIC)

l , c,(XiC).

Since ~ is locally free, the map at the top is injective. Since v is injective and C / A is flat, v i e is injective; hence, c.(vlC), the map on the right, is injective. Therefore, the map on the left v l B is injective.

282 COROLLARY ( 3 . 1 1 ) . Let u: P -~ C be a form, and r > O. Consider a base-change map, T ~ S. Then there exists a unique m a p

B , ( u I T ) --* B , ( u ) x T

(3.11.1)

and it is an isomorphism iff ulBr(u ) x T is r-divisorial.

PROOF: By (3.9)(i), the pullback ulB~(ulT ) is r-divisorial. So, the map (3.11.1) exists and is unique by (3.9)(iii). By the same token, it has an inverse if utB~(u ) × T is r- divisorial. The converse is trivial, because ulB~(ulT) is r-divisorial. PROPOSITION ( 3 . 1 2 ) .

Let u: P --* L be a form, r >_ O. Set B~ := B~ (u) and

S, := P(P®) × P((A2P) ®) × . . . × P((ArP)®). Then: (i) Br is canonically embedded as a dosed subscheme of St. Moreover, for 1 < i < r, ifpi: Br -* ~ ( ( A i P ) ®) denotes the projection, then Jqi(utBr ) : p*0(1). (ii) Denote by U (resp. U') the (maximal) open subscheme o r s on which the m a p A~u: (A~P)® --. £®~ is surjective for 1 < i < r (resp. for i = r), by f : U -* Sr the S-map defined by these surjections, and by U~ (resp. U~r) the (maximal) open subscheme of Br on which i i u[Br is surjective for 1 < i < r (resp. for i = r). Then U = U ~ and Ur = U~r. Moreover, Br is equal to the closure in Sr of f ( U ) , and f : U ~ , Ur. PROOF: (i) T h e assertions are obvious from the definition, (3.8), of Br. (ii) Trivially, U C U' and U~ C U~; the opposite inclusions hold by Laplace expansion. Now, obviously U = Br(u[U) and Ur = B r ( u ) × s U ; hence, (3.11) f : U - , Ur. Finally, B~ - U~ is a divisor in Br: it is the scheme of zeros of A r ulB~, so its ideal is equal to ~ r ( u [ B ~ ) ® (LIBr) ®-~. However, the complement of a divisor is always a dense subscheme: if an ideal vazaishes on the complement, then it must vanish everywhere, because, locally, restriction to the complement is given algebraically by localization with respect to a regular element. Thus, B~ is equal to the closure of U~ in Br, so in S~. PROPOSITION ( 3 . 1 3 ) . L e t u : P ~ L be a form, a n d r > O. Thenbr+l: B r + l ---~B~ is equal to the blowing-up of Br along the subscheme Vr = V r ( u [ B r ) introduced in (2.10). Moreover, in the notation of (3.12),

O(b~_~lVr ) = p*_l 0 ( - 1 ) ®pr 0(2) ®Pr+10(--1)

fir:>2

= p;0(2) ® p;0(--1)

ffr=l

---- L ®p~0(--X)

i f r = O.

PROOF: B y (3.9)(i), u l B r is r-divisorial; so Vr is well-defined. Set Jvii := ~ i ( u I B r ). By (2.8)(iv) with j := 1, . ~ _ ~ J~+~ c .M~ .M~ c £®2~.

283 Hence, if x: .~r-1 ~ follows: (3.13.1)

~®(r-1)IBr denotes the inclusion, then x @ ( A ' u ) factors as

~ r - 1 ~ ( A r ÷ l P[B~)

~~r-1 @ J~+l

'~ ' ~

@ Mr

' L ®::r.

Since ~ r - 1 and ~ are invertible, it follows that b~+l: B~+I -+ B~ is equal to the blowing-up of the scheme of zeros of the composition of the first two maps of (3.13.1). However, this scheme of zeros is obviously equal to V~ in view of the definition of Vr in (2.10). Moreover, the last assertion follows immediately from this same definition and from (3.9)(ii). DEFINITION ( 3 . 1 4 ) . Let F be an arbitrazy sheaf, and £ a bundle. Let H o r n ( F , $) denote the scheme representing the functor whose value at T / S is the set of maps, FT --* ET. Thus, H o r n ( F , ~) = Spec( S ym( F @ ~*) ), and it also represents the functor whose T-points are the maps, (~; @ $*)T --~ OT. DEFINITION ( 3 . 1 5 ) . Let P be a pair, • = ({Qi},q) an r-flag/P. Define Split(O) as the scheme representing the functor whose value at T / S is the set of sequences (P, P l , . . . , Pr) where p: P T ~ QT is a right inverse of qT and Pi : Qi,T --~ Q i - I , T is a right inverse of the inclusion. In other words, a T-point is a direct s u m decomposition P T = P1 $ " " ~ P r @ P r + l such t h a t qT induces an isomorphism from Qi,T onto P1 @" "" @ P i . Let L be a line bundle. Set Ei := ( Q i / Q ~ - I ) ® for 1 min{b + j, t}.) For convenience, whenever it makes sense, set

c'; :-- c L DEFINITION ( 4 . 5 ) .

For an r-div:isorial form u: P --+ £, define/~i = l'i(u) by Li := ~4i@N~_l:

for 0 < i < r.

Denote by ~. = / ~ . ( u ) the following chain of inclusions, which exists by (2.8)(iii): L. := {£ =

Lo ~ £1

~...

~

£,}.

Call it the chain of linking maps of the r-divisorial form u. Also, denote by L . - = /~.-(u), the chain Z:.- :=

and call it the truncated chain of u.

{L:

~-..

~

~.}

289

For 0 < s < r and any j > 0, define a form A~ = A~ (u), called the sth modified j t h exterior power of u, as the form J

At: A P composed of the following maps: the (surjective) bilinear m a p u j : A j P --+ Ny and either the injective map, which exists by (2.8)(iv) and (4.4), ~j ~

~ 8 ® (•8) ®U-8) = L I ® . . . ®1:8® (/~8) ®(j-8) : L~

when j _> s

or the identity maps, .My=£~::£~

whenj<s.

DEFINITION ( 4 . 6 ) . Let u : P --~ £ be an r-complete form. Let a: S --+ B r ( P , •) be the unique section such that a* (w) : u, where w is the canonical form; see (4.3). Define the chain £. = / : . ( u ) and, for 0 < s < r and any j >_ s, the sth modified j t h exterior power A~ : A~(u) of u as the pullbacks of those of the r-divisorial form w defining w , L.(u) := a* (~.(w)) and At (u) : : °* G (w). Similarly, given a projectively r-complete form u on P with r > 1, define the chain L. : L.(u), and for 1 < s < r and any j , the sth modified j t h exterior power A~ : A{ (u) of u as the pullbacks £.(u) := a* ( £ . - ( w ) )

and

A~ ( u ) : = a* At (w).

where cr is the corresponding section of B r ( P ) / S and w is the canonical r-divisorial form. LEMMA ( 4 . 7 ) . Let u : P --* L be an r-complete form, s a y u = (u, u l , . . . , u r ) . 0<s

(B) Let T : T --~ S be a map, and suppose t h a t the forms u and UT are r-divisoriaL Then T*~.(u) = ~.(UT) and **A!(u) = AI(UT) for ~1 s and j. (e) I f u is ~ e r i o r , then L.(u) = L.(~) and Ai(~) = A~(u) for ~ l s and j. (D) The formation of £. (u) commutes with arbitrary base change, and so does the formation of each A~(u). (E) If u is exterior and if u is ~-split where • is an r-flag~P, then the chain of linking m a p s / ~ . ( u ) : L.(u) is equal to the chain (3.3.3). Similarly, given a projectively r-complete form u on P with r > 1, the assertions corresponding to (A) and (C)-(E) hold. PROOF: Assertion (A) follows from the definitions, (4.4)-(4.6). Assertion (B) follows from (3.2)(ii) and the definitions. Assertion (C) follows from (B) and the definitions. Assertion (D) follows from (B), (3.20) and the definitions. Assertion (E) follows from (4.5) and (3.5)(ii). The proof in the projective case is similar.

290

PROPOSITION ( 4 . 8 ) . Let r _> 1. Then: (1) Let u : P -+ £ be an r-complete form, say u = (U, U l , . . . , u ~ ) . Then u - :-( U l , . . . , ur) is a projectively r-complete form on P . Moreover, the Ls(u), eY(u), and A{(u) yield the £ 8 ( u - ) , e3(u-), and A{(u-) by truncation. (2) Let u :---(ul,...,ur) be a projectively r-complete form on P. Let £ be a line bundle, and a : .M 1 --+ £ an arbitrary map. Set u + : = (aUl,ttl,...,Ur). Then u + : P -+ £ is an r-complete form. (3) Let P be a pair, £ a line bundIe. Let w be the canonical form on B r ( P ) . Then:

(a) There is a canonical c o m m u t a t i v e diagram, in which a is the structure map: ttorn(A41(w), £1Br(P))

, B r ( P , £)

°l

l

Br(P)

B,(P).

(b) If say w = ( W l , . . . , wr) and i f a is the tautological m a p on the H o r n , then w + := (awl, W l , . . . , w~) corresponds to the canonical form on B r ( P , ~). (c) T h e O-section o f a yields the embedding of B~(P) in B r ( P , £) of (3.18)(A). PROOF: Assertions (1) and (2) are local. So in view of the definitions, we may assume t h a t the given forms are exterior. Then (1) follows from (4.7)(B) and its analogue for projectively r-complete forms. As to (2), consider H := t t o m ( N 1 , £) and the section a of H i S defined by a. Since H i S is flat, the pullback u g is exterior too by (3.2). Replacing u and a by UH and the tautological map, we m a y assume in addition t h a t a is injective. Then aul is obviously r-divisorial. So u is an exterior r-complete form. Assertions (3)(a) and (b) are a restatement in the language of schemes of the functorial version of (1) and (2) because of (4.3). Finally, (c) is a trivial consequence of (a) and (b) and (3.9)(iii). PROPOSITION ( 4 . 9 ) .

Let u: P --+ f be an r-complete form, u = u, u l , . . . , Ur) say.

Let O < t < s < a < r and j >__O. Then: (1) T h e following diagram is commutative:

A jP

A ~P

L,;(u) 1. Then,

< r.

PROOF: (A) The implications (ii)=>(i) and (iii)=>(ii) follow from (4.11). The implication (i)~(iii) is trivial: the sources and targets of the linking maps are submodules of ~ = G0. (B) The proofs are similar. Alternatively, the assertions may be derived from what was just proved by using (4.8)(1) and (2) with L := 511 and a := 1. DEFINITION ( 4 . 1 3 ) . Let P be a pair of bundles of rank r _> 1. In accordance with the convention of componentwise operations of (1.1), let d e t - l P denote the pair of line bundles whose components are the inverses of the r t h exterior powers of the components of P , and let P* denote the pair of bundles whose components axe the duals of the components of P. Note the following canonical identifications: r--1

P*=det-lp®

A P

and

P** = P .

Let /I be a llne bundle. Define a line bundle/~t = L t ( p ) by L t := ( d e t - l p ) ® ® L®(r-1). For convenience, set L t t := ( £ t ) t ( p * ) = ( £ t ( p ) ) t ( p * ) . Let u : P --+ £ be an arbitrary form. Define the adjugate form ut: P* --+ L ? by r--1

l~Ar_l~

ut : P* = d e t - l P ® A P

~ /~t = ( d e t - l p ) ® ® £®(r-1).

Define the d e t e r m i n a n t of u as the linear map, det(u) : £ - 1 __+/~t, determined by ( d e t P ) ® ® d e t ( u ) : ( d e t P ) ® ® g -1 A " ~ ® l ~ ( d e t P ) ® ® / : t = / ~ ® ~ ® L -1. PROPOSITION ( 4 . 1 4 ) .

A s s u m e t h e c o n d i t i o n s of (4.13). T h e n :

(A) For 1 < i < r, t h e folIowing d i a g r a m is c o m m u t a t i v e : A r-i p

d e t P ® A / P*

At- i~/.i ~e(r--1)

~) ( ~ - - 1 ) ~ ( i - - 1 )

l~(det

u)®({-l)

i

lia / u ?

) (det p)® ® L? ® (£t)®(i-1).

293

(B) Suppose r > 2, and consider the map c : £ --* ~ t t defined by l®(det u) ®('-2) £tt

c: £ = •®(r-D ® (£-1)®(r-2)

( d e t P ) ® ® Lt ® (£t)®(r-2)

Then, for i > O, the following diagram is commutative:

A i p -

i ^ ~ t'

^'~

£®i

A~p**

c® i

~ (£tt)®.

PROOF: (A) The question is local, so we may assume that the components of P are free and that L is free. Choose bases and let M be the matrix of u. Then clearly, in the corresponding bases, the matrix of u? is the adjugate of M, and the matrix of det u is the 1 by 1 matrix on the determinant of M. Now, denote by M[q the matrix indexed by pairs (I, J ) of subsets of { 1 , . . . , r} with i elements, whose (I, J ) t h entry is the cofactor of the corresponding i by i minor. In this notation the adjugate of M is simply M[q. Hence, when (A) is expressed in terms of the ith exterior powers of the dual bases of the components of P*, it amounts to this old fact (cf. [13], ¶175, p. 166): i

(4.14.1)

(det M)~-IM[q = A M [ q "

To prove (4.14.1), note that Laplace expansion of det M yields this: i

A Mtr M[q = (det M ) I . Taking i := 1 here and then taking ith exterior powers yields this: i i A Mtr A MIll = (det M ) i I . Now (4.14.1) follows, first for matrices with generic entries and then for all matrices. (B) It obviously suffices to treat the case that i : 1. To treat it, use (A) with i : : r -- 1 and again with P :-- P* and i : : 1, getting: P

£

d e t P ® At-1 p *

l~(det u)@(r-2)

- -

p**

1

, ( d e t P ) ® ® £t ® (£t)®(r-2) ____, £tt.

294

COROLLARY ( 4 . 1 5 ) . Under the conditions of (4.13), assume that r k 2. Then: (A) T h e following conditions are equivalent: (/) u is regular. (iv) A i r ( u ) i s invertible.

Oi) u has rank r. (v) Aru is injective.

(iii) det u is injective.

Moreover, the conditions are satisfied iff the adjugate form u t satisfies the analogous conditions, (i)t-(v)t. (B) A s s u m e the equivalent conditions of (A) satisfied. Then the map c of (4.14) is injective and, for 1 < i < r, the m a p (det u) ®(~-I) induces an isomorphism,

~r-~(u)

~* ( d e t P ) ® ® ~ ( u t ) ,

and the m a p c ®~ induces an isomorphism,

(c)

The form u is r-divisorial iff the adjugate form u t is r-divisorial. If u is rdivisorial, then the equiwalent conditions of (A) are satisfied, and the map det u: E -1 ---* £ t induces an isomorphism,

- , L.-(ut), where ( £ . - ( u ) ) * is the chain obtained from L . - ( u ) by dualizing all the terms

PROOF: (A) The equivalence of (i) and (ii) is clear from the definitions, (1.7) and (1.10). The equiva]ence ~f (ii), (iii), (iv) and (v) holds by (1.17). Now, (4.14)(A) with i :---~r yields 1 ® (det u) ® ( - 1 ) = Aru t. Hence the equivalence of (iii) and (v) t follows, because r > 1. (B) The assertions are immediate consequences of (4.14)(A) and (4.14)(13). (C) The assertions are immediate consequences of (A) and (13). DEFINITION ( 4 . 1 6 ) . Let P be a pair of bundles of rank r :> 1, and u = ( u ~ , . . . , ur) a proj ectively r-complete form on P . The dual form of u is the sequence fl = (~ 1, • • •, fir) consisting of the following surjective forms (or more correctly, of their associated

quotients of (A P*)®): i

r--i

~: AP* = (det-lP) ® R e THEOREM ( 4 . 1 7 ) .

( d e t - X p ) ® ® .Mr-i(u),

1 < i < r.

Let P be a pair of bundles of rank r ~ 1. Then:

(A) I f u is a projectively r-complete form on P , then the dual form, f3, is a projectively r-complete form on P*, and its chain is given by

-- (L.(u))*.

295

Moreover, the double dual is equal to the original form:

~lzU. (B) / f u is the exterior projectively r-complete form on u: P --+ f., then f2 is the exterior projectively r-complete form on u ? : P* -+ £ t . (C) There are three canonical isomorphisms of schemes: B ~ - I ( P ) = B r ( P ) = B~(P*) = B ~ - I ( P * ) . PROOF: Assertion (B) is immediate from (4.14)(A). The first two assertions of (A) follow from (B) and from (4.15)(C), if u is exterior; whence, by pullback, they hold in general. The third assertion of (A) follows immediately from the definition, (4.16). Finally, in (C) the middle isomorphism exists by (4.3) and by (A) applied to P T for an arbitrary T / S . The extreme isomorphisms exist by the following proposition. PROPOSITION ( 4 . 1 8 ) .

Let P be a pair of bundles of rank r _> 1. T h e n the following

conditions are equivalent and valid:

(a) T h e structure m a p b~ is an isomorphism: br: B~(P) ~ , B ~ - I ( P ) . (b) T h e canonical (r - 1)-divisorial form w on B r - I ( P ) is r-divisorial. (c) I f u ---- ( u l , . . . , u ~ - l ) is a projectively r-complete form on P T where T / S is arbitrary and if u~ : A r P T --+ (A r P T ) ® is the canonical form, then the augmented form (Ul . . . . , u ~ - l , Ur) is a projectively r-complete form. SimiIarly, i f ~ is a line bundle, then the corresponding three conditions on B r ( P , L), etc. are equiwalent and valid. PROOF: If (a) holds, then w is obviously equal to the canonical r-divisorial form on B, (P). Conversely, if (b) holds, then b~ is the blowing-up of an invertible ideal by (3.13); whence, (a) holds. Finally, (a) and (c) are equivalent by (4.3). To verify the conditions, we may obviously work locally. So we m a y assume that both components of P are free on fixed bases. Then (3.18)(C) implies t h a t B , - I ( P ) is covered by the open subschemes W(q~) as q~ ranges over the (r - 1)-flags/P defined by the various orderings of the basis. For each such q~, the components of C o k ( q ) are line bundles. Apply (3.4) to the form w on B r - l ( P ) . The map (3.4.2) with r := r - 1 is injective by (3.16)(8), analogue of (ii). Hence (3.4)(ii) implies that ~ is invertible. Thus (b) is valid. Finally, the corresponding conditions on B~ (P,/~), etc. may be treated similarly. 5.

SPLICING AND STRINGING

DEFINITION ( 5 . 1 ) . Let v : Q -+ £ be an s-complete form on a pair of bundles of rank s _> 0. Let v ' : Q' --+ /28(v) be a t-complete form. Set r := s + t. Using the notation of (4.6), set •i := Li(v) :=

for 0 < i < 8 and f o r s < i < r := s + t .

296

F o r m the chain (5.1.1)

£ =

Go ~

Z ~ ~ - ... ~

£. +- Z.+~

+--... + - £ .

in which the first s m a p s are the linking m a p s of v and the remaining t m a p s are the linking m a p s of v ' . Consider the corresponding modified tensor powers/2~,¢ azid L~ = L i1,c and the m a p s e i of (4.4). For 0 < c < s + t , 0 < k < s, a n d j > 0, construct forms

A~,J: A k q ® A j Q' -~ L~ +i out of the modified exterior powers Ai(~) and Ai(~') of (4.6) as follows: Ak Q ® Ay Q '

A~(v)®4(~,)

k

3

~®e

3

.k+j

respectively, if c < k, k < c < s, or s ~_ c. Set P : = Q G Q ' and define a f o r m A~: A ~P -~ L~ as the direct s u m over k, j for k ÷ j = i of the forms A~,j. Set u := A 1 and u~ := A~ for 1 < i < r. Finally, indicate these constructions by saying t h a t the sequence

u := (A 1, 4 , - . . ,

A:)

has been obtained b y splicing the t-complete f o r m v ' o n t o the s-complete f o r m v. THEOREM ( 5 . 2 ) (SPLICING). A s s u m e the conditions of (5.1). Then the spliced sequence u : P --~ £ is an r-complete form. Its chain of linking maps is the cha/n (5.1.1), a n d A~(u) = A~ fo, o < c < r a n d i >_ O.

PROOF: In view of (4.1) and (4.9)(4), it is obviously sufficient to check the assertions locally. So fix a point p E S. By (4.2), there exists a section a of B , ( Q , L ) / S such t h a t v is the pullback of the canonical form. Replacing S by a neighborhood of p, we m a y by (3.18)(C) assume t h a t the image of a lies in an open subscheme W := W ( ¢ , ~) where • = ( { Q i } , q ) is an s - f l a g / Q such t h a t q: Q , ~ , Q. T h e n a is also a section of W / S , and (3.18)(B)(ii) yields v = a * w where w is the canonical f o r m on W. Hence, (3.16) (A) (ii) a n d (3.4)(iv) yield the first two of the following identities, and t h e third holds because a is a section: ~.(v) = ~*~:.(w) = a *(Q,/Qr-i)w ® = (Q,/Q,-1)

®.

Therefore £ s ( w ) = £ s ( v ) w . Similarly, replace S by a suitable neighborhood of p so t h a t v I = a'*w' where a ' is a section of W ' :--- W(@ t, Ls), where w ' is the canonical f o r m on W ' , mud where O' = ({Q~}, q') is a t - f l a g / Q ' .

297

Form T :-- W Xs W'. T h e proof of (3.16)(A)(i)(ii) shows mutatis mutandis that T carries a canonical r-divisoriai form x: P T ---+ f T , because C o k ( q ) ~ ----0. Further reasoning along the same lines shows that the exterior r-complete form x on x is equal to the sequence obtained by splicing w ~ onto WT; the splicing is possible because

Z , ( w ~ ) = L~(w)~ = Ls(~)r. In fact, such reasoning establishes all the assertions in question for x, w T and w ~ . T h e original assertions for u, VT and v~r follow immediately on pulling back Mong the section T :---- (a, a ' ) : S --* T. THEOREM ( 5 . 3 ) (CUTTING). Let u : P -~ £ be an r-complete form. Let 1 < s < r, and let Q be a pair of bundles of r a n k s. Let q: Q --~ P be a m a p of pairs such that the following composition is surjective (so an isomorphism): U

,(A 8 q): (A ~ Q)*

)

, ~s.

(A sP)®

Set t := r - s. Referring to (1.7), set Q' :-- K e r h(us, q). Then:

(i) q is injective and left invertible.

(ii) h ( u , , q) is surjective and right invertible. (iii) P = q Q @ Q'. (iv) Set v :-- (uq, u l q , . . . , u ~ A S q ) . Then v : Q --+ f is an s-complete form, and A~N) = A~(u) A~q for 0 < k < 8 a n d i h O. (v) F o r O < k < t : - - - - r - s and for i >_ O, set pi+s

(~')~ := ~k+~ ® (L:) -~ and define a form (v')~:

A~q'

-~ ( L ' ) [

pi+s

~o,

~2 ~

by

°

M, ® (v'

: A4s ®

'

•

®

Q'

Y-*

P

--+ ,-k+s.

w h e r e z := (u~ AS q ) - i ® (A~Q')® and y :-- A ( q ® 1) and z := A~_Ss(u). Set v ' := ((v')01, ( v ' ) ~ , . . . , (v')~). Then v : Q ' --+ •s is a t-complete form, and

A~(v') ----(v')~¢. Moreover, At8 @ Im(A~(v')) = Im(Ak+,(u)).~+" (vi) T h e given r-complete form u : P --+ L is equal to the one obtained by s p l i d n g v ' onto v . Conversely, i f u was obtained by splicing a t-complete form v ' 1 : Q ' --+ f8 onto an s-complete form V 1 : Q ~

L, then v' = v~ and v ----vl.

PROOF: B y (4.2), there exist an S-scheme B, an exterior r-complete form w on an r-divisorial form w: P B --+ £B, and an S - m a p S --~ B such t h a t u -- w l S . Clearly the image of S in B lies in the open subset of B on which w s ( f 8 q l B ) is surjective. So we may replace B by this open subset. Then (i), (ii) and (iii) hold for w by (1.18). Moreover, (iv), (v) and (vi) for w are easy to check using (2.7) and (1.15)(i). In fact, the two pieces cut from w in (iv) and (v) are the exterior complete forms on w q and wlQ' where the last form may be viewed as a form w': Q' --+ f s by (2.7)(ii)

298

and (2.8)(iii); the surjectivity required by the definition of complete forms holds by (2.7)(iii) and (ii). Moreover, the last assertion of (v) also holds by (2.7)(iii). (Note that, although w = wq @ w[Q', the hypotheses on the two summands are not the same.) Finally, the assertions about u follow immediately on pulling back along the map S -+ B because K e r h ( u s , q) is equal to the pullback of K e r h ( w s , q[B) since h(w~, q[B) is surjective and right invertible by (1.18). COROLLARY ( 5 . 4 ) .

For an r-complete form u: P -+ ~ with r > 0, the following

conditions are equivalent:

(i) T h e modified power, A~+1 : A r+l P --+/~.+1, vanishes. (ii) T h e quotient, P / I K e r h(ur, P ) , is a pa/r of bundles of r a n k r. (iii) There eMst a pa/r R of bundles of rank r, a surjective m a p p : P -+ R , and an r - c o m p l e t e form v : R --+ ~ such that u = v p . Moreover, i f (iii) holds, then K e r ( p ) = K e r h ( u ~ , P ) and v~: (A ~ R) ®

~

~.

PROOF: Assume (iii). Then K e r h ( u ~ , P ) = p - l K e r h ( v ~ , R ) by (1.8)(ii). Now, R is a pair of bundles of rank r; hence, v~ : (A r R) ® ~ ~ ~ . It follows immediately that K e r h ( v r , R) = 0. Hence (ii) and the last assertion hold. To prove t h a t (ii) implies (i) and that (i) implies (iii), it suffices to work locally; indeed, because of the last assertion, the triple (R, p, v) is determined up to unique isomorphism, so a family of local triples yields a global one. Arguing now as in the proof of (1.19), we may assume t h a t there exists a map of pairs q: Q -* P satisfying the hypotheses of (5.3) with s := r. Hence, in the notation of (5.3), P = Q @ Q', and u is equal to the r-complete form obtained by splicing onto the r-complete form v: Q --~ £ where v := u l Q a certain 0-complete form (v') where v': Q' --*/~. Assume (ii). AS K e r h ( u ~ , P ) is a subpair of Q' := K e r h(u~, Q), there is a surjection P / K e r h ( u ~ , P ) --* P / Q ' = Q. It is an isomorhism by virtue of the hypothesis, (ii). Thus K e r h(u~, P ) = Q'. Hence, in particular, ur = A~ vanishes on the subpalr ( A ~ - I Q ) ® Q ' of A~P. Therefore, the following form, see (5.1), vanishes: Arr-l,l

=

r-I

t: A

-IQ®Q'

r-1

By (4.7)(A), L rr-1 - 1 = Jvlr_l. Hence Ar_I(v ) r - 1 = V r _ l and so it is surjective. Therefore v' = 0. Finally, the definition of spliced form, (5.1), yields (i). Assume (i). Then v' = (v')~ = 0, according to the defintion in (5.3)(v). Therefore, Ag(v') is equal to A i v' = 0 for all j 2> 1. Hence, by the definition of splicing (5.1),

u = (uq)p where p: P --+ Q is the projection. Thus (iii) holds, and the proof is complete. COROLLARY ( 5 . 5 ) . Let u: P -~ • be an r-complete form. Let e , f > r 2> 1. Consider the scheme o f zeros, Vr: Arr+l = 0, and its ideal, Ir := 5it-1 ® Jv[r@-2 ® -rm(A~+l). Set :K := Ker(h(u~,P)IVr). Set R := (PtV~)/K. Then: (i) R is a pa/r o f bundles of rank r, and the form ur induces an isomorphism, ®

299

(ii)

There exists a canonical surjective map, (K) ® -~ ( Zr / Z}) ® E,, and it is an isomorphism if ( K ) ® is locally generated by ( e - r ) ( f - r) elements and if

is l o c a l l y f r e e W r a n k

- r ) ( f - r).

(iii) A s s u m e that the components o f P are locally generated by e, resp. f , elements. Then the components of K are locally generated by e - r , resp. f - r , elements, and I, is locally generated by (e - r ) ( f - r) elements. Moreover, if S is locally noetherian, then cod(Vr, S) < (e - r ) ( f - r).

If equality holds and if S is locally Cohen-Macaulay, then Ir is regular, (5.5.1) is an isomorphism, (K) ® is a bundle of rank (e -- r ) ( f -- r), and P ® [Vr is locally free of rank ef. PROOF: (i) T h e assertion holds by (5.4)(i)=~(ii) if r > 1, and it is trivial if r -- 0. (ii) T h e second assertion follows from the first and from (1.4)(iii). T h e first assertion holds in the case t h a t S = B r ( P , £) and u is the canonical exterior r-complete form by (2.13)(v). In the general case, there exists a m a p S --~ B r ( P , L) such t h a t u is the pullback of the canonical r-complete form by (4.2). T h e formation of Vr a n d / ~ r obviously commute with pullback. So, although the formation of the target of the m a p in question does not commute with base change, there is a natural surjection from the pulled-back target onto the native target. Finally, the formation of IK commutes with base change for the following reason. T h e two R ' s are pairs of bundles of rank r by (i). Hence, when pulled back, I~ remains a subpair of P[V~. Obviously, the pulled-back K is contained in the native K . Therefore, these two subpairs are equal, as claimed, because they b o t h define quotients that are pairs of bundles of rank r. Therefore, the special case of the second assertion induces the general case. (iii) Since R is a pair of bundles of rank r by (i), it is locally a direct summand of P[Vr. Hence, the components of • are locally finitely generated, and their fibers are vector spaces of dimensions at most e - r, resp. f - r. So by Nakayama's lemma, the components of 14: are locally generated by e - r, resp. f - r, elements. Obviously, Ir is locally finitely generated; so, by Nakayama's l e m m a and by (ii), I~ is locally generated by (e - r ) ( f - r) elements, because (K) ® is. (In a local ring with maximal ideal M, an idea] I is generated b y g elements if it is finitely generated and I / I 2 is generated by g elements, because I / I M is generated by g elements.) Hence, if I~ is of this codimension and if S is locally Cohen-Macaulay, then Ir is regular. So, then I ~ / I ] is locally free of this rank. Hence, (ii) implies t h a t (K) ® is locally free of rank (e - r ) ( f - r). So P®[Vr is locally free of rank ef. DEFINITION ( 5 . 6 ) . Let R be a pair of bundles of rank s > 0. Let p : P --* R be a surjective map of pairs, and set R ' := K e r ( p ) . Consider an arbitrary s-complete form v = (v, V l , . . . , vs) : R -~/~ and an arbitrary projectively t-complete form v ' = (v~,.. • , V t) ! on R ' ; here possibly t = 0 and v ' is empty. Set Li := ~/(v)

for 0 < i < s and

£~:=Li-,(v')

fors

i--s

h

R@

a'

- 7r- - * £ s~ ® £ 8i -+- s1 , ~ = £ ~ i

ifi, c>s

w h e r e Vp, R, is t h e canonical surjection considered in (2.12), a n d

:=

A s v

s(

i-s

l+

).

Finally, i n d i c a t e these c o n s t r u c t i o n s b y saying t h a t t h e sequence

u := ( 4 , 4 , . . . , A ; ) h a s b e e n o b t a i n e d b y stringing t h e projectively t - c o m p l e t e f o r m v r on R ~ on after t h e s-complete form v. THEOREM ( 5 . 7 )

(STRINGING).

U n d e r t h e c o n d i t i o n s o f (5.6):

(i) T h e s t r u n g s e q u e n c e u : P ~ ~. is an r - c o m p l e t e form. (ii) I t s chain o f l i n k i n g m a p s is t h e chain (5.6.1); in p a r t i c u l a r , i f t ~ 1 ( t h a t is, i f s < r), t h e n its s t h l i n k i n g m a p , £ s + l --~ •s, vanishes. (iii) Ac(u)i _- Ac~ for all i >_ 0 a n d 0 < c < r.

(iv) AS+ltu = 0. 8 K 1 (v) R ' = K e r h(u~, P ) a n d R = P / R ' . (vi) T h e s - c o m p l e t e f o r m v : R --+ £ a n d t h e p r o j e c t i v e l y t - c o m p l e t e f o r m v ' on R ' are d e t e r m i n e d by u a n d t h e factorizations: u = v p , (5.6.2) w i t h 1 _ O. Let u denote the tautological form on H o l n ( P , £), and Z~ +1 the subscheme of zeros of A r + l u . Let G r ( P ) denote the Grassmannian of quotient pairs of P consisting of rank r bundles, and let p: (P[Gr(P)) --. R be the tautological surjection. Then the following pairs consisting of a scheme and a form on it are canonically isomorphic: (i) V := form (ii) Z : = (iii) G := G.

Vr(w) and ~ = uv, where w = ulB~(u) is the canonical r-divisorial on B~(P, f ) ----B~(u) in accordance with (3.17) and (2.10). B~(uIZ~+l ) and u z . B r ( R , f t G ~ ( P ) ) and vpG, where v is the canonical r-divisorial form on

Moreover, under the isomorphism of (i) and (iii), P v / K e r h ( w r l V , P v ) corresponds to RG.

PROOF: The proof of (3.13) effectively shows that V is equal to the scheme of zeros of A~+1(w). Hence, (5.11) with ]c :-- i and sl := r yields the alleged canonical isomorphism between the pairs of (i) and (iii). By (3.9)(i), u z is r-divisorial. Hence, by (3.9)(iii), there is a (unique) H o l n ( P , £)map from Z to B~ (u) such that u z is equal to the pull back of w. P l a i n l y , /^\r + l u z = O. Therefore A~r+l(uz) = 0. Hence, the map factors through a map Z --~ V. T h e form vpG: P G --* LG defines a map from B~(R, £[G~(P)) into t t o m ( P , f ) such t h a t vpG ----UG. This map factors through Zg +1 ; indeed, R is a pair of bundles of rank r, whence A t + i v ----- 0 and so A r + l ( v p G ) ---- 0. Since v is r-divisorial, so is UG = v p a . Hence, by (3.9)(iii), there exists a unique Z~+l-map from G into Z. Clearly, by the uniqueness of the map defined by a canonical form, this m a p and the map from Z to V defined in the preceding paragraph induce the remaining alleged canonical isomorphisms of pairs.

303

COROLLARY (5.13). Under the conditions of (5.12), there is a canonical isomorp h i s m between the blowup B of X := Z~ +I along Z~ and H := H o m ( R ®, £1G'(P)). Moreover, it identifies the form UB and the composite form xpH, where x is the tautological form on H. PROOF: T h e proof is similar to the last part of the proof of (5.12). Clearly, H is an X-scheme via the map defined by the form z p H . T h e n A r UH factors as follows:

(A P-)®

A

(A" R.)® A" &-,.

Since x is the tautological form on H, obviously A r x is injective. Therefore, the universal property of a blowup yields a (unique) m a p f: H --~ 13 such that f*uB = ZpH. O n the other hand, (1.20) implies that u s is of rank r. Hence, by (1.10) and (1.8)(i), there exist a pair R ~ of bundles of rank r and a factorization ( P B ) ® -+ (R~) ® ---* /~B. Therefore, there is a (unique) X - m a p g: B --* H such that g*(XpH ) = u s . Finally, the universality of B implies that f g = 1B, and the universality of H implies t h a t g f = 1H. REMARK ( 5 . 1 4 ) . For projectively r-complete forms with r >_ 1, there is a parallel theory of splicing and cutting and stringing and unstringing. It may be developed by copying the preceding theory mutatis mutandis. However, it is simplier, more elegant and more enlightening to view the projective theory as part of the affme theory by viewing a projectively r-complete form u on P as an r-complete form u + : P --+ /] whose 0th linking map is equal to 0 for any £ and correspondingly by viewing Br (P) as the fiber of B r ( P , £) over the zero section of H o m ( P ®, £); see (4.8).

6.

THE

CONORMAL

ALGEBRA,

THE

EQUATIONS

LEMMA ( 6 . 1 ) . Let £, 3r, £ and 3q be sheaves, • and ~ invertible. Let f : 3r --* £, u: £ --* 3vt and v: £ --* £ be maps. Set A := • ( £ ) , and denote the tautological surjection by a: £A -~ 0A(1). Finally, let V denote the zero scheme of v, and I the ideal of V. Then:

(i) llg(cok(f)) is equal to the zero scheme of the composition afA : ira ~ 0A(1). (ii) Assume that u: £ --+ At is surjective. Then v: £ --+ £ factors through u iff (v

-,

vanishes, where sw is the switch automorphism o f £ ® E. (iii) T h e following three subschernes o f A are equal: (a) Zo(Cok(C -1 ® a)) where d := (v ~ 1)(1 - sw): £ ® £ -~ £ ® £. (b) the scheme of zeros of the composition,

( l ~ a ) d A : ~A_ 1, the following sequence is exact:

(6.2.n)

Symn-l£ ® £ ®£ ® £-1

g"(l®d) SYrune Sum"w

where izn: S y m n - l ® £ -~ Symn£ is the canonical map. PROOF: The sequence in (i)(a) is just (6.2.n) for n = 1. Its exactness means simply I = Cok(d) ® 17-1. For any n, clearly tym"Cok(d) -- Cok(#n(1 ® d)). The assertion follows. REMARK ( 6 . 3 ) . The map d of (6.2)(i)(a) obviously factors like this: --~

£®

---,£

where 0 fits into the beginning of the Kozul complex A2£ ® £_i

a

Hence the condition (6.2)(i)(a) is equivalent to the vanishing of the Kozul )41. It is well-known that, if £ is a bundle (of finite rank) and if S is locally noetherian, then the vanishing of the Kozul gl implies that the symmetric algebra is equal to the Rees algebra, Condition (6.2)(i)(b). The ususal proof is round about and involved; see [2], rein. §9, no. 7, p. 161. However, the conclusion is valid under the weaker hypothesis t h a t £ be flat, and S need not be locally noetherian. A simple direct proof follows. The proof proceeds by induction on n. Set $ y m - 1 ( £ ) :-- 0. Then (6.2.0) is trivially exact for n = 0. For n _> 0, consider the following diagrax~ tyrnn-l £ ® £ ® £ _ _

Syrnn-l ~ ® £ ® £

i Syrnn£ ® £ ® £ ® L -1

Symn+l£

i ~

1

/~®(n+l).

The upper sequence is Sym"£®(6.2.1). Hence it is exact because, by hypothesis, (6.2.1) is exact and £ is flat (so Sym~$ is flat t o o - - £ is locally a filtered direct

306

limit of bundles (of finite rank), whence so is Sym~E). T h e right vertical sequence is (6.2.n)®£}, which may be assumed to be exact by induction. T h e lower sequence is (6.2.(n + 1)). T h e lower squares are clearly commutative. T h e map h is defined on local sections by

h(x@e@f) = (xe)®f - (xf)®e. Obviously the upper square is commutative, and #n+lh 0. Finally, an easy diagram chase shows that the lower sequence, (6.2.(n + 1)) is exact. =

PROPOSITION ( 6 . 4 ) . Let ~ be a sheaf, ~ a line bund/e. Set H := H o m ( ~ , L) and let t: ~ H --+ ~ H be the tautological map. Then t satisfies the equivalent conditions o~ (~.2). PROOF: T h e question is local. So replace ~ by a module G over a ring R, replace £ by R, and replace H by the symmetric algebra A := Sym G. Then t becomes the canonical map, t: A®RG --~ A and (6.2.n) becomes f(1-1®1®8,~) (6.4.n)

A@nSyrnn-lG@nG@nG

, A®RSym~G -~ A

where f(a®x®y®z) = (az)®(xy). Set SymPG := 0 for p < 0. For fixed n, (6.4.n) is obviously the direct sum over p for - o o < p < oo of the sequences

SymP-IG®RSymn-IG®RG®RG

f(1--1®l®sw)

~SymPG®RSymnG ---+SymV+nG.

Each of these sequences is easily seen to be exact. Thus (6.2)(ii) holds. LEMMA ( 6 . 5 ) . Let ~ be a bundle of rank s, and ~' a sheaf. Set C := ~ @ 8'. Let j, k > O. Then the following diagram is commutative:

A" 9 ® A ~ 9' ® A k ~'

1'

1®^, A~ E ® A j+k ~"

l°

AkE®Aj~'®A ~9 ~®~, AkE®Aj+~C in which ~ is the m a p defined in (2.1) and f := (1 ® p r ® 1)(ST)(1 ® V)(1 ® A) where pr: A j £ --+ A j ~, is the map induced by the projection, £ --~ ~', and sw is the isomorphism switching the first and third factors. PROOF: Omit the projection, and the diagram is plainly commutative. Now, Ker(pr) is equM to ! m ( A j - 1 ~ ® ~ ~ A j ~'). Moreover, A: A j - 1 ~ ® ~ ® A" ~ -~ A j+8 ~ is 0, because ~ is a bundle of rank s. Hence, the diagram is commutative.

307

DEFINITION ( 6 . 6 ) . Let u: P --+ L be a form such that Ms(u) is invertible. Let Q be a subpair of P such that v :-- uIQ is regular of rank s and such that Ms(v) -- M~(u). Then define two pairs Qt and R and a composite map b -- b(Q) as follows: Q':=Kerh(u,Q)

and

R:=Q'®/~Q

b: Ms-1 ® (R) ® l®(u'+l^)

m

where m is the map of (2.8)(iii). LEMMA ( 6 . 7 ) . (A) Under the conditions of (6.6), the map 1 ® (u 8+1 A) is surjective, and the zero scheme of the map b is equal to the scheme Vr(u) defined in (2.10). (S) Let u: P -+ f~ be an r-divisorial form, and e) _-- ({Qi}, q) an r-flag/P such that u is C-split. Then the hypotheses of (6.6) are satisfied with s := r and Q := qQ~, and in the notation of (6.6) and (3.3), Q' -- P r + l and the following diagram is commutative: b

1®1®v t

Mr_I®M~®P~+~ , M~_~®M~®P~ where the isomorphism on the left is induced by v :---- uIQ and the equality on the right is induced by the equality in (3.4)(iv). (C) Let P be a paJr and 4~ an r-fiag/P. Set W :-- W(~, L), and let w: P w -+ £ w be the canonical ~w-split, r-divisorial form; resp., set W := W ( ~ ) , and use w: P w -+ LI,w. Then the map b := b(qQr]W), which is we11-defined by (B), satisfies the equivalent conditions, (i) and (ii), of (6.2). PROOF: The surjectivity asserted in (A) follows easily from (2.7)(ii). This surjectivity then yields the assertion about Vr (u). As to (B), note that there is a canonical isomorphism, r

A: P I ® - . - ® P ~

~

AQ"

Hence by (3.4)(ii)and (i.18)(i),the hypotheses of (6.6) obtain. By (3.5)0) and (l.18)(iii),Q' -- Pr+1. Finally,to prove the commutativity of the diagram, compose the upper and the leftmaps with the m a p (PIG...®P,-1)®®(PI®...®P,+~)®

~, N , - ~ ® R ®,

which is an isomorphism by (3.4)(i), 0ii), and compose the lower and the right map with the inclusion, Then it is clear that the diagram is commutative in view of (3.4)(ii). Thus (B) is proved. By (B), the map b of (C) is isomorphic to a twist of the map v' of (3.3.4). By (3.15) and (3.16)(A)(i), (ii), v' is isomorphic to the tautological map, Cok(q)@w --+ £r. Hence (C) follows from (6.4).

308

THEOREM ( 6 . 8 ) . Let P be a pair, /Z a line bundle, and r >_ O. Let w denote the canonical r-divisorial form on B ~ ( P , £ ) ; resp. on B~(P) provided r > 1. Consider the ideal I := I~(w) and its variety V := V~(w), which were introduced in (2.10). Set K := K e r h ( w ~ [ V , P I V ) i f r _> 1 and K : = P [ V i f r = O. Then: (i) The symmetric a/gebra of I is equal to its Rees a/gebra. (ii) The map n of (2.13)0ii) induces an isomorphism, (6.8.1)

K®

~ , I/I2®.r.r(w).

PROOF: The statements are local. So we may assume that S is MYme, and so by (3.18) we may replace Br(P,/2) with W := W ( 0 , L), resp. B~(P) with W := W ( 0 ) , where ~5 is an r-flag/P. Consider the map b := b(qQ~lW ) of (6.7)(C). By (6.7)(C) the two equivalent conditions (6.2)(i), (ii) hold. So Assertion (i) holds, because it is equivalent to (6.2)(i)(a) by (6.7)(A). Now, by (6.2)(i)(b), the hypothesis of (6.1)(v) is satisfied; hence, (6.1.1) is an isomorphism. However, (6.1.1) is equal, by (2.13)(iv)(b), to Al~-1®At~®(6.8.1). Thus (ii) holds. LEMMA ( 6 . 9 ) .

Under the conditions of (6.6), the following square is commutative:

(A 8-1 Q ) ® Q ' ® A s Q ® R

c

,

51s ® 5t8 ® R ®

(~.®^®A)swl~ N~®(A~,I,sp) ®

~1®1®^ c' , N , ® 5 % N ( A ~ + x p ) ®

where c := [b(u 8-1 ® 1)]®1 and c' := 1 ® [(us ® 1)(~, 1)] and SW1 switches both the first and third paJrs and the first components of the two Q'. PROOF: Consider a local section of the top left term, (6.9.1)

s -] 8 - I (e 8-1 ® e' ® e 8 ® e ,1 ® el,

® f , ® f 8 ® f l, ® f f ) ,

and carry it both ways around the square as follows. Going clockwise, identify 348 ® 5t8 with its image Als • N8 in E ®28. Then the map m in the definition of b is, by definition, the composition of the natural map, N s - 1 ® ~ + 1 -+ ]q8-1 ~ + 1 and the inclusion, N s - 1 N s + l ~-~ ~sJv[s. As Q' = K e r h ( u , Q), therefore b produces this factor:

uS-l(es-1 ® fa-1) uS+X(e, A e s ® f' A fa) = us-X(e 8-x ® f ~ - l ) u(e' ® f') u~(e ~ ® fs)

Therefore, carrying (6.9.1) around clockwise produces this result: (6.9.2)

uS(e ' A e s-1 ® f ' A f , - l ) ® uS(e 8 @ f s ) ® [e~ A e~ @ f~ A f~].

309 Going counterclockwise, assume, as we may, t h a t the first component of Q is free on e l , . . . , e s and that e~ = e 8 = el A . . . A e ~ and e s-1 = el A . . . Ae~_l. Then, in the notation ~ and C{j} of (2.1),

8

(4

----

' A es - l ) + ~ ( - - 1 ) J e ' A e o u } ® ej A e~ A e8-1.

j=l

So, carrying (6.9.1) around counterclockwise produces a corresponding s u m of s + 1 terms. T h e first is this:

uS(ee®fS)®uS(eS®ftAfs-1)®[etAe

® f t 1Af;]"

tI A e s - 1

It vanishes, because Q ' = K e r h ( u , Q) and so the middle factor vanishes. Of the remaining terms, the j t h is this: (6.9.3) (--1)JuS(e s ® f s ) ® uS(e , A e c u } ® f ' A f s - z ) ® (ej A e i A e S - 1 ) ® ( f ; A f~). It vanishes when j < s - 1, because the third factor is then equal to zero. Now, it is easy to see t h a t (6.9.3) with j = s is equal to (6.9.2), and the proof is complete. LEMMA ( 6 . 1 0 ) .

Under the conditions of (6.6), form these two maps:

d := (b® 1)(1 - 1 ® sw): M8-1 ® R ® ® R ® --+ M8 ® M8 ® R ®

D := (u ® 1)(+ ® 1 - 1 ® +): (A

P)® --, M, ® (A 8+1 P)®

where s w is the switch involution ofR ® R. Consider these two sequences:

d

(6.10.1)

x

M~_~ ® R ® ® R ® --+ M~ ® Ms ® R ® --+ Ms ® M, ® Ms+l A,,,1,sP D

(6.10.2)

Ms ® (A s+l P)®

ley) J~s ® J~s+l

where x :---- 1 ® 1 ® ( u 8+1A) and y := l ® u s+l. Then, in both sequences, the compositions are zero, and the right hand maps are surjective. Moreover, if (6.10.1) is exact, then (6.10.2) is also. PROOF: T h e right hand maps are surjective: the second, by definition; and the first,

by (2.7)(ii). For convenience, set D i :--- (u s ® 1)(0, 1) and D2 :-- (u 8 ® 1)(1, ). T h e n D = D i - D2. Consider the following diagram= A a-i Q ®R®R

d(u'-l®l®l)

i (~"@A@A)SWz Ms ® A 8'1'8 P

l@O,

,

M~®Ms@R ®

x

, Ms®Ms®M~+I

ll@l@A M8 ® M8 ® (A ~+1 P)®

u ' J~8 ® J~.8 ® J~s--}-i

310

The top row is (6.10.1), but with d(u ~-1 @ 1 ® t) in place of d. The bottom row is N8®(6.10.2), and SW1 is the switch defined in (6.9). The diagram is commutative. Indeed, the right square is trivially commutative. As to the left square, note that the map at the top, d ( u S - l ® l ® l ) is equal to c - c ( l ® s w ) . Now, replace the map at the top by" c, and replace 1 ® D by 1 ® Di. Then, the square is that of (6.9), so commutative. Next, replace in addition each pair by its transpose, and then transpose the result. In this way, the original pairs are restored, but D1 and SW1 are replaced by their counterparts, D2 and SW2. Now, SW2(1 ® sw) = SW1, because the additional switch of the two/~8 Q is equal to the identity as As Q is a pair of line bundles. Thus the original square becomes commutative also if d(uS-1®l®1) is replaced by c(1 ® sw) and 1 ® D by 1 ® D2. So, the original square itself is commutative. The composition in (6.10.2) is zero by (2.8)(ii). The composition in (6.10.1) therefore is zero because the diagram is commutative and because u *-i is surjective by (2.7)(iii). Assume (6.10.1) is exact. Then the top row in the above diagram is exact, because u *-1 is surjective. A simple diagram chase shows now that (6.10.2) is exact if I ® I ® A is surjective modulo the image of 1 ® D. So it remains to prove that 1 ® A is surjective modulo the image of D. Consider the following diagrarr~ A~Q®Q, QA*p

(u'®l®l)(f,sw) _ _ ,

N~ ® R ®

i (i®a,(i®A)sw) As,~,~p

l i®A D,

.......

.Ms ® (A ~+~ P)®

)

where f and sw are the maps of (6.5). By (6.5), the diagram is it suffices to prove that the counterclockwise composition, DI(1 surjective modulo the image of D. However, modulo this image, equal to/92 (1 ® A, (1 ® A)sw). By (2.7)(iv), the latter is equal v ~ ® A is obviously surjective.

commuatative. So, ® A, (1 ® A)sw), is this composition is to v 8 ® A. Finally,

PROPOSITION (6. i 1). Let u: P -+ £ be a form such that Ha = ~ 8 ( u ) is invertible, s >_ 1. Consider the following sequence: l@u

(A~,,1,Sp)®

D

, ,~8 @ ( A s+l P)(~

6+i

~' ,,~s @ - ~ s + l

) 0

w h e r e D := (u 8 ® 1 ) ( < > ® 1 - 1®). Then ( l ® u S + l ) D = O, and the sequence is exact if ~ 8 - 1 is invertible, and if, locally, there exJsts a subpalr Q of P such that (a) v := uIQ is regular of rank s, (b) J~s(v) = sMs(u), and (c) the map b of(6.g) satisfies

condition (6.2)(i)(a). PROOF: Both assertions may be checked locally. So by (1.19) we may assume that there exists a Q satisfying the hypotheses of (6.6). Then the sequence in question is just (6.10.2); hence, by (6.10) the composition is zero, and the sequence will be exact if (6.10.1) is.

311

Assume the final hypotheses. Then, locally, the hypotheses of (6.6) are satisfied with the given Q. Moreover, by (c), the following sequence is exact: .Ms-1 ® R @ ® .Ms-1 @ R ® --~ .Ms ® .Ms ® .Ms-1 ® R ® ---* .Ms ® .Ms ® .Ms ® .Ms.

Here, the first map is ( b ® l ) ( 1 - s w ) , and the second is l®b. Now, b = rn(l®(uS+ZA)), and m is injective by (2.8)(iii) because .Ms-I is invertible. So, if the second map l®b is replaced by 1 ® (1 ® (uS+~A)), then the resulting sequence is still exact. The resulting sequence is simply (6.10.1)®.M8-1. Thus (6.10.1) is exact, and the proof is complete. PROPOSITION ( 6 . 1 2 ) . Let u: P --* £ be a form such that .Ms and .Ms-1 are invertible for some s _> 1. Consider the ideal %8 := .Ms+l ® .M~-e ® 5%-1 introduced in (2.10). Set A : = ~ ' ( ( h s+~ P)®) and B := Bt(Zs).

Denote the tautological surjection on A by a: ( A s + I P ) ® [ A ~ 0A(1) and denote by C the scheme of zeros of the map ((uSlA) ® a)( ® 1 -- 1 ® ): (A s'~'s P)®IA --+ (.MslA)(1). Then B is canonically embedded in ~(.Ms+l), and P(.Ms+I) lies in C; in short,

B c_ P(.Ms+l) c_ C. Moreover, these three subschemes of A are equal if (1) the symmetric algebra of Is is equal to its Rees algebra, and (2) locally there exists a subpair Q of P such that (a) v := u[Q is regular of rank s, (b) .Ms(v) = .Ms(u), and (c) the map b of (6.6) satisfies condition (6.2)(i)(a). PROOF: It is evident t h a t B C ~P(Jyts+l) and t h a t equality holds if (1) holds. By (6.1)(i), C = JP(Cok(.M~-I®D)), where D is the map d e f n e d in (6.11). Finally, (6.11) implies t h a t ~(.Ms+l) C_ JP(Cok(.Ms-1 ® A)) and that equality holds if (2) holds. DEFINITION ( 6 . 1 3 ) . Let P be a pair, and i , j >_ O. Given two bilinear maps ui: A i P -+ A/i and ui+j: A i + J P ---* A/i+j, define

A(ui, ui+j) := (ui ® ui+y)(~ ® 1 -- 1 ® ~ ) : ( A i'y'i P)® --+ A/i ® A/i+j. THEOREM ( 6 . 1 4 ) . Let P be a pa/r, £ a line bundle, and r >_ O. Then B~ :---B , ( P , E), see (3.17), is equal to the closed subscheme of St(P, L) := H o r n ( P , £) × P ( P ® ) × . . . x P ( ( A r P ) ~ )

defined by the (bilinear) equations A ( ~ , ~ ) = o, A ( ~ I , , ~ )

= o , . . . , A(~ ,_ ~ , u,) = o

312

where u, u l , . . . , ur are the pullbacks to St(P, £) of the tautological maps; in other words, Br is equal to the zero scheme of the indicated maps. Moreover, (uIBr) / = ui]Br. Furthermore, on Br all of the following equations are satisfied:

A(u, u s ) = O

arid A(us, u t ) = 0

forl<s 0. Set BrsYm(P,/2) := Br(U sym) and BrsYm(P) := Br(aSym). By the canonical forms on B~ym(P,/2) and B~ym(P) will be meant the pullbacks of Usym a n d a sym.

315

THEOREM ( 7 . 1 3 ) .

The parallel symmetric versions of (3.18)-(3.20)

are

valid.

PROOF: The proofs are the same mutatis mutandis. DEFINITION ( 7 . 1 4 ) . By an r-complete s~jmwetv~c form (resp. a projectively rcomplete s~vaetri~ form will be meant an r-complete form (resp. a projectively rcomplete form) that is locally the pullback of the exterior form on an r-divisorial form that is symmetric. THEOREM ( 7 . 1 5 ) .

The parallel symmetric versions of (4.2)-(4.3) are valid.

PROOF: The proofs are the same mutatis mutandis. DEFINITION ( 7 . 1 6 ) . Define the chain and the modified exterior powers of an rcomplete (resp. a projectively r-complete) symmetric form in a parallel fashion to the nonsyrnmetric case, namely, as the pullbacks of those of the canonical symmetric forms (which are exterior). LEMMA ( 7 . 1 7 ) . An r-complete (resp. a projectivety r-complete) symmetric form is also an r-complete (resp. a projectively r-complete) nonsyrm~tric form, and its chain and modified exterior powers are the same either way. PROOF: The first assertion follows from the definitions; the second follows from (4.7)(B). PROPOSITION ( 7 . 1 8 ) .


PROOF: The proofs are the same mutatis mutandis. DEFINITION ( 7 . 1 9 ) . Let P be a symmetric pair, L a line bundle, and r > 0. Denote by symBr(P, £) the largest closed subscheme of Br(P, £) on which each component of its canonical r-complete form is symmetric; in other words, each factors through the canonical map (A~P) ~ ~ ( A i P ) sym for the appropriate i. Define symBr(P) similarly. By the canonical forms on ~ymB~(P,/~) and syrnB~(P) will be meant the restrictions of those on B~(P, L) and B~(P). THEOREM ( 7 . 2 0 ) . Let P be a symmetric pair, L a line bund/e, and r :> 0. Then there are canonical isornorphisrr~, which preserve the canonical forms: B~Ym(P, £)

~* symBr(P, £)

and

B~Ym(P)

~, "YmBr(P).

PROOF: The proofs are similar in the two cases. So consider the second. By (7.17), (7.14) and (4.3), the canonical form on B~ym(p) defines a map from B~ym(P) into Br(P) and its image obviously lies in symBr(P). To prove that the induced map is an isomorphism is a local matter. So if say P -- (~', ~'), then we may assume that E is generated by its global sections. Let a be a geometric point of symBr(P). By (7.8), there exists a symmetric r-flag/P, say ¢, such that a lies in the open subset U(¢) of symBr(P). So it suffices to prove that the restriction of the map to the preimage of U(¢) is an isomorphism onto U(~). Now, V(¢) = W(¢) by (3.18)(i). Hence, the assertion follows from the following lemma.

316

LEMMA ( 7 . 2 1 ) . Let P be a symmetric pair, r 7_ 1. (Resp. let P be a s y m m e t r i c pair, ~ a line bundle, r ~ 0). Let q~ be a symmetric r-flag~P, and let symw(ff>) denote the closed subscheme of W ( ¢ ) where all tile canonical forms i

wi:

APw(~)-~ N~

fori= 1,...,r

are symmetric. (Resp. det~ne ~YmW (4, ~) analogously.) Then symw((~) : wsym((~)

(£esp.

symw((I), ~) = w s y m ( ( I >, .~)).

PROOF: By (3.16)(i), on W ( ¢ ) there is a canonical splitting Pw(~,) = P1 ® "'" G P~ • P~+I. By (3.5)(i) and (1.18)(ii), P i = K e r h ( w i - 1]qi,w(~) , Qi-I,w(~)). By (1.18)(ii), a similar formula holds when the decomposition is pulled-back along an arbitrary T-point of W(~). Hence, if the T-point lies in symw(~), then the splitting is symmetric. Also, by (3.5)(ii) with i := r, the pulled-back form C o k ( q ) T = ( P r + I ) T --~ (/~r)T is symmetric. Thus every T-point of symw(~) is a T-point of WSym(¢). The converse is obvious. Thus the schemes are equal. (Resp. the proof is entirely similar.) COROLLARY (7.22). The canonical form on B r ( P , • ) remains r-divisorial when restricted to symBr(P , £), as does that on B~(P) when restricted to symB~(P). Moreover, ~ymBr(P,/~) = B , ( u ~ym) and ~ymB~(P) = B~(u~ym). PROOF: The assertions result immediately from (7.21) and (7.12). THEOREM ( 7 . 2 3 ) . In (4.13)-(4.17), if the pair P , the form u: P -+ ~ and the projectively r-complete form u on P are symmetric, then so are the dual pair P*, the adjugate form u t : P* --~ L t and the dual projectively r-complete form fi on P*. Moreover, there are canonical isomorphisms, B ysym - 1 (P)

=

S~Ym(P)

~

BrsYm(P*)

=

/ ~ s y m~( p . . )" I ~r--1

PROOF: The first assertion is obvious from the definitions, (4.13) and (4.16). The second now follows from (4.17)(C). THEOREM ( 7 . 2 4 ) (SPLICING). Under the conditions of (5.1), the sequence u formed by splicing a t-complete symmetric form onto an s-complete symmetric form is an (s + t)-complete symmetric form. PROOF: The proof is the same as that of (5.2) mutatis mutandis. Alternatively, observe that u is obviously a sequence of symmetric forms, so the assertion results immediately from (7.20) and (5.2).

317

THEOREM ( 7 . 2 5 )

(CUTTING). The parallel symmetric version of (5.3) is valid.

PROOF: T h e proof is the sazne mutatis mutandis. Alternatively, the assertion results from (7.20) and (5.3), because the forms (v')~ are obviously symmetric. LEMMA ( 7 . 2 6 ) . L e t P be a s y m m e t r i c p a l r , a n d u : P -+/~ an arbitraryr-complete form. Let a be a geometric point such that u(cr) is symmetric. Finally, let 1 < s < r. Then, after S is replaced by a neighborhood of a, there exists a symmetric pair Q of bundles of rank s and a s3qnrnetric m a p q: Q --+ P such that the composition is surjective: us(ASq): (ASQ) ® , (A~P) ® , 51s. PROOF: It follows from (4.2)(b) that we may assume t h a t u is exterior. T h e n the assertion follows from (7.8). REMARK ( 7 . 2 7 ) . T h e o r e m (7.20) is equivalent to the first s t a t e m e n t of Corollary (7.22). Indeed, the latter implies by (7.14) and (7.15) t h a t every T-point of sYmBr(P, L) is a T-point of BrsYm(P,/~), and the converse is trivial. T h e case of sYmBr(P ) and B~sym(P) is similar. T h e o r e m (7.20) also follows from (7.26) for s = 1, (5.3) and (7.24); so it thus has a second proof. Indeed, by (7.19), a T-point of symB~(P,/2) is an r-complete form u = (u, u l , . . . , ur) such that the ui are symmetric. To prove t h a t u is a T-point of B~ym(p,/'-), t h a t is, an r-complete symmetric form, we may work locally. T h e case r ---- 0 is trivial, so assume r >_ 1 and proceed by induction. Then by (7.26) with s ---- 1, we may assume that there exists a syrrgnetric pair Q of bundles of rank 1 and a symmetric map q: Q -+ P T such that Q® --+ 511 is surjective. By (5.3), u is formed b y splicing an (r - 1)-complete form v ' : Q' --+ E1 onto the 1-complete form (U, Ul)lQ: Q --+ £. T h e former is an r - 1-complete symmetric form by induction. T h e latter is obviously a 1-complete symmetric form if the map 5tl -~/~ is injective; in fact, it is the exterior form on the symmetric form u. T h e general case may be reduced to this case by replacing T by I-Iota(511,/~T). Therefore u is an r-complete symmetric form by (7.24). T h e case of symBr(P ) and Brsym(P) is similar. COROLLARY ( 7 . 2 8 ) .

The parallel symmetric version of (5.4) is valid.

PROOF: T h e assertion may be proved the same way mutatis mutandis. Alternatively, it may be derived from the assertion of (5.4). COROLLARY ( 7 . 2 9 ) . Let u: P -* £. be an r-compIete syrmnetric form, r >_ 0 . Let e > r. Consider the zero scheme, Vr : A : +1 : o, and its ideal, tr : = 5 1 r - 1 ~ Jvt? -2 - r r n ( i : + l ) . Set K := Ker(h(u~,P)}V~) and R := (PIV~)/K. Then: (i) R is a symmetric pair of bundles of rank r, and u , induces an isomorphism,

i'n ®

(51,iv,).

(ii) There exists a canonical surjective map, (7.29.1)

(K) ~ym --, ( I r / I ~ ) ® f..r,

318

(e-r+l~ and it is an isomorphism if (K) sym is locally generated by ~ 2 / elements and if It~It ~ is locally free of rank (e-2+l) . (iii) Assume that the (two equal) components of P are locally generated by e elements. Then the components of K are locally generated by e - r elements, and Ir is locally generated by (~-~+1) elements. Moreover, if S is locally noetherian, then

c°d(V~'S) < ( e - r2 q - 1 ) If equality holds and if S is tocally Cohen-Macaulay, then Ir is regular, (5.5.1) is an isomorphism, and (K) ~ m i~ a bundle of rank ? - ; + 1 ) . PROOF: T h e proof is the same as that of (5.5) mutatis mutandis. THEOREM ( 7 . 3 0 ) .

The parallel s~zrunetric vers/ons of (5.7)-(5.14) are valid.

PROOF: T h e proofs are the same mutatis mutandis. Alternatively, the assertion may be derived from these results and (7.20). LEMMA ( 7 . 3 1 ) . The parallel symmetric version of L e m m a (6. 7) holds with the parallel symmetric form of Definition (6.6). PROOF: T h e proof is the same mutatis mutandis. Alternatively, the new versions of (6.7)(A) and (B) m a y be derived from the old ones. THEOREM ( 7 . 3 2 ) . Let P be a pa/r, f a line bundle, and r >_ O. Let w denote the canonical r-divisorial form on B~(P, £). Consider the ideal I := It(w) and its variety v : = v r ( w ) , w h i m were i n t r o d u c e d in (2.10). Set K : = Kerh(w~lV, PtV). Then: (i) The symmetric algebra of I is equal to its Rees Mgebra. (ii) The map n of (2.13) induces an isomorphism, K~m

~, z/z~®£r(w).

PROOF: T h e proof is the same as that of (6.8) mutatis mutandis. PROPOSITION ( 7 . 3 3 ) .


PROOF: T h e proofs are the same mutatis mutandis. THEOREM ( 7 . 3 4 ) . Let P be a pa/r, £ a line bundle, and r _> 0. Then Brsym := Brsym(p, L) is equal to the closed subscheme of

s:Ym(P, L) := H o m ( p " m , L) × ~ ( p , ~ m ) × . . . x ~ ( ( A r P ) ~ m ) , defined by the (bilinear) equations (6.13)

~(~, ~1) = 0, zx(~l, u~) = 0 , . . . , zx(~,_~, ~ ) = 0

319

w h e r e u, u l , . . . , Ur a r e t h e pullbacJ~ to t h e p r o d u c t o f t h e tautological m a p s ; in o t h e r words, B~ ym is equal to t h e zero s c h e m e o f t h e indicated m a p s . Moreover, (u[Bsrym) i = ~ ,.1~]c~sym. r" Finally, on -B- r sym all o f t h e following equations are satisfied: A(u, us)=0

and

A(us, ut)=O

forl<s

Algebraic Geometry Sundance 1986

Sundance

Free Resolutions in Commutative Algebra and Algebraic Geometry: Sundance 90

Algebraic Topology. Barcelona 1986

Algebraic Topology, Barcelona 1986