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
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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<_ 8_< ~ ( d - l ) ( d 8) as a vector space over
•
is:
i). dimS(d, O) = i ii). dimS(d, i) = 2 iii). dimS(d, 2) = 3 iv). dimS(d, 8) = 4
for 3 <_ 8 <_ ~(d-1)(d-2) - 2,
v). d i m S ( d , 8) _> $
for 8 = ~ ( d - l ) ( d - 2 ) -
1, (i.e. g = 1),
vi). d i m S ( d , 8) z 2
for $ = ~ ( d - 1 ) ( d - 2 ) ,
(i.e. g = 0).
Proof: W(d, 0) is IPN with a set of codimension 2 r e m o v e d so clearly d i m S ( d , 0) = i. Next, as above f r o m (1.1)-(1.4) we h a v e span(A, B, C, A}
= span{CU, TN, TR, A}
= span{CU, TN, TR, NL}.
35 For 8 = 1, TN = 0 so d i m S ( d , i ) s 2. For 8-- 2, T R - - 0 so d i m S ( d , 2)_< 3. S ( d , i ) is s p a n n e d b y CU and NL. S(d, 2) is s p a n n e d b y CU, TN, and NL. The m a t r i x t h a t r e p r e s e n t s r S ( d , I) --, S(d, 2) w i t h respect to these s p a n n i n g sets is: 0
ii:l w h i c h has r a n k 2. We conclude t h a t if d i m S ( d , 2) = 3 t h e n d i m S ( d , i ) = 2 . S i m i l a r l y S(d, 3) is spanned b y CU, TN, TR, and NL so a m a t r i x t h a t r e p r e s e n t s r: S(d, 2) --, S(d, 3) is
2°il 2
1
0
6
0
0
w h i c h h a s r a n k 3. We conclude t h a t if d i m S ( d , 3) = 4 t h e n d i m S ( d , 2) = 3 . Finally for 8 z 3 a m a t r i x t h a t r e p r e s e n t s r: S(d, 8) - , S(d, 8+1) is o 8-1
0
6
8-2
0
0
w h i c h h a s r a n k 4. We conclude t h a t i f d i m S ( d , 8') = 4 for all 3_< 8'_< 8.
d i m S ( d , $ ) = 4 for s o m e 8_> 3 t h e n
P u t t i n g all we h a v e said t h u s f a r together one concludes t h a t to p r o v e the t h e o r e m it is sufficient to v e r i f y that: a). d i m S ( d , ~ ( d -
1 ) ( d - 2))_> 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 <-2 ? One w a r n i n g note should be sounded: We will show in section 4, following a n idea of Harris, t h a t a general set of d points in p r is schemetheoretically t h e intersection of quadrics, b u t has homogeneous ideal not generated b y quadrics, if 2
<
<[r;l
and these inequalities a r e satisfiable as ~ n as r_>5. As sets of points a r e a l w a y s aritmetically Cohen-Macaulay, this example shows that the analogue of the problem has a negative solution for sets of points. N o w if one of these bad sets of points were the hyperplane section of a projectively normal curve that was cut out scheme-theoretically by quadrics, then the solution to the problem above would be negative. At least in the extremal case
(which is the only one to occur in pS) w e prove in Proposition 4.4 that no such curve can exist. (The related problem of whether the general set of d points in pr is the hyperplane section of a curve, or for that matter of a projectively normal curve, seems open.) In addition to thanking Joe Harris, we would like to t h a n k Jee Koh and Michael Stillman, in conversations w i t h w h o m we first considered t h e problems a t t a c k e d in this note, and Bill Lang and David Morrison, who provided first aid for o u r K3 b u m p s a n d bruises. Also, we a r e grateful to Robert Speiser who b r o u g h t us together for t h e conference on e n u m e r a t i v e g e o m e t r y a t ~undance, a n d t h u s provided a v e r y hospitable setting~ w i t h lots of trail along which t h e problems could be pursued.
54 1) Curves on r a t i o n a l n o r m a l scrolls For t h e b a c k g r o u n d on r a t i o n a l n o r m a l scrolls n e c e s s a r y for this section, t h e r e a d e r m a y consult, for example, t h e book of H a r t s h o r n e [1977] (Ch.5 sect. 4) a n d t h e p a p e r of Eisenbud a n d Harris [1987]. T h e o r e m i.I: Let_C ¢ pr be a curve. If C is s c h e m e theoretically cut out by hv1oersurfaces of deeree e, and if C is contained in s o m e two-dimensional rational normal scroll in pr, then the homogeneous ideal of C is generated by forms of degree ~_ e. R e m a r k : We allow t h e possibility t h a t t h e scroll m a y be singular, a n d r e q u i r e of C only t h a t it be a p u r e l y 1-dimensional s u b s c h e m e . T h e o r e m l.i m a y be applied in several situations. For example, if Cc pr is a n y hyperelliptic curve (including the cases of genera 0 and I) e m b e d d e d by a complete linear series, then, as is well-known, the union of the lines joining points of C which correspond under the hyperelliptic involution is a rational normal scroll, and T h e o r e m I.I applies. It also applies to all nondegenerate curves of degree ! r ÷ l in pr: Proposition 12: If C c pr is a smooth connected curve of degree <_ r+1, not c@.,]atainedin a hvperplane, then C is contained in a rational normal scrollQf dimension 2. The proof of this m o r e or less well-known fact, which w e shall sketch below, goes via the following l e m m a , itselfalmost a special case: L e m m a 1.5: If D c pr+l is the rational normal curve of degree r+l, and L is a line meeting D in (at least) a point, then C = D U L is contained in a Z-dimensional r a t i o n a l n o r m a l scroll. Remark:
There is a unique such scroll with L as a ruling.
Proof of L e m m a 1.3 (sketch): Let I~DrIL, and let q be a n y other point of L. W e m a y write the equations of D as the 2x2 minors of a 2x(r+l) matrix M of linear forms, and after r o w operations w e m a y a s s u m e that all the forms in the first
55 row of M vanish at p. After column operations w e m a y further assume that the first r entries of the firstrow vanish at q. The ideal of 2x2 minors of the 2xr matrix consisting of the firstr columns of M is n o w the homogeneous ideal of the desired scroll. Proof of Proposition 12 (sketch): By Clifford'sTheorem the embedding of C must be nonspecial, and then by the Riemann-Roch Theorem the genus of C must be either 0 or I. Further, in case the genus is 1 the embedding is complete, and the discussion immediately preceding the Proposition m a y be applied (or see the paper of Eisenbud, Koh, and Stillman [1986] for a fairly explicitview of the (all quadratic) gvnerators of the homogeneous ideal of such a curve.) The same arguments (in an even m o r e trivialversion) handle the case where the embedding is complete and the genus is 0. There r e m a i n s t h e case of a s m o o t h rational c u r v e of degree r+ 1 in pr. Such a c u r v e C is t h e projection of a rational n o r m a l c u r v e D in pr+ I f r o m a point p off t h e curve. Taking L to be a n y line joining p to a point of C, we see f r o m L e m m a 1.5 t h a t DUL lies on a scroll, whose projection will be a scroll containing C (in fact this a r g u m e n t shows t h a t C lies on a l - p a r a m e t e r family of rational n o r m a l scrolls.) ~ j ~ We now t u r n to t h e proof of Theorem 1.1. The case e= 1 being t r i b a l , w e henceforward assume that e.>2.
W e will actually prove a sharper result, for which w e need m o r e notation: Let S be the hypothesized scrollcontaining C. By definition,~ is the image of a projectivised vector bundle ~'--P(]~),where ]~ is a rank 2 vector bundle on p 1 under the m a p ~0:p(~)-+pr induced by the complete linear series ]~ a ~ i a t e d to the tautologicaldivisor H on P(]~). Let C' be the divisor on S' which is the total transform of C. Let F be the fiber of the natural projection p(l~)-.p1 The classes of H and F generate the Picard group of S', so w e m a y write C'~aH+bF for some integers a and b. Since C' is effective w e have a = C'Jv ->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 <_ e. ii) C is c o n t a i n e d in s o m e h v p e r s u r f a c e of degree e not containing S, a n d in a neighborhood of s o m e point of ~(C0), C is cut out s c h e m e theoretically b y h v p e r s u r f a c e s of degree e. iii) e_> 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 <_ (e-a)c as required for iii). Finally, suppose t h a t iii) is satisfied. To p r o v e i) it is enough, since t h e h o m o g e n e o u s ideal of S is g e n e r a t e d b y quadrics a n d e _> 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)
HOOs
-~ HOOs~R+H)
is onto. Now it is e a s y to c o m p u t e groups of global sections of line bundles on S'; one has, in general, a n a t u r a l identification H0(S ', 0s<mH+nF) ) = H0(p 1, O p l ( n ) ® S y m m ( 0 p l ( c ) $ 0 p l ( d ) ) ). In t e r m s of this (1) becomes t h e n a t u r a l multiplication m a p
i+j=e-a
HOo i jd-b) 1® [H°
•
i,,j:. 0 i+j=e-~+ 1 i,,jzO
H° Or i jd-b)
57 and one easily checks t h a t this is onto because of the inequalities in i i i ) . ~
2) Curves in p4 and p5 It is easy to check t h a t e v e r y c u r v e in pZ or p5 which is s c h e m e theoretically the intersection of quadrics has homogeneous ideal generated b y quaclrics. We p r o v e h e r e t h e corresponding r e s u l t for all s m o o t h c u r v e s in p 4 for all projeotively n o r m a l c u r v e s in p5 and for m a n y 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 higher dimensional spaces, it becomes false for a r b i t r a r y smooth c u r v e s in p 5 as shown b y t h e example in section 5 below. First, in p4: T h e o r e m 2.1: If C c p4 be a s m o o t h irreducible c u r v e which is s c h e m e theoretically cut out b y quadrics, t h e n the homogeneous ideal of C is generated b y quadrics. Our sharpest positive result in higher dimensional spaces is: T h e o r e m 2.2: Suptx~se t h a t Cc p r is a projectivelv n o r m a l c u r v e which lies on a p r o j e c t i v e l y n o r m a l K5 surface S cut out by quadrics. If t h e r e is a reduced irreducible c u r v e C' on S such t h a t C+C' is linearly e q u i v a l e n t to twice the h._vperplane section on S, a n d C is contained in some quadric not, containing C' (or e q u i v a l e n t l y t h e a r i t h m e t i c genus of C' is >0) t h e n t h e homogeneous ideal of C is g e n e r a t e d b y quadrics. F r o m this we get: Corollary 2.3: If Cc p5 is a proj~tiv~Iy normal c u r w which is the ~ h e m v theoretic intersection of quadrics, then the homogeneous ideal of C is generated by quadrics. To p r o v e these results we will use t h e residual c u r v e C' to C in t h e complete intersection of r-1 general quadrics containing C. The following result shows us (in a m o r e general setting) w h a t we c a n expect of C'. Its first s t a t e m e n t will b e c o m e r e l e v a n t in section 5. (Of course t h e s a m e s t a t e m e n t would hold for complete intersections of h y p e r s u r f a c e s of higher degree.) The complete intersection of 3 quadrics in p5 is a k~ surface, so it also provides t h e link b e t w e e n T h e o r e m 2 2 and Corollary 2.3.
58
Proposition 2.4: If C c p r is a s m o o t h c u r v e w h i c h is t h e s c h e m e t h e o r e t i c intersection of h v o e r s u r f a c e s of degree 2, t h e n a ~eneral set of r-Z a u a d r i c s containing C m e e t in a s m o o t h surface, a n d a general set of r-1 such quadrics m e e t in a c u r v e of t h e f o r m CUe', w h e r e C' is a s m o o t h ( t x ~ i b l y e m p t y or disconnected) c u r v e m e e t i n g C in o r d i n a r y double points. F u r t h e r . i~ C' is not e m p t y t h e n writing d,d' a n d g~g' for t h e degrees a n d g e n e r a of C a n d C' w e h a v e d ' = 2r - l -
d
g' = g + ( r - 3 ) ( 2 r - 2 - d ) , and C meets C' in (r-5)d - 2g + 2 points. If C i$ projectively normal, then so is C', and. in particular C' is then irreducible. (Here w e have taken the genus of C', in case C' is disconnected, to be the s u m of the genera of the components m i n u s the n u m b e r of components plus I, as usual.)
The c u r v e C' of Protx~ition 2.4 is linked to C b y D in t h e sense of t h e t h e o r y of liaison (see for e x a m p l e Peskine-Szpiro 119731.) As an i m m e d i a t e consequence, we see that it is extremely r a r e for a curve in pr to be the ~cheme-theoretic intersection of exactly r quadrics:
Corollary 2.5: I f C ¢ p r is a c u r v e w h i c h is s c h e m e - t h ~ r e t i c a l l y t h e intersection o_ffr linearly i n d e p e n d a n t quadrics, t h e n t h e degree d a n d genus g of C satisfv g = (r-1)d/2 + 1 - 2r-1 In fact, the conditions on C of interest to us are easy to express in terms of the geometry of C', even in a m o r e general situation:
Proposition 2.5: Let C a n d C' be ( n o n - e m p t y ) locally C o h e n - M a c a u l a y c u r v e s w h i c h a r e linked b y a c o m p l e t e intersection of quadrics (respectively bY a c o m o l e t e intersection of a q u a d r i c a n d a p r o j e c t i v e l y n o r m a l t(5 s u r f a c e c u t out by quadrics), We h a v e : i) C is n o n d e g e n e r a t e iff ~oC'(4-r) ~ coC~-l) ) h a s no global sections.
59 ii) C is cut out scheme theoretically by quadrics iff wC~5-r) (resp. ~oC') is ~enerated bv global sections. The ideal of ~ ..... contains exactly hO~oc~5-r) ÷ r - I (resp. hO~o C' ÷ r - I ) independant quadrics. iii) The homogeneous ideal of C is generated by quadrics iff the
multiplication maps H0wc~5-r) ® HOOc~k) -~ H0coC~5-r*k) (resp. H0~C ' @ HOOc~k) -* H0~c~k) ) are surjective for all k>O.
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
63
This r e s u l t follows easily f r o m a general fact, of interest in its own fight: T h e o r e m 42: Let X C p r be a reduced irreducible v a r i e t y , not contained in a h_vperplane, a n d let F c X be a eeneral set of d points. If d ~_ codim X, t h e n F i_~ t h e s c h e m e t h e o r e t i c ~n.tersection of t h e linear space it spans w i t h X. Corollary 4.5: If F ~s a general set of d points in pr, t h e n F is s c h e m e theoretically cut out bv forms of degree e as long as d<_
[71
-1-r.
Proof of CoroUarv 4.5: Apply T h e o r e m 4 2 to p r embedded b y the e th Veronese mapping . ~ T h e o r e m 4.1 suggests a w a y to look for interesting examples: if a set of points in p r satisfying T h e o r e m 4.1 could be "lifted" to a Wojectively n o r m a l c u r v e in pr+ 1, s c h e m e theoretically cut out b y quadric~, t h e n t h a t c u r v e would n o t h a v e homogeneous ideal g e n e r a t e d b y quadrics. It is not known w h e t h e r a general set of points in p r lifts a t all to a c u r v e in pr+ 1. V/e will w o v e a weak non-lifting result, which a t least rules out t h e possibility of finding a n example of a c u r v e in p6 in this way. Proposition 4.4: Let ]~ be a general set of
Ir,ll d:=:2 : points in p r with rzS. There is no linearly normal curve in pr+l, cut out scheme theoretically bv auadric~, whose hvl~erDlane section is F. W e n o w turn to the proofs:
Proof of T h e o r e m 4.1: P is s c h e m e theoretically c u t out b y quadrics b y v i r t u e of Corollary 4.5. Thus we need only show t h a t t h e homogeneous ideal of F is not generated by quadrics. Since t h e points of F a r e general, t h e space of quadrics vanishing on
64 F h a s dimension e x a c t l y
while t h e dimension of t h e space of cubic~ v a n i s h i n g on F is
which, as e l e m e n t a r y c o m p u t a t i o n shows, is > r÷ 1 t i m e s t h e dimension of t h e space of quadrics vanishing on F. Thus the ideal generated by the quadrics in the ideal of IP does not contain all the cubics in the ideal of F. The seognd statement of the Theorem follows triviallyfrom the first. Proof of T h e o r e m 42: It is enough to show t h a t projection of X f r o m (the linear s p a n of) a n y d-1 of t h e points is birational; for t h e n it will be s m o o t h a t t h e last point, w h i c h is t h e desired conclusion. For this it is i n d u c t i v e l y enough to show t h a t projection f r o m a n y general point is birational, t h a t is, t h a t not e v e r y s e c a n t of X is a m u l t i s e c a n t as long as t h e codimension of X is a t least 2. By taking h y p e r p l a n e sections, w e m a y a s s u m e t h a t X is a c u r v e . The result t h e n follows f r o m t h e f a c t t h a t t h e general h y p e r p l a n e section of a n irreducible c u r v e is a set of points in linearly general p o s i t i o n . ~ Proof of Proposition d.4: One checks i m m e d i a t l y t h a t a linearly n o r m a l c u r v e C whose h y p e r p l a n e section is l~ lies on e x a c t l y r ÷ l quadrics in p r + l . Corollary 2.5 gives a f o r m u l a for t h e genus g of C in t e r m s of t h e degree a n d r (note t h a t t h e =r= ~iven m u s t be replaced b y r ÷ l in o u r case). If r_>5, t h e f o r m u l a of Corollary 2.5 yields a v a l u e i n c o m p a t i b l e w i t h t h e inequality g_>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)A<0, so R is effective a n d t h u s b a s e - p o i n t free b y t h e s a m e a r g u m e n t t h a t shows E is. Since RZ=0 it follows t h a t IRI is one dimensional, so E is t h e s c h e m e t h e o r e t i c intersection of t w o quadrics a n d S, t h a t is, of five quadrics, as claimed. To show t h a t t h e homogeneous ideal of E requires t w o g e n e r a t o r s of degree 3 w e m u s t show t h a t t h e multiplication m a p S0Op5(1) ® H0~C(2) -~ H0~C(S) l u ~ 2 - d i m e n s i o n a l cokerneL Since S is t h e c o m p l e t e intersection of quadrics, t h e r e s t r i c t i o n of this m a p to S, H 0 0 ~ A ) ® H0(R) ~
H0(5A-E),
h a s t h e s a m e cokernel. Since IR] is a base point free pencil we h a v e a n e x a c t
69
sequence 0 -40~A-R)
-~ O ~ A ) @ HO(R) - 4 0 ~ S A - E ) -~ O,
from which w e see that the above cokernel is H i Os(A-R). Since A(A-R)=O w e see that neither A - R nor R - A is effective, so the Riemann-Roch formula yields h 1 Os(A-R) = 2, as required.
Proof of Theorem 5.1: First, since the family of nondegenerate elliptic octics in p5 is irreducible, it follows from Theorem 5.5 t h a t the general one will be scheme theoretically the intersection of quadrics. By Proposition P-2 the family of pairs E,S with E c S c p5 and S a smooth surface which is the complete intersection of 5 quadrics (so t h a t in particular S is a K5 surface) is irreducible, and it follows from the existence of the surface guaranteed in Prolx~sition 52 t h a t for the general such pair Pic S is generated by E and the hyperplane section A. Thus Theorem 5.5 describes the generic s i t u a t i o n . ~ References
Bayer, D., and Stillman, M.: Macaulay, a computer algebra system. Available free from the authors for the Macintosh, VAX, Sun, and m a n y other computers (1985). Beauville, A.: Introduction a l'application des periodes. In G~maetrie des Surfaces K3: Modules et Periodes. Societe Math. de France, Asterisque vol. 125 (1985). Castelnuovo, G.: Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rend. Circ. Mat. Palermo 7 (1893) 89-110. Eisenbud, D., and Harris, J.: On Varietie~ of minimal des~ree (a centennial account). To appear in Proceedings of the S u m m e r Institute on Algebraic Geometrv, Bowdoin, 1985, ed. S. Bloch, Amer. Math. ff~c., Providence RI. (1987). Eisenbud, D., Koh, J., and Stillman, M.: Determinant~ equations for c u ~ a ~ of high degree. ]>reprint (1986). A m . J. Math. ,to appear.
Fulton, W.: Intersection Theory. Springer-Verla~ New York, (1984). Green, M.: Koszul cohomology and the geometry of projective varieties. J. Diff. G e o m . 19 ( 1 9 8 4 ) 1 2 5 - 1 7 1 .
70 Hartshorne, R.: Algebraic Geometry~ 8pringer-Verlag, New York (1987). Mattuck, A.: Symmetric products and Jacobians. Am. J. Math. 85 (1961) 189206. Mayer, A.: Families of K5 surfaces. Nagoya Math. J. 48 (1972) 1-17. Merindol, J.Y.: Proprietes elementaires des surfaces KS. In Geometrie des Surfaces KS: Modules et Periodes. Societe Math. de France, Asterisque vol. 126 (1986). Mori, S.: On the degree and genera of curves on smooth quartic surfaces in F 3. Nagoya Math. J. 96 (1984) 127-152 . Morrison, D.R.: On K3 surfaces with large Picard number. InwentMath. 75 (1984) 105-121. Peskine, C., and 8zpiro, L: Liaison des varietes algebriques, Inv. Math. 25 (1973) 271-302. Saint-Donat, B.: Projective models of K5 surfaces, Am. J. Math. 96 (1974) 602-659. Schreyer, F.-O: 8yzygies of canonical curves and special linear series. Math. Ann. 275 (1986) 105-157.
VANISHING
IHEOREMS FOR V A R I E T I E S
Lawrence Department University Box
Let the
X be a s m o o t h
following
(a) (b)
cond
2m-2)/3,
If n>
2m)/3,
linear
and
Most not
of
well
bundle,
We
§1.
In the
rest
is the
following
~m.
(See
Proof. jection
is
of
conjecture open.
in
varieties
in ~m.
In
[5],
Hartshorne
made
for
weaker
paper~
to Zak's
by
Zak.
we'll
But
the
discuss
se-
some
codimension.
are
not
new.
But
they
are
references. and
theorems
by
the
the
about
Le Potier, results
first
and
of
Schneider
studying
paper,
principle
varieties
we
the
low
had
formal
shall
assume
numbers.
X will proof
note
theorem
Throughout
complex
solved
short
future
tangency vanishing
on
intersection. been
section
Peternell,
the
normal.
of small
first
them
some
theorem
has
In this
the
Zak's
the
1.1.(Zak)([4])
1.2.
[11] Let
denote
of the
a smooth
linear
nondegenerate
normality
conjecture
tangency.
Suppose
(h)
By
(1.1),
Let
N be
with
the
Furthermore, for
the
that
P2:
Cx~
normal dim
definition
CX=[(N(-t))
map
P2
dim
linearly
slightly
A key
(m-l-n)-ample.
-1
U.S.A.
L is
a k-plane
(k>n)
in
F m.
fhen
Sing(LNX)
Corollary
X.
n-fold
varieties.
field
in
Theorem
but
of the
base
n-fold
dim
at C h i c a g o
IL 60680~
X is a c o m p l e t e
Independently,
similar
the
X is
include
using
we o b t a i n
neighborhood
Illinois
Chicago,
about
results
2,
eodimension.
that
then
results
known.
obtained
then
remains
the
In s e c t i o n part
of M a t h e m a t i c s
of
projective
normality
conjecture
examples
Ein
conjectures:
If n>
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<(2m)/5.
then
the
N(-I)
property
us
dim
F m be
a finite
is a general
variety
S 4 in F 15 both they
are
X=(2m)/3.
to c o n s t r u c t
satisknown
and
Grassmann
variety
In p a r t i c u l a r ,
varieties
It is well
is ample
6-dimension
Spinor
([3]).
with
will Let
dim
intersection,
10-dimension
X=dim
proposition
classification
X=dim
known
varieties
Proposition (a
dim
if X is a c o m p l e t e X*=m-1.
the
a complete
properties
have
noncomplete
The
a few more
following
examples.
morphism.
element.
Let
g=Tof.
Then
g (b
Assume that X is the G r a s s m a n n variety G(2,5 -I Y=g (X) is not a c o m p l e t e i n t e r s e c t i o n .
(c
Assume not
Proof.
that
X is the
a complete (a)
This
Spinor
variety
S 4 and
and
m=15.
m=9.
lhen
Then
g-1(X)
is
intersection.
follows
from
KJeiman's
transversality
theorem
([6],
3.I0.8). (b)
There
is a 2 - p l a n e
dual
off L i n
span
a 2-dimensional
L in G(2,5)
H8(X) . Let
implies
that
be
and
class
g)
g*[H]
By
are
theorem,
there is a 4 - p l a n e in S 4. -I g (S 4) is not a c o m p l e t e
show
Remark.
that
I learned
Proposition
1.4.
about Assume
g-1(G(2,5))
1.5(a) that
be
the
the
Poincar4"s
Then
[L]
projection
independent
(c) will
[L]
and
[M]
formula,
((a[L]+b[M])n[X]).
8y L e f s c h e t z ' s Similarly,
Let
c I (Ox(1)) 4
in H8(X).
= (deg
g*[L]
([3]).
the
subspace
g.(g*(a[L]+b[M])n[Y]) This
[M]
from
is not
a complete
A similar
b4(Y)=b8(Y)~2. intersection.
argument
as
in
(b)
intersection.
a discussion
n>(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
Furthermore,
and
O<j
if n:m-]
and
Then
X is
m>10,
then
X
intersection.
By 8 a r t h ' s
theorem,
Pic
X=Pic
~m.
Thus
K x = O x ( t O)
for
some
73
(b) By K o d a i r a ' s
vanishing
I
and
arbitrary
Hi(Ox(J))=O
for
1
Macauley. Let
Let
U=Y-{p}
normal
Y be
and
and Wy
see
that
A m+1
larity
By S e r r e ' s arbitrary
affine
cone
projection
the by
Eisenbud
codim
a complete
argument
that
and
that
intersection
the
we
see
with
Then
3 Gorenstein
only
is due
than
Cohen-
codim
its
Thus Using
the
Y is
wy=Oy
and
Y
a theorem ([2]),
of we
noncomplete
point
to seven
in
in Y
3 Gorenstein
to H a r t s h o r n e
vertex.
w x = O U.
or equal
singular
isolated
p as
WU=~
variety
then
less
that
X is p r o j e c t i v e J y
m>10.
intersection,
p is
Hi(Ox(J))=O,
of G r a u e r t .
codimension
showing
is a c o m p l e t e
map.
X=3
on c o d i m e n s i o n
of Y has
Thus
of X in A m+1
a theorem
that
that
duality,
jo
be
contradicts
The
see
the
assume
locus
we
U~X
if Y is not
This
Remark.
Now
and
intersection
j.
and
is r e f l e x i v e
is G o r e n s t e i n . Buehsbaum
~:
theorem,
and
singu-
Ogus
([7],
3,6).
§2.
Let
the
ideal
on X.
Ak+ 1 be the sheaf
Recall
of
that
subscheme the
the
k
p (E) w h e r e Pl
and P2 a r e
striction
map
c I ( L ) E H I ( ~ ~)-
(2.2)
p1 L s)
Proposition
2.3.
by
that
part
I k+1(k>O)
E is a locally
the
p1
When k = l ,
XxX.
Consider
then
the
implies
= pI(L)
® L s-1
The r e s t r i c t i o n
There
is
extension
class
of
Since
f
is
surjective,
sheaf
a natural
2.1)
re-
for
map H i ( p k ( E ) ) ~ H i ( E )
_~ H i ( P I*E I A k + I ) ~ also
given
by
s~O. s surjective
diagram
g is
is
that
ai(plF)
Hi(pk(E))
free
as
(E) + E ~ O.
This
the
I is
we o b t a i n
i.
Proof.
where
of E is d e f i n e d
), Ak+l projection maps o f
bundle,
[I]).
Suppose
principle
pk(E)+P2*(PlElA)=E.
is a line
defined
= p2.(PlEI
0 ~ ~X ® E ~
If E=L
all
diagonal. k'th
1
(2.1)
of XxX
HL(PlEIA ) ~ Hi(E).
surjective.
for
74
Theorem
2.4.
Assume
that
n>(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
O
~ HO(0x(1)) = 0
for
that
1<j
For
is
= O.
O
some
Then
surjective
positive
restriction
integer
map
HO(o
t we (t
have
141(N*(j))
) ~ HO(Ox(t))
is
; O sur-
Fm jective.
e)
Suppose
fior
Proof.
that
l<j
(a)
for
N(-1)
is
Let
V = HO(o
(1)). Fm
V®Ox
1 ~ P (Ox(1)) ¢ 0
HI(N
= O, V *
(1)
By B a r t h ' s
is
theorem,
Thus H i ( O X ( 1 ) )
By i n d u c t i o n , Since
v®HO(Ox(t-1)) ihe
= 0 for
O
Consider
v®o
=
i~3n-Zm,
surjective is
Ox(1
~ O
-*
Ox(1
~- O
is
surjecLive.
= O, l < i < 2 n - m .
we see t h a t
= 0 for
the diagram:
By ( 2 . 3 ) ,
see
we
HO(o
.
Ht(N
(t)) is
by 2.2 similar
Using
V®Hi(OX(1)
fact
[m ( t - l ) )
) is
~ HO(Ox(L-1)
(2 3) we c o n c l u d e
surjective.
Thus HO(O m ( t ) )
(d),
that
Hi(OX(1
= O, u s i n g
is
that ~ HO(Ox(t))
F
and 2 . 3 . to
the ~
1
may assume t h a t
we
~ HO(Ox(t))
proof
-.
H i ( O x)
surjective.
also
= O
Fm
HO(pI(Ox(1)
= 0 for
(e)
Hi(N*(j))
surjeetive.
Hi+I(N*(1))
is
have
N (1) ~
V ~ HO(o x 1 ) )
(d)
(]))
Fm
1 O ~ ax(1 ¢ 0
surjective.
So H i ( N ~ ( 1 )
Then p1(O
=
0 ~ 0 ~m 1 (I)IX
(c)
t we
= [].
O
N (1)
that
integer
Then H J - I ( O x ( t ) )
([11]).
O
Since
positive
(m-l-n)-ample.
by Sommese's t h e o r e m (b)
some
(2
we s h a l l
leave
the details
to
the
rea-
ders. Theorem 2 . 5 . (a)
If n~8,
(b)
HJ(OX(2))
Proof. sider
a) the
Assume t h a t then
Let
HO(OFm(2))
= 0 for
X = 2.
~ IIO(Ox(2))
is
surjective.
l<j
Y be t h e
sequence,
codim
zero
set
of
O -~ A 2 N * ® 0 X ( 3 )
a general -+ N ( 2 )
~
section Iy/x(1)
of -~ O.
N(-1). We may
Conassume
75 that
X is
not
a
a complete
By Z a k ' s
Severi's
variety
Hl(Iy(1))
linear
([9]),
= O. By 2 . 4
2
intersection.
Then A N ® O x ( 3 )
normality
we o b s e r v e (c),
theorem that
H2(IX(1))
HlIy/x(1))
= O. So H I ( N * ( 2 ) )
HO(o m ( 2 ) )
-* H O ( O x ( 2 ) )
is
Y is
and
2.4
normal.
and
some
(d),
of
Hence
= O. We see
(a)
for
classification
linearly
= HI(Ox(1))
= O. U s i n g
the
= Ox(a)
that
we see t h a t
surjective.
F
(b)
Similarly,
Hi+I(tx(1)) Hi(N
(2))
Hi(Iy(1)) = 0 for
-- 0 f o r
= Hi-t(Oy(1))
2
2
= 0 for
2
Thus H i ( I y / x ( 1 ) ) By 2 . 4
(e)
= 0 for
we see t h a t
Also 24i
HJ(Ox(2))
So
= 0 for
l<j
Independently,
served
that
rieties
of
X instead lar
but
Peternell,
k-ampleness
of
low codimension. of
the
slightly
first
N(-1) But
principle
weaker
results
Le P o t J e r , will
give
they
study
part as
in
and S c h n e i d e r vanishing the
bundle. 2.4
(b
formal (c)
ob-
for
neighborhood
1hey h a v e ,
have
theorem
obtained
and 2 . 5
vaof simi-
(a).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Atiyah, M., Complex analytic connections in fiber bundles, trans. A m e r . M a t h . Soc. 85, 1 8 1 - 2 0 7 ( 1 9 5 7 ) . B u c h s b a u m , O. and E i s e n b u d , D . , A l g e b r a s t r u c t u r e for finite free resolution and some s t r u c t u r e t h e o r e m f o r i d e a l o f c o d i m e n s i o n 3, A m e r . J . M a t h . 9 9 , No. 3, 4 4 7 - 4 8 3 ( I 9 7 4 ) . Ein, L., Varieties with small dual varzeties I, [nv. Math. 86, 63-74(1986). Fulton, W. and Lazarsfeld, R., Connectivity and its applications in algebraic geometry, Lect. Notes in Math. 862, 26-92(1981). Hartshorne, R., Varieties in low codimensian in projective space, Bull. Am. Math. Soc. 80, 1017-1032(1978). Hartshorne, R., Algebraic Geometry, Springer~ Heidelberg 1977. Hartshorne, R. and Ogus, A., On the factoriality of local rings of small embedding eodimension, Comm. in Alg. I (5), 415-437(1974). Holme, A. and Schneider, M., A computer aided approach to codimension 2 s u b v a r i e t i e s of Fn; nZ6, preprint.
9. Lazarsfeld, R. and Van de Ven, A., Recent Work of F.L. Zak, DMDSeminar. 10. Peternell, [., [e Potier, J., and Schneider, M., Vanishing theorem, linear and quadratic normality, preprint. 11. Sommese, A., S u b m a n i f o l d of abelin varieties, Hath. Ann. 233, 229-250(1978).
EXPLICIT COMPUTATIONS IN HILB 3 ~2
G. Elencwajg - P. Le Barz Mathgmatiques, IMSP, Parc Valrose 06034 NICE Cedex, France.
It is known since Hartshorne
EIO]
and Fogarty E 7 ] that the punctual Hilbert
scheme Hilb d ~2 is connected and nonsingular of dimension 2d. The variety Hilb 3 ~2 is a natural compactification of the space of (unordered) plane triangles. We have been able to completely compute its ring of rational equivalence CH'( Hilb 3 p2 ) by generators and relations ~ 3
] .
We should like in this survey to convince the non-specialist of the Hilbert scheme that calculations (intersections, multiplicities...) in local coordinates of Hilb 3 ~2 are not insurmountable; this without resorting to the whole general theory ( Grothendieck ~9 ] ). The variety Hilb 3 ~2 parametrizes the tFi~lets
of ~2, i.e. the zero-dimensional
subschemes of length 3. Let us emphasize that these triplets generalize the notion of unordered triangles; this point of view is to be contrasted with that of Roberts and Speiser
E17]
who draw their inspiration from Schubert [19 ] to construct a compac-
tification of the space of ordered equivalence.
triangles and then calculate its ring of rational
77
Let us succinctly give the plan of the article. In O we recall some general facts about the Hilbert scheme in the punctual case. In I we introduce a nonsingular variety H~ilb3 ~2; this is a modification tautological covering of Hilb 3 ~2. To show the smoothness of Hilb
3
~2, we use explicit
charts of Hilb 3 ~2 at the different triplets. We irrtroduce this variety Hilb well as a morphism R : ~ilb 3 ~2 ~
of the
3 ~2
(as
F 2 defined in I!) in order to give a correct de-
finition of certain cycles in CH'( Hilb 3 ~2 ). Ellingsrud and Str~mme
E6 ] have established the additive
CH'( Hilb d ~2 ). We make their results explicit for d = 3 in
structure of
III.
In IV we give a ~-basis for each of the CHi( Hilb 3 ~2 ) composed of cycles with a simple geometric interpretation.
Moreover these cycles are monomials
in the seven
generators: H, A In
E4]
E CH I
,
h, a, p
E
CH 2
,
~
, B ~ CH 3
we shall give the complete table of relations between these seven generators
of the ring
CH'( Hilb 3 ~2 ).
Finally in V we explicitly give some applications of these results to enumerative formulas for families of plane curves. These necessitate delicate multiplicity computations in Hilb 3 ~2. In
[5 ] we shall give other applications
merative formulas, some original, other classical
( Schubert
of our results to enu[19]
).
The base field is ¢.
O) General facts about the punctual Hilhert scheme
a) If X is a projective scheme, recall
[ 9 ] that Hilb d X is the Hilbert scheme as-
sociated to the constant polynomial d ( d ~ N). Set-theoretically by the d-uplets of X, that is more precisely the subschemes ~ C X and length ~ = d (i.e. dime F(X, ideals I ( ~ )
of
O X with:
~)
= d. In other words
Hilb d X is described with dim ~ =
Hilb d X parametrizes
O the
78
Supp ~
finite and dimc~(X , 0 ) =
d , where
0
= ~X/I(~).
Moreover, the scheme Hilb d X comes equipped with a d-sheeted covering (called tautological or universal): ~" :
'Hilb d X
~
Hilb d X .
The main virtue of this covering ~y is to be flat and to verify the: Universal property of Hilb d X: Let S be an arbitrary scheme. If Z of S (via
C
SxX is a flat d-sheeted ramified cover
the first projection), then Z is gotten by pulling back 'Hilb d X through
a unique morphism of schemes f : S
----> Hilb d X.
(Please notice that f has no reason to be flat!)
To hearten the reader,
let us recall
[11]
that in the case of finite morphisms
flatness is easily checked: Flatness criterion: Let
~ : X
---~
T
be a finite morphism of schemes with integral T. Then
is flat if and only if the length of the fibers
~-1(t) is independant of t E T.
Actually 'Hilb d X is a closed subscheme of Hilb d X x X, ~
is the restriction
of pr|. Set theoretically 'Hilb d X is characterized by: (~,m)
K
m C ~
'MilD d X
Example: Take in $2 the triplet
~o of ideal
(x 2, xy, y2) = (x,y)2 in
(~&2
. Of
the two deformations with ~I as basis: ~t of ideal
(x(x-t), xy, y(y-t))
and
~t' of ideal (x2-t 2, xy-t 2, y2-t2), only the former is flat , because the length of every only corresponds to a morphism f : ~1 ~
Hilb 3 ~2 .
fibre is 3; hence the former
79
b) We refer the reader to larrobino
~2]
for a very detailed survey of recent re-
suits about Hilb d X. As for "classical" results, one has i) the open subset Hilb~ X of simple d-uplets (i.e. d distinct points) is not in general dense in Hilb d E, even if
X is irreducible (larrobino
[13]
). If d i m E = n,
Hilb d X may have components of dimension larger than dn or smaller than dn ( larrobino-Emsalem
~5]
).
ii) On the other hand if X is a smooth surface, then 2d and Hilb~ X is dense ( Fogarty ~ 7 ] = Pic (X) @ ~
(Fogarty [ 8 ]
Hilb d X is smooth of dimension
, BrianGon [ 2 ]
, larrobino [14]
)~ moreover, Pic (Hilb d X)
).
iii) Hilb 4 ~3 is singular at the quadruplet of ideal~. 2 w h e r e ~ i s
the ideal of a
point.
c) Coming back to Hilb 3 ~ 2
let us recall ~rian¢on [ 2 ] )
that the different
triplets of the plane are of one of the following forms: i) three simple points; ii) a simple point and a doublet (of ideal (x2,y) in local coordinates); iii) a triplet of support one point, but curvilinear (i.e. of ideal (x3,y-O-x2) in local coordinates). Such triplets with ~ = O will be found in Example 3 below; iv) a triplet corresponding to the sguare of a maximal ideal, i.e. the first infinitesimal neighbourhood of a point (ideal J ~ 2 = (x2 xy,y2) in local coordinates).
We can thus draw these different triplets:
i]
ii)
iii)
iv)
0
0
J
J
80
Case ii) is obviously a specialization of case i). Case iii) is a specialization of case ii): the ideal (x 3, y - 0~x2) is the limit of the ideal If
=
(x2(x-g),
y -o~x2) as
g ---> O. Indeed, for every
E E A I , If
the ideal of a triplet; hence we have a flat family which corresponds
is
to a morphism
&| _.~ Hilb 3 ~2 (see a)). For the same reason, case iv) is a specialization of case iii): here it is the
ideal
(x 2 • xy, y2) which is the limit of (yE - x 2 , xy, y2) = ( Y - x2/g, x3). So intuitively case iv) corresponds to case iii) with an "infinite curvature". Example 1: If C is a curve in ~2 and O
6
C, then C is singular at 0 if and only if
the triplet of type iv) of support {0} is scheme-theoretically precisely,
contained in C. More
if 0 is a nonsingular point of C, the curve C contains only one triplet
of support {0}, obviously of type iii) (hence the terminology Example 2: Two smooth curves C I and C 2 are osculating point 0 if and only if the scheme
"curvilinear").
(but not hyperosculating)
at a
C I ~ C 2 is, in the neighbourhQod of O, a triplet
of type iii). Example 3: The nonsingular point 0 of the curve C is an ordinary flex if and only if C cuts its tangent at 0 along a triplet of type iii).
I) The variety ~ilb3 ~ 2
a) We give here transcendental
charts
for Hilb 3 ~2 at the four types of triplets
seen in paragraph O. Type i):
The triplet ~ is formed of three distinct points P|, P2' P3; we choose local
coordinates
(xi, yi ) for ~2 at P. and the chart for Hilb 3 ~2 at ~ i
is
(x1' YI' x2' Y2' x3' Y3 ) ° Type ii): The triplet
~ is formed of a simple point Po and of a doublet d. Let (Xo,Y o)
be l o c a l c o o r d i n a t e s f o r ~2 a t P
o
and ( x , y ) l o c a l c o o r d i n a t e s f o r
p2 a t Supp d, such
t h a t t h e i d e a l of d be (x 2, y ) . A c h a r t f o r Hilb 2 ~2 a t d i s ( a , b , a ' ,
b') corres-
81
ponding = PoU
to the ideal d is
(x 2 + ax + b , y + a'x + b').
So that a chart for Hilb 3 p2 at
(Xo' Yo' a, b, a', b').
Type iii): The triplet for F 2. A chart
~ has for ideal
(x 3, y - ~x 2) in some local coordinates
for Hilb 3 ~2 at ~ is (a, b, c, a', b', c') corresponding
(x,y)
to the ideal
(x 3 + ax 2 + bx + c , y - (xx2 + a ,x 2 + b'x + c' ) . Type iv): let (x,y) be local coordinates ty of this case as compared of the triplet
is not
for ~2 at the considered
to the former three is that the i d e a l ~
locally
a complete
Hilb 3 ~2 at ~ is (u, v, u' , v',u",
intersection.
v") corresponding
(I) w
The difficul-
2 = (x2,xy,y 2)
In this case a chart
for
to the ideal
I = (x 2 + ux + vy + w , xy + u'x + v'y + w' where the functions
point.
, y2 + u"x + v"y + w")
w, w', w" are defined by
uv' + vv" - u'v - v ,2
=
(2) w ' = u'v'
- u"v
(3) w ''= uu" + u'v '' - u '2 - u"v'. For all these reminders,
we refer to Iarrobino's
We are now able to introduce
b) Recall ring
(cf. § O) that Hilb 3 ~2 comes
'Hilb 3 ~2. Beware
of the simple singular
the variety
that this scheme
point 0 ~ ~2,
point.
as to the chart
Indeed,
Hilb 3
equipped
is sinsular:
the element
(~f~L2,0)
(x,y) of p2 at O. Then
~2
( E12]
, § 1.1)
.
in what follows.
with a tautological if~.
denotes
~ 'Hilb 3 ~2
let us go back to the chart
(3) by the three quadratic
C
3-sheeted
as above
cove-
the ideal
Hilb 3 ~2 x ~2 i~ a
in a) of Hilb 3 ~2 a t e 2
'Hilb 3 ~2 is defined
according
to (I),
as well (2) and
equations
2 x
article
2 +
BE
+
vy
+
uv
r
+
v~f"
-
uWv
-
v v
=
O
xy + u'x + v'y + u'v' - u"v = 0 y
2
+ u"x + v"y + uu" + uVv" - u !
c) To get rid of that difficulty, ,Hilb 3 ~2. tion
It is the subvariety
2
- u"v v = O .
we introduce
a new inciJence
variety
H~ilb 3 ~2 of Hilb 3 ~2 x Hilb 2 ~2 defined
analogous
to
by the condi-
82
(t,d) the inclusion nonsingular, nerically to ~I
being
~
~ilb 3 ~2
-*_ >
in the scheme-theoretic
d
sense.
C
Before
let us remark that the first projection
finite.
However
t,
the fibre above a triplet
showing
that H~ilb3 ~2 is
?r onto Hilb 3 ~2 is 3-sheeted, t o with ideal ~
: it is the set of ( to, to/~ D) where D is a line of ~2 through
point with
O, the closed
ideal
The smoothness
of Hilb 3 ~2
t does not correspond lution
ge--
2 . is isomorphic
is given
to an ideal ~ [ 2
and d o the ideal
I(t)
easy to check at those
(still we advise
(t,d)
for which
the reader to compute;
the so-
in d) below).
Thus let us check
(u, v, u', v',
is relatively
smoothness
at the point
(x 2, y). As a chart
(to,d ° ) where
to has the ideal
for Hilb 3 ~2 at t o we take, of course,
(x,y)
2
the chart
u", v") of a):
( x2 +
=
ux
+
vy
+ w
, xy
+
u'x
+
v'y
+
w'
, Y
2
+
u"x
+
v"y
+
w")
,
where w, w', w" are defined by + vv" - u'v - v ,2
(I)
w
=
uv'
(2)
w'
=
u'v'
(3)
w"
=
uu" + u'v"
As a chart
for
l(d) describing
=
-
u"v
Hilb 2 ~2 at d ( x2 +
ax
+
b
u '2
-
o
_
u"v
we take
, -y
+
a'x
v
.
(a, b, a', b') with +
b'
an ideal close to d o (the minus
)
sign in front of y is a computing
conveni-
ence for what follows). "~ Hilb 3 ~72 of Hilb 3 ~2 x Hilb 2 ~p2 is defined by the incidence
The subvariety tion d C
t. In terms of ideals,
l(t) + I(d) = I(d). But the ideal 2
(x +ux+vy+w or also,
through
, xy+u'x+v'y+w' replacing
(x 2 +(a ,v+u)x+b ,v+w
y
this is translated
by I(t)
2 ,, ,, ,, , y +u x+v y+w , x2+ax+b
by a'x+b'
, -y+a'x+b'),
in the first three generators:
, a'x2+(b'+a'v'+u')x+b'v'+w
into the relations:
l(t) + I(d) =
' , a'2x2+(2a'b'+a ,v ,,+u ,,~ )x+b. ,2 +b. 'v"+w" , -y+a'x+b').
Since I(d) = ( x 2 + ax + b , -y + a'x + b' ), the condition be trans!ated
I(d) or equivalently
l(t) + l(d) can be written
, x2+ax+b
immediately
~
rela-
l(t) +
l(d) = l(d) can
83
(4)
a'v + u
= a
(5)
b'v + w
= b
(6)
b' + a'v' + u' = a'a
(7)
b'v' + w'
(8)
2a'b' + a'v" + u"
(9)
b '2 + b'v" + w" = a'2b
= a'b =
a '2a
These six equations together with (I), (2) and (3) define ~ilb 3 ~2
in a neighbourhood
of (to,d ° ) as a subvariety of ~13 with coordinates a,b,a',b',u,v,w,u',v',w',u",v",w" The equations (I), (2), (3), (4), (6) and (8) allow one to express the six variables w, w', w", u, u', u" as functions of the other seven a,b,a',b',v,v',v"; this presents ~ilb 3 ~2 as a subscheme of the graph of a m o r p h i s m ~ 7
~
~6.
By projecting upon A 7 we are reduced to the subscheme of ~7 defined by the equations (5), (7) and (9) where we have replaced w,w',w",u,u',u" by their values. This yields (s')
2b'v + a(v'-a'v) - v '2 + w "
b
(7')
a'(2b'v + a(v'-a'v) - v '2 + vv")
a'b
(9')
a'2(2b'v + a(v'-a'v) - v '2 + vv")
= a'2b
The equations (7') and (9') are visibly proportional to (5'). To sum up, ~ilb 3 p2 is isomorphic to the hypersurface of ~7 defined by (5'), which is non singular: it is a graph since b is obtained as a function of the six variables a, a'
b', v, v'
v"
It is those six variables that we shall use in the next paragraph as a chart for Hilb
3 ~2
at (to,d ° ) .
d)Let us remark by the way that this definition of H~ilb3 ~2 can be generalized to Hilb
d
X, where X is for instance a nonsingular variety. It is the subvariety of
Hilb d X x Hilbd-Ix defined by (~,~) E
Hilb d X
<
~.
~
~
~
scheme theoretically
It is little plausible that this variety be nonsingular for arbitrary d. However
if
is a curvilinear d-uple (i.e. a subscheme of a nonsingular curve), it is easy to see
84
,a d that (~,N) is a nonsingular point of Hilb X : this is the calculation below ( of no use in what follows...) To simplify the notations, assume X is a surface. Let ~o be a curvilinear d-uple of support 0
E X and (x,y) be local coordinates of X at 0 for which the ideal of o
is (xd, y). The unique (d-l)-tuple
N O that is a subscheme of
~O has as ideal
(xd-I , y). Let us give charts for Hilb d X and Hilbd-Ix at ~o
and qo : a d-tuple ~ near
~o will have as ideal l(~)
(
Xd
+
and a (d-1)-tuple
I a/x d-1 +...+ a d , Y + alxd-1 +...+ a d,)
N near q ° will have as ideal
I(~) = (xd-I + blx d-2 +...+ bd_ / , y + blxd-2 +...+ b~_ I ) The inclusion q t-~ is equivalent to I(~) t- I(N). Looking at the euclidean division
d x
d-1 +
alx
+ ...
(a2-b 2 -bl(al-bl))xd-2
+ ad_lX
d-1 +
ad
x
+
bd_ 1
(a 1 - b l )
+(ad_l-bd_l-bd_2(al-bl))x + (a d - b d _ 1 ( a l - b l )
+...+
+ ,,. x
• ..
d-2 + btx
)
in which the rest must be zero, we already get the first conditions: a2
= b 2 + bl(al - bl)
(*) ad_ I = bd_l+bd_2(al-b I) ad
=
bd_l(a I - bl)
On the other hand, to demand that y + alxd-1
' +...+ ad
E
l(q) is equivalent to
demand that the difference ( y + a~x d-I +...+ a~) - ( y + b~x d-2
+...+
b~[_ 1 )
belong to l(n); this gives
alxdSo we must have
+
(a;-b I ) x d - 2
+'''+ ad'-bd-'I
~
l(q) .
85
a~ (**)
=
b 1
+
a;b I
...
i !
= h _l+a bd_ 1
The equations (*) and (**) locally exhibit Hilb d X as the graph of a ~2d t o ~ 2 d - 2 ;
II)
morphism from
hence the smoothness of ~ilb d X in the curvilinear part.
The residual morphism R
Let N be a (d-1)-tuple scheme theoretically included in a d-tuple ~ : N C ~ . One can then define a closed point Res(N,~) of ~, also denoted abusively
~ - B,
using the following lemma. Lemma: Let A be an algebra of finite rank over a field and I
C
A be an ideal of
(linear) dimension I. Then the annihilator Ann (I) = (0:I) is of codimension I. In our case, if I is the ideal of B in
~
, we define the point Res(n,~) by
its ideal Ann (I) . Warning: Given a point m
C ~,
it is not possible in general to define a residual
(d-1)-tuple: for example look at the triplet ~ = Spec ¢ EEx,y]J /(x,y) 2 We have thus defined a mapping R
>
: ~ilb 3 ~2 (t,d)
:
~p2
~ Res (d,t)
that we shall show to be a morphism. Once again we shall examine R only in the neighbourhood of (to,d ° ) where to has~pL2o for ideal ( as always it is the most complicated case). Resuming the notations of § I we are going to calculate the coordinates (ml,m 2) of m, the residual of d in t. The ideal of m is ~
= (x-m I , Y-m2). So we have
to express the condition =
(l(t)
: (f))
where f = -y + a'x + b', which translates the fact that the doublet d is defined by
86
f = 0. Hence in particular
the following
conditions
(x - ml)(-y + a'x + b' )
C
I(t)
(x - m2)(-y + a'x + b' )
E
I(t)
2 y2 or after replacing x , xy and by their values i
x(u' - a'u + b' - a'ml) x(u" - a'u' - a'm 2)
No
non-zero
affine
must hold:
drawn from I(t)
(see § I) :
+ y(v' - a'v + m|) + w' - a'w - b'ml
+ y(b'
+ v" - a'v'
+ m2)+ w" -a'w' -b'm 2
linear fonction can be in I(t) because
not lie on lines. Hence the two forms above are zero;
triplets
~
I(t)
~
I(t)
close to t
o
do
in particular
v' - a'v + m I = b' + v" - a'v' + m 2 = O . Hence the seeked coordinates
of the
residual
point:
m| = a'v - v'
I
m 2
expressed blished
=
a'v'
-
v"
in the chart
in the
-
b'
(a,a',b',v,v',v")
preceding
paragraph.
of Hilb 3
Of course,
p2
at (to,d ° ) which we had esta-
these explicit
formulas
imply that
R is a morphism.
Iii) The results
a) Ellingsrud equivalence
of Ellingsrud - Str~mme
and Str~mme
[ 6 ] showed that the groups CH i (Hilb d F 2) of rational
are free of finite type and have computed
Their main tool is the following Theorem
(with d = 3)
(Bialynicki-Birula):
action
with only
points
x
finitely
the corresponding
result.
Let X be a smooth projective
many f i x e d
~ X for which the limit
Betti numbers.
points
of t.x
variety and ~
m
x X --~ X an
PI'
"°" ' p P . D e n o t e by Up. t h e s e t o f 1 f o r t - - ~ O i s P . . Then: 1
i)
t h e Up. a r e t h e c e l l s o f a c e l l u l a r d e c o m p o s i t i o n o f X; 1 i i ) f o r t h e i n d u c e d a c t i o n ~m x TpX ---~ TpX. , t h e t a n g e n t s p a c e TpUp. . i s 1 1 11 t o r s p a c e o f TpX w h e r e t h e w e i g h t s a r e p o s i t i v e . 1 In order
to calculate
t h e g r o u p s CHi
(Hilb d ~2),
Ellingsrud
and S t r ~ e
the subvec-
by u s i n g
87 this theorem are led to delicate combinatorial arguments, when d is arbitrary. On the other hand, when d = 3 there is no difficulty and we explain their proof geometrically.
b) Consider on ~2 the action x ~2
)
~2
m
(t,(Xo:X]:X2)) ! The r e a s o n f o r t3x2 w h i l e t2x2
>
(Xo:tX1:t3x 2)
is expected will
a p p e a r a few l i n e s
below. This action
has for only fixed points the three points Po = (1:0:0)
PI = (0:1:0)
P2 = (01011)
This action of ~m on ~2 induces an action on the Hilbert scheme Hilb ~2 = ~p Hilb P In p a r t i c u l a r
v on t h e component ~2 composed o f t h e l i n e s , v
~
x ~2
.,>
this
action
~2
is
~2
m
(t,~o:U1:U2]) where of course lines
for this
[Uo:U]lU2] action
' )
~o:t-lu1:t-3u2 ]
denotes the line UoX ° + ulx ] + u2x 2 = 0
of •
. The invariant
on ~2 a r e m
Lo ° 0:o:o] Similarly
Eo: :o]
the induced action
of •
= E0:o:1]
on t h e component ( i s o m o r p h i c t o p5) o f H i l b ~2
m
composed o f t h e c o n i c s h a s o n l y s i x f i x e d p o i n t s : w h e r e t h e c h o i c e o f t3x2
rather
t h a n t2x2 o c c u r s ,
the conics
as has already
2 a b o v e : we do n o t want t h e c o n i c XoX2 - ax 1 t o be i n v a r i a n t H e n c e f o r t h we o n l y s t u d y t h e a c t i o n
triplets. To apply Bialynicki-Birula's under this ded in
action.
Afortiori
{Po' PI' P2 }
of ~
m
This is
been pointed ou-
when a # O.
on t h e component H i l b 3 ~2 composed o f
theorem let us look for the triplets ~ fixed
Supp (~) i s a f i x e d p a r t
o f ~ 2 , h e n c e Supp (~) i s i n c l u -
. We introduce the following notations:
dOT is the doublet of support {Po} lying on the blets dij
2L. and L. U L . . 1 1 j
(with Supp dij
line PoP~; we so define six dou-
= {Pi} )"
t|2 is the aligned triplet of support
{PI} lying on the line P ~ 2
; we so define
six triplets tij (with Supp tij = {Pi}) 2 P2
is the first infinitesimal neighbourhood of P2 in ~2 ; we so define three tri-
88
plets p2l
(with Supp p2i
= {Pi } )"
P2 ~
P
d _.
L2
~..~2
0
L1
The action of ¢
on Hilb 3 p2 then admits of 22 fixed triplets $; we ~ive them in m
the following table, together with the dimension of the corresponding cell U~ . Fixed triplet
dim U
Fixed triplet
dim U
Po o d10
4
{Po ,PI,P2 }
3
Po U d12
4
to1
6
Po U d20
3
to2
4
Po U d21
2
t10
3
PI ~ do1
5
t12
3
PI u do2
4
t20
2
PI U d20
2
t21
I
PI u d21
I
p2
5
P2 u do1
4
P2 U do2
3
2 PI 2 P2
P2 U d10
2
P2 U d12
2
o
3 O
89
First of all we give proof for the list of fixed triplets. Our argument rests on the two following essential facts: the fixed doublets of support {eo} two
fixed lines through Po are PoPI
and
the fixed triplets of support {Po} other fixed triplet of support {Po} would be of the form (x 2 - a x ~ , not invariant under •
m
(for example) are do1 and do2 , since the only PoP2 ; (for examDle) are toi, to2 and P2o, indeed any
would have a fixed tangent and hence its ideal
x~)or
else
(x I - b x ~ ,
x~ ). But these ideals are
if a and b are non-zero (use of t3x2 rather than t2x2 !)
As for the calculation of the dimension of the cell U~ , we compute case after case, using part ii) of the theorem of Bialynicki-Birula. Let us give two examples among the 22. First example: t12 Local coordinates for ~2 at PI = (O:1:O) are (Xo,X 2) and in those coordinates the ideal
of t12 is (x~ , Xo). A chart for Hilb 3 ~2 at that ideal is given by
(a,b,c,a',b',c') corresponding to the ideal 2 + b'x2 + c' ) xo + a ,x2 ~2 On the other hand the action of @m on near P| is written I = ( x~ + ax~ + bx2 + e
,
t.(Xo,X 2) = t.(Xo:1:x2) = (Xo:t:t3x2) = (t-lxo:1:t2x2) = (t-lxo't2x2) The action of ~
m
on the ideals is an action on equations. It is written therefore
(dually)
t.f(Xo,X 2) = f(tXo,t-2x2 ),
as has already been seen for ~2 (linear equations). Finally the action of @m on Hilb 3 ~2 is written near t12 as t.l = (t-6 x 23 + t-4 ax 22 + t-2bx2 + c , tx ° + t-4 a ,x 22 + t-2b'x2 + c') = (x~
+ t2ax~
The action of •
on m
"
T
+ t4bx2 + t 6 C
, X O + t -5 a ,x 22
+ t-3b'x2 + t-|c ')
Hilb 3 ~2is thus t12
t.(a,b,c,a',bW,c ')
= (t2a,t4b,t6c,t-5a, t-3b, t-lc,).
As there are three positive weights, the dimension of the cell U
is 3. t12
90 2 Second example: P2 Local coordinates for ~2
at P2 = (0:0:1) are (Xo,Xl) and in these coordinates
2 2 the ideal of P2 is (x-~ , XoXl, xl). A chart for Hilb 3 ~2 at that ideal is given by (u,v,u',v',u",v")
corresponding
to the ideal
I = (x 2o + UXo + VXl + w , XoX I + U'Xo + v'x I + w' , x~ + u"Xo + v"xl + w") where w,w',w" are the functions of u,v,u',v',u",v"
seen in § I (equations
(I),(2) and
(3)). ~2
Now the action of gm on
near P2 can be written
t.(Xo,Xl) = (t-3Xo ' t-2xl) and so the action of g
m
on Hilb 3 ~2 near P22
is
t.I = (x2+t-3ou x ° +t-4vxl +t-6 w , X o X 1 + t -2 u ,Xo+t-3v,xl +t -5 w , , x 12+ t -I u ,,X o + t -2 v ,,x 1 + t -4 w ,,. ). So • m
acts on T 2 Hilb 3 ~2 by P2 t.(u,v,u',v',u",v")
= (t-3u',t-4v',t-2u',t-3v',t-lu",t-2v '')
All the weights being negative we have dim U 2 = O. P2 This is the way the preceding table can be justified. The rank of the group CHi(Hilb 3 e 2) is thus obtained by looking at the number of cells of dimension 6-i. Hence the result:
i
0
CH I
1
2
3
4
5
6
g2
~5
26
~5
~
Z
IV) The rin$ structure of CH'(Hilb 3 ~2)
We have announced in can be found in
[3 ]
the following results, the complete proofs of which
[4 ]
a) Explicit basis of CH I and CH 5 Let us introduce the subvariety of Hilb 3 ~2 composed of aligned triplets which are subschemes of a line). The associated divisor will be denoted by A.
(i.e.
91
Now let H be the divisor composed of triplets one point of which lies on a fixed line. Let us be more accurate; we have the diagram
p2
R
~ilb 3 ~2
Hilb 3 2 where the residual morphism R has been defined in § II . If ~ is the hyperplane divisor of
~2,
~ we set H = ~,H
r.~ where H = R*~.
Moreover the following curves lie in Hilb 3 ~2 and have their class in CH 5. - The curve
~I is the set of triplets composed of a moving point on a fixed line
and of two fixed points out of this line. - In order to define the curve
~
o
we take a fixed line, a fixed point and a moving
point lying on this line and a third, fixed point outside this line. Beware that
~ o is not
pairing table obtained via
a specialization of
CH I x CH 5
~
H
A
~' o
1
0
~I
I
I
Since CH I (resp. CH 5) is isomorphic to ~ , {%,
~i})
~I as is shown by the following
the degree map
deg :
(resp.
Make drawings...
CH 6 =
the preceding table shows that
{H, A }
is a Z-basis of it.
b) Explicit basis of CH 2 and CH 4 We denote by h the cycle in Hilb 3 ~2 composed of the following triplets: one point is fixed and the other two are arbitrary. The precise definition of h is given by the formula h = =,(H
where H
has been defined in a).
Analogously, we denote by p the cycle composed of the following triplets: on a fixed line we choose two moving points, the third point is arbitrary. Let us be more accurate: consider the ~orphism
~ : ~ilb 3 @2
v~2
v unique line of ~2 containing the doublet d. Let then ~ and let P
=
which associates to (t,d) the v be the hyperplane class of ~2
~ • The precise definition of the cycle p is then p = =,(P
).
92
Finally, let a be the cycle of triplets aligned with a fixed point; we obviously have a.p = O. It can be checked (via the projection formula) that h 2 consists in: two points, in the triplet, fixed; hp consists in: one point fixed and the other two on a fixed line; ha consists in: three points on a fixed line, one of them being fixed. We give the table of intersection degrees between on the one hand the monomials H2,HA, h,a,p of CH 2 and on the other hand the monomials
H2h, H2a, h 2, ha, hp of CH 4.
H2
HA
h
a
p
H2h
3
3
I
I
I
H2a
6
-3
I
-I
0
h2
I
I
I
0
0
ha
I
-I
0
0
0
hp
I
0
0
0
0
Since this matrix has determinant I, we have the Theorem: I ° ) { H 2, HA, h, a, p }
is a ~-basis of CH 2 ;
2 °) ~ H2h,H2a,h2,ha,hp}
is a ~-basis of CH 4 .
c) Explicit basis of CH 3 Let us introduce in Hilb 3 ~2 the cycle ~ composed of triplets lying on a fixed line, as well as
~ 2) composed of the triplets of which one point is on a
~ = ~,(P .H
fixed line L I and the other two lie on a fixed line L 2. We give the table of intersection
degrees of the pairs of monomials taken in the list H 3,
H3
H2A
Hh
Ha
~
H3
15
15
3
6
I
3
H2A
15
3
3
-3
-I
I
Hh
3
3
I
I
0
0
Ha
6
-3
I
-I
0
0
(~
I
-I
0
0
0
0
B
3
I
0
0
0
I
H2A, Hh, Ha, ~ , B :
93
Since this matrix has determinant I, we have the Theorem: {H 3, H2A, Hh, Ha,
~ , ~ } i s a ~-basis of CH 3 .
V) Examples of enumeratiye ap.~lications
I°) Preliminaries on families of curves a) First of all, let C be a fixed curveof degree d in ~2. Consider the subscheme Hilb 3 C of Hilb 3 ~2 and the associate cycle 8~ilb 3 C] E CH 3. We obtain the following intersection numbers with the chosen basis (IV.c): ~ilb 3 8 . H 3 = d3
Bilb 3 C].H2A = d2(d-2)
B i l b 3 C].Hh = 0
E i l b 3 a.Ha = d(d21)
EEilb 3 C].~
hHilb 3 C].6
= (d)
= d(~)
By solving a linear system, we obtain for the cycle ~iilb 3 C~ the expression:
b) Let ~
be a pencil parametrized by ~1 of curves of degree d in tP2. Associated
to it is a cycle scheme ~
C
~d
~ CH2 in the following manner. This pencil is a relative sub-
p2 x FI over p1. We can thus consider
Hilb 3 (~/~I) C
the relative Hilbert scheme
Hilb 3 F 2 x ~I. We obtain the required cycle
~d through the pro-
jection upon Hilb 3 •2. In the basis studied in IV.b
of CH 2 this cycle is expressed by 2
This is proved
h
+ a + (l-d) p .
as in a), but here one has to introduce some auxiliary cycles. For
details, see E 5 ] c) Finally, to a net X C fore a cycle
~2 x ~ 2 of curves of degree d in ~2 we associate as be-
~d ~ CHI: it is composed of the triplets lying on a curve of the net.
More precisely, we project Hilb 3 (j~/~2) t'- Hilb 3 ~2 x ~2 on Hilb 3 ~2 to obtain Taking up the notations of IV.a, we have ~o" ~ d
= d - I
According to the table (IV.a) we have thus
and
~I" '~d
=
d
~d"
94
~d
2 ° ) Three enumerative
= (d-
I) H
+
A
formulas
Let us give three applications
of the calculation
of
~d
that we have just made
(i .o). a) Let F
be a smooth fixed curve of degree n in ~ 2
family in Hilb 3 ~2 composed of triplets contained
We call y the one-parameter
in F (in the scheme-theoretic
se ) and whose support is one point. The cycle y is in CH 5 and we will compute
senits
degree of intersection with H and A. Set-theoretically each with multiplicity composed
H N
y is composed of n triplets, but they are to he counted
3 (see 3.a below).
set-theoretically
Hence deg H.~ = 3n. Analogously,
of 3n(n - 2) flexes of F (Semple-Roth
of these is to be counted with multiplicity
[18]
A ~ ~ is
,p.78). Each
I (see 3.b below). Hence deg A.y =
3n(n - 2). We deduce from the calculation deg
~d.y
of the class
~d
done in 1.e the formula
= 3n(d + n - 3) .
This formula gives the number of curves of the net
~
osculating
a fixed
of degree n. This formula is a special case of a formula of Schubert's, and Speiser have reproved by the method of ordered triangles
b) Another formula which can be deduced from the class number of curves of the net having a singular point on pose y' the one parameter where~i
~d
in
[I7]
is the one giving the
family in Hilb 3 ~2 formed of the triplets of i d e a l ' ~ triplets
aligned~ one has deg A.y' = O. On the other hand H ~ V' is composed
Consequently
which Roberts
F . Let us call for this pur-
is the ideal of a simple point of r. Tile considered
on n triplets,
curve
but they are to be counted each with multiplicity
2,
never being
set-theoretically
3 (see 3.c below).
the following formula obtains deg ~ d . ¥'
= 3n(d - I).
This formula gives the number of curves of the net a fixed curve of degree n.
~
having a singular point on
95
c)
Let T be the surface in Hilb 3 ~2 composed of the s q u a r e s 2~ "L ' ~
of closed points.
It is clear that T.A = O since such triplets are never aligned. On
the other hand, set-theoretically the ideal of the intersection shows that the intersection
TH 2 is represented by the triplet ~ 2
of two fixed lines. A calculation
is to be counted with multiplicity
If we now consider two nets of curves and their associated ~d2
of the ideals
where~
is
given below (3.d) 9: deg TH 2 = 9. cycles
~dl
and
in CH 2, we obtain thus the degree deg
~ d I. ~ d 2 .
r
= 9(d I - 1)(d 2 - I)
which gives the number of singular points lying on curves of both nets.
3 ° ) Justification
of the multiplicities
a) Let us show that each intersection of H with T counts with multiplicity
3 (see
2.a). We take local coordinates
(x,y) in which F is given by y = f(x) and the line
defining H by x = O. So we consider the triplet of H ~ Local coordinates
with ideal (x 3, y - f(x)).
for Hilb 3 ~2 at this triplet are (a,b,c,a',b',c')
corresponding
to
the ideal (x 3 - 3ax 2 + 3bx - e , y - f(x) + a'x 2 + b'x + c') (The coefficients
are for computational
convenience).
In this chart, y is expressed
by the five equations: I
a' = b '= c' = O which means: c = a 3, b = a 2 which means:
the triplet is on
F;
the triplet hasas support one only point.
(These last two equations are obtained by developing
(×-a)3).
The divisor H is described by the equation c = O, which means: the y-axis.
The ideal of the scheme-theoretic
intersection H ~
the triplet meets
~ is thus, in these
coordinates, (a',b',c,,c,a3,b hence the multiplicity
- a 2)
is indeed 3.
b) Let us show that each ordinary flex counts with multiplicity tion A ~ ~. We take local coordinates
I in the intersec-
(x,y) in which F is given by y = f(x) where
96
val
f = 3 (val O
is the valuation).
With these notations
the intersection A t'~ y is
O
near the origin,
the triplet of ideal
(x3,y). Local coordinates
triplet are given by (a,b,c,a',b',c') triplet
corresponding
for Hilb 3 ~2 at this
to the ideal of a neighbouring
t: i(t) = (x 3 - 3ax 2 + 3bx - c , y + a'x 2 + b'x + e')
In this chart the divisor A of aligned triplets such a triplet
is given by a' = 0 (the line on which
lies is then y + b'x + c' = 0). As seen above
(*)
c = a3
assure us that Supp t I(t) or equivalently By Taylor's
.
is one only point. Let us now express ~
formula this is equivalent f(a) +f' (a) (x-a) +
Hence the last three equations
(**)
that t is on F , i.e.
I(F) = ((f(x) - y)
: f(x) + a'x 2 + b'x + c'
a)+
in a), the conditions
, b =a 2
~f"(a)
E
((x - a) 3)
to
--~f"(a)(x-a) 2 + a'x 2 + b'x + c' = O. defining y in this chart: =
0
b' + f'(a) - af"(a)
= O 1 2 ..... + ~ a ~ ~a# = O .
c' + f(a) - af'(a) In ~ 6 with coordinates
a,b,c,a',b',c',
c,b,c',b'
of a. Thus there remain for the scheme-theoretic
A~y
as functions
(*) and the last two equations
of (**) yield intersection
: I
a' = 0 As val O
and
f" = I, we see that the multiplicity
c) Let us show that each triplet city 3. Let a~ain line defining H by
0
a' + 2f'(a)
i
is I.
of the intersection
(x,v) be local coordinates x = O. We consider
such that
the triplet
and a chart for Hilb 3 p2 at that triplet
H~
y'
counts with multipli-
F is ~iven by ¥ = f(x) and the
t of ideal
(x,y) 2 = (x 2, xy, y2)
is of the form (u,v,u',v',u",v")
ding to the ideal (x 2 + ux + vy + w , xy + u'x + v'y + w' where w, w', w " a r e
given by the equations
The curve y' admits of the
, y2 + u"x + v"y + w")
(I), (2) and
~rametrization
(3) of § I.
correspon-
97
s ~
(x-s ,y-f(s)) 2 = (x2-2sx+s 2 , xy-f(s)x-sy+sf(s)
, y2-2f(s)y+f(s)2)
c o r r e s p o n d i n g to the square of the ideals of the points of F. In our chart the param e t r i z a t i o n yields u = -2S
,
v = 0
,
u' = - f(s)
, v' = - S
, u" = 0
,
v" = -2f(s).
Hence the equations for y' in a n e i g h b o u r h o o d of t: u = 2v'
,
v = 0
,
u' = -f(-v')
,
u" = 0
,
v" = -2f(-v')
N o w w e express the fact that the d i v i s o r H is d e f i n e d by requiring the triplet to cut the y-axis. This means i d e a l - t h e o r e t i c a l l y (vy + w , v'y + w'
. y2 + v"y + w" )
i) On the open subset v # 0 of our chart, (a)
O = - wv' + w'v
(b)
0 = w 2 - w"w
# 0
this means, after replacing y by - w/v,
+ w"v 2.
R e p l a c i n g w, w', w" by their values given in (I),(2),(13) of § I, we get the equations (a')
O = uv '2 - v ~ ' v " + 2u'vv' + v '3 - u"v 2 := F,
(b')
0 = (v' - u)F
.
ii) On the open subset v' # O, after replacing y by - w'/v'
, we get the two condi-
tions (a)
0
=
(c)
0 =
-
wv'
+
w'v
w '2 - v ' v " w + w " v '2 ;
replacing w, w', w" by their values we get (c')
(a') and
0 = - u"F.
Summing up, the e q u a t i o n for H is
F = O
since this is true outside of v = v' = 0
which is of c o d i m e n s i o n 2. Finally the ideal of the scheme-theoretic (u - 2v'
, v , u' + f(-v')
i n t e r s e c t i o n H ~ y' is
v '3
u"
v" + 2f(v'))
and the m u l t i p l i c i t y is 3.
d) We n o w show deg TH 2 = 9 (see 2.c). Let H I be the divisor of the triplets m e e t i n g the y-axis and H 2 the divisor of the triplets m e e t i n g the x-axis. Let us give equations for T n e a r the ideal
(x,y) 2.
98
The
subvariety
T is c o m p o s e d
(x + a, y + b) 2 = (x 2 + 2ax and
so in the c h a r t
u = 2a hence
the
us
,
look
from
=
there
2v'
and
gets
,
,
v = O
=
,
ideal
to t a k i n g and
(***).
of
ideals
+ ay + ab
, y2
+ 2by
in § I c o r r e s p o n d i n g
+ w'
h a n d we -
the
denotes
as c u r v e
v'
chart
, Y
2
+
u"x
to the
v"y
+
+ b2)
w")
+
ideal
,
=
0
,
v"
uv v~
ideal
_
by y'
= b
,
v'
set
T m
v"-
2u'
v'
in c)
the
'' +
2[llvv
w~v
of
the
the
set
F the x - a x i s )
T ~
2u'
v"
~
,
= 2b;
of
the
= 0
0
in the
reduced
, u" = 0
=
.
parametrization
curve , v"
T ~
H I:
= 2b
HI : u"
+
T ~
v t3
(*)
)
of H I : -
u"v 2
HI:
u", v '3) H I (which
o n e has
u"
corresponds
equation
scheme
v, v " - 2 u ' ,
u" = 0
=
this
representation u'
saw
= a
:
of the
, v
(u-2v', If one
v
parametric
the o t h e r
so one
+ v'y
u' = b
the
, the
0
seen
H 1;set-theoretically
(u
On
,
of T in this
at T ~
u = 0
+ a 2 , xy + bx
, xy + u ' x
v = O
equations
to a = O. H e n c e
and
+ w
squares
of T is
u
Let
the
(u,v,u',v',u",v")
(x 2 + u x + vy a parametrization
of
(***) corresponds
T . H I = 3y'
So T . H I . H 2 = 3 y ' H 2 = 9, a c c o r d i n g
to c).
in the n o t a t i o n s
by c o m p a r i s o n
of
the
of 2.b
ideals
(*)
99
REFERENCES
A. Bialynicki-Birula,
Some theorems on actions of algebraic groups, Annals of
Maths 98 (1973), 480-497. J. Brian¢on, Description de Hilb n ¢ { x , y } ~ Inv. Math. 41 (1977), 45-89. G. Elencwajg - P. Le Barz, Dgtermination de l'anneau de Chow de Hilb 3 ~2, C.R. Acad. Sc. Paris, t. 301 (1985), 635 - 638. G. Elencwajg - P. Le Barz, L'anneau CH'( Hilb 3 ~2) des triangles du plan, paraltre. G. Elencwajg - P. Le Barz, Applications gnumgratives du calcul de CH'(Hilb3~2), paraltre. G. Ellingsrud - S. Str~n~ne, On the homology of the Hilbert scheme of points in the plane, Inv. Math.
87 (1987), 343 - 352.
J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511- 521. J. Fogarty, Algebraic families on an algebraic surface II, the Picard scheme of the punctual Hilbert scheme, Am. J. Math. 95 (1973), 660 - 687. 9
A. Grothendieck, Lea sehgmas de Hilbert, S~minaire Bourbaki 221, Paris (1961).
10
R. Hartshorne, Conneetedness of the Hilbert scheme, Publ. Math. IHES 29 (1966), 261 - 304 .
11
R. Hartshorne, Algebraic Geometry, Springer-Verlag
(New-York - Heidelberg -
Berlin), 1977. 12
A. Iarrobino, Hilbert scheme of points: an overview of last ten years, Proceedings of the AMS Bowdoin Conference, 1985.
13
A. Iarrobino, Reducibility of the families of O-dimensional Schemes on a Variety, Inv. Maths. 15 (1972), 72-77.
100
14
A. larrobino,
Punctual Hilbert schemes.
Bull. Amer. Math.
Soc. 78 (1972), 819-
823. 15
A. larrobino - J. Emsalem, nite
16
Some zero-dimensional
generic
algebras having small tangent space, Comp. Math.
J. Roberts,
singularities;
fi-
36 (1978),145 - 188.
Old and new results about the triangle varieties,
these Procee-
dings. 17
J. Roberts - R. Speiser, Enumerative Alg.
12 (1984),
1213-1255
Geometry of triangles
; 14 (1986),
18
J. Semple - L. Roth, Algebraic Geometry,
19
H. Schubert, Anzahlgeometrische 153 - 212.
155-191;
I; II; III, Comm.
to appear.
Clarendon Press, Oxford
(1949).
Behandlung des Dreiecks, Math. Ann.
17 (1880),
Iterated
Blow-ups and Neduli for Rational
Surfaces
Brian Harbourne Department of Mathematics and S t a t i s t i c s University of Nebraska-Lincoln L i n c o l n , ICE 6 8 5 8 8 - 0 3 2 3
This
paper
Workshop
in
conference efforts
is
the
made i t
out
[K] h a s
summer
that
be
they
to
surfaces is
an
of
to
them w i l l applied
for
basic
obtained
and
(cf.
the
based
Sundance
on
Enumerative
[H1].
like
It
to
was
a
Geometry very
nice
t h a n k Bob S p e i s e r
whose
a n d BYU a n d NSF w h o s e s u p p o r t made i t p o s s i b l e .
point
a conference here are
theory.
on t r i a n g l e extending
the
They a l s o
this
be useful. problems
surfaces
of
given
closed
is
are
not
very
closely
Whether
yet
moduli
related
but
Picard
number
field.
n+l)O,
setting
will
is
a is
deeper indicate
are
in
simply
of p2(k),
where k
implicitly
be suppressed
to turn
surfaces,
which
near points
The
this
rational
me
Kleiman
they will
known,
for
let
that
What I w a n t t o do h e r e of
infinitely
[AI].[DM].[M1]) but
geometry,
blowings-up
[RS,S].
work
certainly study
on e n u m e r a t i v e iterated
varieties
to
algebraically
stacks
at
I would
by b l o w i n g up n p o s s i b l y
arbitrary
algebraic
multiple
talk
largely
objects
and S p e i s e r
may b e
particular,
1986,
surroundings,
the principle
essential
understanding how
my
speaking of moduli at
applied
to
of
of
so successful,
work o f R o b e r t s out
account
in beautiful
To j u s t i f y point
an
that
of
in the interest
of
concreteness. In solving variation
of
a moduli problem,
structure
parametrization objects, the
it
way
of
single
jumping
geometrically,
a moduli
uniqueness
a
one a t t e m p t s
a single
parameter
still
that
no
regarding
recover
extra
for
each
but
from
the
beauty that
e v e n s o we c a n a t
us
to
handle
it
the
the beauty of encoding information
work a l l o w s
regarding Since
isomorphism
of a moduli space is
exist,
space.
this
class
automorphisms, nor can it Apart
one a t t r a c t i o n
information
the moduli
point
structure.
space need not often
with
of
to encode all
parametrization,
c a n e n c o d e no i n f o r m a t i o n
information
a
clots
phenomenon
However,
via
deal
o£
of with
encoding
is canonical. cost
of
losing
geometrically,
automorphisms and
jumping
in of
structure. To
see
surfaces, schemes of where
for
how
this
might
be
done,
a
moduli
f o r example, F would he the c o n t r a v a r i a n t finite any
type over an algebraically
such
s c h e m e T,
F(T)
is
p r o p e r and smooth o v e r T whose f i b r e s that F is representable with a universal
f a m i l y Q.
gratefully
the at
b y a s p a c e M; i . e . ,
by a u n i o n X o f o p e n s u b s e t s The a u t h o r
consider
functor
closed set
of
closed
field
The g l u i n g
acknowledges support
For
are
basic
from the category k,
to
basic
Sch of
the category of
surfaces,
data
is
o£ M by f i n d i n g indicated
f r o m a n NSF g r a n t .
Sets,
schemes Suppose
the moduli space for F and it
One c a n g i v e a p r e s e n t a t i o n o f M.
F.
T-isomorphism classes
points
M is
functor
comes
a c o v e r X---$f
by t h e p r o j e c t i o n
102
maps o f t h e f i b r e
product
XXMX t o X.
T h u s we h a v e a d i a g r a m : s
7 ×
XXMX where the family
two m a p s o f Xx~rX t o X a r e
Xfl=X×~Q w h i c h we may a l s o
following
the projections.
denote
as
The m o r p h i s m ~:X---~ i n d u c e s
v~(O).
In
this
situation
we h a v e
a
the
facts:
i)
The family
versality but
- ,M
implies
bear
X~ i s v e r s a l
that
and any family
is
~ is smooth (which of course
with
me) a n d now s a n d
The
space
t are
pullhacks
locally
a pull
is clear
here
of
h a c k o f X~. for other
v by v a n d
so are
But
reasons, themselves
smooth. ii) morphisms
XXMX r e p r e s e n t s
fi:U---*X,i=l,2,
Isom(U)
the is
functor
the
set
Isom where
of
for
any
U-isomorphisms
space
from
fl
U and
(X fl)
to
f2 (x ~ ) Unfortunately, representability with
the moduli functor would imply that
p2 t h e r e
are
nontrivial
our diagram above,
a moduli
for hasic
locally
surfaces
trivial
p2-bundles.
is not representable
families
In such a case
b u t we c a n h o p e t o s a l v a g e
everything
functor F, we will say that a family D£F(X)
are
trivial,
there
since
but already
can be no space M in
else.
In particular,
given
together with the diagram (~)
of morphisms of schemes s
I solves
t h e m o d u l i p r o b l e m f o r F, p r o v i d e d i)
The f a m i l y p i s v e r s a l
f r o m ~;
i.e.,
if
neighborhood
GeF(Y)
for
-~x that
the following
and any family
locally
some s c h e m e Y,
V a n d a m o r p h i s m V--~X s u c h
(~)
then
that
each
the
conditions
for
the etale
point
family
hold: topology
y of
Y has
G×yV---*V i s
comes
an
etale
isomorphic
to
~XxV---*V. ii)
The
morphisms f2~(p).
(I.e.,
a natural the
a
represents Isom(U)
£1xf2
and
t.
smooth (viz.
Versality
While the other present,
of of
I
to
and
) t o Hom(
of
for
,XxX) d e f i n e d
By Y o n e d a ' s
in
where
any
U-isomorphisms
scheme from
from XxX-schemes to sets.)
195].
lemma t h i s the
and second factors
o f p now i m p l i e s
composition
way t h a t
to
We h a v e
by mapping Isom(U) natural
respectively
the standard
U and
fl*(p)
of are
to
transformation which
with
the
the morphisms s
the morphisms s and
t
III.4).
space
a presentation information
Isom
set
functor
XxX [ S B , e x p . first
Corollary
the moduli hand,
functor
now t h e
from Isom( }tom(U,XxX).
o f XxX t o i t s
are
the is
Isom is a contravariant
morphism
projections
to
I
transformation
element
gives
scheme
fi:U---*X,i=l,2,
is
unique
if
can often about
it
exists,
a presentation
be found even if
automorphisms
is
there
automatically
(~)
is not.
is no moduli encoded
On space
into
I.
103
Since there
i s no c a n o n i c a l
To a v o i d r e p e t i t i o n , of
finite
c h o i c e o f X, e x t r i n s i c
we h e r e b y a g r e e
type over an algebraically
that
closed
reasons must guide its
throughout field
k,
this
unless
choice.
paper all
schemes are
specifically
provided
for otherwise.
I.
The f i r s t
order
Iterated
of business
is
Blowings-up
to parametrize
basic
surfaces
of given
Picard
number.
A b l o w i n g - u p o f p2 o f P i c a r d number n + l i s s i m p l y t h e b l o w i n g - u p o£ p2 a t n
points.
A convenient
is
by t h e v a r i e t y
single
closed
point,
Y=Xi_l-variety by Z,
iterated
and
let
fibre
Y-variety
morphism with
product via
blowings-up
blowings-up
X0 b e p2.
X=Xi a s f o l l o w s .
form t h e
becomes a
way t o p a r a m e t r i z e
Xi_ 1 o f
YXzY.
the
Then
By a s s u m p t i o n ,
for
each
o£ YXzY t o
the
let
i>O 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<3, W i s j u s t
E={eo,el,o--,en the
of
index,
PicV to
together
e l - - ~ o - e 2 - e 3, the permutation
with
}
W) b e itself the
e 2 - - - ~ O - e l - e 3. g r o u p on t h e
114
elements of E other is
a
Weyl
t h a n e O.
group,
intersection
it
It
preserves
f o r m on P i c V.
Proposition
V.1.
t u r n s o u t t h a t W h a s some n i c e p r o p e r t i e s ; the
however,
L e t E' be a n o t h e r of
canonical
For u s ,
linear
transformation
order,
t o t h e e l e m e n t s o£ E ' .
itself
blowing-up
n
configurations Proof.
P2 k
at
This is essentially
discussion
see [H2].
Proposition Proof.
V.2.
to
o f E,
points,
in
W d o e s n o t d e p e n d on Moreover, then
o f N a g a t a IN] i n d i f f e r e n t
the
if V is a
exceptional
terminology.
For a
there
the
is
u n i o n o f n o n e m p t y c o m p o n e n t s I w, wEW.
a natural
structure
of
a
transformation
reduced
i n d u c e s a m o r p h i s m I---~ [SB 195].
see
w be t h e
the elements
definition.
general
the
of W is:
{wE={weo,Wel,**-,Wen}:WEW}.
a result
show t h a t
image o f wEW, we s e e t h a t remains
preserves
*
w h e r e we endow W w i t h transformation
it
it
f o r V and l e t
i n W; i n p a r t i c u l a r ,
The scheme I i s a d i s j o i n t
We w i l l
and
by s e n d i n g
used in its
sufficiently
of V are precisely
V,
configuration
defined
Then w i s
configuration
o£
the most important property
exceptional
Pic V to
the choice of exceptional of
class
e.g.,
that
I is a disjoint
I---4q
is
discrete
Hom(
,I)--JrIom(
scheme.
By d e f i n i n g
I
w
,W),
This
natural
to be t h e
inverse
u n i o n o f c o m p o n e n t s I w, wEW, and i t m e r e l y
surjective
to
conclude
that
these
components
are
nonempty. Let Y be a connected Y to
X,
and
ordered
an
V.1
A m o r p h i s m f:Y---*I i n d u c e s m o r p h i s m s s f a n d t r o f
isomorphism ¢ between
blowings-up
Proposition
scheme.
over
and
Y.
Theorem
This IV.3
the
corresponding
induces is
an
families
an automorphism of
element
of
W, g i v i n g
Bsf
and
Btf
o£
P i e ( B s f / Y ) w h i c h by the
desired
natural
transformation. The m o r p h i s m I---~ i s o n t o by P r o p o s i t i o n
By t h e
remark at
has a canonical exceptional
t h e end o f §IV,
structure
of
Corollary
Y.3.
induced exceptional
exceptional Proof.
For
universal
p
L e t V be t h e s u r f a c e
configuration the
property
exceptional U of
the surface
V parametrized
b l o w i n g - u p and
below.
*
by a p o i n t
so comes w i t h
p of X
a distinguished
configuration.
E be
the
ordered
V.1 a n d t h e c o r o l l a r y
first of I.
configuration
in X such
that
parametrized
configuration.
o f X.
Moreover,
statement,
simply
by a c l o s e d
Then p E s ( I w )
wE on V.
apply
the
if
subset
o f X. the the
there
of
wE i s an
I--~W u s i n g
exceptional
definition
p o f X, and l e t
and only
suppose p E s ( I w ) and c o n s i d e r
By Theorem I I I . 5 , to an
point
S(Iw) is a Zariski-open
For the second statement,
wE e x t e n d s
if
is an etale
configuration
neighborhood
( w h i c h we a l s o
115
denote
wE) f o r
the
family
V'
induced
o r d e r e d b l o w i n g - u p on V' and t h u s U---~X, I---*X, a n d U---*I commute; S(Iw).
But
open.
the
on U.
there
Thus wE i n d u c e s
another
i s a m o r p h i s m U--*I . w
in particular,
image o f U i n X c o n t a i n s
the
structure
Clearly
image o f U i n X i s
an open s u b s e t
since
oF
t h e morphism:s
etale
contained
in
morphisms are
•
For a brief
Proposition
discussion
V.4.
of principal
The
morphism
homogeneous s p a c e s ,
s:I
w
---*X i n d u c e s
a
s e e [Mi, p . 1 2 0 ] .
structure
PCL2-homogeneous s p a c e on I w o v e r t h e image o f s i n X, l o c a l l y
of
principal[
for
the etale
trivial
topology. Proof.
We s t a r t
a point
and Z i s
here
the
acting
by d e f i n i n g
action
on
p2, is
each
already
v-equivariant
action
successively,
thus
being
and PGL2 a c t s
factor
P G L 2 - a c t i o n on I ,
an equivariant
on
this Z.
the
the universal
check that
F
property
of
I,
action.
induced
by
Intuitively,
composition
of
I an
be
so
on X×X by
it
lifts
repeated
But
this
p2 and
for
now
to
a
each
n
induces
a
the automorphism
t h e same f a m i l y I×xZ: ~
~Z
I
1
(Vl
~X
induces
that
an action
w h e r e f :X--*X i s
,X
this
is an automorphism,
desired
f t,
~Z
t
X is
X is
we h a v e two m o r p h i s m s o f I t o X, t h e f i r s t
o f ~ on X, b o t h i n d u c i n g
!
can
For n = -1,
For n=l,
diagonal,
generally.
the composition
I
p2.
determines the
construction
action
IXxZ
By
action
preserves
f o r e a c h ~EPGL2,
t and the second b e i n g
i n d u c e d by t h e a c t i o n
This
action This
defining
since
as automorphisms of
defined.
and
P G L 2 - a c t i o n on v:Z--*X.
a morphism
f f=f
F :I--~I and
, and t h a t
parametrizes isomorphism
sF~=s.
isomorphisms,
with
an
and
it is easy
This gives this
isomorphism
action
coming
from
to
the is a
P G L 2 - a u t o m o r p h i s m o f p2. Now l e t the
canonical
structure
exceptional
but also,
neighborhood blowing-up: universal
x be a p o i n t
U of one
property
of s(Iw).
configuration
by C o r o l l a r y x
such
I,
V.3,
that
corresponding of
Then t h e f i b r e
there
m o r p h i s m UXxIw--dJ i n d u c e d by s .
wE.
this
E
Zx o f Z---*X o v e r x h a s n o t o n l y
induced
by
its
By Theorem I I I . 5 ,
UxxZ
also
to
E and
is
therefore
has
another a
two
ordered
structures
corresponding
section
blowing-up
we may c h o o s e a n e t a l e
to
of wE.
a:U---~JXxI w o v e r
ordered By
the
U of
the
116
To see t h a t
this
induces a natural is
a
induces a U-isomorphism of
UXxIw w i t h
whose
equivalent
Composing ~ w i t h
exceptional
to h a v i n g a
configuration
second s t r u c t u r e
of
is
E.
Now
the
f o r t h i s second s t r u c t u r e
yet
blowing-up of
another
~
i s a l s o wE. owing t o t h e morphism E:R-'4JXXIw which f a c t o r s
through the
wE.
Thus we have
This
desired
transformation,
the
two b i r a t i o n a l blowings-up
induces a birational
natural
consider
gx~:R---~GL2XUXxI w.
morphisms P2R*--ZR--~2R f a c t o r i n g corresponding
to
the
exceptional
map of P2R to P2R which i s
therefore
transformation. a
morphism
To
construct
g:R--4~L2 .
This
the
inverse
gives
a
natural
Composing with the PGL2-action morphism PGL2xUxxIw---JJXxIw we
References
[A23 [DM] [EGA]
[H1] [Sa] [Hs]
[K] [Mat] [Mi] [m]
This
morphism
have a morphlsm N:R---~U×xIw giving an inverse natural tranformation as desired.
[A1]
the
exceptional
ordered
h e n c e an automorphism, and so t h e r e i s i n d u c e d a morphism g : R - - 4 ~ L 2.
the
is
P2R whose c a n o n i c a l
of
same b l o w i n g s - u p ,
configuration
of
But ZR has
s e c t i o n a:U--AdxxIw. the
existence
b e i n g wE.
structure
configuration
gives
the
o r d e r e d b l o w i n g - u p of P2 R on ZR,
canonical exceptional configuration
regular,
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t o X i n d u c e s t h e f a m i l y ZR o v e r R. which i s an o r d e r e d b l o w i n g - u p of P2R
canonical
through
we show i t
e q u i v a l e n c e o f f u n c t o r s Hom( ,UXxIw) to Hom( ,U×PGL2).
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projection
UxPCL2.
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3~i(1965), Kyoto,
(A)
3 3 ( 1 9 6 0 ) , 271-293. D.S. Rim, Formal D e f o r m a t i o n Theory, i n Groupes de monodromie en g e o m e t r i e a l g e b r i q u e e x p o s e 6. L e c t u r e N o t e s i n M a t h e m a t i c s 288(1972), B e r l i n - H e i d e l b e r g - N e w York: S p r i n g e r . J . R o b e r t s and R. 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 t r i a n g l e s I , I I , III. Comm. i n Alg. 1 2 ( 1 0 ) ( 1 9 8 4 ) , 1213-1255, 14(1986), 155-191, and t o appear. R. S p e i s e r , E n u m e r a t i n g c o n t a c t s . P r o c e e d i n g s volume f o r 1985 AMS Sunvaer R e s e a r c h I n s t . i n A l g e b r a i c Geometry h e l d a t Bowdoin C o l l e g e , to appear. A. G r o t h e n d i e c k , S e m i n a i r e Bourbaki: Les Theoremes d ' e x i s t e n c e en t h e o r i e £ o r m a l l e d e s modules, 195(1960); Les schemas de H i l b e r t , exp. 2 2 1 ( 1 9 6 0 / 6 1 ) ; Les schemas de P i c a r d , exp. 232 e t 2 3 6 ( 1 9 6 1 / 6 2 ) . C.S. S e s h a d r i , Theory of m o d u l i . (In: P r o c e e d i n g s volume f o r 1974 A r c a t a A/¢S Summer R e s e a r c h I n s t . i n A l g e b r a i c Geometry) P r o c . o f Symp. i n Pure Math. 29(1975), 263-304. J.M. Wahl, Equisin~n~lar d e f o r m a t i o n s o f normal s u r f a c e s i n g u l a r i t i e s . Ann. o f Math. 104(1976), 325-356.
On the embeddings of Projective Varieties Audun Holme U n i v e r s i t y of B e r g e n Norway
Joel Roberts U n i v e r s i t y of M i n n e s o t a Minneapolis Minnesota
Contents I
An absolute embedding obstruction
2
2
E m b e d d i n g i n t o P"~, m _< 2 r - 2
5
3
C o m p u t a t i o n of fl.~(X, f ® 34) a n d "/.~(X, f ® 34)
5
4
E m b e d d i n g b y t h e s m a l l e s t very a m p l e t w i s t
8
5
E m b e d d i n g b y m i n i m a l very a m p l e sheaves
12
6
A p p l i c a t i o n s to c u r v e s
16
7
T h e case of Surfaces
18
8
E m b e d d i n g s of r u l e d surfaces
22
Introduction This paper is devoted to the following P r o b l e m 1 Let X be a smooth irreducible projective variety over an algebraically closed field k.
Put e(X) --
{ min
n
3projectiveembedding } X ~ P'~ = P "
Give a method/or computing e(X) for any X as above in terms of good, computable invariants of X. Our approach here is to continue along the lines of [13] and [14]: In Section 1 we define an
absolute embedding obstruction for X into P'~. While this obstruction solves the above problem in principle, it does not immediately yield a general method for actually computing the number e(X) for specifically given varieties. However, if n < 2r - 2, r = dim(X), then the situation becomes rather simple, see Section 2. For n -- 2r - i it is more complicated, but still managable, sections 3, 4, 5: The main difficulty seems to arise for embeddings into p2r. This, among other things, is explained in the sections 3, 4, 5. Moreover, the final sections on curves and surfaces also illustrate where the main difficulties lie. A central question is the following
119
2 Let D be an ample divisor on X . Let ao denote the smallest integer cr such that oD is very ample. Is it true that the smallest embedding dimension attainable with a multiple of D is attained for trod ? By "the embedding dimension which corresponds to the divisor D ~ we mean the following here: It is the smallest integer n such that X can be embedded into p n by a (generie) projection starting from the embedding defined by D. Problem
T h i s second p r o b l e n is a special case of a m o r e general one. Namely, suppose t h a t D is a very a m p l e divisor a n d D t is ample. A s s u m e also t h a t D q- D ~ is v e r y ample. Is it t h e n t r u e t h a t t h e e m b e d d i n g d i m e n s i o n which c o r r e s p o n d s to D c a n n o t exeed t h e one which c o r r e s p o n d s t.o D + Dr? While we do know this in several interesting cases, we c a n n o t yet draw this conclusion in general. In fact, this seems to be quite difficult a n d will p r o b a b l y require some new techniques, b e y o n d w h a t is available right now. T h e layout of our p a p e r is as follows: In Section 1 we recall t h e definition of t h e relative e m b e d d i n g o b s t r u c t i o n s ~m(X, ~) a n d t h e closely related ~m(X, L), a n d s t a t e t h e i r f u n d a m e n t a l properties: T h e o r e m 1.1 a n d P r o p o s i t i o n 1.2, as well as t h e i m p o r t a n t relations (3), (4) as well as (5). Using this, t h e absolute e m b e d d i n g o b s t r u c t i o n BIn(X) is e s t a b l i s h e d w i t h Definition 1.4 a n d T h e o r e m 1.5. T h e n in Section 2 we use this, t o g e t h e r w i t h t h e classical Barth - Lefshez Theorems to completely d e t e r m i n e w h e n t h e variety X can b e e m b e d d e d into p m w i t h m ___ 2r - 2. Section 3 is devoted to t h e c o m p u t a t i o n of tim(X, ~. ® .M) a n d ~m(X, • @ ~ ) in t e r m s of ~ j ( X , / ~ ) for j _~ m a n d "rm(X,L) for j ~ m , respectively. We use this in Section 5 to show t h a t , at least for m ~ 2r - 1, t h e relative e m b e d d i n g o b s t r u c t i o n s are increasing f u n c t i o n s (wit:h respect to t h e n a t u r a l orderings) on t h e s u b s e t of very a m p l e p o i n t s 1 in N u m ( X ) , T h e o r e m 5.1. U n f o r t u n a t e l y t h e s i t u a t i o n is consideably m o r e c o m p l i c a t e d for m = 2r, w h e r e we only o b t a i n p a r t i a l answers, in T h e o r e m 5.3 (see also R e m a r k 5.2). T h e m a t e r i a l in Section 4 c a n b e viewed as a special case of this: We s t u d y how t h e relative e m b e d d i n g o b s t r u c t i o n s b e h a v e o n t h e set of n u m e r i c a l l y v e r y a m p l e multiples of some fixed a m p l e divisor. A g a i n t h e case m = 2r t u r n s o u t to be t h e difficult one, where we so far only have p a r t i a l answers in T h e o r e m 4.1. In t h e final sections we a p p l y t h e t h e o r y to curves a n d surfaces. For curves we find our p o i n t of view p a r t i c u l a r l y useful in d e t e r m i n i n g how big a m u s t be for a P to be n u m e r i c a l l y equivalent to a very a m p l e divisor. For surfaces the question of e m b e d d i n g into p a c a n be answered comletely, whereas t h e case of p 4 is m o r e difficult. T h e m a i n p r o b l e m is to decide numerical very amplene.~s for a given a m p l e divisor. T h u s this case reflects the o u t s t a n d i n g questions in t h e general case: Progress here will hopefully provide a powerful (and at this t i m e m u c h needed) general tool for the classification of smooth surfaces in p 4 , where so far only ad hoc m e t h o d s seem to be available.
A n absolute e m b e d d i n g o b s t r u c t i o n
1
Let X be a s m o o t h variety over a n algebraically closed field k, of d i m e n s i o n r, a n d let • be a n invertible s h e a f on X. Let m E [ r , . . . , 2r t be an integer. Following [13] we p u t m--r
(11
-
m-,-3
) d~-
5=0
Here d = deg (X) is t h e degree of X w i t h respect to t h e e m b e d d i n g ~L: X ~
P (g ° (~))
1A point in Nurn(X) will be called very ample if the corresponding divisor class contains a very ample divisor. A divisor which is numerically equivalent to a very ample one will be called numericallyvery ample.
120
and dj = deg (s i (X)) similarly denotes the degree of the j th. Scgre class of X, which is defined by 1
s ( X ) - c ( X ) - l + sl (X) + . . . + s n ( X ) e A ( X ) ,
where A (X) denotes the Chow ring of X , c (X) being the total Chern class of X in A (X). The following theorem determines the smallest possible embedding of X by means of ~L a n d a projection from P ( H ° (£)): T h e o r e m 1.1 The following are equivalent:
(a) (b)
tim (X, Z) = 0 X can be embedded into p N by ~o£ and a projection from P (H ° (ft.)) or a linear embedding of P (H ° (£))
While relatively easy to deduce from the embedding obstruction in [13], this simple and elegant form was first noted by Dan Laksov in his very interesting and readable article [18]. [13] has fl2r = . . . . tim = 0 instead of (a). However, letting m+l--r
(2)
"~m ( X ' '~) :
m+l--r--j
)
d3
j=o
we find that
(3)
flra--1
- -
/~m = ~rr,--1
Interpreting ~,~ as the degree of the ramification locus of the generic projection Pm : X ----* p m
we note that -~,~ >_ 0, and the simplification of the embedding obstruction follows. Actually, it is a consequence of the Fulton - Hansen Connectedness Theorem that
(4)
]~rn,--1 ~ 0 ~
"]trt--1 ~ 0
This implication was conjectured by K. Johnson in [15] and proven by W. Fulton and J. Hansen in their fundamentally important paper [8]. The relation (3) was observed as a relation among the corresponding elements of the Chow ring, whereby/3m corresponds to (is the degree of) the double point scheme of Pro, see [15] An alternative form of (3) should be noted: Since/3r = d 2 - d, repeated use of(3) yields the following, which conversely implies (3): (5)
tim = d 2 - d - ~/T - . . . .
~/,~-1
We can consider tim ( X , / ' ) as a relative embedding obstruction: It is the obstruction to embedding X into p m by a projection starting from a given projective embedding. This depends of course on the corresponding very ample s h e a f / : . The aim of this section is to define an absolute embedding obstruction: We give a set of data depending only on the smooth variety X , which determines whether or not it has an embedding into a given pro. A first observation is the following: P r o p o s i t i o n 1.2 tim(X, L) and "Yra(X, L) depend only on the numerical class of L,
Nurn ( X)
[/~] c
121
Proof. Let £ = Ox (D) where D is the divisor which corresponds to £. By definition [•] == [D] E N u m (X). The claim is immediate by the definiton of numerical equivalence, since d = (D r) = f D r
dj = (sj ( X ) . D r-5) : / sj (X) D r - j We will use the following concepts: D e f i n i t i o n 1.1 An element ~ E N u m ( X ) is called (very) ample if there exists a (very) ampi'e divisor n such that ~ = [n]. The corresponding subset of N u m (X) are denoted by Amp (X) and
Vet (x). By Proposition 1.2 the expressions (1) and (2) give well defined functions on Vet (X). In fact, the expressions make sense on all of N u m (X). Define
~m : N u m ( X ) ~
Z
"Ym : N u m (X) --+ Z by the following formulas, where ( = [D]: r~--r
j=O
(7)
n ~ (~1 =
~
~ +~_, _ j
j=O
~,~ (~) = ~ ~2~ i f m = 2~ ( •~,~ i f m < 2r - 1
(8) Further, put
(9) Having introduced the above notation, we may state the following, which is an immediate consequence of Theorem 1 and (4): T h e o r e m 1.3 X can be embedded into p m if and only if Bm (X) = O. The above definition of Bm (X) is clearly not the simplest one which would have yielded a theorem of this type: In fact, /~,n (~) could have been replaced by fl,~ (~) in (9). But the point here is to have a definition of Bm (X) which has the property of the theorem and at the same time has nice computational properties. This is to some extent achieved with the definition given here: D e f i n i t i o n 1.2 Bm (X) is called the m t h . (absolute) projective embedding obstruction of X
122
2
Embedding intoPm,
m_<2r-2
For embeddings into p m with m ~ 2r - 2 the obstruction is particularly simple. We have the following result, which we formulate only for m = 2r - 2, since the modification needed to cover m _< 2r - 2 is obvious: Theorem
(a) (b)
2.1 Assume that k is of characteristic O. Then:
If N u m (X) ~ Z then X can not be embedded into p~r-2. Assume that N u m (X) = ZG ~ = [D] being the ample generator. Then Z can be embedded into p 2 r - : if and only if D is very ample and 3t2r_2 (X, Ox (D)) = O.
Proof. In view of the observations from section 1, the claim is immediate from the following Barth type theorem: T h e o r e m 2.2 If X c p m is smooth of dimension r, and m <_ 2 r - 2, then N u m ( X ) generated by the class of a hyperplane section.
~ Z,
Proof. For k = C this is part of the classical Barth Theorems, {3]. An algebraic proof, valid whenever k is of characteristic zero, is given by A. Ogus in [20]. Although there are several results in the direction of theorems of Barth type in characteristic p as well, they do not seem sufficient to prove the above results in characteristic p. See for instance R. Speiser's interesting article [27] for some details. In the situation of T h e o r e m 2.1, if N u m (X) = Z~, and ~ = [D] for D very ample, then we have in fact for rn < 2r - 2, ~3m (X) = flm(~) = "/.~ ( X , D ) This will be shown in the next section. In the following, whenever convenient, we shall write for instance fl.~ (X, L) instead of fl.~ (X, £), where L = c1 ( £ ) .
3
C o m p u t a t i o n o f / 3 m ( X , ~ ® ¢M) and "Tm(X,~ ® ~)
We consider the following elements of the Chow ring A ( X ) , the degrees of which are the integers tim and 7m- Let ~ be an invertible sheaf on X , and L = c1(£) C A I ( X ) = Pie(X). Define
b.,(x,c)=(
/ L~IL"-~- ~
)
.,_~_~"+~ ~,(X)L . . . .
3'=0
for m = 2 r , . . . , r
and rn+l--r
gm(X,~) =
sj(X) Lm+l-r-j
m+l-r-j j=o
for rn = 2r - 1 , . . . r. If/~ is very ample and X is embedded into P ~ by £, then these elements are the classes of the double point locus, respectively the ramified locus, of the generic projection pm : X - - ~ pro, see for instance [15], [14[. P u t t i n g g r - l ( X , L ) = 1, these definitions may be written as (1)
(2)
b(X,/~) = ( / L ~ ) L ( L ) Y
- T(L)s(X)
gCX,,~) = VCL)sCx)
123
We also write b(X, L)and g(X, L)whenever this is convenient. Here
(brI br+l
b =
( r-]-llL
1
,L(L) =
:
0
)
...
• °°
O
"'"
O
b2r
81 S
gr
U(L) =
L
+1
"'"
0
• -"
0
=
i
'g=
i
'
...
The following useful relations are verified immediately: (3)
T(L + M) = T ( L ) T ( M ) , U ( L + M) = U ( L ) U ( M ) , T ( O ) = U(O) = E
(4)
T ( L ) U ( - L ) = V(L) where V(L)=
1
0
...
0~
L
1
...
0
...
1
L ~ L ~-1
)
Thus solving for s in (2) and substituting into (1) we get the relation b ( Z , L) = ( f Lr)L(L) - V(L)g(X, L) J
(5)
When L is very ample this is the well known expressions for the double point classes in terms of the ramification classes. We next note the P r o p o s i t i o n 3.1 The following formulae hold: (6)
b ( Z , L + M) = ( f ( L + M)~)L(L + M) - T ( M ) ( f Lr)L(L) + T ( M ) b ( X , L ) J J
and moreover g(X, L + M) = U ( M ) g ( X , L)
(7)
Finally we have (8)
b(X, aL) = T ( ( a - 1)L)b(X,L) + 8 ( b ( P r , a L ) )
where ~ = f (L r) and b ( P r , a L ) denotes the element o / A I ( X ) obtained by formally substituting L / o r H, the class o/ a hyperplane, in the vector b ( P ~ , a H ) of elements from A(pr). In particular, i l L is spanned by global sections, then we obtain a morphism p : X ~ P~ induced by a projection from P(H°(/~)). Then b ( P r, aL) = p* (b(P ~, all)) and thus (8) aquires a geometric interpretation.
124
Proof To show (5), first solve (1) for s(X):
(9)
s(X) = T ( - L ) ( ( / L r ) L ( L )
- b ( X , L))
which we get using (3). Substituting this into (1) with i + i is similar. As for (8) we have
and using (3) again yields (6). (7)
b(X, ai) = arSL(aL) - T(aL)s(X) = a r 6 L ( a L ) - T ( a L ) ( T ( - L ) ( 6 L ( L ) - b ( Z , L))) = T ( ( a - 1)L)b(X, L) + 6 ( a " L ( a L ) - T ( ( a - 1)L)L(L)) which is easily obtained by (3). To finish the proof of (8) we need only to observe t h a t s ( P ~) ----T ( - H ) L ( H ) which yields (10)
b(Pr,aU)
= a r L ( a H ) - T ( a H ) s ( P r) = o % ( a H ) - T ( ( a - 1 ) H ) L ( H )
This completes the proof, the last assertion of the proposition being obvious. The following was proved by J.-C. Vignal in [28]: C o r o l l a r y 3.2 ( J . - C . V i g n a l ) Let L be a very ample divisor on X , and let cr >_ 2 be an integer. Then 13:~(X, aL)
#
0
unless X : p r , a : 2 and L is the class of a hyperplane. Proof Since f~2r vanishes if and only if the cycle class b2~ is zero, the claim follows by (8):
i:0
Since L is very ample, this is non zero unless all terms vanish. Thus if •2r(X, crL) : 0, then b~ (X, L) : 0 so X : P~ and L = H, a hyperplane, h simple computation shows that ~2r ( p r , a l l ) > 0 unless a = 1, 2: 2r
pr
1)g) :
2r /=0
=--
{~
2r
i:0
2r
i
\
i~l
/:rj-I
) (t 2 r - i
t~+l)
/=0
Thus the polynomial ~(t) = fl2~(P ~,(t + 1)H) has the roots t = 0, 1 and ~(t) > 0 for all t > 1. This completes the proof of the corollary. The observation at the end of section 2 is also immediate from the proposition: We have the C o r o l l a r y 3.3 Let L be an ample divisor and assume that crL is very ample. Let r <<m < 2 r - 1. Then we have for all a > ao "~m(X, aL) > ~ 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) <_11 r = 4 and ao(L) <_4
R e m a r k 4.2 (d) and (e) can be continued to higher dimensions as well, but the results obtained become rapidly weaker.
Proof. (a) follows from the proof of Corollary 3.2, and (b) is Corollary 3.3. (c), (d) and (e) (as well as the remark ) is proved by means of a version of the Castelnuovo bound for the genus of space - curves which is due to J. Harris, see [9] or [5] : Let C be a smooth, non degenerate curve in p N of degree d and genus g. T h e n
(Ii)
g<M(d-
where
M+I(N._I)_I) 2
126
Let X ¢-* p 2 r + l be a smooth connected subvariety of dimension r, which we may assume is not contained in a hyperplane. We let X p denote the intersection of X with a generic linear subspace P , in particular XH be the generic hyperplane section of X. Elementary considerations (e.g., a direct computation) then yields f l r + l ( X ) = ~,.(XH) . . . . .
~2(Xp) = d(d - 3) - 2(g - 1)
where P is a generic linear subspace of dimension r + 2, and consequently X p is a smooth,connected and non degenerate curve in p r + 2 , of degree d and genus 9. We note in particular that (12)
d > r + 2
Indeed, a subvariety of p N of degree d and dimension p always lies in a linear subspace of dimension at most d + p - 1. We a p p l y ( l l ) to C = X p and find
([r~ d-i t +1)(~+i)-2)-2
29 - 2 _< [r~l(2dand thus (13) Here M
~r+l(X) >_(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 <- M <- ~ r + l + I) + r + 3 - 2°~6). Since a ~ S~+1
Put # : M(M(r
1
(17)
U > -~(°3 -
we h a v e t h e e s t i m a t e
- 1)(°~- 2)
r+
U s i n g (17) in (16) finally yields l)t r +
] (18)
(2r+l) ,._
~
1
,-~(o~
- 1)(o~ - 2)r--~
+oo~,(t) T h i s final i n e q u a l i t y is t h e key to t h e r e m a i n i n g p a r t of t h e proof. Since~o2(t) = t 4 - t + 4 ( t 3-t2),weget
We first t a k e r
=
2:
10 A s > 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 <_ 4, as c l a i m e d in t h e t h e o r e m . To c o n t i n u e this to h i g h e r values of r, t h e m e t h o d will have to be r e f i n e d s o m e w h a t : In fact, after a finite n u m b e r of s t e p s as a b o v e t h e e s t i m a t e (18) can only be i m p r o v e d to an e s t i m a t e a r b i t r a r i l y close to
where
¢,(t)=~r(t)+
~ rr+ l
tr+ r~
r-1
W h e n r is large~ this e s t i m a t e ceases to yield useful i n f o r m a t i o n .
129
5
E m b e d d i n g by m i n i m a l very a m p l e sheaves
In this section we show that the relative embedding obstructions are increasing functions on This ties in with the results of the previous section. First recall that an invertible sheaf is called nef or in an older terminology pseudo - ample, (see [17], [11]) if for all closed subschemes Y of X of dimension
V e t ( X ) under the orderings induced by the usual orderings on N u m ( X ) .
s:
l,...,r-I
] [ Y I L r-s > 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 -<, respectively, if these relations are carried over to < , < and <, respectively. The main problem we are dealing with in this section is the following Conjecture
5.1 The functions "~m, /gin and ~m are all increasing with respect to <, <_ and
~:
Of course this conjecture implies Conjecture 4.1. Moreover, if we knew the assertion above, then it would follow that in order to compute the absolute embedding obstructions BIn(X), one would only have to consider extremal (minimal) points of V e t ( X ) . The results below do no,; completely settle this question, but nevertheless it gives what we want in some cases, or with certain additional assumptions. First of all, for m < 2r we obtain the best possible answer: Theorem
5.1 For m <_ 2r - 1, ~m is increasing with respect to < and <
Proof. By (7) in Proposition 3.2 we get for any L1 and L2 g(Z, L1) : U ( L 1 - L2)g(X, n2), and thus m4-1--r
m+l-r-{
(L1-
Yr+i--l(,L2)
2j
i-----0 m--r
:
m+l-r-i
-
2)
yr+i-l~
, L 2 ) + g m ( X , L2)
i:0
Now
"~,~(X, L1) : / gin(X, L1)L[ r - ' - ' ~ :
(2)
m-r (
m+l
)/
L'm+l-r-iL 2r-l-m-
i:0 "Jr ]f
L 21 r - l - m
g m ("X , L2)
'XL'
130 A s s u m e L2 < L1, i.e. t h a t L1 - L2 is nef. Since L 21r - - l - - m gr+~-l(X, L2) is t h e class of a n ( m + 1 - r - i) - d i m e n s i o n a l s u b s c h e m e of X , we get t h a t all t e r m s in (2) are n o n negative by (1). Thus
~m(X, L1) >_ f L 2~-I-m 1 g,~[~X , L 2]~ By 2r--l--m
L 21 ~ - 1 - , ~ = (L2 + L i - L2) 2 r - l - m
~ ~r 22r--l--ra 4-
E
( 2r-l, - rn ) L g r - l - m ( L 1 _ L2)i
i=1
we finally get
/-2~ 1~ 1
-t
gm[-A,
L2) _> z m ( X , L2)
Analogously we show t h a t L2 < L t implies t h a t ~fm(X, L1) > "Tin(X, L2), using the Nakai Criterion: In fact, if "Tin(X, Lt) = q m ( X , L2), t h e n all t e r m s in t h e s u m of (2) would be zero. In p a r t i c u l a r we would get
(L1 - L2)'n+I-~L 2~-1-'~ = 0 which is clearly impossible w h e n L1 - L2 is ample. Remark
5.2 It follows from the theorem that if L2 < L1 are both very ample, then
~ 2 , - l ( X , L1) > 0 In fact, at least w h e n k = C this conclusion also holds u n d e r the weaker a s s u m p t i o n t h a t L1 a n d L2 are b o t h very ample a n d M = L1 - L2 is nef a n d numerically fractional, i.e. t h e r e exists a n integer k such t h a t k M is numerically equivalent to a n o n zero effective divisor: Namely, we m a y a s s u m e t h a t N u m ( X ) is not cyclic w i t h very a m p l e g e n e r a t o r , since t h e c l a i m is obvious in this case. T h u s X c a n n o t be e m b e d d e d into p 2 , - 2 by v i r t u e of T h e o r e m 2.1. Hence
q2r-2(X, L2) > 0 Since M is nef, all t e r m s in (2) are n o n negative, hence in p a r t i c u l a r we find t h a t "72r-l(X, L1) = 0 would imply f Mg2r-2 (X, L2) -- 0. B u t if k M is numerically equivalent to the effective non zero divisor D , t h e n f D g 2 ~ - 2 ( X , L2) > 0, since we m a y move t h e center of projection so t h a t t h e induced m o r p h i s m X --+ p 2 r - 2 h a s a ramification cycle which p r o p e r l y intersects D. T h u s we get a c o n t r a d i c t i o n , a n d t h e claim follows. We now t u r n to t h e case m = 2r, where t h e results are less complete. We prove t h e Theorem
5 . 3 Assume that L1 and L2 -< LI are very ample divisors on X . Then:
If fl2,(L1) > O, then f12,(L2) > 0 If M = L 2 - L1 is such that ~t = O x ( M ) is spanned by global sections, then fl2r(L2) > 0, unless X = p r and L1 is a hyperplane. Proof. (i): M = L1 - L2 is effective, so t h e r e is a n exact sequence
0 ~
Ox - - * )4 = O x ( M )
which w i t h o u r usual n o t a t i o n yields 0 - - - ~ ~ 1 -----+ ~_2
131
and thus 0 ~
H°(£1) ~
H°(£2)
So the embedding defined by £1 can be regarded as a projection of the embedding defined by/~2, from a center which meets X precisely along the divisor M . It follows t h a t the secant variety of the first embedding is the closed image under this projection of the secant variety of the second embedding. T h e claim follows from this. (ii): We have £2 = L1 ® )4. Letting ~ : X ---o p m =
p(H0()4))
be the morphism defined by M, we give X a new projective embedding by composing the graph with the Segre embedding a: X ~
F(~) C X × prn ~_~ p n × p r o
~
pN
where "~(x) = ( x , ~ ( x ) ) on k - points, and ¢ is the product of the embedding given by /~1 and the identity. Since the pullback of OpN(1) to X is /~1 @ )4 = £2, we compute ~ 2 r ( X , £ 2 ) as the relative embedding obstruction given by this embedding of X into p g . The claim follows from the Lemma
(a) and
(b)
5.4 If x , x I are ponts of X such that
x ¢ tx,z' and x I ~ tx, z,
~(~) # ~(~:')
then Span(tx,a(z,~(x)), tx, a(z,,~(z,))) has dimension 2r + 1 Here tx, z and t x , a(z,~o(x)) denote the embedded tangent spaces in P " and p N , respectively. In our situation we can find k - points x and x' e X which satisfy (a) and (b): (a) since X is not a linear subspace of P'~, (b) is seen as follows: ~ ( X ) spans pro, so to show is t h a t m _> I. If m = 0, then H ° ( ) 4 ) = rk. But )4 is sparmed by global sections, hence r : O x ~ )4 is an isomorphism in this case. Thus M = 0, contrary to our assumption. Now both (a) and (b) are (Zariski-) open conditions on X × X , hence we have a non e m p t y open subset of X × X such t h a t (a) and (b) hold simultaneously. It is immediate t h a t the l e m m a implies the claim, by means of the following useful fact known as Terracini's Lemma. See [7] and [41 for proofs and details. P r o p o s i t i o n 5.5 ( T e r r a e i n i ' s L e m m a ) (i) Let X C P ~ = P'~ be a reduced and irreducible closed subvariety over the algebraically closed field k of any characteristic. Let x , y be distinct points of X and let z E ~y be any point of the secant ~y. Then
tx,z, tX,y C tsec(X),z (ii} If k is of characteristic zero, then there is a dense open subset U C S e e ( X ) such that lot any z 6 U and (x,y) 6 X × X - A x /or which z E ~y , S p a n ( t x , z, tx,~) = tSec(X),.
132
Of course we only need part (i) of Terracini's L e m m a here, but we have included the full statement for completeness. It now remains to prove L e m m a 5.4. For this, it suffices to show that tx,a(=,,~(=) ) f3 txcr(=, ~(=,)) = 0
Since a ( X x pro) D X under this embedding of X in p N , it therefore suffices to show that ) = 0
ta(xxP~),o(=,y ) N t ( x x P ~ ) , a ( = , , V
P u t p n = P ( V ) , P ' ~ = P ( W ) and let v,v' E V be the coordinate vectors for x,x'; and p,p~ E W ditto for y, y' E p m Further let L, L t E V be the subspaces of coordinate vectors for points in tx,= and tx,=,, respectively. T h e n to(x×p~),o(~,u) corresponds to the subspace
v@ W + L @ p c V @W Thus to show is t h a t
(v @ W + L ® p ) N (v' ® W + L' ®p') = (0) Since dim(v @ W + L @ p) = m + r + 1, this will follow if we can show that
dim(Span(v, v') ® W + L ® p + L' ® p') = 2 m + 2 r + 2 To show this we consider bases
{p,p, .... ,p(m)} o f W {u0 . . . . ,uk, v = vk+l . . . . . vr} of L
{uo . . . . . u/c,v' =
. . . . . v',} o f L'
where {u0 . . . . , uk} is a basis of L N L' Now (L + L') ® W has a basis consisting of decomposable tensors, and Span(v,v') ® W + L ® p + L ~ @ pt is spanned by the following vectors:
Vk+l ® p , Vk+l ®pl ... ,Vk+l ®p(m)
(from v ®W)
Vtk~=l ® P~v'k-~-i ® pt ... , v'kq-1 ® p ( ' 9
(from v' ® W )
uo®p ..... uk®p, ve+l®p,...,vr®p uo ® p ' , . . . , uk ® p', v~+ 1 ® p' . . . . , v', ® p'
(fromL®p) ( f r o m L' ® p')
The list has 2(m + 1) + 2r + 2 entries, and any selection of distinct vectors from the list is linearly independent. Since the only duplications are vk+l ® p, which is listed in the first and the third row, and v~+ 1 ® p', which occur in the second and the fourth row, the dimension of the space spanned must be 2m + 2r + 2, as claimed. This completes the proof of the theorem.
133
6
A p p l i c a t i o n s to
curves
In t h e two r e m a i n i n g sections we discuss curves a n d surfaces in light in t h e p r e v i o u s sections. We s t a r t w i t h curves. Let X be a s m o o t h curve over k, of genus g(X) = g. If g = 0 e m b e d d i n g dimension e(X) is 1. If g = 1 t h e n X is a n elliptic curve. As deg(3P) -- 3 -- 2g + 1, 3 P is very ample. Moreover, for a >_ 0 t h e _> 2g - 2 -- 0, hence it is n o n special, so by the Riemann - Roch t h e o r e m
of t h e t h e o r y developed t h e n X ~ p 1 a n d the Let P E X be a poinl;. divisor a P is of degree
dim(H°(Ox(aP)) = a + 1 - g = a Hence in this case we h a v e
ao(P)
=
3,
e(X) = 2,
a n d moreover if ~ -- [P] t h e n a ~ E N u m ( X ) = Z~ is very ample for all a >_ ao(P) = 3. A s s u m e g _ 2. T h e p r o b l e m is to d e t e r m i n e w h e n we have e(X) = 2. Clearly a necessary c o n d i t i o n is t h a t g can be w r i t t e n as ( d - 1)(d - 2)
g--
2
for some integer d > 3. T h i s is equivalent to
' (3 ÷ W i t h ~ = [P] we have N u m ( X ) = ~Z. So by Corollary 4.2 we know t h a t if X can l:,e e m b e d d e d into p 2 , t h e n it is via the projective e m b e d d i n g given by a divisor numerically equivalent to a o ( P ) P . T h u s we have the Theorem
(i) 60
6 . 1 Let X be a smooth curve with g(X) >_ 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<
where equality holds if a n d only if t h e curve is planar. Moreover, if the curve is n o n - planar, t h e n t h e C a s t e l n u o v o B o u n d holds: g<
-4
ld2-d+
l
L e t t i n g ao = ao(P) we t h u s have 1 2 g-< ~ao-ao+l
if t h e e m b e d d i n g is n o n p l a n a r , i.e.
oo ~ 2(i + v ~
134 Finally, recall the classical result of HaIphen (see [10], Proposition 6.1 of Chapter IV ): A smooth curve X of genus g > 2 has a non special very ample divisor of degree d if and only if d :> 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 <<_11 and d < 11.
Proof. We have (K~:) = 9 - m and X ( 0 x ) = 1 . Thus, if X is not a plane, a cubic in p a , or a projection of the Veronese s u r f a c e , we obtain d2 - 1 5 d 4 - 3 4 , 4 , 2 m = O This equation c a n n o t be satisfied if d > 13 since d 2 - 15d + 34 > 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 <: k < 5(g + 1), t h e n we would have: 16(g+1)+8= 10k(g+l)-k 2_> 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 <_ 6. The case y -- 3. In this case, e q u a t i o n (8) reduces to 1 0 k ( g + 2 ) - k 2 = 36(g+2)+27. T h e s m a l l e s t feasible value is k -- 4. If t h e r e were a solution of (8) w i t h k -- 4, we would have 40(g + 2) - 16 = 36(g + 2) + 27, which is clearly impossible. T h u s , t h e r e is no solution w i t h k = 4. If t h e r e were a solution of (8) with 5 < k < 5(g + 2), t h e n we would have: 3 6 ( g + 2) + 27 = l O k ( g + 2) - k 2 >_ 5 0 ( g + 2) - 25
This would imply t h a t 14(g + 2) <_ 52, so t h a t g _< 1. Therefore, w h e n g _> 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.
146
[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<
a line
P. U P. as one l ] P ( S y m 2 U *) ~ G ( 3 , 5 )
by
meeting
a fixed
)(1))
z = ci(0P(Sym2U,
intersecting
Lemma
that
this
c H ~
(1.6.1)
.
2-plane],
Note
a fixed
of [C}~]
5(x 2 - y) +
intersection). .
y = ~*
that
for
[2 p l a n e s
[conics
2 plane
As
Generators
Finally,
meeting
in a line]
+ z) £ - 10(2y
for w h i c h
+ xz)
conics,
situation
we
can
[Kal,§2].
these
correspond
conics
should
in
D
.
G
-
:
So the n u m b e r Remark: will
The
be
since
such
final
to c o u n t
the
success
computations consult
2 plane;
are
x = ~*
line],
twice]
We a p p l y
we de-
inter-
conics are
a fixed
2 plane .
locus
= ~*
(1.4.4)
[2
and
line
L C
to
C
- 48,600
, L' c H
.
.
We s t a r t for
the l o c u s
for w h i c h (at
These
Hence
the n u m b e r
~ L)
from
51,
i-L)
trisecant conics
c
(L'
; there L ~ C
there
are
locus
is cut
are
],600
has
out
on
of
conics
test
conics of
for
on
Ft
is 1 8 7 , 8 5 0
of t h i s m e t h o d
needed
by d e g e n e r a t i n g
the m e t h o d
the
depends
about
case
before
+ 163,200
proceeding
to 5 h y p e r p l a n e s . (at p r e s e n t )
twisted
of lines.
+ 258,200
among
such
lines
plane;
163,200
such
as l i m i t s
is
cubics
in
This
test
on the a b i l i t y p3
.
The
= 609,250.
to t w i s t e d
QED
cubics
is c r u c i a l to r e d u c e
interested
V
This
102 nodes; =
arising
CO
in
However, f" D)
(102)(1,600) conics
are
those
to a g e n e r a l
the p r o j e c t i o n
of r e d u c i b l e
conics
by c o u n t i n g
the
of e a c h
(L'
L' C
genus
= 258,200
.
formulas
, project
C .
10z 2 - 25y
+ 0 - 6400
, L • G
C
has
+
163,200.
to c o m p u t a t i o n s [Ka3]
the
ring
(:;~,2 x c3,2 ) • [V] = i .
those
meets
Since
Hence,
on
]From the
, since
L
to t r i s e c a n t s
163,200
Chow
a fixed
= xZ - y
× C(2,5)
[Kal,~2],
include
when
19 curve.
to be e x c l u d e d .
326,400
not
only
For e a c h
a degree
in [CMW], the
: 4 5 x ± + 30xz
C O = L O L' V c C(2,5)
to
of l i n e s
arise
w e get
can
the
is t r i s e c a n t
locus
of c o n i c s
1,020o3, ? x 32003, 2 = 3 2 6 , 4 0 0
these
L
count
by a s u b v a r i e t y
L'
the
to the v a r i e t y
we m u s t
parameterized
by
a linear
F
to o b t a i n
10(2x
restricting
[L] and
, p.
meeting
313,200 after
p3
by d e g e n e r a t i n g
lip. has n o r m a l c r o s s i n g s . In the limit, the d o u b l e p o i n t i to the l o c u s of c o n i c s m e e t i n g LIPi twice (counting a transverse
with
planes
Alternatively,
that
parameterized [2 p l a n e s
.
all
reader
155
References [CMW] [F] [Kal] [Ka2] [Ka3] [KI] ILl
A. Collino, J. Murre, G. Welters. On the family of conics lying on a quartic threefold. Rend Sem. Mat. Univ. Pol. Torino 39 (1980) 151-181. W. Fulton. Intersection Theory. Springer-Verlag, Berlin 1984. S. Katz. Degenerations of quintic threefolds and their lines. Duke Math. J. 50 (1983) i127-I135~ S. Katz. On the finiteness of rational curves on quintic threefolds. Comp. Math. 60 (1986) 151-162, S. Katz. Lines on complete intersection threefolds with K = 0 . Math. Z. 191 (1986) 293-296. S. Kleiman. Multiple point formulas I: Iteration. Acta Math. ]47 (1981) 13-49. P. Le Barz. Formules multisecantes pour les courbes gauches quelconques. Enumerative Geometry and Classical Algebraic Geometry, Birkhauser, Boston 1982.
ENUMERATIVE GEOMETRYOF NODAL PLANE CUBICS
Steven L. Kleiman Department of Mathematics, 2-278 MIT Cambridge, Massachusetts 02139, USA and Robert Speiser Deparment of Mathematics, 292 TMCB BYU Provo, Utah 84602, USA
This paper continues the determination of the characteristic numbers for all plane cubics, begun in [KS], which studied the cuspidal ones. Our treatment emphasizes the underlying geometry of the space of nodal cubics and their duals, and in so doing, advances, clarifies, and deepens the conceptual approach initiated by [KS]. The 8-parameter family of nodal plane cubics has nine characteristic numbers, denoted by NS,0, NT, 1. . . . . NO,8. By definition, each N~,8 counts the nodal cubics which pass through c¢ general points in p2, and are tangent to ~ general lines. The N~,I~ were originally found by Maillard [M] and Zeuthen [Z] in the 1870s, by essentially heuristic methods. They are also discussed in Schubert's book [Sch] of 1879, and their rigorous justification is part of Hilbert's 15th Problem. Our determination of the N~,~, as well as further characteristic numbers, vindicates the pioneers' heuristic outlook, by justifying their methods as well as their conclusions. At the same time, as a study of the underlying parameter spaces, some of our results are closely related to recent work of Diaz and Harris ([DHI] and [DH2]) on the Severi varieties, in particular to their study of the Picard group. We proceed, however, wtth a different conceptual emphasis, employ different general methods, and sometimes go beyond the scope of their results. Section I ls devoted to the basic facts about nodal cubics and their codimenston I degenerations in pg. We show that there are two of these. A curve of tvBe • , in the terminology of [Z] and [Sch], is a cuspidal cubic, while a curve of type ~, notation as in [Sch] (here [Z] writes ~) is the union of a smooth conic and a secant line. The section ends with
157
examples of such degenerations, showing, for later use, the behavior of the flexes and the nodal and inflectional tangents. In Section 2, we pass from plane curves to their conormal schemes. We parametrize the conormal schemes of the nodal cubics, together with the conormal schemes of the curves of type )' and ~, by normalizing the locally closed subscheme of p9 which parametrlzes them. The resultlng normaIIzatlon, denoted N, Is smooth, but ramifies over the locus of curves of type 7. Further, N has 2 branches over the locus of curves of type ~t, because each curve of type ~ can be the limit of a nodal cubic in two essentially different ways. These results are proved by restricting to explicit curves in N. In the terminology of [DHI], the ~pace N i5 a Severi variety, and its behavior illustrates some aspects of the general picture. Our exposition, however, is concrete and self-contained, based on an explicit construction of N due to Sacchiero [Sa]. In section 3 we extend the Ramification Formula of [KS] to general families of plane curves, of any degree, with arbitrary singularities, and with possibly nonreduced special fibers. Specializing to nodal cubics, we obtain two key relations in Pic(N). Applications include important classical results [Z, formulas ( 1), (2), p. 605] about 1parameter families whose general member is a nodal cubic and whose special members are all curves of type 7 and ~. The extra generality is needed, even in this special case, because nonreduced special fibers turn up in the family parametrizing the dual curves, where each degenerate curve of type ~ contributes a quartic with a double line. In Section 4 we continue the study of Pic(N), establishing further basic relations. Our derivations involve sharp use of modern versions of Schubert's incidence and coincidence formulas. A crucial issue here is the determination of multiplicities;we address it by restricting to explicit curves in N, and inspecting directly. Applications include the classical formulas [Z, (3),(4) ,(5), loc.cit.]about the behavior of the nodal and inflectional tangents of a good l-parameter family of nodal cubics. Combining our results, we obtain the following identities [Z,(6),(7), loc.cit.],valid for any l-parameter famlly, over a smooth, complete base curve, whose general member is a nodal cubic, and whose degenerate members are all curves of types )" and ~: p. = 712
and
IJ.' = (2~ + 7)13,
where i~ (resp. I.L')denotes the number of curves in the family through a given general point (resp. tangent to a general line),and 7 (resp. ~) is the number of degenerate curves of type 7 (resp. of type ~) in the family. As in [Z}, the characteristic numbers I~x~8for nodal cubics can now be obtained directly from those for curves of types y and ~, by applying the two formulas above to the
158
one-parameter family obtained by imposing 7 of the 8 conditions used to define each given N~(~. In Section 5 we obtain, with mild restrictions on the characteristic, precisely the classical results [Z, p.606]: Ns,O, NT,I, ... , NO,8
= 12, 36, 100, 240, 480, 712, 756, 600, 400. In the final Section 6, we pass beyond the partial compactification N, to treat conditions represented by classes of codimension 2. We apply our results to obtain further characteristic numbers, also discovered classically, which will be needed (elsewhere) to treat nonsingular cubics. Here we construct a new compactification of N, denoted N++, by attaching conormal schemes as well as nodes, and we investigate its intersection theory in codimension 2 by restricting the ramification theory of Sectlon 3 to suitable local complete intersections, based on a prescient and fundamental observation of Zeuthen [Z, p.607]. We expect that our results about the boundary of Nwill generalize, to give information about the boundaries of other Severi varieties. The work In Sectlon 6 draws on much of our previous work, both here and In [KS], and it motivates many choices we have made along the way. It also distinguishes our approach from those of [Sa] and the forthcoming [XM2]. tn [Sa], key relations are deduced from a double-point formula. In [XM2], the authors study a different parameter space, based on ideas of [5ch], so as to work In codlmenslon I. Finally, a few words about the base. All our results are valid over the spectrum of an algebraically closed ground field of characteristic ~ 2 or 3. This restriction prevents the loss of key derivatives, and hence guarantees, for example, that the famllles we study will be reflexive. It also ensures that the PtQcker formulas will take their usual forms.
1. Basics
The nodal cubtcs in p2 are parametrlzed by a locally closed 8-fold, denoted N, in the p9 of all plane cublcs. For a given nodal cubic N, denote by b the node, by Pl and P2 the nodal tangents, and by v l , v 2 and v 3 the inflectional tangents. Define O-cycles on the dual plane I~2 as follows:
159
P=PI+P2, V = V l + v 2 + v 3. Although the order of the subscripts, for a given N, is plainly arbitrary, the cycles p and v are canonical. We can pararnetrize the 0-cycles p and v by parametrlzing the corresponding divisors (reducible conics and cublcs consisting of unions or perhaps multiple lines) in the dual plane. Denote by Z2 (resp. Z3) the space of reducible conics (resp. cubics) in the dual plane which are unions of lines. Then (resp. v) defines a morphism N-, Z2 (resp. N 4 Z3). We define the sinoularitv confiouration T.(N) of the nodal cubic N to be the point (b,p,v) c P2×Z2×Z3. Exactly two types of degenerate nodal cubics are parametrized by 7-folds on the boundary of N. First there are the cuspidal cubics, pararnetrized by a locally closed subscheme, denoted K, tn pg. Following Zeuthen [Z], and Schubert [5ch] we shall call these curves of type 7. As the nodal cubic N degenrates to a curve NO ~ N, the node of N degenerates to the cusp of NO. Then there are the degenerations of tvDe $ each the union of a nonsingular conic with a secant line whtch ls not a tangent. (Here we choose Schubert's symbol; Zeuthen used =;E.) These are parametrized by a locally closed subscheme, denoted X, in pg. Here, however, as a nodal cubic degenerates to a given curve No ~ X, the node can degenerate equally well to either singular point of N0. However, when we speak of a degeneration of type ~t, one singular polnt of the limit curve ls implicitly distinguished as a llmit node, while the other singular point (sometimes called, in the old literature, a virtually nonexistent double point ) is not. Denote by G the group PGL(2) of linear automorphlsms of p2. PROPOSITION 1.t. The group G acts transitively on ~ Kand X Further, /
160
EXAMPLE 1.2. (Degeneration of type ~f.) Denote Spec(k[t]) by A I. For each parameter t c A I , define Nt to be the curve in I}2 given in homogeneous coordinates by the equation Nt:
Ft(x,y,z ) = y2 z - x2(x-tz) = O.
For t ,, O, the curve Nt is a nodal cubic, with node b - (0,0,1 ) and nodal tangents Pi:
x = ±,f~"
When t=O, we obtain
NO:
y2z = x3,
which is a curve of type ~/. (When t= I, we obtain the folit~'nof Descartes.) Now it'seasy to check that the point at ,,,on the x-axis is a flex for every Nt, with the line at ~ as inflectional tangent. For general t, the remaining 2 flexes, which are always at finite distance, must limit to the cusp of N 0. To locate the flexes, consider the Hessian of Ft(x,y,z). After cancelling a constant factor, we flnd
Ht(x,y,z) = 3xy 2 - t(y2+t x 2 )z. For general t, the vanishing of H t defines an irreducible nodal cubic C t, with the same node as Nt, but with different nodal tangents. (The assertion about the nodal tangents follows trivially,by setting z = I and inspecting the leading form. Since H t is invariant under the substitution y -, -y, with a node at (0,0, I), the irredicibilityis also clear. Indeed, if the curve were reducible, it would contain one line, hence, by invariance, two. These would have to be the nodal tangents, which is absurd.) Hence C t and N t meet 6 times at their common node, and 3 times elsewhere, plainly at the flexes of N t. Direct inspection shows that one flex lies at (0, 1,0), with the line z=O as inflectional tangent. To locate the other 2 flexes, pass to affine coordinates with z = I. Then solve F t for y2 and substitute into Ht: it follows that the x-coordinate common to the remaining flexes (their y-coordlnates differ by a sign, by the Invarlance under y-~ -y) satisfies the quadratic equation 3x 2 - 4tx + t 2 - t = O. As t-,O, It ts clear that x40, so y4o by Inspection of the defining equation of the family. Hence both of these flexes approach the origin. It follows easily that the corresponding inflectional tangents converge to the cuspidal tangent of No.
161
EXAMPLE 1.3. (Deaeneration of type ~.) For each parameter t ~ A 1 as before, define Nt to be the curve tn P2 defined in homogeneous coordinates by the equation Ft(x,y,z) = x3 + ty3 - xyz = O.
Nt:
For t ,, O, Nt is a nodal cubic, with node b = (0,0, I ), and nodal tangents defined by x=O and y=O. When t=O, we obtain No:
x(x2 - yz) = O,
which Is a curve of type ~/: the union of the nonsingular conic yz = X2 and the y-axis. Hence NO has node b = (0,0,1 ) and double vertex e = (0, 1,0). (When t= I, we obtain the classic example x3+y3,- xyz.) To see that the flexes of N t limit to e, we can inspect the Hessian. The latter, up to a constant factor, ls given by Ht(x,y,z) = 3(x3 + ty3) + xyz. Again, Ht=O defines a nodal cubic for general t. To flnd the flexes, set z=O In Ft: we obtain x3+ty 3. Then set z=O In Ht: we obtain 3(x3*ty3), which is a product of distinct linear factors. Hence, for general t, the restricted polynomial function HtIN t has 3 distinct zeros on the line z=O. As t-,O, these zeros approach the double vertex e=(O, 1,0) on CO. Since the Hessian curve Ht=O has a node coinciding wlth the node or Nt but with different nodal tangents, the intersection there has multiplicity 6, so the 3 zeros on the line z~O are simple, and occur at the 3 flexes of Nt. Finally, it is easy to check that the nodal tangents atl approach the line z=O, which ts tangent, at e,to the conic yz = x 2 in NO.
2. ParametrlzlnG the Conormal Schemes v
Denote by N
the normalization.
, NUKUX
162 PROPOSITION 2.1. ( 1) The normalization N is smooth. (2) The normalizaton map n is bijective on n - l ( / v ) and n - I ( / o , and restricts to a connected double cover from n - l ( X ) to X. (3) Under the natural G-action on N, the orbits are n - I ( N ) , n - l ( / O , and n - l ( x ) . Proof. Our argument is based on a construction due to Sacchiero [Sa]. Denote by V the image of p2 in p9 under the Veronese embedding defined by the sections of 0(3), and denote by CV the conormal scheme of v In pg. A point of V consists of a point x of V, together with a hyperplane H of p9 tangent to V at x. The intersection HrlV pulls back to a singular cubic curve C in p2, in general a nodal cubic, and x pulls back to a singutar point of C. Conversely, each singular cubic C, with a given singutar point, corresponds to a unique point of CV. Since nodal and cuspidal cubics have unique singular points, we see that CV is naturally a compactification of N, maps to the closure of N in p 9 and maps bijectively over /VUK. Since a curve of type ? has two singularities, the restriction of CV is a double cover of X. Further, for a curve of type ?, the 2 singular points can be exchanged by an element of G which carries C to itself, so the double cover of X is connected because G is connected. Finally,CV is smooth because V is smooth. Hence the restriction of CV to N U K U X is a normalization, and all 3 assertions follow. PROPOSITION 2.2. The map N .+p9 is nonramified on n- I(A/UX), but ramifies on n - I ( K ) . Proof. By 2. I, the restriction n-I(11/)+N is an isomorphism and n-l(x)+x is a double cover. Hence, by G-linearity, it suffices, for the first assertion, to find a line in P' which meets the divisorA/UX exactly twice at one point, for then both branches of N U X at the given point will be smooth. We choose the line in Pg given by the I-parameter family of plane cubics { C t }, for tE T= A', given In afflne coordinates by the equation F(x,y,t) = 0, where F(x,y,t) = x~- ( I-t)xy - t(x + y~). Hence for t = 1 we have the elllptic curve y2 = x3_ x, while for t = 0 we have the curve x3 = xy of type xp. on the total space AZx A I, the parttal derivatives Fx and Fy, together with F, define a subscheme Z whose direct image cycle on A I is the intersection of the line T with NUX. Indeed, Z is the Jacoblan locus of the family, and, since the Jacobian locus commutes with base change, Z is the restriction of the Jacobian of the family of all plane cubits. Since the direct image of the latter is the closure of N, and since formation of the associated cycle commutes with restriction to the line T, our assertion about the intersection of NUX is clear. Now for the computation. We have
163
Fx = 3xz- ( 1-t)y - t and
Fy = (t- 1)x- 2ty. Denote by J the ideal (F,Fx,Fy) of k[{x,y,t]]. We claim that J=(x,y,t2). Indeed, write A for 2 t / ( t - 1), and let =- denote equivalence modulo J. Then Fy= 0 gives x - A y , and a direct calculation gives 3F -xFx = ty(4 + 4(l-t)-i -3y).
Hence ty=O, and so x-Ay=O. Combining the last congruences with Fx-=O, we obtain -y-t-O, hence y=--t, so t2=O. Conclusion: J ~ (x,y,t2). But t-O is impossible, since T intersects a singular point of NUX when t=O, so indeed J=(x,y,t2). Since Z is defined by (F,Fx,Fy), it follows that the cycle [Z] pushes down to the doubled point 2-[0] In A', and this proves the flrst assertion. The second assertion follows by a similar approach; since we do not need it in the sequel, we shall omit the proof. Using N as a parameter space, we can study the behavlor of the singularity configuration T.(N) as a nodal cubic N degenerates. PROPOSITION2.3. Suppose given a degeneration N -~ NO, where N is a nodal cubic in I}2. Then: (1) If NO of type ~', then 2 of the 3 flexes of N llmit to the vertex of N0, while thelr tangents approach the cuspidal tangent of No. The remaining flex approaches the unique flex of NO, while its tangent approaches the inflectional tangent. (2) If NOis of type ~, the 3 flexes of N all approach the double vertex of No, whlle the 3 Inflectional tangents all approach the tangent to the conic in No at the double vertex. As the node of N approaches its limit, denoted bo, on NO, the nodal tangents approach the tangents to the 2 branches of NO at bO. Proof. As N varies, the assignment N -+ T_(N) defines a rational G-map T. N
, p2xZ2xZ3.
Denote by C the complement of the domain of T.. Since ]C is a G-map, C is a union of Gorbits; since N is normal, these have codimension ) 2. By 2. I, there are no such orbits, so Z is a morphisrn. It follows that ;C assigns a canonical singularity configuration to each point of N. Because n - I ( K ) and n - l ( x ) are orbits, and because T. is a G-map, our
164
assertions can be checked by inspecting any particular degeneration of the given type. Since Examples 1.2 and 1.3 exhibit the required behavior, the proposition follows. As the nodal cubic N varies in N the dual curve I~ a quartic with 3 cusps and a bitangent, traces an 8-dimensional orbit 117on the p14 of quartics in the dual plane. When N degenerates to a curve of type 7 or ~, we shall need to study the behavior of ~. Denote by 8 the rational G-map N -~ p14 assigning to each nodal cubic N its dual curve 1~. Replacing I}14 by the Hilbert scheme of the incidence correspondence of points and lines in p 2 we have a rational G-map, denoted by C, assigning to each nodal cubic its conormal scheme. PROPOSITION 2.4. In the situation above, we have: ( I ) The rational maps ~ and C are morphisms. (2) As a nodal cubic N degenerates to a curve K of type 7, the dual curve I~ degenerates to the union I~ of a cuspidal cubtc and a line. The cuspldal cubic in t~ is the usual dual of the cuspidal cubic K, while the line in ~ corresponds, under projective duality, to the cusp of K. (3) As a nodal cubic N degenerates to a curve X of type ~, the dual curve I~ degenerates to the union ~ of a nonsingular conic and a double line. The conic is the usual dual of the of the conic In X, whlle the double llne corresponds under projective duality to the unique singular point of X which is not the limit of the node of N. Proof. As in the last proof, we see that 8 and C is are morphisms; so ( 1) holds. The dual curve I~ is a quartlc, so the same must hold for any degeneration. As N degenerates to a cuspidal cubic NO, the dual curve ~ degenerates to a curve containing the dual of N0, which ls also a cuspidal cubic. Hence I~ degenerates to the union of a cuspidal cubic and a line. Since the line cannot correspond to a smooth point of NO (by [RS2, Th.3.2,p. 175 ] or [K]) it corresponds to the cusp; so we have (2). As for (3), the node b of N clearly degenerates to one of the singular points of a degeneration NO of type ~ . Reasoning as in the proof of (2), the dual curve here degenerates to the union of a conic and two lines. To locate these lines, consider the ratlonal G-map from N to the p9 parametrtztng the cublcs In the dual plane, which assigns to each NoN the dual curve I~. As in the proof of the last proposition, this is a morphism, and, since n - l ( x ) is an orbit, we can reduce to an inspection of an example. The next example will prove (3), and hence the proposition. EXAMPLE 2.5. Denote by {Nt} the family of nodal cubics, parametrized by t~A 1, given in affine coordinates by the equation (~)
xS-t = xy.
This is actually the family of Example 1.3, but with the variables y and z exchanged, on the
165
affine plane with z = 1. Here No is a curve of type ~. The effect of this coordinate change is to move the common node of the Nt to the point at ~ on the y-axis, while placing the virtually nonexistent double point of No, which we claim is a double vertex, at the origin. Here is an old and elegant way to find a defining equation for { I~t}. In the affine (x,y)-plane, represent each line y=mx*b by the point (m,b), so that m and b are affine coordinates for the dual plane. Then set y=mx+b in (*). We find x3-mxZ-bx-t,, O. The discriminant of this cubic polynomial in x is A= bZ(m2+4b)-t(4m~+18mb+27t), and 4 = 0 , the condition that the line y=mx*b meet Nt twice, defines the dual family {NtV}. When t=0, we have the union of the nonsingular conic m2 = -4b and the double line b2=0. The factor bz shows that the origin is a double vertex on No, as we had claimed. NOW we return to the main discussion. Following Zeuthen's terminology (compare [Z] and [KS]), a point of p2 corresponding to a line in special member ~t of the dual family will be called a vertex of Ct. Hence a curve of type )' has a single vertex at its cusp, white a curve of type ~t has a double vertex at its virtually nonexistent double point. The 2 sheets of the double cover n-1 (X) -, X correspond to the different ways the node and vertex of a curve of type ~ can be chosen. The next result is clear from Proposition 2.4. COROLLARY 2.6. The map N 4 pl4 is injective. We now prove a strengthened version of 1. I. Denote by N+ the graph, in pgxpt4, of the correspondence,S, which assigns the dual quartic to each nondegenerate nodal cubic. Clearly N+ ~ N. Denote by K+ and X+ the pullbacks ofKand X t o the closure of N+. By 2.4, we have a G-bijection N4N+UK+UX+;hence K+ and X+ are G-orbits of dimension 7.
PROPOSITION 2.7. The group G acts transitively on N+, K+ and X+. Further, K+ and X+ are the only 7-folds in the closure of N+. Proof. The proof is the same as for 1.1 until we consider the dual curves associated to degenerate nodal cubics which are unions of lines. So let CObe a degenerate nodal cubic which is a union of lines. If the iines are distinct, we have an orbit of dimension 6, since
166
vertices can only appear at singularitiesof C 0. Next suppose C O is of the form 21*m, for not necessarily distinct lines I, m. The dual curve is a union of 4 lines, corresponding to vertices on C O, and these are on sing(CO) = I. Hence the 4 lines of the dual curve meet at the point dual to I. Since the characteristic ~ 2 or 3, the family is reflexive, so C o is the dual of its dual curve, via the conormal scheme. Hence I and m both correspond to singular points of the dual curve, by [K]. Therefore, if I,,m, it follows that 2 lines on the dual curve must coincide, giving a double vertex on C O at I Fl re.Hence the orbit of such a CO has dimension ~;6. If I=m, we have 4 vertices on a line, so again the dimension is ~;6,and the proof is complete.
3. Ramification Formulas
We begin by generalizing the ramification formula [KS,Th. 5.8, p.250} to families of curves which can have nonreduced special members, and we sharpen it to a relation among divisor classes on the parameter space, which can now have any dimension. (Familiarity with the constructions and results of [KS, sec.5] will be helpful but not essential.) Then we specialize to families of nodal cubics and their duals. First we establish some notation. Let {xtlt ~ T} be an r-parameter family of plane curves of degree d, satisfying the following hypothesis: the general X t is reduced, irreducible and reflexive, and the parameter scheme T is smooth and irreducible. Fix a general point L. Denote by B the blowup of pz at L, and denote by T[:B-,P' the morphism induced by the projection from the center L. Denote by D the total transform on B ×T of the total space X_of the family {Xt}. Denote by f:_D-, p lxT the restriction of T[xT. Finally, denote by R the ramification scheme of f. The argument of [KS, pp. 247-8] applies to show that R is a divisor on D: in fact, it shows that R is a divisor whenever T is normal. Fix a general line A through the center L, and also denote by A its strict transform on B. Denote by i the inclusion i:A xT--, BxT; plainly i is a regular embedding. Denote by p the projection p: AxT--,T. Conceptually, the ramification formula unites two distinct analyses: the evaluation of the global divisor class p,i*[R] on B, and the determination of the local contributions to It. The global class. Denote by I~ the projective space of plane curves of degree d, and denote by h the map h:T--~ pN which determines the family {Xt}. Let H denote the class of a hyperplane in pN, and set
M = h*H.
167
PROPOSITION3.1. We have the following identity of divisor classes on T: p,i*[R] = (2d - 2)M.
Proof. The verificationis similar to that of [KS, Prop. 5.7, p.249]. Let k:D--,B~T denote the inclusion. By [KS, Cor. 5.3, p.247], the class of R on _D equals the pullback k*8 of a certain divisor class 8; moreover, i*8=((d-2)[L],[M]). Now a well-known expression [I(5, 5.4, p.247] for the sheaf OB×T(D__) yields i*[_D]= (diLl,[M]). Therefore we have i*[R] = i*k,k*~ = i*([D_]-8) = i*[D].i*~ = (2d-2)([L]x[M])+ [A] x[M]2.
Applying p. yields the assertion. The local contributions: (I) tanGencles. We want to study the part of the divisor p,i*R supported on the locus of t~T such that Xt ls tangent to A. To do so, we constder the dual curves ~t. Let d' be the degree of the general ~t, let pN' be the projective space of curves of degree d', and let ~-:T--,P N' be the rational map assigning ~t to a general t~T. Since T ls smooth, • Is a morphism off a closed subset of codimension 72. 5o, slnce we are concerned about a relation among divisor classes, we may assume without loss of generality that ~- is a morphism.
Denote by A' the point of the dual plane i~2 corresponding to A, and by A + the hyperplane in pN' of curves passing through A'. Set M'=T.A +. We shall prove that 1"1'has the following properties: (i) A general point toM' corresponds to a curve Xt such that (a) Xt is tangent to A at a unique and general (so simple) point, (b) Xt has no singular point on A, and (c) Xt does not have A as slngular tangent. (11) M' Is the part or p.I*R In question. To prove (1), denote by CX the conormal scheme of the fatally; it embeds canonically tn p2xi~2xT. The Image X' of C'~ in l~2xT corresponds to the map T:T--,P N'. Hence M'=P'2, p'I*(.A.'), where P'i denotes the 1u~projection from x'; cf. [KS, Lerr~a 5.6, p.248]. The map v2: CX_-.X_.ls birattonal, as a general Xt is reflexive. Therefore M'=q3,q2*(A'), where qi is the i th projection from C~. By dimensional transversallty, since A' ls a general polnt of !~2 every component of q2*(A') meets any given dense open subset U of CX_. Take for U the intersection of the
168 preimages of the smooth loci of X/T and )('IT. Then (a) holds; indeed, A is not a bitangent of Xt because A' is a simple point of X't. Similarly, shrinking U if neccessary, we obtain (b) and (c). To prove (II), denote by Pi the iu~projection from X. The projection vI:CX--+X Is birational; in fact, it is an isomorphism over the smooth locus of X_. Hence we have P2*1Jt~q2"(A') =M'. Let (P,A',t) be a general point of a component of q2"(A'). By (i), we have P,~L, and locally at (P,t) the cycles vl.q2*(A') and i*R have the same support. Therefore, to prove (ii), it suffices to show that the two cycles have the same multiplicity at (P,t). To finish the proof or (liL we shall need a general result, which we shall now recall. Denote by I the incidence correspondence of points and lines on i}2. Schubert's principal incidence formula is the following identity of cycles on I: the intersection of the cycle of pairs (Q,II) such that Q lies on a given line with the cycle of (Q,II) such that I-I contains a given point is equal to the sum of he cycle of (Q,11) such that Q is a given point and the cycle of (O,I1) such that I1 is a given line. The formula is well-known and elementary; it is also part of the explicit description of the intersection ring of I to be given in the next section. Continuing the proof of (il), take A for the given line and L for the given polnt. Then Schubert's identity implies that at (P,A',T) we have the cycle equality q2*(A' ) = q I*(A)-q2*(L'), where L' is the line of I~2 corresponding to the point L. Now, by the projection formula, we have Vl.(qi*(A)-q2*(L')) = (AxT)-vl.q2*(L'). Hence it suffices to prove that at (P,t) the multiplicity of vi.q2~(L ') is the same as that of R. Work in a neighborhood of (P,t). Choose local coordinates x,y,s so that P is the origin, A is the y-axis, and L is the point at infinity on A. Say X_is defined by F(x,y,s)=0. Then, near (P,t), the map cI2Vl-l:X--~ I~2 carries the point (xo,Yo,So) to the line (y-yo)Fy(xo,Yo,So)+ (X-Xo) F,(xo,Yo,So )= 0. Hence lJiwq2~(L') is cut out of X by the equation Fy=0. On the other hand, R is defined by the Fitting ideal FoQ~. So it may be computed using the presentation
I
o.
169
where I is the ideal of D. Now, near (P,t), we have ll(xo,Yo)= xo, and f = (ll× 1) I x. Thus (ii) holds. In passing, note that In characteristic 0 the divisor M' is reduced. Indeed, by (i), a general point toM' is covered by at most one point (P,A',t) of the cycle q2*(A'), and it is general in its component. So it suffices to notice that the cycle is reduced by the theorem of transversality of the general translate. The local contributions: (2) isolated sjhgular points. Denote by E_I,E_2,... the irreducible components of the closure of the locus on D_traced by the isolated singular points of the general Xt. For each i write e1 for the multiplicity of the component of the ramification dlvisor R supported by Ei, and set El = P,I*EI. Then ~.eiEI Is the part of p.I*R supported on the locus of toT such that X t has an isolated singularity on A. To identify the multiplicity el, fix a general point (P,t) of E I. As in the preceding case, choose local coordinates centered at (P,t) and an equation X: F(s,y,s)=O so that R is the divisor on X defined at (P,t) by t!heequation Fy=O. Hence the trace R t of R on the plane curve Xt ls a divisor, which is defined at P~Xt by the equation Fy=O. Since A is not tangent to Xt at P, by (i)(b) above, it follows (with some work) that the multiplicity of Rt at P is
equal to the multiplicity ep(Xt) of the Jacobian ideal FIC21vt.,, This description is of theoretical interest, because it shows that the multiplicity is an intrinsic invariant of the curve X t. In practice, however, it is often easier to work with the equation Fy=O. For example, suppose that X t has a single branch of multiplicitym at P, which is given parametrically by the power series x = T m + ... and y = T n+ --- with m
F(x,y) = yrn_ xn + ~,, ai.jxtyJ i+j)m Moreover, we have Im+Jn;~mn, wlth equality If and only If l=n. Hence we rind Fy(T) = mT(m- 1)n + .... Therefore the multiplicity in question equals (m- 1)n, if m is not divisible by the characteristic. In particular, if Xt has an ordinary cusp (m=2, n=3) at P, then, as the characteristic is ,~ 2 or 3, we find that the multiplicity is 3. If Xt has several branches Zl, Z2.... at P, then the multiplicity of Fy on Xt is obviously the sum of its multiplicities on the branches Zi. Say Zi is defined by Fi=O and that F= I-IF i. Then Fy= FlyH + FiHy where H = ~i,,tFi. Hence the multiplicity of Fy I Zl = Fly I Zi is the
170
sum of the Jacobian multiplicity ep(Z1) and the sum, for i~l, of the intersection numbers i(P,ZI.Zi). For instance, if X t has an ordinary m-fold point at P, then ep(Xt)= m(m-I). Normally, ep(Xt)= e i. Indeed, always ep(Xt)= ei-pfi,where pfl is the order of Inseparability of E i over its image In B under the projection from B×T, because, since t~T Is a general point, the trace of E_i on Bx{t} is pfi.P. However, normally we have pfi = I. For exarnple, since P # 2 or 3, it follows from the conclusions of the last few paragraphs that ei =2 at an ordinary node and ei =3 at an ordinary cusP. The local contributions: (3) nonreduced special fibers. Denote by XI, X_2, ... the irreducible components of codlrnension 1 of the locus on D traced by the nonreduced components of the various Xt. For each i, write r i for the multiplicity of the component of the ramification divisor R supported by X_i, and set Xi = p,l*V~i.
Then ~riX i is the part of p,i*R supported on the locus of t~T such that Xt has a nonreduced component. If T is 1-dirnensional, then r i may be identified by considering the curve traced by DI on the smooth surface A×T, and using local equations as described in Cases ( I ) and (2). In general, we can reduce to the 1-dimensional case by making a base change to a suitable curve on T or one mapping into it. Every component of R has now been taken into account. Indeed, since L is a general point, the set of toT pararnetrizing the Xt that have as component a line through L or that have L as a "strange" point (a point on infinitely many tangents of Xt) is a subset of codimension > 1 on T. Combining our previous conclusions, we obtain the main result of this section. THEOREM 3.2 (the Ramification Formula). Let {Xt I tET} be an r-parameter family of curves of degree d such that the general Xt is reduced, irreducible and reflexive, and the parameter scheme T is smooth and irreducible. Then we have the following identity of divisor classes on T:
(2d-2)[M] = [M']+ ~ e i I E i ] + ~ r j [ X j ] . i
I
The classes are those of the divisors discussed above: M parametrizes the curves through a general point; M' those tangent to a general line; the Ei those with an isolated singularity on a general line; and the Xi those with a nonreduced component.
171
Families of nodal cubics. Here T will be N, the variety discussed in Section 2. In the sequel, we shall primarily be concerned with divisor classes, rather than divisors, on the parameter spaces. Hence It wlll be convenient to omlt the square brackets In the notation for divisor classes, but we shall do so only for classes on the parameter space T (or N), and only when the context makes confusion unlikeiy. With mis understanding, we shall show how Theorem 3.2 yields the following statement. THEOREM3.3. Let {Nt I teN} be the canonical famlly of nodal cublcs and their first-order degenerations. Then we have the following identities of divisor classes on N: (I)
4M = M' + 2B;
(2)
6M' : M + 3F + 4,
where M and M' denote the classes of the divisors on N parametrizing the N t through a general point L and tangent to a general line A (defined precisely as the pulIbacks of hyperplanes under the canonlcal maps from N to p9 and pi4) and where B, ~ and F are, respectively, the classes of the reduced divisors parametrizlng the N t whose node lies on
A, having an inflectional tangent through L, and which are of type ~. PrQof. First of all,note that the general Nt is reduced, irreducible and reflexive, and that N is smooth and irreducible by the results of Sections I ans 2. Hence we may apply Theorem 3.2 to the family N t. Similarly we may apply it to the family of dual quartics. Thus we obtain formulas ( I) and (2) up to the asserted multiplicitiesof [B], [~] and [F],which we now check. Consider [B]. The proof of Proposition 2. I shows that N is the conormal variety of p2 embedded in p9 by the third Veronese embedding. In particular, the total space NIN has a canonical section, which assigns the node, and the corresponding map v:N4p2 is smooth (in fact, it is a I)6-bundle). Hence there is only a single cycle E I, and p,i*E_I_=V*A=B. Finally, el=2 by the last line of Case (2). Thus fomula (1) holds. Consider [F]. The 3 cusps on the general dual quartic ~It trace an irreducible set. (Equivalently, the 3 flexes of the general nodal cubic trace an irreducible set. This is wellknown, but for a proof, see the remark below.) Its closure, on the total space of the dual family, Is a prime cycle El, which Is generlcaIIy a 3- I cover of N. However, I'El Is generically a I- I cover of its image F, because the t such that I~t has 3 distinct cusps, 2 of which lie on A, form a subset of N of codlrnenslon > I, and because F contains neither q/nor the prime divisor of cuspldal cublcs. Hence P,I*E i=F. Finally,we have el =3 by the last line of Case 2. Finally, consider [~]. By Proposition 2. I (3), ~ is an orbit. By Proposition 2.4(2), the dual of a curve of type ~/is the union of a conic and a double line. The various double lines trace a prime divisor X_I, and i'X_I is a I- I cover of ~. Hence, to finish the proof of
172
Formula (2), it remains to show that r I = 1. To do so, It suffices by the discussion In Case (3) to restrict to a suitable l-parameter family. The following does the trick. EXAMPLE 3.4. Our example will be the dual of the family {Nt} of nodal cubics, parametrized by t~A 1, given in affine coordinates by the equation (*)
x 3 - t = xy.
This is the family of Example 2.5. Recall that the dual family, denoted by {l~t}, ls defined by the vanishing of the discriminant A = b2(m2-4b) + t(4m3-18mb-27t), that when t~O the curve Nt is a nondegenerate nodal cubic, and that when t=O, the curve No was found to be the union of the nonsingular conic mz = 4b and the double line bz =0. Now an argument like than at the end of Case ( 1) above shows that r 1 = 1. Remark. (From a conversation with BillLang.) To see why the 3 flexes of the canonical family of nodal cubics trace an irreduciblecycle, we use the l-parameter family in Example 1.2. There one flex is fixed at infinityon the y-axis, while the other 2 go to the origin as Nt degenerates. We claim that the 2 latter flexes define an irreducibledouble cover of the t-line. This is clear, in fact, because the discriminant of the quadratic equation in x at the end of the cited discussion is not a square in k(t). So suppose the flexes of the canonlcal family of nodal cublcs trace a reducible cover. Choose a nodal cubic N such that 2 flexes lie on unique and different irreducible components. Then choose affine coordinates so that these flexes are at finitedistance, the third flex is at infinity,and N is the curve N I of Example 1.4. As t varies in that example, we obtain a l-parameter family where the flexes at finitedistance trace an irreducible 2-sheeted cover, as we have seen above. This irreduciblecover, however, is a restriction of the cover given by the canonical family, where the nodes at finitedistance are on different irreduciblecomponents, so we have a contradiction.Therefore the 3sheeted cover of N given by the flexes of the canonical family Is irreducible.
4. Coincidence Formulas
Prelininaries. Again we denote by ! the incidence correspondence of points and lines In p 2 so that a point of ! ts a pair (p,L)~p2xl ~2 with I~L. We shall abuse notatlon In Schubert's style [Sch, Ch. 1] by writing p (resp. L) also for the pullback to I of the class of a
173
line in p2 (resp. ~2). The intersection ring A-~i is generated by the classes p,L~A 1I, subject to the obvious relations p3=L3=O, together with the third relation pL = p2+L2. The latter, Schubert's incidence formula [Sch, toc. cir.], was employed in the last section; here we express it tn terms of the generators p and L. (For modem proofs, see [RS2, Prop. (2.1) p. 166], [F, p. 189] and [G].) Denote by B the blowup of p2x p2 along its diagonal. We can view B as the set of ordered triples (p,q,L)E p2x p2x ~2 SuCh that p,qEL. (Cf. [Kl,p.368].) Denote by E the exceptional locus of B: clearly E identifies with I. Extending our usage, we shall also write p (resp.q, resp. L) for the class of the pullback to B of a line in p2 (resp. p 2 resp j~2). Since p and L on B restrict to the previous p and L on I=E, there is little danger of confusion. Schubert's coincidence formula is the relation
[E] : p+q-L in PIc(B). (For a fast proof and further references, see [KS,pp.238-240.].) In the sequel, we shall view the incidence (resp. coincidence) formula as an identity of intersection operators, valid on any space mapping to I (resp. B). The formula for P. Let {Ct} be a family of plane curves, given by a map h:T4N, whose general member is a nodal cubic. (Here N is the partial compactification of the space of nodal cubics Introduced In Section 2.) Assigning the nodal tangents gives a map T-,Z2, where Z2cP5 is the space of singular plane conics. We define P to be the pullback of the class of a hyperplane in p5 to PIc(T). Hence P represents the condition that a nodal tangent pass though a given point. We shall contlnue to omit the square brackets in the notation for a divisor class on the parameter space. With this understanding, B (resp. M) denotes the class of the divisor on T representing the condition that the node of Ct lie on a general line (resp. that Ct pass through a general point) as introduced in Section 3. The next result generalizes and sharpens [Z, (3), p.605]. THEOREM4. I. Supposegiven a family {Ct} as above. Then we have
P = B+M in PIc(T).
174 Proof. We can prove this on N. For a point NoN, we shall also write N for the plane curve it parametrizes. Denote by A the closure of the following reduced locally closed subset: {(b,d,L,N)cBxNl b is the node of N, and d#b is also on N}. Rephrase the defining condition: L contains the node of N, and d is the unique third point of intersection of L with N. Atlowing d and b to come infinitely near, so L becomes a nodal tangent, we obtaln the remaining points of A. We have projections to N and B, denoted p:A-~N, and q:A4B. The proof is based on the concidence formula [E} =b+d-L, where we regard the terms as intersection operators, giving an identity of divisor classes in any space mapping to O. Now we require L to pass through a general point z~P 2. Pulling back the line In the dual plane corresponding to z, we obtain a subscheme, denoted by A, in A. HenceA represents the intersection class L. Denote by e the restriction ~ = [EllA; hence, we have
£ = bL+dL-L ~ in A2A. The theorem follows if, on N, we can show: ( I)
p.(bL+ dL-L2) = B* M;
(2)
p.(e) = P.
To check ( 1), we first pull the incidence formula back to A along the projection A4 I, which selects the components (b,L). By the Incidence Formula, we have bL: b2÷L 2. Similarly, we find dL:d2+L 2. Hence, for (I), it suffices to show that p.(d2)=M, that p.( b2)=O, and that p~( L2)=B. Flrst we Investigate d2. Denote by U the total space, In p2×N, of the universal family of nodal cubics. The projection (b,d,L,N)4(d,N) defines a morphism, denoted ~(:A-,U. We observe that (z is birational. Indeed, denote by U0 the open subscheme {(d,N)cUId,,b(N)}, where b:N4P2 is the node map. Define L to be the unique line from d to b=b(N), and we have constructed an inverse map Uo-~A. (Off UO, the map ec is clearly 2-1, since b=d allows both nodal tangents as choices for L.) The projection A-,P 2 factors through ~. Hence we shall abuse notation and denote by (F the pullback to U, so d~=e(*d2. Now, applying the projection formula for ec, we find = . d ' = = . = * d L - d~.= .[A] = ~ U] = d'.
175 By [KS, Lernrna 5.6], the class d2 on U pushes down to M; hence, p.(o~):M. Next we examine b2. The obvious 7-fold on A which represents b2 is the pullback of the the 6-fold on N of nodal cubics with fixed node. Hence p.(b 2) = 0. On the other hand, consider L2. We have projections q:A-,I, where I = {all (b,L)IbEL}, s:A -, ~2={all L}, compatible with the projection r:l ,p2. Observe that the following square P A
PN
m p2 r
ls Cartesian, where b is the node map. Since N is smooth, and r Is smooth, It follows that b and q are local complete intersection maps of the same codimension. Hence, by [F, i 7.4.1, p.327], and the I.c.i. version of [F,6.2(a),p.98] for any cycle Y on I, we have pwq~[Y]=b~r~[Y]. (An alternative proof, using the smoothness of I and P~, but not that of A and N, may be based on [KT]. Indeed, b*[I] = b*r*[P ~]= p'b*[ p2]= p*[ N] = [A]; hence, b~ll=q*.) Now take Y to be the pullback of the point Lc i52 to i. Then q*[Y]=L 2. On the other hand, since r.[Y] equals the class on I~ corresponding to L, the right side is B. Hence p.(L2) = B, and this proves ( 1). Now we check (2). Denote by N 1 the open subscheme of N where b is not the fixed general point z, and denote by A 1 (resp. A 1) the pullback of N 1 to A (resp. A). It suffices to check (2) on N1, because N-N I has codimension 2. This helps, because the projection A~-, N, is an isomorphism. Denote by 11 the pullback to A, of the exceptional divisor of B. Denote by e the pullback divisor of N 1 which represents the condition that a nodal tangent passes through z, so [Q]=P. Since the projection A I-, N I maps II bijectively onto e as sets, we only need to show that both [e] and the direct image of [ll] are reduced cycles. To do so, choose a curve NO in the image of 1-I. To prove this, it suffices to pull back to a suitable curve T
176
which maps to A 1, such that NO is in the image, and check that that the pullbacks of [~] and {e} are reduced. EXAMPLE4.2. Given NO, choose affine coordinates x and y so that we have
NO:
x3+y3= xy,
so z falls on the x axis, but not at the origin, which is the node b. Then take T =A I, and set Nt:
x3+y3= x(y-tx).
The general Nt is nodal, with node b at the origin, and the nodal tangents are the lines Pl: x=O and P2: y=tx. We map T into A 1 via the assignment t-,(b,d,L), where L is the xaxis, and d ls the Intersection point (t,O). Now t4(b,d) meets the diagonal of LxL transversally. However, Lx L is the preimage in B of the point LE I~2, and E restricts to its diagonal. Thus T-,B meets E transversally, so T-,N meets the direct image of [l]] transversally. Now for the pulback of [0]. Since the tangent cone at the node of Nt consists of the a fixed line off z, together with a second line which moves linearly as a function of t, it follows directly that [0] also has a reduced pullback to T, which proves 4.1. The formula for P+F. Here we generalize and strengthen Zeuthen's formula [Z,(4) p.605]. THEOREM4.3. Suppose given a family {Ct} as in 4.1. Then we have P+ F =M'+ 3M
in Pic(T). Proof. This time let A denote the closure of {(x,y,L,N)~BxNtL is tangent to N at x, and y,'x on L~N}. (Since the characteristic ,, 2 or 3, each N is reflexive, so the the general tangent is an ordinary tangent; hence the set above is nonempty, In fact a 9-fold.) Denote by oc the projection from A to the conorrnal scheme CUc Ix N of the total space IJ of the universal family on N, given by the assignment (x,y,L,N)4(x,L,N). Since y, the unique third intersection point of L with N, can be recovered uniquely from any (x,L,N), it follows that (x is an isomorphism.
177
To prove 4.3, we have the coincidence formula [E]=x+y-L on A. Again we require L to pass through a general point z~P 2. Here we shall interpret the formula ~=xL+yL-zL, where is the class of the pullback to A of the line in 1~2 dual to z~P 2. We need to show ( 1) that p.(xL+yL-zL)=M'+3M and (2) that p,,(~)=P+F, both on N. For ( I ), using the incidence formulas for the 3 summands, we are reduced to evaluating x2+y2+L 2. Via oc, we can study each square as a class on CU. Now x 2 is represented by the pullback of a point of p2 to CU. We claim that x2 pushes down to M. Since CU4U is birational, by the projection formula it suffices recall, from [KS,5.6, p.248] that x2 on U pushes down to M. Dually, we find p.(L2)=M'. Finally, to treat y 2 define a new map CU-,U by assigning to (x,L,N) the pair (y,N), where y+2x Is the Intersection cycle L.N. This map ts finite of degree 2. (Indeed, if y ls general on a nodal N, then there are 2 tangent lines L through y touching N at points x other than y, because the dual curve a quartic.) Now, applying the projection formula for this map CU-,IJ, we find by [KS,5.6], that p~y2=2M. Combining our results, we obtain ( 1). Now we prove (2). As before, the natural representative of £=[E]L pushes down ~ t theoretically to a representative of P+F, because a tangent at x~N through z meets N three times at x only if it is a nodal or inflectional tangent. To establish (2), it suffices to construct two specific I -parameter families, the first (resp. the second) meeting a representative of p.(~) once at a given point of a representative of P (resp. of F). EXAItPLE 4.4. (To check the pushdown over P.) Choose affine coordinates x and y in p 2 and start with the foli~n N:
y2 = x3+ x 2,
equipped with the standard pararnetrization, which we write as
p(t) = (t2- I, t(t2-1 )). We have the squared "norm", tl(x,y)ll 2= x2*y 2, and the squared "distance", d((x,y),(z,w))2 = II(x-z,y-w)U 2. For each t~T, the tangent line to N at p(t) is deflned by the parametrlzatlon; denote It by Lt. We have Lt:
y - ( t 3 - t ) ==m(x-(t 2- I )),
where m = (3t 2-1 )12t. Now the general Lt, as well as meeting N twice at p(t), meets N at a unique addltlonal point, denoted by q(t). Further, for general t, a direct calculation shows that we have q(t)=p(u) for exactly one u~T, namely
178
u = -( I+t2)12t. To construct our example, we are going to move N through a l-parameter family of rigid motions. Set T = A I=Spec(k). W e set up an orthogonal frame at p(t), for each toT, as follows. The coordinates in the frame will be X,Y. The X-axis will be L t, and the Y axis will be the unique line through p(t) normal to Lt, with respect to the euclidean inner product in (x,y)-space. W e transport coordinates to thls frame by the Euclidean motion which slides the (x,y)-axes to the new ones. (Granted, this motion is unique up to multiplicationby -*I, but we can make a coherent choice near any t we want to study, by taking the "normalized velocity vector" to define X, and then choosing the unit vector in Y to give a determinant= I.) Writing N in the (X,Y)-coordinates,we obtain, as t varies, a family of nodal cublcs, denoted {Nt}. Each is tangent to the X-axis, and, when t=O or I, the X-axis is a nodal tangent. Denote by h:T-~N the morphism corresponding to thls family, and let h':T'4A denote the liftof h under the base change A4N. W e are going to show that T' meets the pullback of E once in A, at t= I. (By the projection formula, this will show that cL pushes down to P in the proof of Theorem 4.3.) Observe first that T-,N liftsto A, via t-~(p(t),q(t),Lt,Nt), where we now write our coordinates in X and Y, that is, after applying the Euclidean motion to pass from N to N t. In particular, the X-coordinate of q(t), squared, is d(p(t),q(t))2,independent of the coordinate frame. So we compute in x and y, and obtain d(p(t),q(t)) 2 = (u-t) 2 [(u2÷tu+t 2-1 )2 + (u+t)2]. Now, to see what happens when t= I, set s=t-1 and reparametrize. Since u-t is a unit at s=O, we need only study the expression in square brackets. W e have u = - I- ( 1/2)s2- ..., hence we find (u2+tu+t2-1)2= s2 + .... and (u+t)2 = s2 + ..., so that d(p(t),q(t)) 2 vanishes to second order in s. It follows that the X-coordinate of q(t)ENt approaches the (X,Y)-ortgtn p(t) linearly. Hence T-~A meets E exactly once at t= 1. But, we need to show that T' meets E once over t= I. It does. Indeed, since T' is the fiber product TxNA , the map T-,A lifts to a section of T'-,T. Denote this section by o: it is easy to see that the only point of T' over t = 1 which maps into E is o~ I ). Hence T'-,A meets E once at t = I, as was to be shown.
179
EXAMPLE4.5. (To check the pushdown over F.) This time T will be Spec(k[[t]]). Choose x,y as in the last example, but now take a nodal cubic N which has a flex at (0,0) with the xaxls as inflectional tangent. Analytically, N is equivalent to y=x3 near the origin. Parametrize N with x=t, y=t3, and then set up a frame iX,Y) at pit) as before, this time over our analytic local T. We obtain a family of curves Nt, where the general member meets the X-axis twice at the origin, but the speclat member meets the X-axls 3 tlmes. Define the third point q(t) as before. Immediately, we find qit) = p(u), where u = -2t. Then it's easy to check that the squared distance is 11(-7t3,-t)ll -" t 2+ .... so again we meet E once in A, at least for the family constructed from y=x 3. But our conclusion holds in general, because the analytic Isomorphism of the plane at the origin, which transforms N to the curve we have studied, preserves the multiplicity of the pullback of E to T at the closed point of T. Hence 4.3 is true. The formula for ]'. Finally, we generalize and strengthen the result [Z,(5), p.605]. THEOREM4.6. Let {Ct} be a family as in 4.1. Then we have F = 2P-2B in Pic(T).
We would like to apply the coincidence formula to the nodal tangents p I and P2, but these cannot be ordered consistently over a complete T. Hence we need a symmetric version of that formula. Denote by B the variety of complete plane conics (compare [K3] or [K4]), and denote by R the subvarlety of B parametrizing the reducible complete conics (line pairs) and their llmlts (double lines wlth double vertices). Denote by I) the divisor on R parametrizing the double lines with double vertices. Denote by B v the blowup of ~2× 1~2 along its diagonal. A point of 8 is a configuration (L,M,p) where L and M are lines, and p is a point on both. We map 8 onto R by sending each (L,M,p) to the complete conic consisting of L+M, provided with a double vertex at p. Now we push the coincidence formula [El = L+M-p down from 8 to R. Let h be a line in ~ 2 and denote by A 1 the pullback of the divisor h x 152 c 152xl52 to 8. Similarly, denote by A 2 the pullback of l~2xh. Hence L,-[A 1], and M=[A2]. Both A1 map to the same image in R, and it's easy to see that they do so birationally. We define x to be the class, in Pic(IR), of this image. Fix a line, denoted by g,in I}2. We represent the class p by the divisor on 8, denoted by l'I, parametrizing the (L,M,p) with I~g. Clearly IS maps to its image by a map of degree 2; we denote the reduced Image divisor by ~. Finally, the exceptional set E maps lsomorphlcally to D; we denote by ~ the class of this image. Then, pushing down the coincidence formula, we obtain the following result.
180
PROPOSITION 4.7 (Cycle Coincidence Formula). We have = 2;,,-2~
in Pic(R). Proof of 4.6. Assigning the embedded tangent cone at the node of each curve in the universal family of nodal cublcs on N, we obtaln a morphlsm N4R. Pulling the formula 4.7 back, we obtain the right hand side of 4.6, and all we need to show is that ~ pulls back to F. Now a nodal cubic degenerates to a cuspidal cubic exactly when its nodal tangents coincide. Hence all we need to show is that the pullback of D to N is reduced. We do this by reducing to an example. EXAMPLE 4.8. Given any cuspidal cubic N, choose affine coordinates so that the equation of N is x3+y3= y2. Set T=A I, and define Nt:
x3+y3 = y(y-tx),
which is nodal for t,'O. Here the node is fixed at the origin, and the right hand slde defines Its tangent cone at the node. By definition, B is the blowup of the p5 of conics along the Veronese surface, denoted V, parametrizing the double lines. Hence we are done if the pullback of V, along the map T-~ p5 via t-~conic y(x-ty)=O, is reduced. Now But the incluslon V4 p5 factors through the diagonal I~2--~ I~2 x I~2, and so does T-~P~, by choosing an ordering of the components of the tangent cone. Then the result follows, because it is clear that the diagonal has a reduced pullback to T. Zeuthen's Identities. For a family {Ct} as in 4. I, the identities in Plc(T) found in this section and the last are easily solved for M and M' in terms of F and 4. We obtain a generalization and strengthening of Zeuthen's key relations [Z, (6)-((7), p.605]. THEOREM 4.9. For a family {Ct} as in 4. I, we have
M = FI2 in Pic(T).
and
M'=
(2~+r)13
181 5. Characteristic Numbers.
We obtain the characteristic numbers of the curves of types y and (~ first, and then derive those for nodal cubics from them. Curves of type "y. Denote by r the inverse Image of /d In N. Hence r parametrlzes the curves of type ~'. Denote by K the compactification of the space of cuspidal cubics employed in [KS, Sec. 8]. (The characteristic numbers determined in [KS] were shown to be Independent of the compactiflcatlon, so now we choose one for which we have all the extra structure we shall need. In [KS], this compactification was denoted by ~.) We have a natural G-map I:F-~K, defined by assigning to each curve of type 7 the associated cuspldal cubic. In fact, 1 ls an open Immersion. Indeed, on the open subscheme I(F) of K, we can construct an inverse by assigning a vertex to the cusp of each nondegenerate cuspidal cubic. Thus K compactifies r . For pep2 (resp. L~ 1~2), denote by Hp (resp. HL) the hyperplane of p9 (resp. pl4) parametrizing the plane cubics through p (resp. the quartlcs in 1~2 through L). We have natural maps f: 1(4 p9 and g: 1(4 p14 assigning the underlying curve and dual curve. PROPOSITION 5. I. The divisors f- I(Hp) and g-l(H L) on IF are generically reduced. Proof. Identify I" In K. Then f-I(Hp) ls generically reduced by [KS, 7.3]. Now look at the pullback g- I (HL)" The map g can be constructed as follows. Map I" to the 1~9 of dual cubics by assigning the dual cuspidal cubic, and map r to p2 by the cusp map. Then map r into p29 via the Segre embedding. Composing with an appropriate projection to p14 we obtain g. Hence g-I(H L) ls the union of the pullback of a hyperplane under K-,I~9 and the pullback of a line under the cusp map. The first is generically reduced by [K5,7.3]. The second is generically reduced because it is so after restriction to the one-parameter family, Ct: y2= (x-t) 3, with cusp map c(t) = (t,O)~A 2. Indeed, the pullback of {x=O} ls {t=O}. The proof ls now complete. Denote by D (resp. E) the divisor class [Dp] (resp. [EL]) on K. Hence D (resp. E) represents the condition that a curve of type 7 pass through a polnt (resp. be tangent to a general line). For nonnegattve integers cx,13with oc+13= 7, we define the characteristic number r¢,13 to be the intersection number
=I K
[K].
182
These numbers are represented by intersection schemes in the open orbit r on K, and hence are independent of the cornpactification of r. We now compute the r~,p. To facilitate the process, we need to rewrite the powers of D and E. Denote by d and e the pullbacks to K of the first Chern classes of the pullbacks Dp and EL of Hp and HL, as in [KS, Sec. 7], and denote by c the pullback of class of a line in I~ under the cusp map. Pulling Hp and HL back to K, first as sets, and then applying (5. I ) for the scheme structure, we obtain the identities (5.2)
D= d
and
E = e+c.
The second identity says in precise terms that a curve of type y is tangent to a given line if and only if the underlying cuspidal cubic is tangent to the line or its cusp lies on the line. Abuslng notation, we shall write (cp) for the class of the codlmenslon 2 subvarlety of K parametrizing the cuspidal cubics with cusp at a given point (defined as in [KS, Sec. 8]), and for its pullbacks to F and K. Then we clearly have (5.3)
c 2= (cp)
and
c5=0.
PROPOSITION 5.4 (Characteristic Numbers for Curves of Type 7). We have F7,0 . . . . . FO,7 = 24, 72,200, 240,960, 1424, 1512, 1200. Proof. The case of r3, 4 will serve to illustrate the technique. Using (5.2) and (5.3), we obtain F3, 4 = j'd3(e+c)4 = J'd3(e4+4e3c+6e2c2+O) = N3,4 + 4j'd3e3c + 6NCP3,2, where the first and last terms of the bottom tlne are the characteristic numbers of [KS, Secs. 7 and 8]. Now the integral appearing in the middle term is the lnvartant c of the elementary system ~-3,3 of [KS, Sec. 7], so, by the formula 4~=1~'+3c of [KS, 5.9], where p,=N4,3 = N3,4 = I~', we find F3, 4 =960 by the evaluations [KS, 7.6 and 8.5]. To see that the calculation above is legitimate, note first that the orbit of curves of type y on N identifies canonically with the open orbit r of K +. In particular, the node map from N restricts to the cusp map on F. Further, the condition for NcP of [KS] Is the complete intersection of two divisors, each representing the condition that the cusp lie on a given line. Hence our condition for NnP restricts, on F, to the condition for NcP of [KS].
183 Finally, by dimensional transversality, NcP3,2 is easily seen to be independent of the compactification of £, so the results of [KS] may be applied. Remark. We cannot use the formula 4tJ.=~'+3c to evaluate the integral of d3e2c2 in the third term above. Indeed, denote by {Ct} a corresponding l-parameter family of cuspidal cublcs (through 3 general points, tangent to 2 general lines, with cusp on a third general line). This is not an elementary system, so {Ct} may contain degenerations other than curves of type o, so the formula might not apply. In fact, this is the caseJ We shall find, for example, degenerations of type 52, in the terminology of [Sch,p. 112]. By definition, a curve of type ~2 consists of a double line b, together with a single other line a, the union equipped with a double vertex at the intersection polnt, plus a simple vertex elsewhere on b. The cusp falls at an arbitrary point of b, but off the vertices. To obtain such a degeneration, take a cuspidal cubic C, choose a point S~C, off the cusp and flex, and take a homolography [FKId] from 5. The resulting plane cubic curves fill out a 5-fold, denoted A2, in p9. Including the cusp, one shows directly, by constructing homolographles, that we obtain a 6-fold, denoted D2, in K +, which dominates A 2. Now return to the condition d3e2c. It is easy to find a curve d~A 2 through 3 points, tangent to 2 lines. Choose a third line x; assigning a cusp at the point where x meets the double line b of the curve d, we obtain a curve of type 52 in the 1-pararneter family {Ct} representing d3e2c. We conclude that the 6-dimensional subvariety D2 of K ÷ meets the parameter curve of the family {Ct} above, so the formula 41J,=1~'÷3c, indeed, does not apply, since there are degenerations other than curves of type (I. For a further perspective, suppose we were to solve the last equation for c, substitute, and evaluate NcP3,2, using the values of the characteristic numbers for cuspidal cubics [KS, $ec.7]. We would obtain NCP3,2=72, in contradiction to the correct value, 20, found In [KS, 5ec.8]. We claim that a term representing D2 must therefore appear in a more general vesion of the formula in question. Indeed, the contradiction 72,"20 shows that some additional term is needed. To explain why, observe that the intersection giving the 72 is in fact proper. Indeed, whenever we have a cusp map, the set of curves with cusps on a glven line must be a divisor, C, the pullback of a line under the cusp map. Moreover, C meets each 6orbit properly, because we can translate a cusp off any given line. Hence our numerical contradiction must be due, at least, to the presence of the class, denoted 62,of D2 in an extended version of the formula 4~=1~'÷3c. In fact, th|s extended version is already implicit in Schubert's d|scuss|on [Sch, pp. 1 l Off.I: we actually have 4~ = IJ.'~3c+ 252. In addition, Schubert's subsequent numerical results show that the number of curves of type 82 in the family {Ct} above is 78. A modern treatment of Schubert's full picture, including these results, will appear in [Xril ].
184
Curves of type "~. Denote by ~ the inverse image of X in N. Hence ~ parametrizes the curves of type ~t- We have a projection L:~4 ~2 which assigns to each curve of type "~ the unique line which appears as a component. Our first goal is to define a natural compactification, ~+, of ~. To pave the way, we first recall the natural compactification S of the space of curves of type a given in [I<S,5ec.6, p.25 I]. By definition, a curve of type a is the union of a nonlngular plane conic C wlth one of Its tangent lines L. Each curve of type o comes equipped with a dual curve, also of type a, consisting of the conic dual to C, together with the line in the dual plane corresponding to the point of tangency of C and L. To construct S, we again use B, the space of complete conics recalled in Example 4.8. Denote by C the open subscheme of B parametrlzlng the nonsingular conics. Let C c I x C denote the conormal scheme of the universal family of nonsingular conics. A point of C__is a triple (p,L,C) where C is a conic, I~C, and L is the line tangent to C at p. Hence C parametrizes the curves of type o (and their duals). The compactification 5 is defined to be the closure of the natural image of C In IxB. We now construct ~+. To each curve C of type ~, we can assign a curve of type a by assigning to the unique conic component of C a simple vertex at the vertex of C. (Recall that as curve of type ~, C has a double vertex at one of the intersections of the line component L(C) wlth the conic, and a node at the other.) Thls defines a projection p:~ 4 5. We define the compactification ~ + of ~t to be the closure of the image of the map Lxp:~ -, 1~2xS. Again denote by Dp (resp. EL) the pullback of the hyperplane sheaf on pg (resp. P 14). Hence Dp (resp. EL) represents the condition that a curve of type 9 pass through a point (resp. be tangent to a line). Denote by mp (resp. nL) the pullback to ~+ of the divisor of complete conics through p (resp. tangent to L), and denote by aL the pullback to ~+ of the divisor of pointed conics with point on L. Finally define Lp to be the pullback to ~+ of a tine
In i~. PROPOSITION5.5. The divisors mp and Lp are generically reduced, and we have Dp = mp+Lp. The dlvlsors nL and aL are generically reduced, and we have EL= nL+2aL. Proof. The divisors mp, Lp, nL and aL are generically reduced as pullbacks of known generically reduced subschemes under bundle maps. The proof of the assertion about Dp is similar to that for 5. !. To investigate EL, we map to P'" by projecting the product of P~ and the Veronese surface of double lines, and work as before. This verifies 5. I.
185
For nonnegative integers oc,lBwith oc+13= 7, we define the characteristic n ~ b e r ~a~8 to be the intersection number
q/a.l~ = I DaEI3" We begin by rewriting D and E. Denote [mp] by m, denote [Lp] by L, etc. Then we have (5.6)
D= m+L
and
E = n÷2a.
PROPOSITION 5.7 (Characteristic Numbers for curves of type ~). We have ~7.o .....~o,7 = 42, 114, 260, 480, 588, 422, 144, O. Proof. Let's work out ~3,4 to illustratethe procedure. Integrating on ~+, we find qL3.4= SD3E4 = j{(m+L)3(n+2a)4 Expanding and omltting terms (llke L3 and m3n 4) that are obviously 0 by reason of dimension, we obtain • 3.4= .[(72m2n2a2L + 3mn4L2 + 24rnn3aL2 + 72mn2a2L2Now we push this down to S. Denote by p the projection ~+-, S. First of all, one checks directly as in [KS, Lemma 6.2] that we have (5.8)
p,L = [5]
and
p,L 2=a.
By the projection formula, this time integrating on S,we find • 3,4 = J"(72m2n2a2 + 3mn4a +24mn3a2)(Of course, for the last term of the previous integral, we have p,(mn2a2L 2) =mn2a3 =0.) This last integral is evaluated by pushing down to B as in [KS, Sec.6]; we then find 43.4 = 588. Nodal Cubics. We begin with some preliminary results on I-parameter families. So denote by {Nt] a 1-parameter family of nodal cubics, parametrized by a smooth, complete
186
curve T, such that the only degenerate members are of type 7 or ~. The family determines a map h:T4N. We define the numerical invariants 7 and ~ to be the integrals
7 = Jh*[r]
and
T
"~ =
Ih*[~]T
Hence )" counts the number of t such that Nt is of type 7, and ~t counts the number of t such that Nt has type ~. Similarly, we define the numerical invariants p, and F' to be the integrals p.= I[M]
and
T
I~'= J[M'], T
where M and M' are the divisors on T defined in Section 3. Hence tJ. counts the number of t such that Nt goes though a given point, and g' counts the number of t such that Nt is tangent to a given line. By Theorem 4.9, we obtain the fundamental identities (5.9)
I• = " / 1 2
and
p.' = (2~+7)13
of [Z, p.605]. The consequence we shall need first is the following.
COROLLARY 5. I O. Let {Nt} be a family as above, such that every Nt is a nondegenerate nodal cubic. Then {Nt} is a constant family. Proof. Suppose {Nt} were nonconstant. Then iz must be nonzero. By the first identity, T would meet r, a contradiction. To define the characteristic numbers, first we compactify the open orbit n- 1(#) of N, which parametrizes the nodal cubics and their duals. Our compactification, denoted N +, is defined to be the closure of the image of the induced morphism
N 4 CVxp14 where CV is the conormal scheme of the image V of p2 in p9 under the Veronese embedding (compare the proof of 2. I). In keeping with the abbreviated notation introduced at the end of Section 3, we shall write M and M' for the puIIbacI<sto N+of the hyperplane classes on p9 and p14. The maps N , P 9 and N-~P 14 factor through N +, so this does no harm. Now let ~+IB--dim(N)=8. We define the characteristic numbers for nodal cublcs to be
187
N i~= IM~NIB,,[N+]. ii°
For ~,13 as above, denote by Z~,8 the intersection of oc general translates of the pullback of Hp to N+ and 13general translates of the pullback of HL to N+. THEOREM5. I 1. In the situation above, we have: ( 1) the intersection scheme Zoc,t3 is O-dimensional, lies in the open orbit n- 1(N) of N, and we have
N~,I~: J'[Z~,I~], where the integral can be taken over N+, or over any other compactification of n- 1(N); (2) in characteristic 0, the scheme Z~,I3is reduced, and N~j3 counts the nodal cubics through ~ general points, tangent to I~ general lines, each with multiplicity I. pro.0f. The argument is just like that for [KS,Th.7.4], where the assertions of the next Lemma play the parts of [K5,7.2 and 7.3]. LEMMA 5. i2. ( 1) For I~P 2 (resp. Lc I~2) the pullback to N of the divisor Hp on p9 (resp. HL on pi4) is generically reduced. (2) The restrictions of M and M' to r (resp. ~) are D and E. Proof of the Lemma. To prove ( I ), suppose lip (resp. H L) had a nonreduced pullback, denoted by Dp, to N. Choose an irreducible component Y of Dp. We claim first that Y meets n-l(N). Indeed, suppose not. Then Y is a component of the boundary of n-1(N), and dim(Y) = 7. The boundary is a union of orbits, each orbit irreducible, so Y must De an orbit. Since any curve through a point p can be translated off p, this is Impossible, so Y meets n- I(N). Next, consider the image Z of Y in p9. Denote by T the intersection of 7 general translates of Z. Then T is a complete curve in NUKUX, by the standard translation argument, because K and X are the only 7-fold orbits In the closure of N, by I.I. By 5.7, we know that T meets K or X (or both), so Y meets lr or ~, Now the pullbacks of Hp to [" and ~ are generically reduced, by 5.1 and 5.4, and since each component Y of Dp meets one or the other, it follows that Dp is geneMcalIy reduced. A dual argument handles the pullback of H L, and completes the proof of ( I). As for (2), the assertion clearly holds for the underlying sets. But we have observed above that all components of the pullbacks of Hp and H are generically reduced on [" and ~ , so (2) holds at the scheme level as well. Hence the Lermna is true.
188
We define, for nonnegative integers o~,~, this time with o~+t3=7, the elementary system of nodal cubics, denoted by N_~,I~,as follows. For p~p2 (resp. L¢ 1~2), denote by Hp (resp. H'L) the hyperplane in p9 (resp. in pI4) parametrizing the cubics through p (resp. the duals of cubics tangent to L). Choose o~ general points Pi and I~ general lines Lj. Denote by Hi (resp. H'j) the pullback to N of Hpi (resp. H'Lj). Then an argument, based on dlmenslonal transversallty (this time uslng 2.7) Ilke that for the analogous construction [KS,7.5 ] shows that the intersection of the Hi and the H'j is a complete curve on N; denote this curve S~,IB. Then N_~,8 is defined to be the pullback to S ~ 8 of the universal family of nodal cubics on p 9 Similarly one obtains the dual family. We now define the numerical invariants i~, t~', y, and ~ of the system ~ , 8 , whose parameter curve S ~ 8 may not even be reduced. Denote by S~,8(1) the irreducible components of S~,8. For each i, denote by n~,13(i) the multiplicity of S~,8(i) in S~,8. Denote by p., It', )', and ? the corresponding Itnear combinations of the tl, I~', ~', and ~ of the families pararnetrized by the normalizations of the S~,I3(i). We define these linear combinations to be the numerical invariants i~, t~', )', and ~ of the elementary system N_~,I3. The identities (5.9) hold, by linearity, for each N_~. Of course, each H1 (resp. H'j) above represents M (resp. M') ~ Pic(N). Each elementary system EI~,I3 maps Into N and, as such, has the numerical invartants i~= N~+ 1~ and p.'= N~,8+ 1. The intersections on [" and ~ are just what we computed in 5.4 and 5.7. Hence we deduce the next result directly from 5.4, 5.7, and 5.9. THEOREM 5.13 (CharacteristicNLcnbers of Nodal Cubics, Zeuthen [Z, p.606]). For nonnegative integers ~,~, such that (x+~=8, we have Ns,0 ..... No,8 = 12, 36, 100, 240, 480, 712, 756, 600, 400.
6. Further CharacteristicN~l~bers.
Now we study conditions which are represented by subchemes of codimension 2. Our approach wtll stronger than that of [KS, Sec. 8], In that here we shall establish relations among cycle classes on a suitable compacttfication of N. Conormal Schemes. For each nondegenerate nodal cubic N, denote by CN Its conormal scheme. Hence CN Is a closed subscheme of the point-line flag variety I, and we can vlew CN as a point of the Hilbert scheme Hilb(I).
189
We shall need to recompactify the space N of nodal cubics so that each point of the compactification has a well-determined node and conormal scheme. We have a rational map from N+ to Hltb(I) which assigns the conormal scheme to each nondegenerate nodal cubic. Denote by N+÷ the normalization of the closure of the graph of this rational map. Clearly N*+ compactifies the space of nondegenerate nodal cubics, and it dominates N÷. From here on, N÷÷ will be our basic compactification. Since the characteristic numbers already found were shown to be independent of the compactification, our earlier determinations remain valid. Denote by by N_ the pullback to N *+ of the total space of the universal family of plane cublcs, via the projection N~+4 pg. Denote by CN the pullback to N** of the universal family on HIIb(1). For any point N~N ++, we shall write CN for the corresponding fiber or C ~ and call it the conormal scheme of the curve N. The classes (p), (I), (rip) and (pl). Fix a point p (resp. a line 1) in p2, and denote by Dp (resp. EI ) the pullback to N÷÷ of the hyperplane, in the p9 of cubics, of those through p (resp. of the hyperplane, in the P 14 of quartics, of those through the point of ~2 corresponding to I). Define (p), (I)c AI(N ++) to be the bivariant cohomology classes corresponding to these divisors. Denote by n the node map N÷+-. p2 Set G(np)=n - lp. This subscheme of codlmenslon 2 on N ÷÷ is a complete intersection because N ++ is normal. Define (np)~ A2(N ++) to be the corresponding blvariant cohomology class. Obviously (np) represents the condition that a nodal cubic have a node at p. 51nce a nodal cubic cannot have 2 nodes, we expect that (np)2=O in the cohomology ring. And indeed, we have (np)2=n-1(p)2=O, because (p)cA2(p2). Now consider a flag (p,l)~I. The total space CN of the family of conormal schemes is a closed subscheme of IxN *+. Let ¢:CN 4 N +÷ denote the structure map. Since (p,I) Is regularly embedded in I, it defines a class in A3(1), and hence a class c(p,I)~A3(CN). Therefore, since (~ is proper and flat, we obtain a bivariant cohomology class
(pl)=(1)w(c(pl))(l)WEA2(N*+). Obviously (pl) represents the condition that a nodal cubic have a given tangent line at a given point. Define N " ( p , I) to be the isomorphic image of the scheme CNFI((pl)xN ++) under (h. Obviously it represents the cycle class
(pl),-,[N+÷](A6(N"+). To prove both Prop. 6.2 and Th. 6.3 below, we shall need more information about N++(p,I) and (pt).
190
On N++, denote by VXp the locus of curves N whose conormal scheme CN has a fiber of dimension > 0 over the given i~P 2. (Equivalently, such that N has a vertex at p.) By semicontlnuity of fiber dimension, VXp is closed in N~+, and it clearly has codimenslon •2. Denote by VX(p,I ) the unlon of VXp and the locus, on G(np), of curves N wlth I as nodal tangent at p. Again one shows directly that VX(p,I) is closed, of codimension • 2. Set U(p,I ) = N++ - VX(p,I ).
Consider the following sequence of embeddings: x y z N+*(p, I)NU(p, l) --' (Dp- G(np)) n U(p, I) --, DpA U(p, i)--,U(p, 1), of which x and z are closed embeddings and y is open. Denote by e:N4N ++ the structure map, and by d(p)EA2(N_) the class defined by the inclusion of p in p2. Then we have
z" = (e.d(p)e*)I U(p,I). (cf. [KS, Lemma 5.6, p.248].) It follows, if we denote by c(p)~A2(CN) the class defined by the inclusion of the fiber Ip over p into I, that we have
z" = (¢,,c(p)dp") I U(p,i). Assigning to each curve N the tangent at p defines a morphism (Dp-G(np))nU(p 'I)"~ Ip. Moreover, x Is the embedding Induced by the Incluslon of (p,I) In Ip. 51nce Dp Is reduced (as a generically reduced divisor on the normal scheme N ++) and since I is generic, x is therefore the embedding of a divisor, whose class equals that induced by the embedding of (p,I) into Ip. Finally,observing that (p,I) c Ip¢ I, we conclude that
(pl) I U(p,l)
= (zyx).xwy*z*.
tn suTn, we have proved the following result.
LEI~'IA 6. I. The scheme N++(p, l)NU(p,I) is dense in N+*(p, I),and it is a complete intersection of codimension 2 in U(p, I)- Moreover, the corresponding class in A2(U(p, I))is equal to the restriction of (pl)EA2(N*+). The following cycle class identity empowers Zeuthen's argument [Z,p.607] for Cor. 6.6 below.
191 PROPOSITION6.2. In A6(N+*), we have the following identity: (p)2~,[N *+] = (pl)~,[N *+] * 2(np)~[N**]. Proof. Apply the analysis of Sec. 3, taking A = 1, to the family of cubics {Nt} parametMzed by T=N**. Although N** Is probably not smooth, nevertheless the ramification scheme R ts a divisor on the total transform D_of the total space N. The argument of [KS, pp. 247-8] applies because D_has no embedded components, since N+* has none. To obtain the identity of 6.2, pull R back via the section of B×N** over N** defined by p. The left side of the identity comes from the global class [R], via the expression for i*[R] displayed in the proof or Prop. 3.1. The right side of the identity comes from the two local contributions to R. The analysis in 5ec. 3 applies in a neighborhood of a general point of each component of R, because such a polnt projects Into N, which ls smooth. The part of R defined by the condition of tangency to I is proved in Sec. 3 to be given by the pullback of (the point representing) I under the map assigning the tangent line. Hence, by the analysis leading to Lemma 6. I, this part of R pulls back to (pl),~[N+*]. The remaining part of R was shown at the end of 5ec. 3 to be equal to 2E, where E ls the fundamental cycle of the locus of nodes on D. the image of the node section. Since p is regularly embedded in p 2 the pullback of E as a cycle (t) is equal to the class of G(np), which represents (np),~[N*+]. This completes the proof. Remark. The last proof follows the same general plan as that for [KS, (8.2), p.263], which was given for 2-parameter families. The reasoning there was incorrect, however, when it was implicitly argued that the invariants K and M, corresponding to our classes (np),~[N ++] and (p),~[N++], enumerate distinct points of T. Even in characteristic O, this will not be the case for all families satisfying the conditions of [KS, Sec. 5], because there p and I are general relative to the family, but here they are not. A more general version of [KS, (8.2)], in the style of Prop. 6.2, is easily obtained by a similar argument. Characteristic n~.bers. Let ~, 13and 8 be non-negative integers, such that oc.~. 28 = 8. We define the characteristic number N¢,8,6 to be
= J(p)a(I ~(pl)S,~[N**]. Ii*' Now let ~ and ~ be non-negative integers such that oc+13= 6. We define the characteristic number NnP¢,8 to be
192
I(p)~(1)P(np),~[N++].
NnP IB =
N~
Hence Ncx.8,6counts the weighted number of nodal cubics through (~ general points, tangent to 13general lines, and tangent to 8 other general lines each at a given general point, while Nm=3 counts the nodal cubics through oc general points, tangent to 13general lines, with a given general point as node. To determine these characteristic numbers, we first establish some notation. Here we let o(, p and 8 be any non-negative Integers. Choose general points P l . . . . . p(x~ p 2 general lines 11 1~ i~2 and general flags (ql,ml),---,(q~,ms)~ i. Denote by 1(~,13,8) . . . . .
the following intersection subscheme of N'+:
I(c(,p,~) =
(("~Dpi)I"1(~'~IEIj)f'l(Aw .÷(m,m~)). iI1
j-I
k-I
THEOREM6.3. (Characteristic Numbers for the Condition (pl), Zeuthen [Z, p.606]). We have: N6,0,1 . . . . . No,6,1 = 10, 28, 68, 136, 196,200, 148; N4,0,2, ... , NO,4,2 = 8, 20, 40, 56, 56; N2,0,3. . . . . N0,2,3 = 6, 12, 16.
p
Proof. In each Noc,8,6above, we have either c( or >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 <_k < 5; let 15 be a basis of Elk(w*), and let "rS" be the basis of ~6"k(w*) dual to 13. If m e 15, we denote by m" the unique element of 13' such that j'm'm = 1 and m ' n = 0 forall ne13, n ~ m . EroDositiQn (6.1). [Sch, formulas (61), (67), (74)] Let 1 <_k_< 5, let S be a purely k-dimensional family of triangles, and let S' be a purely (6 - k) - dimensional family of triangles. Assume that there are only finitely many triangles common to S and S'. Then the number of common triangles, counted with suitable multiplicities, is x =m,~1 (,r[s].m) (f[s'] m' ), where 13 is a basis of Qk(w*) and the other notation is as above. Proof. Since the paidng o-k(w *) x G-6- k(w*) --> Z iS represented by the identity matrix relative to the bases 15 and 13', we have [S]=Z (~[S]*m)m' me13
and
[Sl=Z (j'[S']m')m. rn~13
The conclusion of Proposition (6.1) follows immediately from this. Schubert obtained formulas in §§4,5,6 of [Sch], which are all of the special cases of Proposition (6.1). For instance, in the case where dim(S') = 1 and dim(S) = 5, Schubert's formula (61) can be written in our notation in the form
2 2 + (f[S'].a3) (f[S].a~a27) X = (~[S'].a,) (f[S]-a~a~'/)(f[s'].a2)(f[S] + "ala3Y)
+
,I,s.,.oH,s,. o 40)+
where d = c + cq + ~ + cc3 = y+ a 1 + a2 + a3 and the other notation is as in Theorem (5.1). We obtain Schubert's formula (67) in a similarly straightforward way from Theorem (5.3) above. The process for obtaining Schubert's formula (74) from our Corolrary (5.8) is based on similar ideas, but the calculations are rather messy. This situation might be improved somewhat if one could find a better pair of dual bases for Q3(W*).
206 Schubert proved his formulas only under the hypothesis that the families of triangles are ~ . Saying that a family S Is ordinary means that all of the intersections of S with the various sets of degenerate triangles, viz. S~C*, SnD*, SnDij*, SnC*nDij*, SnD*nDij*, SnX*, Sc~C*nX*, and Sc~D*nX*, are dimensionally correct. (Here, we are using the notation of [RS1, §4], as in §3 above.) This hypothesis is not used in our proof of Proposition (6.1). Modern versions of Schubert's proofs, which will be explained in §7, do not use that hypothesis either. Thus, the use of a different approach to the proof is not the best explanation of Schubert's reason for using that hypothesis. I suspect that the correct explanation of Schubert's use of this hypothesis is that Schubert probably considered certain intersection numbers in the formulas to be meaningless without that hypothesis. Consider, for example, the case where S is 2-dimensional. It is easy to justify, using rather elementary methods, that .f[S].al 2 is well defined. Indeed, this expression counts the triangles in S such that the first vertex coincides with some fixed point of p2, and we can guarantee the finiteness of that number by choosing the fixed point in sufficiently general position. On the other hand, there seems to be no comparable approach to showing that certain other intersection numbers, such as .[[S].ac, are well defined. Indeed, c = [C*], where C* is the locus of concurrent triangles, and there would seem to be no enumeratively meaningful cycle to which C* can be moved. If the machinery of modern intersection were not available, one might consider it prudent to avoid this difficulty by assuming that the families of triangles are ordinary. Schubert's proofs of his formulas (61), (67), and (74) can be considered as being based largely on the intersection theory of the (nonsingular) variety W' which parametrizes "triangles with one side omitted", i.e. the following type of configuration:
X3J ~.3"~/'~.2
"•X2
It is easy to check that W ' is a nonsingular variety of dimension 6. There is a birational morphism W* --> W', given by forgetting the third side and the family of conics. Under the hypothesis that S and S" are ordinary and in general position, one does a calculation in O.'(W') which yields an expression for the number of pairs of triangles s E S and s' e S' such that s and s' have the same image in W'. This expression counts the triangles common to S and S', but it also counts some pairs of triangles, of two specific degenerate types, which are not actually common to S and S'. Schubert corrects for this by subtracting expressions which count these pairs. One method of obtaining such expressions will be explained in the next section.
207 7. The intersection pairino of a blowuo. Let Y be a smooth n-dimensional variety. We say that the intersection pairing on Y is ~ if the pairings Q~(Y) x Qn4(y) ._~ Z induce isomorphisms Qn-i(y) = Homz (Qi(y), Z). We will prove in this section that if the intersection pairing on Y is unimodular and ~ is a locally free sheaf on Y, then the pairing on P(~) is also unimodular. We will also prove that if X is a nonsingular closed subvariety of Y and q( is the blowup of Y along X, the intersection pairing on Y is unimodular if and only if the intersection pairings on Y and on X are both unimodutar. We will apply the first result to describe the intersection pairing on W', the parameter variety for triangles with one side omitted. We will briefly explain how one can apply the second result and a construction of W* due to Speiser, Kleiman, and Harbourne to describe the intersection pairing on W*. Finally, we will use these results to explain Schubert's proof of the results of §6.
Let Y be a nonsingular variety, let ~. be a locally free sheaf on Y of rank r, and let p: P(C-)-->Y be the structural morphism. There is an exact sequence of locally free sheaves
on P(~): (*)
0 -~ :£ --> p*(~) --> O(1) ---> 0.
By dualizing this, we obtain:
(*-)
0---> (~(-1) --> p.(~v) -->
--> 0
It is well known that Q'(P('~)) is a free CI.'(X) - module with basis {1, ~ . . . . . ~r-1 }, where ~=c1(O(1)), andthat p,(~i)=0 for i = 1 ..... r-2, while p,(~rq) = 1• C1.'(X). Using (*), we see that there is an identity of total Chern polynomials c(~'") = p*c(~")(1- {) -1. Thus: c i ( ~ v ) = P, Ci(~V)+P, Ci(.~v).~+
... + ~ i
for every i. Therefore, {1, c 1(~v) ..... c t . l(~v)} is an Q.'(X)-module basis of Q'(P('~)), and p*(c i (~")) = 0 for i = 1 ..... r-2, while p*(c,. 1(~")) = leC['(X). ProPosition (7.1L If Y and ~ are as above then the intersection pairing on P(~.) is unimodular if and only if the intersection pairing on Y is unimodular. Specifically: (1)
Under the intersection pairing on P('£.), £~'(X).c i (~v) is orthogonal to if j ~ i. In other words, if a and b are homogeneous elements of Q'(X) such that p*(a)ci(~ "v) and p*(b)~ r-j- 1 are of complementary codimensions, then O'(X).~r-j-
1
J'p(¢) p*(a)c i (;P0-p*(b)~ ' J 1 = 0. (2)
fp(~.) p*(a)c i(;V).p*(b){ r-i- 1 = ;y ab
elements of complementary codimensions.
when a,b • Q'(Y)
are homogeneous
208 Proof. We deduce the conclusion about unimodularity from statements (1) and (2), using the direct sum decompositions of the groups CLP(p(~.)) obtained from the two Q ' ( Y ) - m o d u l e bases of Q'(P(~)). To prove statements (1) and (2), we must calculate j'p(t:) p*(a)c i (iTv)P*(b)~ r.j- 1, where a and b are homogeneous elements of Q'(X) such that p*(a)c i (~v) and p*(b)~ r-j- 1 are of complementary codimensions. We have: j'p(~.) p'(a)c i (;Tv)' p*(b)~r i-1 = j'y p.(p.(ab)c i (~.v)~r- ;- 1) = ] y ab.p.(c i (~-v)~r. j- 1). Therefore, the conclusion is a consequence of the following: Lemma (7,2). With the assumptions and notation as above, we have: p.(ci(~.v).~r.j.1)=
{01 ifj=iif j~i
proof. The conclusion is already known in the case where j = r -1, so we proceed by descending induction on j. From (**) we see that c i (~-") - c i. l(~v)~ = p*c i (-~v), so that C i+l(~v)~r'j'2
. C i('~'v)~r'j'l
= p*(ci+ 1 (~v))~r'j'2
The refo re: p.(c ~(~-v).~r-j-l) = (p.(c i + 1 (~.v).~r-j-2)). c i (.~V)p.(~r-j-2). Since we may assume that j < r - 2, the second term on the right side of this equation vanishes. We apply the inductive hypothesis to complete the proof. Corollarv ('7.3L With the hypotheses as above, assume that rank(£) = 2, that Clk(Y) is free with basis {a 1..... ap}, and that Qk- l(y) is free with basis {b 1..... bq}. Let n = dim(Y). Suppose that Gn- k(y) has a basis {a 1', .... ap' } which is dual to the given basis of Qk(y), andthat CLn'k+l(Y) has a basis {bl", .... bq'} dual tothegiven basis of CLk-I(Y). ~hi-~ means that fai.aj'= 1 (resp. 0) if i = j (resp. i s j), and similarly for b i and bj'.] Then Qk(p(~)) has a first basis {a 1..... ap, bl~ ..... bq ~} and a second basis { bl~l ..... bqq, a 1..... ap } where q = c 1(~''), while cLn- k + l(p(~.)) has a first basis { b 1",.... bq', a 1'~ ..... ap'~ } and a second basis { a 1'T1..... ap"q, b l',...,bq' }. The second basis of Qn-k + l(p(~)) iS dual to the first basis of Gk(P('~)) and the first basis of G n k + l(p(~)) is dual to the second basis of Qk(P(Z)). ExamDl(# (7.4.1). Let X (E P2× 152 be the point-line incidence correspondence. Thus, X_= P(~) as a scheme over p2, where ~ is a locally free sheaf on P2 which fits into an exact sequence 0 ~ Op2 (-1) --) ((~p2) 3 ---> ~ ---)"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 <_r-2. proof. Statement (2) is obvious. To prove (1), let a e G ' ( Y ) , b e K e r ( g * ) be homogeneous elements such that f*(a) and j.(b) are of complementary codimensions. Using the projection formula, we find that : f*(a).j.(b) = j.(j*f*(a).b) = j.(g*i*(a).b) e j.(Ker(g.)).
Now, an element of Ker(g.) is a sum of terms of the form g*(x)~ p, where p_
• *a fJ,(g()
c ~( .;.v ))t.(g( . * b )~ r-j -2) =.fj.(g.(a)ci(~-v)),g.(b)~r-j-1) =- f g.(a)c i (~- v). g.(b)~r-j-1 X
I
~ab
=
-
X
if i ~ j
if i=j
211 D.9_[~LE.~_LL~. Let Y be a nonsingular projective variety, and let X be a nonsingular closed subvariety of codimension 2. With the remaining notation as above, assume that i*: Q ' ( Y ) ~ Q ' ( X ) is surjective, Suppose that Ok(Y) is free with basis {a 1..... ap}, that Qk'l(X) is free with basis {i*(bl) ..... i*(bq)}. Suppose that Qn-k(y) has a basis {al', .... ap'} which is dual to the given basis of C~k(Y), and that Qn-k-l(X) has a basis {i*(bl' ) ..... i*(bq')} which is dual to the given basis of C[kI(X). Let "q = [X]. Then the pairing Qk(q() xQn-k(~') -->Z is described relative to the bases {a I ..... ap, f*(bl)'q ..... f*(bq)q} of CLk(~) and {al', .... ap', f*(b l')'q ..... f*(bq')'q} of Qn-k(~) by the diagonal matrix with entries 1..... 1 (p entries),-1 ..... -l(q entries). Proof. We observe that f*(b)'q = f*(b)-j,(1 ~) = j,(j*f*(b)) -- j,(g*i*(b)) for any be Q.'(~). Thus, the conclusion fotows immediately from Theorem (7.5). ExamPle (7,7,1). Let Y 2 c p2x p2x !~2 be the variety which parametrizes pairs of points on a variable line in p2. It is well known that Y2 =-15&&(P2xP2), the bTowup of p2 x p2 along its diagonal. Let a 1 and a 2 be the pullbacks of the hyperplane classes from the p2 factors, and let 5 be the class of the exceptional divisor. Since a18 = a25, we denote this element as a& The pairing Q,I(Y2) x Q.3(Y2) --> Z is described by the matrix diag(1,1 ,-1) relative to the basis { a 1, a2, 5} of QI(Y2) and the basis { ala22, a12a2 , a28} of C[3(Y2). The pairing (12(Y2) x C~2(Y2)--~ Z is described relative to the basis {al 2, a22, ala 2 , a5 } by the matrix consisting of the diagonal blocks:
Examole (7.7.21. Let W' be the variety (discussed above) which parametrizes triangles with one side omitted. It is possible to show that we can construct W' by starting with (p2)3, blowing up (p2)3 along A12 = {(x 1,x 2 ,x3)l xl= x 2 }, and then blowing up 1-~.~] 2 (p2)3 along ~13, the strict transform of A13. TO check this, consider the obvious morphism W'---> (p2)3, show by local considerations that the universal property of the blowup provides a morphism W'~'1~,~z&12((p2)3), and repeat the process to obtain a morphism W' ~ 15&~ ( ~ (p2)3). By studying the fibers of these different morphisms and A A 13
12
using Zariski's Main Theorem, we show that this morphism is an isomorphism.
Using this construction, we obtain bases for the groups
G.k(W ')
consisting of
monomials in a 1, a 2 , a3 , (~12,513, where 812 and 513 are two exceptional divisors coming from the blowing up processes. Thus, the pairings O. i (W') x CI.n" i (W') --> Z can described by diagonal matrices relative to the explicit bases, as explained below.
212 First of all, QI(W')
has the basis {al, a2, a 3, 612,813}, while QS(W') has the basis 2 2 2 2 2 2 , a~a2812 2 2 { ala2a 3 , ala2a 3 , ala2a 3 , a2a 613 } .
where aSii denotes either a;Sij or aiS~j. Relative to these bases, the pairing between QI(W') and ClS(w ") is described by the diagonal matrix with entries 1,1, 1, -1, -1. For the pairing between Q2(W') and Q4(W'), we note that C12(W') has the basis: 2 2 2 { a 1 , a 2 , a 3 , ala 2 , ala 3 ,a2a 3 , a812, a3812, a813 , a2813,812(~13 } , while Q4(W') has the basis: 2 2 2 2 2 2 2 2 2 2 a3a2~12,8~ a2a2813 2 {a2a3, ala 3 , ala 2 , ala2a 3 , ala2a3, ala2a3, a3a812, a813, , a 612613 }, Relative to these bases, the pairing Q2(W') x Q4(W')-+ Z matrix with entries 1, 1 , 1 , 1 , 1, 1, -1, -1, -1, -1,1. Finally, 2 2 {ala22 , a,a~, a~a 2 , a2a32, a~a3 , a2a 3 , ala2a3 , a 512,
is described by the diagonal Q3(W') has the basis: 2 a3512, a3a512 ,
a28t3, a ~ l 3 , a2a813, a812813 } • Relative to this basis, the pairing on C13(W') is described by the matrix of diagonal blocks:
Examole (7.7.31. Speiser, Kleiman, and Harbourne have proved [HKS] that W* can be constructed from W' by a sequence of blowups. In terms of the construction of W' discussed in (7.7.2), we can describe this construction as follows. First, let z~23' be the strict transform on W' of A23 c (p2)3, where z~23= {(x 1, x 2 , x3) I x 2 = x 3 }, and let W" be obtained by blowing up W' along A23'. (In terms of the original definition of W', we can describe A23' as {(x1, x2, x3, "~'2, ~3 1 x2 = x3 and ~'2 = 63}' ) Let 6 " c W" be the strict transform of the small diagonal ~, = {(x, x, x)} c (p2)3, or equivalently the strict transform on W" of the "locus of concurrent result is that W* is isomorphic describe this is to observe that correspondence discussed above,
triangles" on W', viz. C': = {(x, to the blowup of W" along W ' = Y2xp2Y2, where Y2 is regarded as a scheme over p2
x, x, 62 , ~'3)} c W'. The A". Another way to the point-line incidence by projection to the first
p2 factor. Then W"---13~.A(Y2xp2Y2); this variety is called X 3 in [Sp] and is related to Kleiman's iteration scheme, discussed in [K1]. With this notation, W* is isomorphic to the blowup of X 3 along the set of pairs ((X,X,62),(X,X,,~3)). It is not difficult to use this construction of W* to describe the pairings Qi(W*) xQsi(W*) ~ Z. We do not need to work this out in full detail, but I will briefly describe the intersection pairings and present the details of one special case.
213 Let 15o be a basis of CI.k(w'). If g: W* ~ W' is the structural morphism, then we obtain a basis 1"5 of £l.k(w *) consisting of g*(150) and the following additional elements: k=l {)23, c (these divisor classes are defined as in §2). k=5: m023, nc where m = g*(m'), n = g*(n'), and m'~O-4(W'), n'~O-4(W ') restrict to positive generators of (~4(/k23') and G4(C') respectively. k = 2,3,4: me23, - nc, - Pe23c, with m = g*(m'), n = g*(n'), p = g*(p'), where m',n'~O.k'l(W') and p ' ~ G k 2 ( W ' ) run through sets of elements which restrict to bases of Gk'1(A23"), Cl,k'l(C'), and clk-2(A,") respectively. -
If 150' is the basis of G6"k(w ') dual to 15o, we obtain similarly a basis 15" of C]-6"k(W*), dual to 1~, consisting of g*(150') and the following additional elements: k=l: k = 5: k = 2,3,4
-e23, -c. - me23, - nc where m = g*(m'), n = g*(n'), and m'~ G4(W'), n'E G4(W ') restrict to positive generators of G4(A23') and G4(C') respectively. - m823, - nc, - pe23c, with m = g*(m"), n = g*(n"), p = g*(p"), where m", n"E GSk(w') and p'~GS'k(w') run through sets of elements which restrict to bases of C].5"k(A23'), QS~k(c'), and G6k(A ") respectively, dual to the bases of Gk'l(z~23' ), Gk'I(C'), and Gk'2(A ") mentioned above.
If k = l , we can take 15o = { a 1 , a 2 , a 3,(x 2 , e ~ } . Then 15={a 1 , a 2 , a 3 , ( x 2,e~ 3 , e 2 3 , c } is a basis of GI(W*). By Example (7.4.2), the basis dual to /~0 is: 1~0"= {ala2a3(x2or. 3 - a12a2a3(:x2- a12a2a3(x 3 , a12a3o~2c~3, a12a2(x20~3 , a12a2o~j(a3-(~2), a12a3o~2(a2-.c(3)}. Since a12a2(x 3 ~ Q4(W') restricts to a generator of G4(A23'), and since a12(~zo.3 = (122(I32 restricts to a generator of G4(C'), the basis dual to /5 is: 2
2
22
22
15'= { ala2a3(:z2(x3 - ala2a30{2 - a 1a2a3(x3, a3(x2(:~ , a2~.2(~3 , 2 2 2 2 2 ala2(x3(a 3 - cc2), ala3(x2(a2- (x3), - ala(xe23, - (~2c~3c}. Let S, S'~W*, with dim(S') = 1 and dim(S) = 5. Computing with these dual bases, we obtain: .f[s']-[S] = (.f[s].(ala2a3(~2o~3 - a12a2a3~2- a12a2a30c3 )cf[s'].a 1 ) + (j'[S].a3c(22(x32 )(f[s,].a2 ) + (i'[S].a2(~22(x3 2 )(f[s,].a3 ) + (j[S].a12a2(~3(a3-cc2))(.f[s'].(z 2 ) + (.l'[S]-a12a3c~2(a2-c~3))(.1"[S'].c{3 ) - (f[S].a12ac~e23)d[s'].e23 ) - (f[S]-(x220~32c)(J[S'].c). This
is
Schubert's
introduced by our
formula blowing-up
(60).
The
process,
negative correspond
terms to
the
at the
end,
correction
which were terms
which
214 Schubert introduced to account for irrelevant pairs of degenerate triangles counted by the expression j'w,g*[S3.g*[S]. Schubert deduces his formula (61) from this by some rather elementary calculations. In doing this type of calculation, Schubert was guided by the criteria of making his formulas (i) as invariant as possible under permutations of the various factors of (p2)3 and (p2)3 and (ii) balanced with respect to switching p2 and I~2 factors.
In this section, we have used the construction of W* as a blowup largely for purposes of understanding some of Schubert's proofs. It would appear, however, that this type of construction could be of interest in other situations. Indeed, the constructions used here, to construct W' by a sequence of blowups starting from (p2)3 and to construct W* by a sequence of blowups starting from W', could be imitated in the situation where Y is an arbitrary nonsingular variety and the starting point is Y x Y x Y. The variety constructed by that process would appear to be closely related to the variety of complete triples which Le Barz [L] recently constructed by using Hilbert scheme methods.
8. ComDarino W* and B. Since Qk(w*) and Gk(B) have the same rank for every value of k, it seems interesting to ask whether or not W* and B are isomorphic. Speiser and I observed in [RSl] that W* and B are not isomorphic as schemes over W. In this section, we begin by observing that there are natural actions of the symmetric group S 3 on W* and B. The main result says that there is no isomorphism of W* and B which is compatible with those actions. To describe the action of S3 , recall that W c (p2)3x (p2)3 is defined to be the set of all points (x 1 , x2, x3 , & l , ~ z , ~ 3 ) such that xie ~.j when i ~ j . Clearly, S 3 acts on W; just let ~ • S 3 send (x 1 , x 2 , x 3 , ~-1, ~z, ~3 ) "-> (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 (
~
~
/ ~=
~ = ( c°nlcst2hr°Th }
~i ~ : : ~ : ~ - I ~ _ _
a i " x.1 E f i x e d
o<,
line I~.k
&, ~ f i x e d
fixedpoint
point
T. r conicsthrough xk } =~ l::angentto&kat Xi= Xj
~ j x k xi = x j / ~ ~
@ x-x ij
i
~=~J
j
and ~.=£. 1
,, "~ /.. J
I conicswitha"l = singularity at,~> Xl=X2=X3
Xl
]
j/ variableline 4---'/(secondcomponentof Z ) I ~"X~ , "~
\
.
"~1= '~ 2 = [ :~ X3
i~ conicsofwhich"l
Z=~ ~i=~'2= ~3 is~
J
I,.a component .)
tf: c o n c u r r e n t
sides
c: col linear
points
218
The exceptional set X* The diagram shows a generic element (×l,x2,×3,#.l,~.z,~.~,:E) or X*.
(" / = / k.
conics l;angentto ~. at x, haying second 1. order contact with a I~ fixed conic .2
x l = x 2 = x:~
=~-,1 = ~-z = &3 '~x,
= x 1 = x2= x 3
In §2, X* is detinedas q~(X), where qw is the natur~.lprojection of W ~ onto W. The codimension of X* in W* is 2. The rational equivalence class [X*] is useful in the description of the intersection pairings, in §5. The intersection theory of the nonsingular varieb/ X* is studied in [RS 2].
219 REFERENCES
[B1]
A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973) 480-497.
[B2]
A. Bialynicki-Birula, Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Polon. Sci. Ser. Sci. Math .... 24 (1976) 667-674.
[CF]
A. Collino and W. Fulton, On the space of plane triangles, to appear.
[F]
W. Fulton, Intersection Theory, Springer-Verlag, 1984.
[HKS] B. Harbourne, S. Kleiman, and R. Speiser, work in progress. [K1]
S. Kleiman, Multiple point formulas I Iteration, Acta Math. 147 (1981) 13-49.
[K2]
S. Kleiman, Intersection theory and enumerative geometry: a decade in review, Proc. Symp. Pure Math. (Bowdoin conference proceedings), to appear,
[L]
P. Le Barz, La variete des triplets complets, preprint, 1986.
[RS1] J. Roberts and R. Speiser, Enumerative geometry of triangles, 1, Comm. in Algebra 1__22(1984) 1213-1255. [RS2] J. Roberts and R. Speiser, Enumerative geometry of triangles, II, Comm. in Algebra 1._44(1986) 155-191. [RS3] J. Roberts and R. Speiser, Enumerative geometry of triangles, III, Comm. in Algebra, to appear. [Sch]
H. Schubert, Anzahlgeometrische Behandlung des Dreiecks, Math. Ann. 17 (1880) 1213-1255.
[Se]
J. Semple, The triangle as a geometric variable, Mathematika 1 (1954) 80-88.
[Sp]
R. Speiser, Enumerating contacts, Proc. Symp. Pure Math. (Bowdoin conference proceedings), to appear.
[-ry]
J. Tyrrell, On the enumerative geometry of triangles, Mathematika 6 (1959) 158-164.
COMPUTING CHOW GROUPS F. Rossell6 Llompart and S. X a m b o Descamps *
Dept A.lgebra i Geomema, Fac. Mat., Univ. Barcelona, Gran Via 585, Barcelona 08007, Spain
1. Introduction 2. Notations and conventions 3. Good closed filtrations 4. Chow group of some fibre spaces 5, A.(Hilb3pn) 6. The case of positive characteristic To Dian
Fos$cy,
in
entmoriam
1. I n t r o d u c t i o n In this paper we prove two results concerning Chow groups. The first gives information about the Chow groups of schemes (see the conventions below) that have a "sufficiently nice" filtration (see T h e o r e m 1). This theorem implies, in particular, that the Chow groups of a scheme that possesses a cellular decomposition are free with basis the closures of the cells (see C o r o l l a r y to T h e o r e m 1). This result seems to be well known in characteristic 0; we include a proof in general because we have not found one in the literature (see, for instance, Fulton [1984], 1.9.1, where it is proved that the cells generate the Chow groups), The second result (Theorem 2) gives information about the Chow groups o f " n i c e fibrations" in terms of the Chow groups of the base and the Chow groups of the fiber. Part (i) of this theorem generalizes substantially the statement 1.I0.2 in Fulton [1984], while part (ii) gives a toot for computing Chow groups that seems to be more effective than alternative methods that are available, such as those derived from the results of Bialynicki-Birula [1973,1976] about actions of the multiplicative group on complete smooth schemes with finitely many fixed points. These methods have been used by Ellingsrud and Stromme [1984] to compute the Chow groups of Hilbkp 2, the Hilbert scheme of k-tuples of p 2 for all integers k (for the case of Hilb3p 2 see Elencwajg and Le Bar-z [ 1985a]). * The authors were partially supported by the CAICYT
221
One of the motivations of the present work was the study of Copkp n (and in particular Hilb3pn), the scheme which parametrizes, in the sense of Hilbert scheme, k-tuples of coplanary points in pn. The aim of such an study is to establish enumerative formulae for multisecant planes similar to the multisecant formulae for lines, especially those obtained by Le Barz. But it turns out tilat Copkp n, for k>3, among other pathologies, is singular, and that ~oopkp n, the scheme that parametrizes pairs formed by a k-tuple of points in pn and a plane that contains it, is a natural desingularization of Copkp n, and so many of the formulae we are seeking can already be obtained from the knowledge of the Chow groups of ~opkP n. The computation of these groups using the method of Bialynicki-Birula [ 1973,1976] appears to be much more intricate than for the case treated by Ellingsrud and Str0mme, which makes it desirable to have a more convenient method at hand. This computation has been done, using the methods introduced in this paper, by Rossell6 [1986], regarding ~'oopkpa as a fibration over the Grassmannian of'planes. In this paper we give an independent computation of the Chow groups of Hilb3p n (see T h e o r e m 3) which has interest in itself. As far as the determination of the multiplicative structure of A.(H-ilb3p3) goes, as well as enumerative applications of it, they will appear elsewhere. For the relevance in enumerative geometry of knowing that certain Chow groups are finitely generated and free, we refer to the articles of Kleiman [ 1976,1979].
2. Notations and conventions By scheme we shall understand an algebraic k-scheme of finite type which can be embedded as a closed subscheme of a smooth k-scheme of finite type, where k is an algebraically closed field. The hypothesis of finite type for the schemes is in order to apply the intersection theory as developed in Fulton [1984]. Our restriction to a field k comes from the fact that in our arguments we use an homology theory satisfying properties (a) to (d) below. In the characteristic 0 case, it is the homology with locally finite supports, or the Borel-Moore homology (see Fulton [1984], Ch. I9; Fulton-MacPherson [1981], Ch. III; Iversen [1986], Ch. 10), and if k has positive characteristic p then the homology theory is def'med as some suitable relative l -adic cohomology, l a prime number different from p (see lversen [1986], Ch. 9 and Laumon [1976]). If the quoted properties of intersection and homology theories could be guaranteed under more general hypothesis, then our proofs would be also valid in such a generality. Notice also that (i) of T h e o r e m 2 does not involve any homology arguments. By a closed f'dtration of a scheme we shall understand a Finite filtration by closed subschemes. We shall let H i denote an homology theory, that is, a functor from schemes to abelian groups that is covariant for proper maps and contravariant for open embeddings, and which, moreover, satisfies the following statements (see [F], Ch. 19):
222
(a) Let X be a scheme, Y a closed subscheme and U = X-Y. Then' there exists a long exact sequence ... --o Hi+I(U ) --~ Hi(Y ) --4 Hi(X ) -~ Hi(U ) ~ ... (b) For any finite disjoint union of schemes UX i , and for all k, H k ( U X i) = @Hk(Xi).
(c) For all schemes X and all integers k there exists a map C/x : Ak(X) ~ I-L2k(X) that commutes with push-forward by proper morphisms and with restrictions to open sets.
In characteristic 0 we shall say cl X that is an isomorphism if cl X is an isomorphism and H2k+I(X)=O for all k. In characteristic p > 0 we shall say that el X is an isomorphism if cl X : Ak(X) ® Z 1 ~ H2k(X)
is an isomorphism for all k, and H2k+I(X ) = 0 for all k. (We do not know whether "cl X is an isomorphism for all k " implies "I-I~+I(X) = 0 for all k ".)
(d) If X is a scheme such that cl X is an isomorphism, then given any projective bundle P ---) X the map clp is an isomorphism.
For convenience of the exposition we shall first develop in detail the characteristic 0 case and in Section 6 we will explain the slight modifications of the proofs that are required in characteristic p>O. Now combining properties (a)-(d) we prove a simple lemma which plays a key role in the proof of our theorems. Lemma
Let X be a scheme such that cl X is an isomorphism. Then for any fiber bundle
E--*X we have that cl E is an isomorphism.
223
Proof We shall use induction on n. The case n = 0 is a direct consequence of the hypothesis and the fact that X 0 = Z 0.
Assume now that the theorem is true for n-1. Then we have a commutative diagram
0 --~ H2k(Xn_l) --~ H2k(Xn) ---->H2k(Z n) --->0
?clkx~., Ak(Xn_l) ~
$clxk
$cl~
Ak(Xn) --4.Ak(Zn) ~ 0
In this diagram, clx~.l is an isomorphism by induction and cl7_~ is an isomorphism by hypothesis, so we see, by (a) and the definition, that the top row is exact. The bottom row is also exact. Therefore, clxn is an isomorphism and hence we have en exact sequence
(*)
0 --')'Ak(Xn.l) ~
Ak(Xn) ~ Ak(Z n) -'#0.
In thisexact sequence, by induction,Ak(Xn.l) is a finitelygenerated frcc group such that the classes of thc imagcs in Xn. 1 of cyclc representativesof givcn bases of the Ak(Zi), for i < n, form a basis. Sincc Ak(Zn) is free by hypothesis, the exact scqucncc (*) is split and from this the theorem follows.
•
We say (see Fulton [ 1984], Ex. 1.9.1) that a scheme X has a cellular decomposition if there exists a closed filtration X = X n ~ Xr, q ~... D X o D X _ 1 = ~ such that Z i = Xi-Xi. 1 is a disjoint union of locally closed subschemes Zij isomorphic to affine spaces A mij, The Zij wilt be referred to as cells of the cellular decomposition. These notations will be used henceforth.
Corollary Let X be a scheme and assume that X admits a cellular decomposition. Then Ak(X) is,for all k, a finitely generated free group for wtu'ch the classes of the closures of the k-dimensional cells forrn a basis.
224 Proof Let ~ = P(E~I) be the projective completion of E, so that we have an open embedding j:E-~P such that P - E = P, where P is the projective bundle associated to E. Therefore we have, for all k, a commutative diagram 0 ~ H2k+I(E) ~ H2k(P) ~ H2k(P) ~ H2k(E) --) 0
?cl~
+cl~
$ct~
0 ---->A k(P) ~ A k(P) --) A k ~ ) ----)0 k in which the rows are exact and clp,cl~ arc isomorphisms. For the latterw c use (d), which then implies that thc odd homology groups of P (and of P) arc zcro, so that by (a) we gct the exactncss of the top row. For the cxactncss of the bottom row, see Fulton [1984], 1.8. Moreover, the map Ak(P) --) A k(P) is injective (See Fulton [1984], Theorem 3.3 and its proof.) N o w by a little diagram chasing we easily get that cl Ek is an isomorphism and that H21:+ICE) = O.
*
3. Good closed filtrations Theorem 1
Let X be a scheme and let X = XnD Xn_ l D... ~ X 0 ~ X_ 1 = ~
be a closed filtration of X. Set Z i = Xi-Xi_ l
and assume that for all i (i) Ak(Zi) is a finitely generatedfree group, and (ii) clzi is an isomorphism.
Then cl X is an isomorphism and, for all k, Ak(X) is finitely generated free group. Moreover, the union of the classes of the images in X of representative cycles of free bases of Ak(Zi) is a free basis of Ak(X ).
225
Proof With the notations explained before, we shall show that the conditions (i) and (ii) of Theorem 1 are fullfilled. That (i) is satisfied is a direct consequence of the definitions and Fulton [1984], 1.3.I and 1.9. Moreover, the classes of the k-dimensional cells of Z i form a basis of AI~(Zi). Similarly, the lernma above and the properties of cl imply that clzi is an isomorphism.
*
Remark 1 This proof also gives that cl X is an isomorphism under the conditions of the corollary, which is the statement (b) in Ex. 19.1.11 of Fulton [1984]. Remark 2 The above C o r o l l a r y implies Theorem 4.5 in Bialynicki-Birula [1973], which says that the number of cells of given dimension in any two cellular decompositions of X is the same. Let us also remark that the proof in Bialynicki-Birula [1973] is purely combinatorial.
4. The Chow group of some fibre spaces Theorem 2 Let X be a scheme which adrmts a cellular decomposition and let
f : X'--~ X be a morphism such that for all cells ZSj of the decomposition
f-l( Zij ) ~ Zij × F where F is a fixed scheme. Then
(i) For all k there exists an epirnorphism
(*)
~) Ar(X) ® As(F) --) Ak(X') r+s=k
(ii) l f cl F is an isomorphism and Ak(F) is free for all k, then (*) is an isomorphism for all k and cl x is an isomorphism.
226 Proof |
Let X i = f-l( Xi ), so that !
x'= X ~
!
X~_ I~...~ Xo~X_' I = ~ I
I
I
is a closed filtration of X'. We shall write Z i = X i - Xi. l = f l ( Z i ) .
For all i, j we fix an isomorphism hij : Zij × F -_- f-l( Zij ).
T o prove (i) we shall proceed by induction on n. If n = 0 then
Ak(X g) = Ak(f'l(z0)) @ hod • ~j Ak(f'l(Z0j)) < "-'-'
Ak(Z0j x F)
= ff)j Amoj(Z0 j) ®Ak-m0j(F)
= ff)j( ~r+s=k Ar(Zoj) GAs(F))
=
(~
A.r(7--,0j) GAs(F)
j. r+s=k
= ~ Ar(X 0) GAs(F). r+s=k
Notice that the resulting isomorphism
h0 : ~
Ar(X 0) ®As~) --~ Ak(X0)
r+s=k
is such that
ho dZojJ®tvl)
-- 1%(z~ × v)i,
for all j and any [V] in Ak.rnoj(F).
227 Assume now by induction that (i) is true for n-l, i. e., that we have an epimorphism
Ar(Xn. 1) ®As(F) -+ Ak(Xn.l)
hn_l: •
r+s=k suchthat
hn.l([Zi,j,] ®[V]) = [hi,j,(Zi,d,×V)] for i' < n, any j' and any [V] in Ak_mi,j,~).
Now reasoning as in the case n=0 we see that there exists an isomorphism
gn : ~ Ar(Zn) ®As(F) --d.Ak(Z~)
r+s=k such that
g°(tz m] ®[vl) = [hm(Z.j×v)] for any j and any IV] in Ak.m~j(F). From these facts and the exact sequence
° hk(X~_l) ~ hk(Xn)~
A
' k(Z~)~0
one can easily construct an epimorphism
hn : •
Ar(Xn) ® A s ( F ) ~
Ak(X'n)
r÷s=k
such that
hn([Zi,,j, ] ®[V]) = [hi,j,(Zi, j,xV)l for i' < n, any j' and any IV] in Ak_mrj,(F). This ends the proof of (i). In order to see (ii) we shall prove that the filtration
t
I
X t = X n ~ Xn_ 1 3... ~ ~-0~ X -1 t =0
satisfies the statements (i) and (ii) of T h e o r e m 1.
228
The statement (i) has already been proved above (isomorphisms g). Notice, moreover, that if for all k we have bases
of
Ak(F) then we have bases {Y- mj,ct[hij'(Zij'xVj,ct )] }ct~j'
of Ak(Zi). Now to see that statement (ii) is satisfied, notice that by the lemma cl~jxF is an isomorphism because cl F is an isomorphism. From this it follows that cl f.t(Zij) is an isomorphism and so
cl Z~ = e c t f-t(zij )
is an isomorphism. Thus we can apply Theorem 1, which gives that cl)c is an isomorphism and that Ak(X') is finitely generated free group for which the elements
{~ mj, ot[hij,(Zij,×Vj,a) ] }ot,i,j'
form a basis. This implies that the epimorphism h n is an isomorphism. •
Corollary (of the proof) In T h e o r e m 2 (ii) if we assume that cl F is an isomorphism, then (*) is an isomorphism for all k, up to torsion, and cl)c is an isomorphism. *
5. A.(Hilb3p n)
Let Gr(2,n) be the Grassmannian of planes in projective n-space pn, n~.3. Let A13pn be the subscheme of HAIb31~ that parametfizes triples of colinear points and let i be the closed embedding
229
of A13p n in Hilb3p n. Let U' = Hilb3p n -AI3p n. We have that the map which sends a given triple of non-colinear points to the unique plane that contains it, U'--~Gr(2,n), is locally trivial with fiber U = Hilb3p 2 - AI3p2. Proposition The Chow groups of U are given by the following table."
i
0
1
2
Ai
0
Z/3Z
Z2~Z/3Z
3
4
5
6
Z3
Z3
Z
Z
and cl U is an isomorphism. Moreover, these groups are determined by the following table of generators and (abetian) relations (we use notations and conventions of Elencwajg and Le B ~ z
[ 1985b], which for convenience of the reader we list at the end of this paper): i
Generators
6
U
5
H
4
H2, h, p
3
H3,Hh,~
2
H2h, h2,hp
1
Hhp
Relatiens
3H2h -6h2-6hp = 0 3I-thp = 0
Proof The sequence (*)
Ak(A13p2) ~
A k(Hilb3p2) ~ A k(U) --4 0
is exact by Fulton [1984], 1.8. Now we first compute A.(AI3p2) using the fact that Proposition 2 in Le Barz [1987] actually gives Z-bases of Ak(AI3p2). Consider the divisors
v
1 v
on AI3p 2' Then we have that A 5 = Z, with basis [AI3p2], A 4 = Z 2, with a basis given by V and V', A 3 = Z 3, with a basis given by V 2, VV' and V '2,
230
A 2 = Z 3, with a basis V 3, V2V ', VV '2 A 1 = Z 2, with a basis V3V ', V2V '2, and A0= Z.
Now we know a basis of A.(Hilb3p 2) (see Elencwajg and Le Barz [1985b1),
i
Ai
Bases
5
Z2
H, A
4
Z5
H2,HA,h,a,p
3
Z6
H 3, H2A, l-[h, Ha, ct, 13
2
Z5
H2h, H2a, h2,ha, hp
1
Z2
Hha, Hhp
0
Z
h3
An straightforward computation shows us that i.([A13p2]) = A. i.(V) = HA,
i.(V') = a.
i,(V 2) = H2A,
i,(VV') = Ha,
i,(V '2) = ct.
i.(V 3) = 3H2h + 6H2a - 6h 2 - 18ha - 6hp. i.(V2V ') = H2a,
i.(VV '2) = ha.
i.(V3V ') = 3Hhp + 3Hha,
i,(V2V '2) = Hha.
i.(V3V '3) = h 3.
From these relations we infer on the one hand that Ak(U) are the groups given in the statement, and on the other that io is a monomorphism. Finally by an argument similar to that used in the proof of the L e m m a in Section 2 we conclude that cl U is an isomorphism.
Remark
#
3
The scheme U provides an example of a scheme in which cl U is an isomorphism and A.(U) has torsion. Unfortunately, in order to compute the Chow groups of U', it is not possible to apply T h e o r e m 2 as it stands. Notice, however, that the C o r o l l a r y to T h e o r e m 2 applies and so Ak(U')Q ~ ~i=0:k Ai(U)Q®Ak-i(Gr(2'n))Q and cl U, is an isomorphism, whence
231
0 ---) Ak(A13pn)Q ---) Ak(Hilb3pn) Q ~ Ak(U')Q ~ 0 is a split exact sequence, hence bk(Hilb3pn ) = bk(Al3pn)+Ei=0:kbi(U).bk.i(Gr(2,n)) =Y-i=0:kbi(P3).bk.i(Gr( l ,n))+'Fi=0:k(bi(Hilb3p2)-bi(AI3p2)).bk_i(Gr(2,n)) Finally A.(Hilb3p n) is free (apply Bialynicki-Birula [1973]) and so we have obtained a formula for rgZAk(Hilb3pn). For a combinatorial expression of this formula, see Rossell6 [ 1986]. Remark 4
The expression of bk(Hitb3p n) given in Rossell6 [1986] is different from the one given above, but it is not difficult to see, again using Theorem 2, that they agree. Remark 5
Theorem 2 can also be apptied to determine the Betti numbers of varieties of ordered triangles in projective space. Def'me WD*, for r~>3, as the closure in (pn)3xGr(1,n)3×Gr(2,n)xGr(2,P(Sym2En*)) of {(xl,x2,x 3,
ll,12,13,rc,~,)lxl,x2,x 3 distinct points, xielJ for all j~,
rc the plane spanned by xl,x2,x 3, and E the 2-dimensional system
of conics in ~ that contain the points x~,x2,x3}. Then one can see that the projection from W * to Gr(2,n) is a fibration that satisfies the hypothesis of Theorem 2, with fiber the triangle variety W* of Schubert, Semple, and Roberts and Speiser, and thus one can obtain that Ak(Wn*) is a finitely generated free abelian group of rank
bk(Wn*) = Zi=0:kbi(Gr(2'n)bk-i(W*) and the class map is an isomorphism for all k. In particularone easily sees that
bk(Wn* ) = b3n_k(Wn*).
232 Here is a table companng Betti numbers for W3* and for Hflb3p3: k
0
1
bk(W3*)
1
8
I
2
bk(Hilb3p3)
2
3
4
5
6
8
9
25 47 63 63 47 25 8
1
6 10 13 13 10
7
6 2
1
6. The case of positive characteristic We shall indicate briefly how to modify the proofs of the characteristic 0 case when the charateristic of the ground field is p>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
Sr := e ( , r ® 7 ) x e ( A ~ , ~ ' o A ' ~ )
x... x ~(A",r®A".,:).
In the present work too, the base scheme S is arbitrary, and 2 is invertible in the symmetric case. However, we set P := (E, F), and consider forms u: P -+/~ rather t h a n maps u: ~' ~ F* ® L; in this way, E and F are given equal treatment, and they can, for the most part, be arbitrary quasi-coherent sheaves, not simply bundles of finite rank. It is i m p o r t a n t to distinguish between the s t u d y of all the forms u: P ---+ /~ and the s t u d y of the nowhere vanishing forms u up to scalar multiple. In the second case, the associated linear map u: P ® -+ /~ is surjective, where P ® := £ @ F; also, u is considered equivalent to v if there is an isomorphism a: /~ ~ ~ ~ such that au = v. For convience, call the first case the aj~ne case, and call the second case the projective case. In the affi_ne case, a family parametrized by T / S is defined by an a r b i t r a r y form u: P T ---+ /~T; in the projective case, a family is defined by a nowhere vanishing form u: P T --+ ill, where ~M need not be the pullback of a line bundle on S. O t h e r authors have explicitly considered only the projective case. It is, after all, the case of the geometry: a correlation, a collineation, and a quadric hypersurface are each representable by a nowhere vanishing form; two forms represent the same geometric object iff t h e y differ by a scalar multiple. However, the s t u d y of the projective case leads inevitably to the study of the affine case, much as the s t u d y of the projective space leads to the s t u d y of the a/Kne space. Indeed, there are two inseparable b u t distinct theories: a theory of all r-complete forms and their associated parameter space B r ( P , / ~ ) and a theory of projectively r-complete forms and their associated p a r a m e t e r space B r ( P ) . Moreover, see (4.8), B r ( P , / ~ ) is a geometric line bundle over B r ( P ) ; whence, via the zero section, B r ( P ) is canonically embedded in B r ( P , / ~ ) . T h e aft-me pieces of Br (P) of Tyrrell and Laksov may be viewed more abstractly and generalized as follows; for more details, see §3. Let Qi be a pair of free bundles of rank i, and let q: Qr -+ P be the map defined by the first r elements of the reordered bases of £ and F on S ~. More generally, the Qi m a y be the members of any chain 0 ----Q0 c Q1 c ... c Qr such that each quotient Q i / Q i - 1 is a pair of line bundles. Set ¢ :-- ({Qi}, q), and consider the scheme S p l i t ( C ) that parametrizes the
257
splittings, or right inverses, of the inclusions Q / - 1 ~ Qi. In the case of Tyrrell and Laksov, Split(@) is simply the space of the various pairs of matrices (L, U). Let /~i denote the tensor product ( Q ~ / Q i _ I ) ® of the two components of the quotient. In the case of Tyrrell and Laksov, /~ is simply the structure sheaf 0 s . Set V(@) := Split(@) W(@) := V(@)
x
x
H o r n ( L 2 , 12i) × . . .
x
Hom(Lr,/~r-1),
H o m ( C o k ( q ) ®, Lr).
In the case of Tyrrell and Laksov, l=Iom(12i+l,/~i) parametrizes the successive quotients qi+l = d~+l/di of the diagonal elements di of D, and l=Iom(Cok(q) ®, Lr) paraznetrizes the f - r by e - r matrices, which appear, multiplied by dr, in the lower right corner of D. In any case, W(@) is canonically isomorphic to an affine open subscheme U(@) of B r ( P ) . In fact, U ( ¢ ) is the maximal open subscheme on which all the compositions A i Qi --~ ] ~ P --~ H i vanish nowhere for I < i < r, w h e r e / ~ is applied coordinatewise. Moreover, if Y(@) is embedded in W ( ¢ ) via the zero section of the second factor, then its image is the trace of the blowup center Vr. T h e r e is a similar t h e o r y in the affme case. Set V ( ¢ , / ~ ) := V ( ~ ) × I - I o m ( L i , L), and W ( ¢ , / ~ ) := W(@) × t=Iom(~l,/~). T h e n W(@,/~) is canonically isomorphic to an affme open subscheme of B r ( P , £), etc. Note that, even in the projective case, the scheme t t o m ( C o k ( q ) ®,/'~r), which appears as a factor in W ( ~ ) , parametrizes a/l the forms; thus the s t u d y of the projective case leads to t h a t of the affme case. T h e tautological m a p / ~ i + i IW(O) --~/~ IW(@) is equal to the restriction of a canonical global map on Br (P), called the ith linking map of the canonical form w. In the affme case, the situation is parallel. T h e theory of linking maps will be discussed next. It occupies a large portion of the present work, and it is one of our main contributions. Consider an arbitrary r-divisorial form u: P --~/~; by Definition (3.1), u is a form whose minors H i = Hi(u) := I m A ~u are invertible. Following (4.5), define invertible sheaves/~i ---- /~i(u) by
Li :---- .Mi®.M~Ji
for 1 < i < r,
L0 :-- L.
Then, for 0 < i < r, there is a natural map L i + i -~ 12i, called the ith linking map of u. For i = 0, the map is simply the inclusion of H1 in /~. For i = 1 , . . . , r - 1, the map is obtained by taking i := i - 1, j := 1 and s := i in the following inclusion:
Hi Hj+s C_ .Ms Hj+i
for i , j , s > O and i < s.
T h e inclusion may be established locally, because both sides lie in /2®~+i+°. Then, with i := i - 1, j := 1 and s := i, the inclusion expresses the divisibility property di+l = qi+ldi if 1 ~ i < r and dh,j = qhjdr if i = r of the m a t r i x D of Tyrrell and Laksov. T h e general inclusion itself is established in (2.8)(iii) u n d e r the sole assumption t h a t $t8 is invertible, and in (2.6)(ii) under the sole assumption that one
258
of the two c o m p o n e n t s ~, F of P is of r a n k j + s. Related results are already in the literature, see (2.9). An r-complete f o r m u : P -* ~ is defined in (4.1 / as a sequence u = (u, u l , . . . , u r ) where u: P --+ /~ is an a r b i t r a r y f o r m and ui: A ~ P -* H i is a nowhere vanishing form, normaJJzed so t h a t H i = ~ ( u ) is a quotient of (A i P ) ® . T h e sequence is required to be locally the pullback of a sequence consisting of a suitable r-divisorial f o r m and its exterior powers; m o r e precisely, for each point of S, there m u s t exist a neighborhood U of the point, a scheme S ~, an r-divisorial f o r m v : p r _ , / ~ on S ~, and a m a p a : U ---* S ' such t h a t u]U -- a* (v) and uilU = a* (v i) where v i : A i P ' --* H i ( v ) is the exterior power A i v with its target reduced to its image. This condition is r a t h e r weak: p l a n d Z~~ are not required to be the p u l l b a c ~ of P a n d L; in fact, S ~ is not even required to m a p to U. In practice, this local definition works out well; t h e definition is convienent, and t h e ui play the proper role of exterior powers. If ui is equal to the "reduced" exterior power ui: / k i P ---* ~4i(u), then u is said to b e exterior. E v e r y r-complete f o r m u : P -* £ is the pullback of the canonical exterior f o r m w on B r ( P , / ~ ) via a unique section, according to a m a i n theorem, (4.2). It follows, see (4.3), t h a t the pair ( B r ( P , / ~ ) , w ) represents the functor of r-complete forms, u : P T ---+ /~T. In fact, these two results are clearly equivalent, given a n o t h e r i m p o r t a n t result, (3.20); it asserts t h a t the f o r m a t i o n of B , ( P , / ~ ) c o m m u t e s with b a s e change. T h i s result is also the key to proving t h a t u is induced by w . Indeed, it clearly reduces the m a t t e r to proving t h a t , in the notation above, the r-divisorial f o r m v: p t __+ Z~ is t h e pullback of the canonical r-divisorial f o r m on B ~ ( P ' , ~/)/S I via a unique section. Finally, this section exists because by (3.16) (A) (iii) it does locally, on each afFme W ( ~ , L'). T h e projective case is similar; see (4.1)-(4.3), (3.20) and (3.16) (B) (iv) . A projectively r-complete form u on P is defined as a sequence u = ( U l , . . . , U r ) where ui: / k i p --~ )vii is a (normalized) nowhere vanishing form. T h e sequence is required to be locally the pullback of the sequence of '~reduced" exterior powers of an rdivisorial form. E v e r y projectively r-complete f o r m u on P is the pullback of the canonical exterior f o r m w on B~(P) via a unique section, and the pair ( B r ( P ) , w ) represents the functor of projectively r-complete forms u on P T . T h e p r o o f is local, and the key is the result (3.20) which asserts t h a t the f o r m a t i o n of Br (P) c o m m u t e s with base change. In the present work, B~(P, Z~) and B~(P) are not defined as closures nor by patching b u t as blowups. In fact, given an a r b i t r a r y f o r m u: P --* L, a scheme Br = Br(u) is defined in (3.8) as follows: B0 is defined to be S, and Bi+l to be t h e blowup of Bi along the scheme of zeros o f / k i + l uiBi" It is easy to prove, see (3.9), t h a t uIB r is r-divisorial and that, given any T / S such t h a t ulT is r-divisorial, there is a unique Sm a p T -* Br; in other words, B~ renders u r-divisorial in a universal way. It follows, see (3.11), t h a t for any T / S , there is a unique base-change m a p Br(ulT ) -* B~(u) x T. It is obvious f r o m the definition t h a t the invertible quotients ~i(ulBr) define an e m b e d d i n g of B~ = Br (u) into S,. := P ( P ® ) x
p((/k2p)®)x . . -
x
P((/krP)®).
It is not h a r d to prove, see (3.12)(ii), t h a t the image is isomorphic to the closure of the section of S r / S defined on the open subscheme of S w h e r e / k r u is nowhere vanishing.
259 It is also not hard to prove that B~+I is equal to the blowup of B~ along the '~right" center, namely, the subscheme whose ideal is
L := (~,--1 -~,-+1) ® ~ - 2 . It is not obvious t h a t L is an ideal, b u t this follows from the key inclusion of (2.8)(iii) discussed above. All of this yields Schubert's formula for the class of the exceptional divisor in terms of the pullbacks of the hyperplane classes, see (3.13). Finally, B r ( P , L) is defined in (3.17) as Br(u) where u is the tautological form on I t o m ( P ®,/:), and B r ( P ) is defined as Br(a) where a is the tautological form on • ( P ® ) . Note that, since a is nowhere vanishing, B I ( P ) = B0(P); whence, B r ( P ) m a y be constructed via r - 1 blowing-ups. T h e preceding discussion shows t h a t these schemes have the s t a n d a r d basic properties. Any form v: P --*/: is the pullback of the tautological form u on H o m ( P ®, L)/S via a unique section. Hence, if u is r-divisorial, then v is the pullback of w := ulB~(P ,/:) via a unique section of B r ( P , ~:)IS. Thus the universality of w for rdivisorial forms has been proved again. This proof avoids (3.18)(C), the theorem that says t h a t B r ( P , L) is covered by open sets of the form W(~P,/:). On the other hand, (3.18)(C) is used in the proof of (3.20), which asserts t h a t the base change map from B,.(PIT, LIT) to B~(P, L) x T is an isomorphism: the formation of W(qh, L) obviously commutes with base change. T h e situation is entirely parallel for B r ( P ) . T h e linking maps L i + l --* Li of an arbitrary r-complete f o r m u : P --~ • are defined in (4.6) as the pullbacks of the linking maps of the canonical r-divisorial form w on B r ( P , L). Say u --- (u, ul,... ,ur). It is not hard to prove, see (4.7)(C) and (4.12), that u is the exterior form on u iff u is r-divisorial, iff all the linking maps of u are injective; moreover, if these equivalent conditions hold, then the linking maps of u are equal to those of u. T h e key to the proof is the following result, (4.11): Ca) the pullback of .Mi(w) is equal to H i ( u ) ; (b) there is a natural surjection H i ( u ) --~ Hi(u); and (c) the surjection is an isomorphism if J~li(u) is invertible or if the first i linking maps are injective. T h e linking maps are characterized in (4.9)(4) on the basis of the preliminary work in §2 as the unique maps fitting into certain functorial diagrams. This uniqueness justifies an alternative definition of the linking maps, see (4.10): by definition, u is locally the pullback of an exterior form, which has natural linking maps; b y the uniqueness, the various pulled-back linking maps patch and the result is independent of the choice of local representation. This alternative avoids B~ (P, L). Of course, a parallel theory exists in the projective case. T h e joining and splitting of complete forms is discussed in §5. T h e idea is this. First, suppose P :---- Q @ Qi where Q is a pair of bundles of rank s, and consider an s-complete form v : Q - - * / : and a t-complete form V : Qt __,/:s(v). T h e n v ~ may be spliced onto v to give an r-complete form u : P --+ /: where r = s + t. Its chain of linking maps is /: = / : o *-- 1:1 *-- -.- ~- Ls ~ - L~+I ~ - . . . ~
£~
where the first s maps are the linking maps of v and the remaining $ axe the linking maps of v I b u t reindexed so t h a t / : i :=/:i-s(v ~) for s < i <: r. This result is proved in
260
(5.2) using the universality of B , ( Q , / : ) and B t ( Q ' , / 2 8 ) and their local descriptions. In (5.1), u is defined as an explicit sequence u := (u, U l , . . . , Ur). However, the formulas for u and the ui are a little complicated, and they involve some t e r m s introduced in earlier sections. In particular, certain m a p s , called modified exterior powers, are crucially involved. For an exterior form, these powers are defined in (4.5) as the usual exterior powers b u t with carefully chosen targets; for an a r b i t r a r y form, they are defined in (4.6) as the pullbacks of the powers of the universal form, which is exterior. T h e reverse of splicing is called cutting. Let u : P -~/Z be an r-complete form, Q a pair of bundles of r a n k s, and q: Q --+ P a m a p of pairs such t h a t the induced f o r m A 8 Q -~ J~8 is nowhere vanishing. T h e n (1) q is an embedding; (2) Q has a canonical c o m p l e m e n t Q ' , which is a s u b m o d u l e of P such t h a t P = Q @ QI; (3) u induces an s-complete f o r m v : Q --~ ~ and a t-complete f o r m v ' : Q ' -+ L s ( v ) , and u is equM to the f o r m o b t a i n e d b y splicing V onto v; and (5) if u is also equal to the f o r m obtained by splicing a t-complete f o r m v~ : Q ' --+ ~ onto an s-complete f o r m Vl : Q --* L, then V = v~ and v = v l . These results are proved in (5.3) b y reducing to the case t h a t u is exterior and t h e n applying a n u m b e r of preliminary results proved in §1 and §2. This result and some m o r e preliminaries f r o m §1 yield a characterization of the case in which there exists a pair R of bundles of r a n k r, a surjection p : P --* R , and an r-complete f o r m v : R --* /" such t h a t u ----v p ; see (5.4). Moreover, (5.4) gives a simple global description of R in t e r m s of the f o r m ur : A r P -+ H r . This description makes it clear t h a t R is uniquely determined and m a y be t r e a t e d locally. Stringing is a n o t h e r way to join complete forms. It is m o r e general t h a n splicing in t h a t it s t a r t s not with a direct s u m decomposition b u t with a short exact sequence, 0 ----~ R I ---* P
---+ R
---+ 0 ,
in which R is a pair of bundles of r a n k s. Stringing is m o r e special in t h a t a projectively t - c o m p l e t e f o r m V on R ' , r a t h e r t h a n an a r b i t r a r y t-complete f o r m v~ : Q ' --~ C, is strung on after an s-complete f o r m v : Q ~ C to give an r - c o m p l e t e f o r m u : P --+ L where r = s + t. T h e chain of linking m a p s of u is :
CO +-
C 1 +--- . . -
+--- C a +--- C a + I
¢"- ...
+-" Cr
where the first s m a p s are the linking m a p s of v, the last t - 1 m a p s are the linking m a p s of v I b u t reindexed so t h a t Ci :-- C i - 8 ( V ) for s < i _< r, and the middle m a p C~+I ~ C~ is equal to 0. In (5.6), u is defined by explicit formulas; they are much simpler t h a n the corresponding formulas in (5.1) for a spliced f o r m because of the vanishing. Conversely, if u : P ---+ C is an r-complete f o r m whose s t h linking m a p vanishes, t h e n u m a y be unstrung: there are a canonical short exact sequence and forms v , V as above such t h a t u is formed by stringing V on after v . T h e s e results are proved in (5.7) and (5.8) by observing t h a t t h e y m a y be proved locally a n d t h a t locally the short exact sequence is split and t h a t then u is obtained b y splicing the f o r m (0, V ) onto v . Stringing a n d unstringing yield a description of an a r b i t r a r y T - p o i n t of a principal orbit closure Or [I] of B r ( P , / : ) where as before I={Sl,...,sk}
and
So:=O<Sl<...<sl¢<Sk+l:=r.
261
(The group scheme acting is Aut(P) := Aut($) × Aut(7). By definition, an orbit closure is a closed stable subscheme that is not the union of two others.) Indeed, given T / S , repeated stringing and repeated unstringing establish a bijective correspondence between the set of r-complete forms u: P T ~ /~ whose s~-th linking map vanishes for 1 < i < k and the set of pairs (F, U) where F is a flag of subpairs P i of P T such that (PT)/P~ is a pair of bundles of rank si for 1 <: i < k and U is a set consisting of an si-complete form u(0): ( P T / P i ) --+ £ and of a projectively [si+i - s~]-complete form u'(i) on (P~/P~+i) for 1 < i < k, where sk+i := r and P k + l := (0, 0); see (5.9). Reformulated geometrically, that statement says this, see (5.11): the subscheme Or [I] of Br(P, ~), which is defined by the vanishing of the linking maps indexed by I, is canonically equivariantly isomorphic to the product
B~I(PF/Pi, LF) ×r B82-8i ( P i / P J ) X F ' ' " ×F B~k--~k_, ( P k - i / P k ) XF Br-sk (Pk) where F = F : ( P ) is the (partial) flag scheme and {Pi} is the universal flag. In particular, the r-complete forms u: P T ---* ~ whose si-th linking map vanishes for 1 < i < k and whose remaining linking maps are injective correspond bijectively to the set of pairs ( F , U ' ) where F is a flag of subpairs P i of P T as above and U is a set consisting of an exterior si-complete form u(0): ( P T ) / P i ) --~ /~ and of an exterior projectively [s~+i - s~]-complete form u'(i) on ( P j P ~ + i ) ; see (5.10). These pairs (F, U') are essentially the r-complete forms considered by Laksov. Thus, his assignment of a T-point of Or [I] to each of these special r-complete forms has been recovered. Moreover, it has been reversed: the T-points so obtained have been identified among all T-points, and to each one of them the corresponding pair (F, U') constructed. Hence the pullback along T' --~ T of one of Laksov's special T-points is a special T'-point iff the nonzero linking maps of the pullback are injective; and if so, then the flag of the pullback refines the pullback of the original flag. These special Tpoints are, as Laksov observed, sufficient to characterize Or [I], because the universal r-complete form on Or[I], which is the restriction of that on Br(P,/~) and which corresponds to the identity map of Or [I], is special. Vainsencher essentially considered the case that the nonzero linking maps are isomorphisms; the corresponding T-points of Or [I] are just those of the open orbit Or [I]. An extension of the preceding analysis shows the following, see (5.12) again: the subscheme of the blowup center Vr that is defined by the vanishing of the s~-th linking map for 1 < i < k is equivariantly isomorphic to the product B81 ( P F / P i , LF) XF B82-~1 ( P i / P J ) X F ' ' " XF Bsk+~--sk ( P k / P k + i ) . In particular, when k = 0, the above isomorphism identifies Vr with Br(R, LIGr(P)) where G r (P) is the Grassmannian of quotient pairs of P consisting of rank-r bundles and 1% is the universal quotient pair. With a little more work, see (5.12), Vr may be identified with Br(ulZ~+i ) where u is the tautological form on I-Iom(P ®,/~) and Z~+i is the subscheme of zeros of A r+i u. Then Vr may be identified with the proper transform of Z~+i in Br(P,/~). Similarly, see (5.13) as well as [8] (5.2) and [17] (2.1), the blowup of Z~+i along Z~ may be identified with t t o m ( R ®,/~IGr(P)).
262
T h e conormal algebra of the blowup center Vr in B r ( P , ~), which is defined as the restriction to V, of the Rees algebra of its ideal 2", is described in (6.8) as follows: (i) the algebra is equal to the symmetric algebra Syrn(I/2.2); in fact, the Rees algebra of 2. is equal to the symmetric algebra of 2"; (ii) there is a canonical isomorphism n:K ®
~,
I/I2®~.r(w)
where w is the canonical r-divisorial form on Br (P,/~) and K is equal to the pullback of the universal pair of subsheaves on the Grassmannian G r (P). Actually, in (6.8), K is described directly in terms of w; the two description are equb,~alent b y the last assertion of (5.12). T h e m a p n is defined in (2.13) for an arbitrary r-divisorial form, and some other conditions t h a t imply t h a t it is an isomorphism are given in (5.5). In the present case, (i) and (ii) are checked locally on affine open sets W(CP,/~) in (6.5)-(6.7); thus the issue is reduced to proving the corresponding statements about the conormal algebra of the zero section of an a/~lne scheme Spec(Sym(,r)). Since ~ is not necessarily locally free nor of finite type, there is need for a suitably genera] theory of regular maps ~ --~/~ with /~ invertible. Such a theory is developed in (6.1)-(6.4). In particular, (6.3) contains a new and remarkably simple proof of this fact: if the Koszul ~/1 vanishes and if ~ is flat, then the symmetric algebra of the image of ~ ® L -1, an ideal, is equal to its Rees algebra. T h e standard proof is round about, and it requires ~ to be of finite t y p e and S to be locally noetherian; see [2]. One incidental question arises: if -~ ~ is regular, what conditions on a second map ~r _ . ~ with the same image guarantee t h a t it too is regular? For example, if S is locally noetherian and ~ is a bundle of rank e, then the usual condition is that ~ too be a bundle of rank e. T h e r e is a natural system of bilinear equations defining B r ( P , / ~ ) in S~(P, 1]) := H o m ( P , 1]) x P ( P e )
x..,
x
P ( ( A ~P ) e ) .
T h e equations involve the following composite map 0 =- 0 i'j'k, which is introduced in (2.1) for an arbitrary sheaf 8: i
j+k
i
j
k
k
o: Ac®Ac l®V,Ac®Ac®Ac
j
i
k
j+~
°,Ac®Ac®Ac l® ,Ac®Ac;
here V is the exterior coproduct map, sw is the canonical isomorphism, and A is the exterior p r o d u c t map. Now, for P ----(~, 3r), set
i,j,k
AP:=
(~
j+k
k
j+i \
c®
Given forms ui: A ~ P --. 3¢i and ui+j: A~+JP --* Jqi+j, set
A(ui, ui+j ) : = (ui ® ui+j)('~ ® 1 -- 1 ® ~ ) : (A i'j'i P ) ® --~ .gi ® .A/i+j.
263 Then Br(P,/2) is defined in St(P,/~) by the equations A(u,
= o, z x ( u , , u : ) = o , . . . ,
= o
where u, U l , . . . , u, are the pullbacks to S~(P, £) of the tautological maps. The equations vanish on B , ( P , L) for basically the same reason that justifies the inclusion proved in (2.8)(iii), which was discussed above and used in the definition of the linking maps: a certain natural diagram given in (2.5) is commutative. The equations are shown in (6.14) to define Br(P, £) using the same preliminary work needed for the description of the conormal algebra plus some additional technical work done in (6.9)-(6.12). The equations yield the following pretty characterization of an r-complete form: it is a sequence (u, U l , . . . , ur), where u: P ~ / ~ is an arbitrary form and ui: A i P ---+.Mi is a nowhere vanishing form, such that the equations are satisfied. For symmetric forms on symmetric pairs P = (8, E), there is a complete parallel theory, provided that 2 is invertible in F(S, 0s); see §7. Establishing the symmetric theory by adapting the asymmetric theory is for the most part a simple straightforward matter, and a few indications suffice: replace P® := ~" @ ~ by psym : : Syrn2~, etc. However, a few issues do require special attention. One such issue is the open covering of the space of r-complete symmetric forms Brsym(P, L). The open subschemes are, of course, indexed by the symmetric ¢ ---- ({QI}, q). Each subscheme is just the following closed subscheme of W(~):
w"Ym(¢) := Split"Ym( ) × Hom(122,
X'''
×
Hom(£r, Lr-1) × Itom(Cok(q) "ym, £r)
where the first factor parametrizes the symmetric splittings and last factor parametrizes the symmetric forms. However, it is a little tricker to prove that there are enough to yield a covering; fortunately, see (7.8), it is not hard to adapt the usual device employed when diagonalizing a symmetric matrix. The definition of B~"ym(p,/~) itself requires attention. The parallel definition is that Brsym(P, ~) := Br (UsYm), where/2, syrn is the tautological form on H o m ( P sym, ~), and this definition is essentially the usual one. However, a second natural definition is that Br"ym(p, L) :---- "ymB~(P, L), where sYmBr(P , L) denotes the largest subscheme of B,(P,/~) on which the components w and w~ of the universal form w are symmetric. A main theorem, (7.20), asserts that these two definitions are equivalent: symBr (P, L) = B~ym(p, £). The proof is based on a lenmm, (7.21), which asserts that the corresponding subschemes symw(~) and wsym(~) are equal. A second proof is given in (7.27); it proceeds by induction on r via splicing. The equations above, which cut Br(P, •) out of St, also cut B~Ym(P, £) out of Srsym(p, £) := I t o m ( P "ym, L) × p ( p s y m ) × . . . × p((Arp)sym);
264
see (7.34). T h e parallel proof fails! However, the equations obviously cut out symBr (P,/~) ----B r ( P , £) N Srsym(p, •). Notice t h a t the restricted equations are not independent on -.q,rsym , because the m a p
(i @ <> -- <> @ i): (A ~P ® A ~+1 P)® --* (A ~P ® A ~+I p)sym factors through A=(A ~ c ® A ~+1 ~). For the projective case, there is a parallel theory of splicing, cutting, stringing, unstringing and applications, see (5.13); of the conormal algebra, see (6.8); of the equations, see (6.14); and of s y m m e t r i c forms, see §7. T h e t h e o r y of duality was p r o m i n a n t in the 19th century, b u t it has been p r e t t y m u c h neglected by m o d e r n writers on r-complete forms. It is discussed in (4.13)(4.17) and (7.23), and here is the idea. Assume t h a t P is a pair of bundles of r a n k r _> 1, and let u = ( u l , . . . ,ur) be a projectively r-complete f o r m on P . T h e n the dual form fi on the pair P * of dual bundles consists of the following nowhere vanishing forms: e~:
A~p. =
(det_lp)
®A~_~p le~,_, ( d e t _ l p )
® ® M r - i ( u ) for 1 < i < r.
Obviously, if P and u are synunetric, then so are P* and ~. Now, T h e o r e m (4.17) asserts this: (a) fi is projectively r-complete; (b) its linking m a p s are the duals of those of u and a p p e a r in reverse order; (c) ~ = u; and (d) there are canonical isomorphisms, Br-l(P)--sym
B~-I(P)
=
B~(P)
=
B~(P*)
=
B ~ - I (P*)
Brsym(P)
=-
BrsYm(P*)
---
~r--1 ~-- J"
][:~sym[]=),
T h e first and third equalities of each line also hold in the affme case and are proved in (4.18). Here are two alternative proofs, carried out for the first equality to illustrate the ideas. Obviously S~(P) -- S~_~(P) since P is a pair of bundles of r a n k r. Hence B ~ ( P ) is cut out of B ~ - I (P) by the equation A ( u r _ l , u~) ----0. However, this equation is always satisfied, because it contains the factor <~i,l,i ® 1 - 1 ® ~i,1,~ and <>i,l,i __ 1 when i ---- r - 1 by (2.2)(ii). Alternatively, B ~ - I ( P ) = B~(P) because b o t h schemes are equal to the closure of the s a m e section of S ~ ( P ) / / P ( P ) defined on the open subscheme of S where A ~ u is nowhere vanishing. T h e second line of the last display confirms a bit of c o m m o n knowledge: the space of complete plane conics over a field of characteristic not 2 m a y indeed be constructed f r o m ZP5 via a single blowing-up, and it is self-dual.
i.
FORMS
AND
EXTERIOR
POWERS
SETUP (1.1). Work in the category of quasi-coherent sheaves £, 9",... on an arbit r a r y ground scheme S. Also, work with pairs P = (8, 9"),... and with c o m p o n e n t wise operations such as direct sum, tensor product, exterior power, etc. Given T/S, denote the pullback of ~ by $T or by CIT. Set d'* = )¢orn(E, Os). By convention, a bundle will be a locally free sheaf of finite rank, and a line bundle will be an invertible sheaf.
265
LEMMA ( 1 . 2 ) . Associated to any m a p v: ~ --+ jr and i , j >_ O, there are two (dual) commutative diagrams, whose maps arise from the ith, j t h and (i + j ) t h exterior powers of v and from exterior and interior multiplication:
A ~ jr .....
(1.2.1)
, ~om(A; jr, A ~+~ jr) - -
T
i
Ai ~
A ~ ~' ~.... (1.2.1")
~orn(A ~ jr, A ~+J jr)
, ~orn(A i ~, A~+3• ~)
(A ~ e)* ® A ~+j e
i A ~ jr., ........
, ~orn(A ~ ~, A~+3" jr);
(A j e)* ® A ~+j
g (A j jr)* ® A ~+j jr ,
(A j jr)* ® A ~+j ~.
PROOF: T h e diagrams are commutative because they express in another way the functorial properties of exterior product and coproduct. REMARK ( 1 . 3 ) . Assume ~ and Jr are free of rank r. If i := I and j :--- r - 1, then the diagrams of (1.2) simply express in abstract terms the classical fact that a square m a t r i x multiplied on the left, resp. on the right, by its adjugate is equal to the determinant times the identity matrix. More generally, if j := r - i, then the diagrams correspond to the Laplace expansion of the determinant. LEMMA ( 1 . 4 ) .
Let v: ~ --+ Jr be a m a p . Then:
(i) I f v is surjective, then A i v is surjective for all i. (ii) If jr is a bundle of rank f a n d if A i r is surjective, then A i r is surjective for all i. (iii) I f ~ is Iocedly finitely generated by e elements, if jr is a bundle of rank f = e, and if A j v is surjective for some j , 1 <_j <_ e, then A i v is an isomorphism for alli. (i*) I f ~ and Jr are bundles and v is injective, then A i v is injective for alI i. (ii*) If ~ is a bundle of rank e and A e v is injective, then A i v is injective for a/l i . (iii*) If ~ and Jr are bundles of the same rank e and if A j v is injective for some j , 1 < j <_ e, then A i v is injective for all i. PROOF: (i) T h e assertion is trivial. (ii) T h e assertion is trivial if i > f . If i _< f , set j : = f - i and consider the diagram (1.2.1"). At the b o t t o m , the first map is an isomorphism because Jr is a bundle, and the second m a p is surjective because A ! v is. Hence A i v is surjective too. (ii*) T h e assertion is trivia] if i > e. If i ~ e, set j := e - i and consider the diagram (1.2.1). At the bottom, the first m a p is an isomorphism because £ is a bundle, and the second m a p is injective because A e v is injective. Hence A i v is injective too. (i*) Set e : = rk(,¢). B y (ii*), it suffices to prove t h a t A ~ v is injective. T h e question is local, so we may assume that ~ is free and t h a t the base is atYme. Assume that A e
266
is not injective. Then, because A e £ is of r a n k 1, there is a nonzero scalar a such t h a t a A e v -- 0. Choose p minimal such t h a t a A P v -- 0. T h e n a A P - l v ~ o. Therefore, since £ is free, there is a direct s u m m a n d ~ of r a n k p - 1 such t h a t p--1
a
A(vl,.~) # 0.
Replacing £ with a direct s u m m a n d of r a n k p t h a t contains ~ as a direct s u m m a a d , we m a y a s s u m e t h a t p = e. Consider the d i a g r a m (1.2.1") with i := 1 and j := e - 1. T h e m a p at the top is an isomorphism. T h e m a p on the right, when multiplied b y a, remains nonzero; indeed, the dual of the p r o d u c t is equal to a A e - l v , which is nonzero. However, the m a p at the b o t t o m , when multiplied b y a, becomes 0, because a Aev =- o. Hence v is not injective. (iii) (resp. (iii*)) It is well known t h a t d e t ( A j v) is a power of det(v). Hence, since A j v is surjective (resp. injective), so is get(v) by (i) (resp. (i*)). Now, a surjective m a p of invertible sheaves is obviously an isomorphism. For all i, therefore, A i v is an i s o m o r p h i s m b y (ii) and (ii*) (resp. injective by (ii*)). LEMMA ( 1 . 5 ) . Let v : £ ~ jr be a map between bundles of the s a m e r a n k r. I f v is injective, then so is its dual v* : £* --+ jr*. PROOF: If r = 1, then the assertion is obviously true. T h e general case follows, by virtue of (1.4) (i*) and (ii*), because A r ( v *) -- (A ~ v)*. DEFINITION ( 1 . 6 ) .
Let P = (£, jr) be a pair. Define the transposed pair by
Pt~ := (jr, £) and define the associated sheaf by
P®:=£®F. Let ~ be a sheaf. In keeping with the convention of coordinatewise operation, define H o r n ( P , .~) := ( )~om( £ , ~ ), )tom(jr, ~ ) ). T h e s a m e symbol u will be used to denote a (linear) m a p u: P ® --* ~ and the associated bilinear m a p u: P -~ .~. If ~ is an invertible sheaf, t h e n u will be called a form to indicate this hypothesis. Finally, in the obvious way, define the transposed
bilinear map DEFINITION ( 1 . 7 ) . Let i _> 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 <_ r. T h e n t h e ith exterior p o w e r A ~ u is regular iff u is regular. PROOF: Clearly, h ( A i u, A i P ) = A i h(u, P). The assertion follows from this equation because of (1.4)(iii*). PROPOSITION ( 1 . 1 4 ) .
I r a form u: P -+ £ is o f r a n k r, then the i t h exterior p o w e r __. £®i is o f r a n k (~), and its regularization is equaJ to t h e i t h exterior p o w e r o f t h e regularization o f u,
Aiu: Aip
i
i
i
( A u ) r e g = A ( u r t g ) : A ( l = ' / K e r h ( u , P ) ) --~ £®~. PROOf: By (1.13), A~(ureg) is regular; whence, it is equal to the regularization of A ' u by (1.8)(ii). Obviously, A'(ur~g) is of rank (r). Hence, A~u is of this same rank. LEMMA ( 1 . 1 5 ) .
L e t u: P --* f. be a bilinear map such that u = v (3 w. T h e n :
(i) (orthogonality) A i u -- A' v@ [(A ~-1 v ) ® w ] • [(A ~-2 v ) Q ( A 2 w)] @ . . . @ A ~w. (ii) I f v is regular, then w is u n i q u e l y determined; in fact, w ----u ] K e r h ( u , D o m a i n ( v ) ) . PROOF: The assertions are easily checked. DEFINITION ( 1 . 1 6 ) . Let u: P ---* £ be a form, and 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 <_ i, then A4k 34j+~ C_ .M~ .Mj+k. PROOF: (i) Take k = 0 in (2.5). Then, in the diagram, (0, 1) ---- ( A, 1). This map is surjective. Hence, as the diagram is commutative, the first inclusion holds. The remaining two inclusions are trivial. (ii) Say P = ( ~, jr) where ~¢ is a bundle of rank j + i. Then, in the diagram of (2.5), ( 0, 1) is surjective by (2.2)(ii). Hence, as the diagram is commutative, the asserted inclusion holds.
273 PROPOSITION ( 2 . 7 ) . Let u: P ---* ./~ be a f o r m such t h a t M e ( u ) is invertible. L e t Q be a subpedr of P of b u n d l e s of rank s. Consider v := u l q . A s s u m e M s ( v ) -- Me(u). Set Q' :-- K e r h ( u , Q ) a n d v' := ulQ'. Then: (i) T h e r e is a decomposition u = v @ v t, and if also u -- v @ w, then "w = v ~. (ii) For j > O, M e + j ( u ) -- M e ( v ) M j ( v ' ) = M e ( u ) M j ( v ' ) . (iii) For 0 < i < s, M i ( u ) = M , ( v ) . (iv) For 0 < i < s and j , k >_ 0, t h e following diagram is c o m m u t a t i v e : A ~q ® A ~ Q, ® A k p
,
A~,3,k p
11~A
l (ui~l)(1,O)
A~Q ® Aj+k P
v'@l M~® (Aj+k P)®
w h e r e t h e m a p at t h e top is t h e natural m a p (1 ® A, (1 ® A)sw), the i s o m o r p h i s m s w i t c h i n g t h e first and third factors.
w h e r e s w is
PROOF: (i) The assertions are simply restatements of (1.18)(iv) and (1.15)(ii). (ii) Since v is regular of rank s, its source is a pair of bundles of rank s. Moreover, u : v • v' by (i). Hence, (1.15)(i) yields 8+j
8
s--1
.~"
j-~l
8-~-j
--[lAy) ® (A,,')] eEIA v)®(A v,)] e---e A v'. Take images, use (2.6)(i), and take images again but with j :-- 0, to see that
Me+j(4) = -~s(V) J~j(V t) + 3~s--I(V) .Mj+I(V/) --~..---~ d~s+j(V t)
J~e(V) 3~3"(V' ) -~- ,~e--l(V) .,~1 (V/) J~3"(V') -~-"""-~- 3~e(V') Mj'(V t) Me(~) M~(v') = Ms(v) Mj(~') c_ Me+~(~). Thus (ii) is valid. (iii) Note that, by (i) and (1.15)(i), i
A,, =
i
i--1
1
i
• [(A v) ® (Av)] e... ® A,y.
Take images, use (2.6)(ii), take images again but with i := s, to see thal;
Me(,) M~(~) = MJv) M~(,) + Me(~) M~_I(~) MI(~') + . . - + Ms(,) M~(,')
c M~(~)[Me(~) + Ms-l(~) MI(,') + . . . + Me_,(,) M~(~')] _c M~(~) Me(u) c_ M,(~) Me(u). Since M8 (v) = Ms(u), the inclusions are equalities. Since Me (v) is invertible, therefore (iii) is valid. (iv) Denote the second components of P , Q, Q' by )4, K, ~. Apply (2.3) with £ :---94, i :---- 0, j :----j , k := 0, rn :---- k, n :----i. View the commutative diagram as the second component of the corresponding diagram of pairs. Note that the image ! obstructing commutativity in (2.3) is of no consequence here because of "orthogonali ty", (1.1 5) (i) applied to the decomposition u = v @ v ~, which is valid by (i).
274
PROPOSITION ( 2 . 8 ) . Let u: P --* /~ be a form such that 34~ is invertible, s > 1. Let 0 < i < s and j >_ O. Then: (i) As indicated, these two bilinear maps are equal: 1,j,1
(U ® u J + l ) ( l , ~l,j,1) = (U @ u J + l ) ( ~ l'j'l, 1): i
P --*/~ ® 34f+1.
I f i = s or i f j + i = s, then as indicated next, the two bilinear m a p s are equal:
®
=
®
1): i
P
®
(ii) The bilinear map, (u ~ ® 1)(1, ¢ ) : A i'j'~ P --* 34i ® ( i j+8 P)®, is surjective. (iii) These inclusions hold: 34i 34j+8 c_ 34834j+i C f®0'+i+s). Moreover, there exists a unique m a p m m a k i n g the following diagram commutative: A i,i,s p
n i,j,8 p
(u':®uJ+') (1,~' J") l
l (u' ®uJ+') (~"5'* ,1)
34i ® 34j+~
m
~ 34s ® 34j+~.
I f 34i or 34j+, is invertible, then m is injective. (iv) These inclusions hold: 34{-134j+8 C_ 34~+1 C £®8U+1).
PROOF: (i) Since, in each case, one of the two factors of the target of the map in question is invertible, the natural map of the target into the appropriate power of/~ is injective. So the assertion follows from (2.5). (ii) The question is local. So, by (1.19) we may assume that u -= v ~ v' where v is regular of rank s and 348(v) = 348(u). Then, (2.7)(iv) applies. The map in question appears on the right in (2.7)(iv). The map on the left, 1 ® A, is surjective because Q' has corank s in P . The map at the bottom, u ~ ® 1, is surjective because of (2.7)(iii). Hence, (ii) holds. (iii) It follows from (ii) that (u i ®uJ+~)(1, ~) is surjective; whence, the uniqueness. Since 34s is invertible, the natural map from 34s ® 34j+i into/,®(f+i+s) is injective. Since (u i ® uf+8)(1, ~) is surjective, (2.5) yields 34i 34j+8 C_ 518 34i+~Hence, the natural map from 34i ® 34j+, onto 34i 34j+8 yields rn. If 34i or 34j+s is invertible, then the natural map from 34~ ® 34j+s onto 34~ 34f+8 is an isomorphism; whence, rn is injective. (iv) The question is local. Hence, by (1.19), we may assume that u -- v ~9 v', where v is regular of rank s and 34s(v) = 348. By (2.6)(i), 34s-F1 ® 34~-1 ~ ~. So, by (iii), 34s-l(34s+l ® 347 1) C_ 518. Then, by (2.7)(ii), by inspection, and by (2.7)(ii) again,
348+ =
34,(.') c 34, 34 (¢)J = 34s [34 +1 ® 34:1] ;.
The previous inclusion now yields the assertion.
275
REMARK ( 2 . 9 ) . inclusion,
Let u: P -~ £ be a form, and i , j , s > 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 <:i< s. Then (2.6)(i), inspection, and (2.7)(ii) yield
•~s--i(V) .~j+i(V') ~ J~s--i(v) J~i--I(V') J~I(V t) ~ j ( V t)
~- .~a_l(.Msq_l ® .~:l)(2~ta-4-j ® .Ms 1) = .r~~t~+3. Hence, all summands but the first of the induced direct s u m decomposition of u ~+j vanish when restricted to V~. Thus (ii) is proved. LEMMA ( 2 . 1 2 ) . Let s > 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 <_ r, the natural surjection )~i(U)T --~ ~ i ( U T ) is an isomorphism; iff, for 1 < i < r, the pullback of the inclusion 3~i(U)T ~ £ ~ i iS still injective. PROOF: The assertions are obvious.
278 DEFINITION ( 3 . 3 ) . Fix a pair P , and r >_ 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 <_ r, then J ~ ( u l % ) = J~. PROOF: Assertion (i) obviously holds. Now, by (1.15)(i), i
.i-IJI J_cO,...r} j ~ J
.
/
Therefore (ii) holds. Consider (iii). The source of the map in (3.4.1) is an invertible sheaf. Hence the map is injective, iff its image is invertible. So (ii) yields (b) <=~ (c) and (a) ¢~ (f). Trivially (a) ==~ (b). Clearly (b) ==~ (c) > (a). Obviously (c) ~ > (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 <: j _< r - 1.
Let q: Q r ~ P be the map defined by adding to (e, f ) the subset of r - 1 elements found above. T h e n clearly, q Q y + l = P1 @ q ' Q i
for 1 < j _< r - 1.
T h e argument t h a t gave (3.7.1) here gives (3.7.3)
~ y + l ( u l q Q y + x ) = )¢tl(vx) ~y(v21q'Qy)
for 1 < j _< r - 1.
Combining (3.7.1), (3.7.2) and (3.7.3) yields ~ ( u l q Q ~ ) = )q~(u) Hence U = S.
for 1 < i _< r.
281
DEFINITION ( 3 . 8 ) . Let u: P --+ • be a form. Define B0 = B0 (u) simply as S, and for i >_ 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 <: i < r. Define V ( 0 ) := Split(O) × t t o m ( E 2 , ~1) × " " × I-Iom(/~r, Lr-1),
w ( o ) := v ( , ) × v(,, := v(o) ×
om(Cok(q)®, Lr),
om(L1,
W(O, L) := W(O) × I-Iota(L1, ~) = V(O, L) × H o m ( C o k ( q ) ®, Lr). View the first three schemes as closed subschemes of the last via the zero-sections of the additional factors. LEMMA ( 3 . 1 6 ) .
Assume the conditions of (3.15). (A) Set W := W ( 0 , L). Then:
(i) W carries a canonical Ow-split, r-divisorial form w: P w -* Lw. (ii) The chain (3.3.3) and form (3.3.4) of w are isomorphic to the tautological chain and form, (3.16.1)
L w *-- L1,w +--"" +- •r,w
and
C o k ( q ) w -* Lr,w,
whose maps are the puflbacks of the tautological maps. (iii) Given any T / S and any OT-split form u: P T --+ f,T, there is an S-map T --+ W such that u = WT. Moreover, this map is uniquely determined if u is rdi visori al.
284
(iv) These two subschemes of W are equal: V(O, ~) = Vr(w). (v) Consider a direct sum decomposition P : P1 (9 • .. (9 P~ (9 P , + I in which each P~ is a pair of line bundles. Set
Qi := :P1 (9 "" " (9 P i
for 0 < i < r.
Let q: Q~ --+ P be the inclusion. Set • : : ({Qi}, q). Then r+l
S p l i t ( 0 ) = I ] H o m ( P q Q~-I). i=1
(B) The analogues of (i)-(iv) with W := W(~), with w: P w --+ f . w replaced by w: P w -+ £.l,w, with u: P T --~ f-T replaced by u: P T --~ f~I,T and required to be surjective, and with (3.16.1) truncated at £1,w are also valid. PROOF: (A) Because W sits over Split((I)), there is a canonical decomposition, P w = P1
•
•
"" (9 P r (9 P r + l
such that q w induces an isomorphism from Q~,w onto P1 (9" ' •(gPi. Plainly £~i,w ~ P ~ for 1 < i < r and Cok(q)~w - , P~+l- So (3.16.1) yields forms, v~: P~ --* £ w . Set w := Vl (9 . . . (9 v~+l. Obviously w: P w --* £ w is ~w-split, and its chain and form are as given in (3.16.1). Clearly the linking maps of the chain in (3.16.1) are injective. Hence w is r-divisorial by (3.4)(iii) (d)=~(f). Thus (i) and (ii) hold. In (iii), u is CT-split; so by (3.3), there are a decomposition of P T and a chain of linking maps. However, W ( ~ , £) represents the functor of just such data. Thus the asserted map, T -* W, exists. It is unique if u is r-divisorial, since the d a t a are unique by (3.5)(i), (ii). Assertion (iv) follows from (3.4)(iv) using the definition, (2.10), of V~(w). Assertion (v) is obvious from the definitions, (3.14) and (3.15). (B) The proof is nearly the same as that of (A). DEFINITION ( 3 . 1 7 ) . Let P be a pair, ~ a line bundle, and r _> 0. Let u denote the tautological form on t t o m ( P ®, £), and a t h a t on ~ ( P @ ) . Referring to (3.8), define B~(P,/~) := B~(u)
and
B r ( P ) :-- Br(a).
By the canonical forms on Br(P,/2) and B~(P) will be meant the pullbacks of u and a.
Let ¢ be an r-flag/P. Referring to (3.6), define U(¢, £) :-- U ( ¢ t B r ( P , £ ) , u l B r ( P , £)) C_ B~(P, £) U(¢) := U ( ¢ I B r ( P ) , aIBr(P)) THEOREM ( 3 . 1 8 ) .
_C B~(P).
A s s u m e the conditions of (3.17). Then:
285
(A) There is a canonical embedding of B~ (P) in Br (P, £) as the fiber over the zerosection of H o m ( P , £), and [u[Br(P, Z)]I[B~(P) = a[B~(P). (B) Let • be an r-flag/P. Then: (i) There are canonical isomorphisms, U(~, £.) = W ( ~ , £) and U(~) = W ( ~ ) . (ii) The pullback u[U(~, £) is equal to the canonical form w on W ( ~ , £), and the pullbac& aIU((I)) is equal to that on W ( ~ ) . (![ii) The embedding of Br(P) in Br(P, •) of (A) restricts to the embedding of W ( ~ ) in W ( ~ , £) induced by the zero-section of H o m ( £ 1 , £). (C) Suppose P is generated by its global sections. Then every point of Br (P, £) lies in U(,I),/~) for some ~, and every point of B r ( P ) lies in U (~ ) for some ~. In fact, gP may be taken of the form • :---- ({Qi},q) where Q0 :-- (0,0) and Qi := Q i - 1 (9 (Os, Os) for 1 < i < r and where q: Q~ --* P is the map defined by an ordered subset of r elements (e, f ) of the cartesian product of two given systems of generators of the components of P. Moreover, the trace of trace of •)) on U(¢, L) (resp. the trace of Vr(alBr(P) ) on U( ~ ) ) is equal to the zero scheme of the pullback of the tautological f o r m on
Horn(Cog(q)®, L,).
(D) If P is a pair of trivial bundles of ranks e and f , then B~ (P, £) is covered by (er) " (It) open subschemes, each equal to the a ~ n e el-space over S (resp. Br(P) is covered by (~) . (~) open subschemes, each equal to the aJFme ( e l - X)-space over S). PF,'OOF: (B)(i)(ii) For convenience, set U :-- U((I), £) and W := W((I),/Z). By (3.7)(i), u[U is (I)-split. Hence, by (3.16)(A)(iii), there exists a map f : U ---*W such that u]U -~ f*w. Now, by (3.16)(A)(i), w is r-divisorial, and by the universal property of u, there exists a map W --+ I-Iom(P ®, L) such that u l W = w. Hence, by (3.9)(iii), there exists a map g: W --~ U over t t o m ( P ®, E). The latter condition implies that g* (u[U) = w. Finally, f g -- 1 by (3.16)(A)(iii), because w is r-divisorial by (3.16)(A)(i); moreover, g f = 1 by (3.9)(iii), because u is r-divisorial by (3.9)(i). Thus the first assertions of (i) and (ii) are proved; the proofs of the second assertions are virtually the same. Moreover, the fiber of U over the zero-section of t t o m ( P ®, L) is plainly equal to the subscheme W(¢I,) of W. (C) The first two assertions follow from (3.7)(ii): the various ~ are defined on S, but (3.7)(ii) is applied to their pullbacks on B~(P, L) and Br(P). The last assertion follows immediately from (B)(i)(ii) and (3.16)(iv). (A), (S)(iii) To prove that there exists a unique isomorphism from the fiber of B~(P, £) in question onto Br(P) such that the pullback of a]B~(P) is equal to the r~.triction of the form [u[B~(P, L)]I[B~(P), it suffices, because of (3.9)(i)(ii), to work locally; the isomorphisms, established locally, must agree on the overlaps by the uniqueness assertion of (3.9)(iii). So assume that P is generated by its global sections. Then by (C) the various U(¢, L) cover Br(P,/Z), and the various V(q~) cover B~(P). Now, on a given U(O, L), the desired isomorphism exists by (B)(i)(ii) and the final statement of their proof, because obviously the restriction wl[W(O) is equal to the cauonical form on W(O). Thus (A) and (B)(iii) hold.
286
(D) Fix a basis of each component of P. Then proceed as in the argument of the last part of the preceding paragraph, and use (3.16)(A)(v) to identify the open subschemes as affine spaces. COROLLARY ( 3 . 1 9 ) . Assume the conditions of (3.17). Then: (A) Given a map x from a coherent sheaf to a bundle, let Zx denote its scheme of zeros. Then there exists a canonical embedding of Br(uIZ /~r+l u) in Br(P, £) the sub~hen~ V~(~IB~(P, L)). S i ~ 1 ~ l y there is a canonic~ embedding of Br(~tZ A ~+~ a) in B~(P) as the sub.heine V~(~IB~(P)). (B) Let ~P be an r-flag/P. Then the embedding of V(~, £) in W(¢, L) and that of v ( ~ ) in w ( ~ ) , the embeddings of (A), and the c=onic~ i s o m o ~ h i s ~ of
(3.18) (B) (i) in duce canonical isomorphisms, U(¢, £ ) A B ~ ( u I z A ' + l u ) = v ( ¢ , £)
u(o) ~B~(arZA'+la) = V(O). PROOF: In its notation, (3.12)(ii) implies that Br(ulZA r+l u) is equal to the closure of f(U N Z A ~+1 u) and that
Z(u N ZA'+'u) = v, A ZAr+'u. The right side is obviously isomorphic to U~ N Vr in view of the definition of U~ and that, (2.10), of V~ = V~(ulB~(P , £)). It remains to prove that U~ N V~ is dense in V~. This question is local, so assume that P is generated by its global section. Set Y :-- V(O, L). Then by (3.18)(B)(i), (ii), (C) and (3.16)(A)(iv), it suffices to prove that U~ N V is dense in V. However, by (3.4)(ii) and (3.16)(A)(ii), the former is obviously the complement of the union, for 1 < i < r, of the zeros of the maps, £i+lIV -* LilV. Since this complement is a divisor, the density holds (see the proof of (3.10)(ii)). Thus the first half of Assertion (A) and that of (B), which concern B~(ulZ A i u), are proved. The other halves, which concern Br(alZ Ai a), may be proved similarly. THEOREM ( 3 . 2 0 ) . Under the conditions of (3.18), the formations of Br(P, £), of Br (P), of Br (u IZ A i u), and of Br (a IZ A i a) corrtmute with any base-d2ange. PROOF: By (3.11), there is, in each case, a global base-change map, and it suffices to check that this map is an isomorphism locally. By (3.19)(B) and (3.18)(B)(i),(C), locally these schemes are equal respectively to W(¢, L), W(~), Y(¢, •), and Y(¢), and the formation of each of these obviously commutes with base-change.
4.
DEFINITION (4.1)0
COMPLETE
FORMS
Let P be a pair, let £ be a line bundle, and let r _> 0. Let
u = (u, ul,...,ur) be a sequence where u: P --~ £ is an arbitrary form and each
287
ui: A i P --+ ~ i is the f o r m associated to an invertible quotient ~ i = ~ ( u ) of ( A i P ) ®. T h e n u : P --* f will be called an r-complete form if, for each point s of S, there exist a neighborhood U of s, a scheme S', an r-divisor/hi f o r m v: P ' --+ £ ' on S ' , and a m a p a : U -* S ' such t h a t u[U = a*(v) and always uilU -- a*(v~), where vi: A / P --* 34i(v) is the m a p associated to the exterior power A i v in (1.16). If u is r-divisor/hi a n d always ui = u i, then u will be ca/led an exterior r-complete form, or
the exterior r-complete form on u. Similarly, let u = (Ul . . . . , ur) be a sequence, e m p t y if r = 0, where each ui : /~i p --+ aMi is t h e f o r m associated to an invertible quotient A4i ---- ~ i ( u ) of ( A i P ) ®. T h e n u will be called a projectively r-complete form u on P if, for each point s of S, there exist a neighborhood U of s, a scheme S ' , an r-divisor/hi f o r m v: P ' --+ £ ' on S ' , and a m a p a : U --* S ' such t h a t always uilU = a*(v/). If there exists an r-divisor/hi f o r m u a n d always ui = u i, then u will be called an exterior projectively r-complete form, or the exterior projectively r-complete form on u. For convenience, define u0: A ° P - * Os as the identity, and set N0 -- A40(u) =
:= Os
(the s t r u c t u r e sheaf)
:= £-1.
By the canonical r-complete form w on Br(P, f ) , resp. the canonical projectively r-complete form w on B r ( P ) , will be m e a n t the exterior f o r m on the canonical rdivisor/hi f o r m w, which was introduced in (3.17) PROPOSITION ( 4 . 2 ) .
In the setup of (4.1), the following conditions are equivalent:
(a) The sequence u is an r-complete form, resp. a project/rely r-complete form a n d r _ > 1. (b) T h e r e exists a section a o f B r ( P , f ) / S , resp. of B r ( P ) / S and r >_ 1, such that u -- o * ( w ) .
Moreover, if (b) holds, then a is unique. PROOF: Trivially (b) implies (a). Moreover, if the section a exists, t h e n it is unique by (3.9)(iii). Conversely, assume (a). By the uniqueness, the existence of a is a local question. Therefore we m a y assume t h a t there exists a scheme S ' , an r-divisor/hi f o r m u' : P ' -* £ ' , and a m a p r : S --* S ' such t h a t u = ~* ( u ' ) where u ' is the exterior f o r m on u'. Now, u' is the pullback of the tautological f o r m on H o m ( P ' , £ ' ) via a unique section ~ of H o m ( P ' , £ ' ) . So, by (3.9)(iii) with S := H o m ( P ' , £') and T := S', t h e r e exists a unique lifting of a~ to a section a ' of B ~ ( P ' , £') such t h a t ( a ' ) * ( w ' ) = v ' . By (3.20),
B,(P, L)
= B,(P',
Hence, t h e S t - m a p a'r defines a section of B r ( P , / ~ ) . Clearly, this is the required section. Resp., the proof is entirely similar, if (3.2)(i) is kept in mind.
288
THEOREM ( 4 . 3 ) . In the setup of (4.1), the pair ( Br (P, ~.), w ) represents the functor whose value on T / S is the set oft-complete forms, u : P7" ---* £T. Resp., i f r >_ 1, then (Br (P), w ) represents that of projectively r-complete forms on PT. PROOF: Given an r-complete form u on T / S , there is a unique section of Br(PT, ~.T) such t h a t u is equal to the pullback of the canonical form. T h e assertion follows because the formation of the pair ( B r ( P , / 2 ) , w ) commutes with change by (3.20). Resp., the proof is entirely similar. DEFINITION ( 4 . 4 ) .
Given a chain of line bundles and maps, called linking maps,
define by induction on j line bundles, called modified tensor powers, by
G°, : =
Os
ffi+:
~,8 := Z;{,,, ® ?-.m~n{,~+i,,,}.
whenever this definition makes sense. Given /2b and Lt such t h a t b _< a and t <_ s, define by induction on j maps ej = eja,8,b,t between the modified tensor powers,
eJ: L~,8 ~ L jb,t' called extended linking maps, by e ° := 1
(the identity of the structure sheaf)
ei+: := ei ® c where c: /2mln{a+/,8} - - ~ ~rrfin{b+j,t} is the indicated composition of the given linking maps. (Note that min{a + j, s} _> 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
(u) :
=
Let
for 0 < i < s , a n d for
>
(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) <', d(u). (2) The following diagram is commutative: At,a, 8 p
A t,j,* p
(~,®A~+')(:,O) i ® a +; =
®
t (~,®A':+')(O,1)
® a+')
l~e'-*®l
)
5t8®£{+~ = (~,® £8-t~ ~,sj® £~+t
291
(3) T h e following m a p is surjective: ( u t ® l ) ( l ® + ) : (A t'j'~ P)® ~-+ ~ t ® ( A 8+j P)®. (4) T h e f a m i l y o f linking m a p s el: £i+1 -~ £i a n d the f a m i l y o f modit~ed exterior p o w e r s At: A j P -+ £~ are characterized by the c o m m u t a t i v i t y o f the diag r a m s in (1) and (2) and these initial conditions: AJo = A j u and A t = uj for O<j<_s. Similarly, given a p r o j e c t i v e l y r - c o m p l e t e f o r m u on P with r >_ 1, the corresponding assertions to (1)-(4) hold.
PROOF: Assertions (1)-(3) and the initial conditions in (4) follow from the definitions and from (2.5) and (2.8)(ii). The part of (4) about the linking maps follows from (3) and from (2) with a := r, s := i, t :--- i - 1 and j := 1. The part about the modified powers follows from (3) and from (2) with a := s, t := s - 1 and j := j - s, by induction on j. The projective case m a y be proved similarly; alternatively, it may be derived from what was just proved by using (4.8)(1) and (2) with £ = ~ 1 and a = 1. REMARK ( 4 . 1 0 ) . The uniqueness statement of (4.9) (4) yields an alternative definition of the linking maps and modified powers of a complete form, resp. a projectively complete form. Namely, by Definition (4.1) the form is locally induced by an exterior form, which has canonical linking maps and modified powers, and it is natural to define the linking maps and modified powers of the complete form by patching the pullbacks. The uniqueness implies t h a t the patching can be done and t h a t the result is independent of the choice of local representation. Moreover, this procedure avoids the use of (3.20), a relatively deep result. The uniqueness statement of (4.9)(4) also makes (4.7)(D) and (4.8)(1) and (2) obvious. COROLLARY ( 4 . 1 1). Let u : P -+ L be an r - c o m p l e t e form, u -- (u, Ul, . . . , ur) say. Then, for 1 < s < r, the e x t e n d e d 1inldng m a p e ~ : f.~ -+ f.~ induces a surjective map, --.
a n d it is an i s o m o r p h i s m i f t h e first s linldng m a p s are injective or i f J~ s (u) is invertible. Similarly, given a p r o j e c t i v e l y r - c o m p l e t e form u on P , the corresponding assertions hold.
PROOF: Consider the commutative diagram of (4.9)(1). By (4.9)(4), A~ = u8 and so it is surjective. Again by (4.9)(4), A~0 = Aiu. Therefore Irn(e s) = Ms(u). Now, it is obvious from the definition (4.4), that e s is injective if the first s linking maps are. Finally, the surjection is an isomorphism if ~8(u) is invertible, because Ms(u) is invertible. The projective case m a y be proved similarly; alternatively, it m a y be derived from what was just proved by using (4.8)(1) and (2) with £ = N1 and a = 1.
292
COROLLARY (4.12). (A) L e t u: P --+ f. be an r - c o m p l e t e say. Then, the following c o n d i t i o n s are equivaJent: (i) u is the exterior r - c o m p l e t e f o r m on u : P --+ L. (ii) The f o r m u : P -+ f, is r-divisorial. (iii) A/1 the link/ng maps, Li --+ Li-1, are injective, 1 < i (B) L e t u = ( u l , . . . , u r ) be a p r o j e c t i v e l y r - c o m p l e t e f o r m the foI1owing c o n d i t i o n s are equivaIent: (i) u is the exterior r - c o m p l e t e f o r m on Ul : P -+ 511. (ii) The f o r m ul : P --+ 511 is r-divlsorial. (iii) A/1 the l i n k i n g maps, Li --+ g i - 1 , are i n j e c t i v e , 2 < i
form, u
=
( U, U l , . . . , Ur )
< r. on P w i t h r _> 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
300
F o r m t h e chain (5.6.1)
£ = £0 +-- £~ ~
.-. ~
£ s +-- £8+1 ~ - . - .
~
LL
in which t h e first s m a p s are t h e linking m a p s of v , a n d t h e r e m a i n i n g t m a p s are t h e linking m a p s of
v '+ := (0,v'): R'
£8
Consider t h e c o r r e s p o n d i n g modified tensor powers £~ of (4.4). For 0 < c < r a n d i _> 0, define a c o m p o s i t e form, A / : P -+ L~ as follows: i
(5.6.2)
A~:
AP
(5.6.3)
A ie : = 0
i
~
AR ....... A~(v) , £ ~
if i < s < c
i (5.6.4)
'h
Ac:
ifc<s
s P
>
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 <_ i = c <_ s a n d (5.6.4) w i t h s < i --- c <_ r.
PROOF: Assertions (i)-(iii) m a y be checked locally. So, to check t h e m , we m a y a s s u m e t h a t t h e r e exists a m a p q : R -* P such t h a t p q -- 1. T h e n , it is e a s y to see, t h e s e q u e n c e u is equal to t h e sequence o b t a i n e d b y splicing t h e t - c o m p l e t e f o r m (v) ' + ---- ( 0 , v ' ) : R ' ~ •8 o n t o t h e s - c o m p l e t e f o r m v : R ~ / ] ; indeed, o b s e r v e t h a t , in t h e n o t a t i o n of (5.1), A~ ,i ---- 0 for j > 1 b e c a u s e e j = 0. Hence, (i)-(iii) follow f r o m (5.2).
301
Assertion (iv) holds if s < r, because, by (4.7)(B) with s : = s + 1 and t := s, the m a p A : +1 factors through the linking map, L~s+l ~ Ls, which vanishes by (ii). Assertion (iv) holds if s = r, by (iii) and (5.6.3). Assertion (v) follows from (iv) and from (5.4) applied to the t r u n c a t e d s-complete form (u, u l , . . . ,us): P---* L. As to (vi), v : R --~ L is determined by the factorizations u = v p and (5.6.2), because p and A i p are surjective. Finally, vs : (A s R ) ® ~ Ms is an isomorphism, because it is a surjective map between invertible sheaves. So, as vj. = A;;i(v') for 1 ~ j < t, it is determined by (5.6.4). So (vi) holds. TttEOREM ( 5 . 8 ) (UNSTRINGING). Let u : P --~ ~ be an r-complete form. 0<s
Let
= o.
(b) P / I K e r h ( u s , P )
is a pair of bundles of rank s.
(2) I f s < r, then (a) is equivalent to the following condition: (c) T h e sth linking map, e 1 : f s + l --* f s , vanishes. (3) A s s u m e (a). Set t := r - s. Set R ' := K e r h ( u s , P ) . the canonical surjection b y p : P -* R . Then:
Set R := P / R ' , and denote
(i) T h e factorizations u = vp, (5.6.2) with 1 ~_ i = c <_ s and (5.6.4) with s < i = c _~ r define an s-compIete form v : R ---* L~ and a projectiveIy tcomplete form v ~ on R I. (ii) u : P --*/~ is equal to the r-complete form obtained by stringing v r onto v. PROOF: T h e proof is similar to that of (5.7). Applying (5.4) to the t r u n c a t e d scomplete form (u, U l , . . . , u s ) : P --* • yields (a)~=~(b) and the assertion about v in (i). T h e assertion about v ' in (i) and assertion (ii) may be checked locally, and locally they follow from (5.3)(v)(vi) applied to a map q: R ---* P such that p q = 1. Finally, if s < r, then (a) and (c) are equivalent, because A ] + l ( u ) ----elvs+l by (4.9)(1), (4) and because v8+1 is surjective. COROLLARY ( 5 . 9 ) . Let 0 < s 1 < ..- < sk ~ r. Stringing and unstringing establish a bijective correspondence between the set of r-complete forms u : P --~ £ such that A~ +1 -- 0 for 1 < i < k, and the set of pairs ( F , U ) where 1~ is a flag of subpairs P i of P such that P / P i is a pair of bundles of r a n k si for 1 < i < k and where U is a set consisting of an s l - c o m p l e t e form u(0): ( P / P 1 ) -+ £. and of a projectively [si+1 - si]-complete form u ' ( i ) on ( P i / P i + l ) for 1 < i < k, where sk+l := r and P k + l := (0, 0). PROOF: T h e assertion results directly from (5.7) and (5.8) by induction on k. COROLLARY (5.10). Let 0 _~ s 1 < " ' ' < : S k ~ r. Stringing and unstringing establish a bijective correspondence between the set o f r-complete forms u : P ---* L such that A~ +1 ----0 for 1 < i < k and such that the linking m a p s £ j + I -* L j are injective for all j in { 0 , . . . , r -- 1} \ { s l , . . . , sk }, and the set of pairs (F, U0) where F is a flag of subpairs P i of P such that P / P i is a pair of bundles of r a n k si for 1 < i < k
302
and where U 0 is a set consisting of an sl-divisorial form u(0): (P/PI) -~ f of rank sl and of an [si+1- si]-divisorial form u'(i): (Pi/P~+1)--+ A4(i) of rank [s~+1- si] and associated to an invertible quotient ~4({) of (Pi/Pi+1) ® for I < i < k, where 8k+ 1 :~- r and P k + l := (0, 0). PROOF: T h e assertion results immediately from (5.9), (4.12) and (1.20). COROLLARY ( 5 . 1 1 ) . Let 0 _< Sl < .-. < sk _< r. Let w denote the canonical r-complete form on B r ( P , / ~ ) , and V = V{s,} the dosed subscheme of zeros of Ag~+1 for 1 < i < k. Let F ----F{8,}(P) denote the flag scheme of subpaJrs of P, and {Pi} the tautological flag, so that P f /Pi is a pair of bundles of rank si. Then there exists a canonical isomorphism from V onto the product
B81(PF/PI, £f) × F Bs~-sl (Pl/P2) ×F''" × F B~k-~,_1 (Pk-i/Pk) × F Br-~k (P;¢) and under it, the restriction w I V corresponds to the r-complete form obtained by stringing the various canonical projectively complete forms on one after the other onto the string starting with the canonical s l-complete form. PROOF: T h e first assertion is simply a geometric reformulation of the equivalent version of (5.9) formulated for an arbitrary T / S . COROLLARY ( 5.1 2 ) . Let P be a pair, £ a iine bundIe, and r >_ 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<~ ~A---+ f - A ( ~ A - - + f.A(1) • (c) the subscheme representing the condition that the pullback of v factors through the restriction of a.
304 (iv) T h e m a p v: £ --* £. defines an e m b e d d i n g o f ~ ( £ ® I ) into the three equal subschemes of (iii), and v induces a surjective map, (6.1.1)
~v --+ ( I / I 2) ® £.
(v) Assume that the following sequence is exact: (6.1.2)
L -1 ® E ® $
£-~@d
v ~ ~ --+ .~.
Then ~ ( £. ® ~) is equ~ to each of the t h ~ ~ubs~emes in (iii), and (~.~.1) is an isomorphism.
PROOF: (i) Obviously, for any T / S , a map w: ET -"* N factors through Cok(f)T iff the composition, f T : FT ---+ET --+ X, vanishes. The assertion follows immediately via the characterization of the T-points of a / P ( ~ ) as the invertible quotients of ~T. (ii) The question is local, so assume S is affme and ~t = 03. Then obviously (u ® ~ ) ( 1 - , w ) = (1 - ~ ) ( u
® u) = 0.
Hence, also (v@u)(1 - sw) = 0 if v factors through u. Conversely, suppose that ( v ® u ) ( 1 - s w ) : O. Fix a section e of ~ such that u(e) = 1. Then v(e) defines a map w: Al --~ L, and obviously,
v(x) = u(~) v(x) = u(x) ~(~) = w(~(x)). Thus (ii) is proved. (iii) The subschemes (a) and (b) are equal by (i). The subschemes (b) and (c) are equal by (ii). (iv) Obviously, v induces a surjection, E ----* ~ ® I . Restricting it to V yields (6.1.1). Now, consider this sequence, obtained from (6.1.2): (6.1.3)
~ ' ® £ ~d L@~"
I®'-',[email protected]..
The composition vanishes for the same reason as in (ii). Clearly, Ira(1 ® v) is equal to/~ @/~ ® I . Hence, there exists a canonical surjective map, (6.1.4)
Cok(/~ -1 ® d) -+ ~ ® I .
So v induces an embedding o f / P ( £ ® I ) into the subscheme (a) in (iii). (v) Since (6.1.2) is exact, so is (6.1.3). Hence (6.1.4) is an isomorphism; whence -~(~ @ I ) is equal to the subscheme (a) of (iii). Restricting the isomorphism (6.1.4) to V gives (6.1.1), because it commutes with the formation of the cokernel and it makes the map d vanish.
305
LEMMA ( 6 . 2 ) . The following two conditions 0), (ii) on a map w: £ --~ L a r e equivalent: (i) This condition has two parts. (a) The following sequence is exact: (6.2.1)
d
® ~
w
® •--1 __+ £ __~ £
where d := (1 ® w ® 1)(1 -- sw ® 1)
where sw is the switch involution of £ ® £. (b) Set I : = Irn(w) ® f-1. Then the symmetric algebra Sym(1) of I is equal to its Rees a/gebra ;~ ees( I ) :---- G n l n. (ii) For every n >_ 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
The corresponding statement in the projective case hoIds if r > 1; name/y, B r ( P ) is e q u a / t o the dosed subscheme of S~(P) : = / P ( P ® ) × ~P((A2P) ®) × . . - × E'((ArP)®), defined by the (bilinear) equations = 0,...,
= 0
where u l , . . . , u~ are the pullbacks of the tautologicaI maps; etc. PROOF: If r = 0, then Br is defined as H o r n ( P , £), and there are no equations. If r = 1, then B~ is defined as the blowup of the zero scheme of the tautological map on EIom(P,/~). Hence, by (6.4) and (6.1)(v), Br is equal to the closed subscheme of St(P,/~) defined by ®
1)(1 - s w )
= 0.
However, this equation is just A(u, Ul) = 0. Assume r _> 1. By virtue of (3.12)(i) and (3.9)(i), (ii), Br is canonically embedded in S~(P,/~), and ulB~ is r-divisorial, and (u]B~) ~ = u~lBr for 1 < i < r. By (3.13), B~+I is the blowup of 2"r. By (3.18)(B)(i), (3.18)(C) and (6.7)(C), all the hypotheses of (6.12) axe satisfied for Br and u]Br. Hence, by (6.12), B r + l is equal to the subscheme of B~ x ~ ( ( A ~ + I P ) ®) defined by the equation, f~(u~, u~+l) = 0. Therefore, if the assertion is valid for r, then it is aiso valid for r + 1. Furthermore, all the additionai equations are satisfied by (2.8)(i). The proof of the corresponding statement in the projective case is entirely similar.
7.
SYMMETRIC FORMS
SETUP ( 7 . 1 ) . Throughout this section, assume t h a t 2 is invertible in F(S, Os). DEFINITION ( 7 . 2 ) . A pair P will be called symmetric if P = p t r ; t h a t is, P is symmetric if P is of the form P = (£, £). If P is symmetric, set psym := Sym2~. A bilinear map u : P --~ 9 will be called symmetric if u = u t r . If u is symmetric, then p = p t r and the same symbol u will be used to denote the associated linear map U: p s y m --+9" A m a p of pairs q: Q --+ P will be called symmetric if q -- qtr; that is, if Q ~- Qtr, p __ ptr, and if the two coordinates of q are equal.
313
PROPOSITION ( 7 . 3 ) . (1) I f w: A ~P -~ ~ is a symmetric bilinear m a p an d q: Q --+ P is a symmetric m a p of pairs, then the map h(w, q) --- h(w, Q) of (1.7) is a symmetric m a p of pairs: i-1
h(w, q): P --+ Hom(A Q®
i
AP,
(2) T h e direct sum of symmetric of bilinear maps is synm]etric. (3) An exterior power of a symmetric form is symmetric. PROOF: Assertions (1) and (2) are obvious. Assertion (3) holds because the value of a determinant is invariant under transposition. REMARK ( 7 . 4 ) . It would be good to know if the version of (1.19) with u and Q symmetric is true or not. It is obvious t h a t even if P is symmetric, A ~'i'k P need not be. PROPOSITION ( 7 . 5 ) . In (2.13), if u is symmetric, then so are K , R and (the form associated to) )q-[1 ® n. PROOF: T h e assertion is obvious. DEFINITION ( 7 . 6 ) . Let ¢ ----({Q~},q) be an r - f l a g / e . It will be called symmetric if the m a p q and the pairs Qi are syrmnetric. PROPOSITION ( 7 . 7 ) . Let iI~ be an r-flag/P, and u: P --+ £ a gP-split, r-divisorial form. I r e and u are symmetric, then so is the corresponding direct sum decomposition (3.3.1). PROOF: Apply (3.5)(i) and (7.2). LEMMA ( 7 . 8 ) . Let P = (£, £) be a symmetric pair, and u: P --~ £ an arbitrary r-divisorial form. Let a be a geometric point of S such that the pullbacks u l ( a ) , . . . , u ~ ( a ) are symmetric. Set Qo :-- (0,0) and Qi :-- Q i - 1 • (Os, Os) for 1 < i < r. Then given a set of generators e of ~, there exists an ordered sequence of Z[1/2]-linear combinations f l , . . . , fr of the e's generating the same submodule as the e's such that, if qr : Q~ --* P denotes the syrmnetric m a p defined similarly on each component by the r sections f ~ o f F , then the open subscheme U := U ( { Q i } , q r ) contains the point a. PROOF: T h e case r ----0 is trivial. Proceeding by induction on r, suppose t h a t r _> 1. Since ~ 1 is invertible, there exists a pair (e,g) of the given generators such t h a t u(e ® g) generates ~ 1 at the given point a. Set f + : = (e + g ) / 2 and f - := ( e - g)/2. Then u(e ® g)(a) = u ( f + ® f + ) ( a ) -- u ( f - ® f - ) ( o ) . So one of the summands is nonzero. Accordingly, take f l := f + or f l := f - as the first linear combination. Proceed to finish up along the lines of the proof of (3.7)(ii). However, instead of using v2 := uIP2 directly, use the isomorphic form v~:P/P1
~
P2-~
314
because P / P 1 is synm~tric. By hypothesis, the ui(a) axe syrmnetric, and by (3.5)0) and (1.18) (ii), P2 (a) = K e r h ( u 1(a), q l (a)). Hence, P2 (a) is symmetric; whence, so are the v~ (a) and the v~2i (a). PROPOSITION ( 7 . 9 ) .
Let u: P --* L be a symmetric form, r > O. Set
Srsyrn : : J~i°(Psym) x ~°((A2P)~Ym ) x . . . x ZP((A~P)~Ym),
which is a subscheme of the scheme Sr of (3.12). Then the canonical embedding of Br (u) in Sr embeds it in .q,sym
PROOF: Use (7.3)(3). DEFINITION ( 7 . 1 0 ) . Under the conditions of (3.15), suppose that P and G are symmetric. Define Split~ym(G) as the closed subscheme of Split(G) representing the subfunctor of those sequences (P, p l , - . . ,pr) such that p and the Pi are syrrmaetric. Also, define v~ym(G, £) and w~ym(G, £), etc., by replacing in the definition of V(G,/2) and W(G, ~), etc., the terms Split(G) and Cok(q) ® by Splitsym(G) and Cok(q) sym. LEMMA ( 7 . 1 1 ) .
Assume the conditions of(7.10). Set W := W(G, £) a n d W sym :=
w s y m ( G , ~). Then: (A)
(i) Consider the canonical form w: P w -+ F.w on W , see (3.16)(A)(i). Its restriction is a Gw,y=-split, r-divisorial symmetric form w sym :--- w I W sym. (ii) The chain (3.3.3) and the form (3.3.4) of w sym are isomorphic to the tautological chain and form, Ew,ym ~-- Ll,W,rm +--''" ~-- Lr,W'Y=
and
C o k ( q ) w . y = -+/2r,wor,,,
whose linking maps are the pullbacks of the tautological (iii) Given any T / S and any O-split symmetric form u: P T sym . Moreover, an S-map T --~ W sym SUCAtlthat u = w T determined if u is r-divisorial. (iv) These two subschemes o f W Sym are equal: v~ym(G, L) =
maps. --* LT, there exists this map is uniquely V~(wSym).
(B) The analogues of (i)-(iv) in the projective case, similar to (3.16)(B), are also valid. PROOF: The proof is similar to that of (3.16).
DEFINITION (7.12). Let P be a symmetric pair, £ a line bundle. Denote by u sym the tautological form on I-Iom(P Sym,/2) and by a Sym t h a t on tP(PSYm). Let r _> 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 ) .
The parallel symmetric versions of (4.8)-(4.9) are valid.
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 ) .
The parallel symmetric versions of (6.10)-(6.12) are valid.
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
T h e corresponding s t a t e m e n t in t h e projective case holds; namely, B~Ym(P) is e q u a l to t h e closed s u b s c h e m e o f SrsYm(p) :____j~:,(psym) >< Z p ( ( A 2 p ) s y m ) x - . .
x ZP((Arp)sym),
defined by t h e (bilinear) equations n ( U l , U 2 ) = 0,...,/~(~tr_l,Ur)
= 0
where U l , . . . , Ur are t h e pullbacks o f t h e tautological m a p s ; etc. PROOF: T h e p r o o f is t h e s a m e as t h a t of (6.14) m u t a t i s m u t a n d i s . A l t e r n a t i v e l y , t h e c o r r e s p o n d i n g a s s e r t i o n s for s y m B r ( P , £) a n d sYmBr(P ) follow i m m e d i a t e l y f r o m (6.11). So t h e a s s e r t i o n s t h e m s e l v e s hold b y (7.20).
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