Algebraic Geometry: Summer Meeting, Copenhagen, August 7–12, 1978 PDF
Preview Algebraic Geometry: Summer Meeting, Copenhagen, August 7–12, 1978
Lecture Notes ni Mathematics Edited yb .A Dold dna .B Eckmann 732 ciarbeglA yrtemoeG Summer Meeting, Copenhagen, August 7-12, 1978 Edited yb .K Lensted galreV-regnirpS Berlin Heidelberg New kroY 1979 Editor Knud Lensted Kebenhavns Universitets Matematiske Institut Universitetsparken 5 DK-2100 Kebenhavn AMS Subject Classifications (1970): 14-XX ISBN 3-540-09527-6 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09527-6 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data Copenhagen Summer Meeting in Algebraic Geometry, 1978. Algebraic geometry. (Lecture notes in mathematics ; 732) Bibliography: p. Includes index. .1 Geometry, Algebraic--Congresses. .I Lensted, Knud, 1942- .1I Title. .1II Series: Lecture notes in mathematics (Berlin) ; 732. QA3.L28 no. 732 0A564 510'.8s 512'.33 79-17367 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210 PREFACE These Proceedings contain the papers contributed to the Copenhagen Summer Meeting in Algebraic Geometry 1978, held August 7-12, 1978, at the H. C. ~rsted Institut of the University of Copenhagen. In addition to the papers presented at lectures this volume also includs a few ones by other participants, as well as a couple by some who were pre- vented from participating. It is the editors hope that the present volume will offer a repre- sentative coverage of the actual activities in algebraic GEOMETRY, and that it may be found useful to workers in this field. The meeting was finanically supported by the Danish Natural Science Research Council under grant no. 511-10092. Matematisk Institut and the H. C. ~rsted Institut offered hospitality and practical help. On the behalf of the organizers, who also included H.-B. Foxby, A. Thorup and N. Yui, I should like to express our gratitude for this, and also to all the participants for their collaboration, which made the orga- nization much easier than expected. Special thanks go to Mrs. A. Tar- nov and to Ms. Ulla Jacobsen, whose assistance during critical moments of the meeting and at the final preparation of the manuscript proved to be invaluable. Knud L~nsted LIST OF PARTICIPANTS M. ARTIN A. MAGID H. BASS G. MALTSINIOTI S J. F. BOUTOT Z. EiY BKHOU T R.O. BUCHWEI TZ J. Y. MERINDOL F. CATANESE A. MILH~J P. CHERENACK C. MITSCHI .N CHIARLI Y. MIYAOKA A. COLLINO .L NESS D. CORAY H.A. NIELSEN D. COX N. 0. NYGAARD M. DESCHAMPS .L D. OLSON G. ELLINGSRUD F. OORT F. ELZEIN .S PEDERSEN H. ESNAULT U. PERSSON D. FERRAND R. PIENE R. FOSSUM H. PINKHAM H. B, FOXBY H. POPP .S GRECO M. vdPUT M. HAZEWINKEL A. RAMANATHAN A. HOLME M. REID J. HUBBARD F. SAKAI C. H~STMAEHLI NGEN M. SCHAPS B. IVERSEN C. .S SESHADRI J. .P JOUANOLOU T. SHIODA J. M. KANTOR J. .R STROOKER M. KATO U. STUHLER T. KATSURA S.A. STR@MME Y. KAWAMATA M. TEICHER G. KEMPF A. THORUP H. KLEPPE R. TREGER J° KLEPPE E. VIEHWEG A. KURIBAYASHI .S USUI D. LAKSOV VAJNRYB O. A. LAUDAL L. VERMEULEN E. LLUIS C. H. WACHMANN .S LUBKIN .G E .WELTERS A. LUBOTZKY N. YUI K. L~NSTED .S ZUCKER CONTENTS F. CATENESE: Surfaces with K 2 = pg : I and their period mapping ............................... I P. CHERENACK: Internal hom-sets in an extension of affine schemes over a field .................. 3O D.A. COX: Solutions of Weierstrass equations ........... 43 V.G. DRINFEL'D and JU.I. MANIN: Instantons and sheaves on { ~3 ............... 60 D. FERRAND: Set theoretical complete intersections in characteristic p > 0 ..................... 82 H.-B. FOXBY: Intersection properties of modules ........... 90 S. GRECO and P. VALABREGA: On the theory of adjoints .................... 98 M. HAZEWINKEL: Infinite dimensional universal formal group laws and formal A-modules .............. 124 A. HOLME: On the dual of a smooth variety .............. 144 S. IITAKA: Symmetric forms and Weierstrass semi- groups ....................................... 157 V.A. ISKOVSKIH and V.V. ~OKUROV: Biregular theory of Fano 3-folds ............. 171 J.P. JOUANOLOU: Singularit~s rationnelles du resultant ....... 183 Y. KAWAMATA: On the classification of non-complete algebraic surfaces ........................... 215 G. KEMPF and L. NESS: The length of vectors in representation spaces ....................................... 233 H. KLEPPE and D. LAKSOV: The generic perfectness of determinantal schemes ...................................... 244 A. KURIBAYASHI and K. KOMIYA: On Weierstrass points and automorphisms of curves of genus three ..................... 253 D. LAKSOV: Deformation and transversality ............... 300 S. LUBKIN: Finite generations of lifted p-adic homology with compact supports. Generalization of the weil conjectures to singular, non-complete algebraic varieties .................................... 317 IV A. LUBOTZKY: On a problem of Grothendieck ................. 374 A.R. MAGID: Faithfully representable analytic groups ....................................... 384 Z. MEBKHOUT: The Poincare-Serre-Verdier duality ........... 396 L. NESS: Mumford's numerical function and stable projective hypersurfaces .............. 417 L.D. OLSON: The trace of Frobenius for elliptic curves with complex multiplication ........... 454 F. OORT: Abelian varieties: moduli and lifting properties ........................... 477 U. PERSSON: A family of genus two fibrations ............. 496 R. PIENE: Ideals associated to a desingula- rization ..................................... 5O3 M. van der PUT: Schottky groups and Schottky curves ....................................... 518 A. RAMANATHAN: Moduli for principal bundles ................. 527 M. REID: n I for surfaces with small K 2 .............. 534 F. SAKAI: Symmetric powers of the cotangent bundle and classification of alge- braic varieties .............................. 545 T. SHIODA: Supersingular K3 surfaces .................. 564 R. TREGER: Rational singularities in dimension > 2 .............................. 592 S. USUI: Deformations and local Torelli theorem for certain surfaces of general type ................................. 605 N. YUI: Formal groups and some arithmetic properties of elliptic curves ................ 630 SURFACES WITH K 2 = pg = 1 AND THEIR PERIOD MAPPING. Fabrizio Catanese - Universit~ di Pisa - Harvard Unlverslty • . + Introduction. Recently a result of Kynef (~4) drew attention on minimal surfaces S with K 2 = Pg = 1 : he constructed a quotient of the Fermat sextic in ~3 by a suitable action of ~/6 ' with these invariants, such that the differential of the period mapping (see 7 , ~J ) is not injective at it, thus answering negatively a problem posed by Griffiths in 8 . One may remark however that the local Torelli theorem (injectivity of the infinitesimal period mapping) fails , for curves, exactly when one has an hyperelliptic curve (7J), though the global Torelli theorem holds. So one is motivated ~study these surfaces and their period mapping. They were first considered by Enriques in 1897, who proved their existence in ~ (see also 6~ l pag. 305 ) ; Bombieri (~, pag. 201) proved rigorously that for these surfaces the tricanonical map is birational. )+( The author was partly supported by a N.A.T.O.-C.N.R. fellowship during his stay at Harvard University. Here we prove that the bicanonical map ~ = ~2K is a morphism and that ++) any such surface si a weighted complete intersection of type (616) in the weighted projective space ~(1,2,2,3,3) (see ~J , ~5J about the theory of weighted complete intersections). We also show that these surfaces have equations in canonical form: this is a first step towards an explicit description of their moduli space, that we hope to accomplish in the future. Then we describe a geometric construction giving all the "special" surfaces,i.e, those for which ~ is a Galois covering (and it turns out that the Galois group is ~/2 + ~/2 ") Using this explicit description of our surfaces we prove that they are all diffeomorphic and simply connected, and that when K is ample the Kuranishi family is smooth of dimension 18 sa( their local period space): our main result is that the differential of the period mapping is inverti- ble outside an hypersurface, so that the period mapping is generally finite. The 12 dimensional subfamily parametrizing "special" surfaces is strictly contained in the subvariety where the rank of the differential drops by 2 (the maximum possible amount) and we prove, by means of a more general result on deformations of cyclic coverings, that the restriction of the period mapping to this subfamily is locally I-I : this suggests that the period mapping might have no positive dimensional fibres, but we have not yet pursued such investigation. One last remark is that our results on the failure of the local Torelli theorem for ~eighted complete intersections (w.c.i.) show that the ++) The proof which appears here of this result is due to collaboration with Miles Reid. 3 restrictions put by S. Usui in his work ~OJ cannot all be eliminated. I would like here to thank P. Griffiths for suggesting this research and I. Dolgachev for useful conversations. Notations throughout the paper: S is a minimal smooth surface with pg = K 2 = 1 o x o~H (S,O(K)) the unique (up to constants) non zero section C = div(Xo) the canonical curve R the graded ring C EXo,YI,Y2,Z3,Z4 , where deg Xo = 1 , deg i.Y = 2, deg Z. = 3 (i = 1,2, j = 3,4) i W = C EYI,Y2,Z3,Z4 as a graded subring of R Rm,W m the graded parts of degree m of R , resp. W Q = Q(1,2,2,3,3) = proj (R) oo o R(S) = ~. H (S, ~(mK)) the canonical ring of S m=o hl(S,L) = dim Hi(S,L) if L is a coherent sheaf on S § I . STRUCTURE OF SURFACES S WITH K 2 = P =I . g LEMMA I. Pm=h°(S,~(mK)) = I m(m-1) + 2. Proo£. Pm= 1~ m(m_l)K2+X( ~s) (see 1 pag. 185, or ~1) and by Theorems 11,15 o£ 1 O=q and S has no torsion, so X: =2. One can choose there£ore yl,Y2, Z3,Z 4 such that 2 o 3 x°'Yl 'Y2 are a basis o9 H S( ~(2K)) and Xo xoYl,Xoy 2 5 % O are a basis o£ H (S,O (3K)). Write now C=div(xo)= ~+ Z , where K'F =I, K.Z=O. AMMEL 2. I£ De ~2K~ and D > ~ , D=2C. Proof. Write D=D'+ P and let D" be the movable part O£ 'D I : D"-K=I , so by the index theorem either D" is homo- logous (hence linearly equivalent, as S has no torsion) to K, or D"2< -1, hence in both cases h°(S,(9 (D"))=I. Y~/ALLOROC 3. H (S,(9(4K))= xo-H (S, O(3K))@(C (~ylY2(~ ). Proo£. Because P4=8 it is enough to prove that the two vector subspaces have no common line. Supposing the contrary, O there would exist a section s ~ H (~(3K)), and costants k1,~1,k2,~2 such that x o. s= (llY1+~lY2)(k2Y1+~2Y2) .
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