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Lecture Notes in Mathematics 1842 Editors: J.--M.Morel, Cachan F

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Lecture Notes in Mathematics 1842 Editors: J.--M.Morel, Cachan F Lecture Notes in Mathematics 1842 Editors: J.--M.Morel, Cachan F. Takens, Groningen B. Teissier, Paris 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Trygve Johnsen Andreas Leopold Knutsen K3 Projective Models in Scrolls 13 Authors Trygve Johnsen Department of Mathematics University of Bergen Johs. Bruns gt. 12 5008 Bergen, Norway e-mail: [email protected] Andreas Leopold Knutsen Department of Mathematics University of Oslo Box 1053, Blindern 0316 Oslo, Norway e-mail: [email protected] LibraryofCongressControlNumber:2004103750 Mathematics Subject Classification (2000): 14J28, 14H51 ISSN 0075-8434 ISBN 3-540-21505-0 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEXoutputbytheauthor SPIN: 10999523 41/3142/ du - 543210 - Printed on acid-free paper Preface The cover picture shows a smooth quartic surface in space, the simplest ex- ample of a projective model of a K3 surface. In the following pages we will encounter many more examples of models of such surfaces. The purpose of this volume is to study and classify projective models of complex K3 surfaces polarized by a line bundle L such that all smooth curves in |L| have non-general Clifford index. Such models are in a natural way contained in rational normal scrolls. These models are special in moduli in the sense that they do not repre- sent the general member in the countable union of 19-dimensional families of polarized K3 surfaces. However, they are of interest because they fill up the set of models in Pg for g ≤ 10 not described as complete intersections in projective space or in a homogeneous space as described by Mukai, with a few classificable exceptions. Thus our study enables us to classify and describe all projective models of K3 surfaces of genus g ≤ 10, which is the main aim of the volume. Acknowledgements. We thank Kristian Ranestad, who suggested to study certain projective models of K3 surfaces in scrolls that had shown up in connection with his work on varieties of sums of powers (see [I-R1], [I-R2] and [R-S]). This idea was the starting point of our work. We are also grateful to M. Coppens, G. Fløystad, S. Ishii, S. Lekaus, R. Piene, J. Stevens, S. A. Strømme, B. Toen and J. E. Vatne for useful conversations, and to G. M. Hana for pointing out several mistakes in an earlier version of the manuscript. We also thank Alessandra Sarti who made us the nice picture we have used on the cover. Most of this book was written while the authors were visitors at the De- partment of Mathematics, University of Utah, and at the Max-Planck-Institut VI Preface für Mathematik, Bonn, in the year 2000. We thank both institutions for their hospitality. The second author was supported by a grant from the Research Council of Norway. Bergen/Oslo, March 2004 Trygve Johnsen Andreas Leopold Knutsen Contents 1 Introduction ............................................... 1 1.1 Background............................................. 1 1.2 Relatedliterature........................................ 7 1.3 Howthebookisorganised................................ 7 1.4 Notationandconventions................................. 9 2 Surfaces in Scrolls ......................................... 15 2.1 Rationalnormalscrolls................................... 15 2.2 Specializing to K3 surfaces............................... 17 3 The Clifford index of smooth curves in |L| and the definition of the scrolls T (c, D, {Dλ}) ...................... 19 3.1 GonalityandCliffordindexofcurves ...................... 19 3.2 TheresultofGreenandLazarsfeld ........................ 20 3.3 Clifforddivisors......................................... 21 3.4 Gettingascroll ......................................... 28 4 Two existence theorems ................................... 31 5 The singular locus of the surface S and the scroll T ....... 35 5.1 The singular locus of ϕL(S) ............................... 35 5.2 The singular locus of T andperfectClifforddivisors......... 36 6 Postponed proofs .......................................... 47 7 Projective models in smooth scrolls ........................ 59 8 Projective models in singular scrolls ....................... 63 8.1 Blowing up S ........................................... 64 8.2 The smooth scroll T0 ..................................... 67 8.3 Techniques for finding Betti-numbers of the ϕL(Dλ) ......... 70 8.4 Resolutionsforprojectivemodels.......................... 82 8.4.1 Pushingdownresolutions........................... 86 8.5 Rolling factors coordinates . 89 8.6 Someexamples.......................................... 92 VIII Contents 9 More on projective models in smooth scrolls of K3 surfaces of low Clifford-indices ............................. 99 9.1 Projective models with c =1..............................100 9.2 Projective models with c =2..............................105 9.2.1 An interpretation of b1 and b2 .......................108 9.2.2 Possible scroll types for c =2 .......................109 9.3 Projective models with c =3..............................114 9.4 Higher values of c .......................................117 10 BN general and Clifford general K3 surfaces ..............121 10.1TheresultsofMukai.....................................121 10.2Notionsofgenerality.....................................123 10.3 The case g =8..........................................124 10.4 The case g =10.........................................127 11 Projective models of K3 surfaces of low genus .............129 11.1 A new decomposition of R ................................130 11.2 Perfect Clifford divisors for low c ..........................131 11.3Thepossiblescrolltypes..................................135 11.4Someconcreteexamples..................................138 11.5Thelistofprojectivemodelsoflowgenus...................143 12 Some applications and open questions .....................155 12.1 BN generality..........................................155 12.2ApplicationstoCalabi-Yauthreefolds......................156 12.3AnalogieswithEnriquessurfaces..........................157 References .....................................................159 Index ..........................................................163 1 Introduction 1.1 Background A K3 surface is a smooth compact complex connected surface with trivial canonical bundle and vanishing first Betti number. The mysterious name K3 is explained by A. Weil in the comment on his Final report on contract AF18(603)-57 (see [We] p 546): Dans la seconde partie de mon rapport, il s’agit des variétés kähler- iénnes dites K3, ainsi nommées en l’honneur de Kummer, Kodaira, Kähler et de la belle montagne K2 au Cachemire. It is well known that all K3 surfaces are diffeomorphic, and that there is a 20-dimensional family of analytical isomorphism classes of K3 surfaces. How- ever, the general element in this family is not algebraic, in fact the algebraic ones form a countable union of 19-dimensional families. More precisely, for any n>0 there is a 19-dimensional irreducible family of K3 surfaces equipped with a base point free line bundle of self-intersection n. Moreover, the family of K3 surfaces having ≥ k linearly independent divisors (i.e. the surfaces with Picard number ≥ k, where the Picard number is by definition the rank of the Picard group) forms a dense countable union of subvarieties of dimension 20 − k in the family of all K3 surfaces. In particular, on the general algebraic K3 surface all divisors are linearly equivalent to some rational multiple of the hyperplane class (see [G-H, pp. 590–594]). Apair(S, L) of a K3 surface S and a base point free line bundle L with L2 =2g − 2 will be called a polarized K3 surface of genus g.Notethat g = h0(L) − 1 and that g is the arithmetic genus of any member of |L|.The g sections of L give a map ϕL of S to P , and the image is called a projective model of S.WhenϕL is birational, the image is a surface of degree 2g − 2 in Pg. It is also easy to see that a projective model of genus 2 is a 2:1map S → P2 branched along a sextic curve. A very central point in the theory of projective models of K3 surfaces is that by the adjunction formula every smooth hyperplane section of a projec- T. Johnsen and A.L. Knutsen: LNM 1842, pp. 1–14, 2004. c Springer-Verlag Berlin Heidelberg 2004 2 1 Introduction tive model of S (these are the images by ϕL of the smooth members of |L|) are canonical curves, i.e. curves for which ωC OC (1). The first examples of projective models of K3 surfaces are the ones which are complete intersections in projective space. Using the fact that for a com- − n plete intersection
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