Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

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1 Lecture Notes in Mathematics 1842 Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

2 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

3 Trygve Johnsen Andreas Leopold Knutsen K3 Projective Models in Scrolls 13

4 Authors Trygve Johnsen Department of Mathematics University of Bergen Johs. Bruns gt Bergen, Norway Andreas Leopold Knutsen Department of Mathematics University of Oslo Box 1053, Blindern 0316 Oslo, Norway LibraryofCongressControlNumber: Mathematics Subject Classification (2000): 14J28, 14H51 ISSN ISBN Springer-VerlagBerlinHeidelbergNewYork 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 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 TEX outputbytheauthor SPIN: /3142/ du Printed on acid-free paper

5 Preface The cover picture shows a smooth quartic surface in space, the simplest example 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 represent 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 P g 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 Department of Mathematics, University of Utah, and at the Max-Planck-Institut

6 VI Preface für Mathematik, Bonn, in the year 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

7 Contents 1 Introduction Background Relatedliterature Howthebookisorganised Notationandconventions Surfaces in Scrolls Rationalnormalscrolls Specializing to K3 surfaces The Clifford index of smooth curves in L and the definition of the scrolls T (c, D, {D λ }) GonalityandCliffordindexofcurves TheresultofGreenandLazarsfeld Clifforddivisors Gettingascroll Two existence theorems The singular locus of the surface S and the scroll T The singular locus of ϕ L (S) The singular locus of T andperfectclifforddivisors Postponed proofs Projective models in smooth scrolls Projective models in singular scrolls Blowing up S The smooth scroll T Techniques for finding Betti-numbers of the ϕ L (D λ ) Resolutionsforprojectivemodels Pushing downresolutions Rolling factors coordinates Someexamples... 92

8 VIII Contents 9 More on projective models in smooth scrolls of K3 surfaces of low Clifford-indices Projective models with c = Projective models with c = An interpretation of b 1 and b Possible scroll types for c = Projective models with c = Higher values of c BN general and Clifford general K3 surfaces TheresultsofMukai Notionsofgenerality The case g = The case g = Projective models of K3 surfaces of low genus A new decomposition of R Perfect Clifford divisors for low c Thepossiblescrolltypes Someconcreteexamples Thelistofprojectivemodelsoflowgenus Some applications and open questions BN generality ApplicationstoCalabi-Yauthreefolds Analogieswith Enriquessurfaces References Index...163