K3 Projective Models in Scrolls
The exposition studies projective models of K3 surfaces whose hyperplane sections are non-Clifford general curves. These models are contained in rational normal scrolls. The exposition supplements standard descriptions of models of general K3 surfaces in
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Trygve Johnsen Andreas Leopold Knutsen
K3 Projective Models in Scrolls
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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]
Library of Congress Control Number: 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, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 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 specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10999523
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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 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. C
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