K 3 Projective Models in Scrolls

Abstract

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 T E X output by the author SPIN: 10999523 41/3142/ du -543210 -Printed on acid-free paper PrefaceThe 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.

Half Nikulin Surfaces and Moduli of Prym Curves

“…Assume that Y is non-standard. By [23], all smooth curves in [25,27]. Conversely, for any decomposition H…”

“…Let X d be a half Nikulin surface as in § 4, keeping the notation therein. Let A be a fixed smooth anticanonical divisor on X d , recalling Remark 4.2 and in particular (25).…”

Brill–Noether general K3 surfaces with the maximal number of elliptic pencils of minimal degree

“…This implies (b) by either a direct computation using the definition of Brill-Noether generality or invoking, e.g., [17,Prop. 10.5] and [32], or [14,Lemma 1.7].…”

Some degenerations of $G_2$ and Calabi–Yau varieties

“…The Proposition 1.2 follows directly from Proposition 6.4 and the fact that in the more degenerate case we clearly get a higher Picard number due to the decomposition of C. Remark 6.6. The K3 surfaces obtained as sections of considered varieties fit to the case g = 10, c = 3, D 2 = 0, and scroll of type (2, 1, 1, 1, 1) from [10] (Observe that there is a misprint in the table, because H 0 (L − 2D) should be 1 in this case). The embedding in the scroll corresponds to the induced embedding in the projection of Ĝ2 from the distinguished plane.…”

“…By the Mukai linear section theorem (see [14]) we know that a generic polarized K3 surface of genus 10 appears as a complete linear section of G 2 . A classification of the nongeneral cases has been presented in [10]. The classification is however made using descriptions in scrolls, which is not completely precise in a few special cases.…”