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Polarized K3 surfaces of genus 18 and 20

Published online by Cambridge University Press:  06 July 2010

G. Ellingsrud
Affiliation:
Universitetet i Bergen, Norway
C. Peskine
Affiliation:
Université de Paris VI (Pierre et Marie Curie)
G. Sacchiero
Affiliation:
Università degli Studi di Trieste
S. A. Stromme
Affiliation:
Universitetet i Bergen, Norway
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Summary

A surface, i.e., 2-dimensional compact complex manifold, S is of type K3 if its canonical line bundle Os(Ks) is trivial and if. An ample line bundle L on a K3 surface S is a polarization of genus g if its self intersection number (L2) is equal to 2g - 2, and called primitive if implies k = ±1. The moduli space Fg of primitively polarized K3 surfaces (S, L) of genus g is a quasi-projective variety of dimension 19 for every g ≥ 2 ([15]). In [12], we have studied the generic primitively polarized K3 surfaces (S, L) of genus 6 ≤ g ≤ 10. In each case, the K3 surface S is a complete intersection of divisors in a homogeneous space X and the polarization L is the restriction of the ample generator of the Picard group Pic XZ of X.

In this article, we shall study the generic (polarized) K3 surfaces (S, L) of genus 18 and 20. (Polarization of genus 18 and 20 are always primitive.) The K3 surface S has a canonical embedding into a homogeneous space X such that L is the restriction of the ample generator of Pic XZ. S is not a complete intersection of divisors any more but a complete intersection in X with respect to a homogeneous vector bundle V (Definition 1.1): S is the zero locus of a global section s of V. Moreover, the global section s is uniquely determined by the isomorphism class of (S, L) up to the automorphisms of the pair (X, V). As a corollary, we obtain a description of birational types of F18 and F20 as orbit spaces (Theorem 0.3 and Corollary 5.10).

Type
Chapter
Information
Complex Projective Geometry
Selected Papers
, pp. 264 - 276
Publisher: Cambridge University Press
Print publication year: 1992

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