Polarized K3 surfaces of genus 18 and 20

Summary

A surface, i.e., 2-dimensional compact complex manifold, S is of type K3 if its canonical line bundle O_s(K_s) 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 F_g 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 X ≃ Z 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 X ≃ Z. 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).