Curves, K3 Surfaces and Fano 3-folds of Genus ≤ 10

Abstract

A pair (S, L) of a K3 surface S and a pseudo-ample line bundle L on S with (L2) = 2g − 2 is called a (polarized) K3 surface of genus g. Over the complex number field, the moduli space Fg of those (S, L)s is irreducible by the Torelli type theorem for K3 surfaces. A general K3 surface of genus 6 ≤ g ≤ 10 is still a complete intersection in a certain homogeneous space The homogeneous space X is the quotient of a simply connected semisimple complex Lie group G by a maximal parabolic subgroup P. This chapter describes the application of this to the description of birational type of Fg for g ≤ 10 and the study of curves and Fano 3-folds.