Moduli Space of Quasi-Polarized K3 Surfaces of Degree 6 and 8

Abstract

In this paper, the authors study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and they have natural GIT constructions for the corresponding moduli spaces and they show that the K3 surfaces with at worst ADE singularities are GIT stable. They give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion. They compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga’s compactifications from arithmetic perspective. One of the main ingredients is the study of the projective models of K3 surfaces in terms of Noether-Lefschetz divisors.