Moduli space of quasi-polarized K3 surfaces of degree 6 and 8

TL;DR

This paper investigates the moduli spaces of quasi-polarized K3 surfaces of degrees 6 and 8, using geometric invariant theory to describe their structure, stability, and compactifications, with connections to Noether-Lefschetz theory.

Contribution

It provides a GIT construction of the moduli spaces, describes their boundary and stability conditions, and computes the Picard group using Noether-Lefschetz theory.

Findings

K3 surfaces with ADE singularities are GIT stable.

Boundary of the degree 6 moduli space is explicitly described.

Picard group computed via Noether-Lefschetz theory.

Abstract

In this paper, we 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 we have natural GIT constructions for the corresponding moduli spaces and we show that the K3 surfaces with at worst ADE singularities are GIT stable. We give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion. We 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.