Mori dream K3 surfaces of Picard number four: projective models and Cox rings

TL;DR

This paper explores the geometry of 14 specific K3 surface families with Picard number four, providing projective models, Cox ring generators, and establishing unirationality in some cases.

Contribution

It offers explicit projective models and Cox ring generators for these K3 surfaces, and proves unirationality of their moduli spaces in certain instances.

Findings

Explicit projective models for 14 K3 surface families.

Identification of Cox ring generators for these families.

Proof of unirationality for some moduli spaces.

Abstract

In this paper we study the geometry of the $14$ families of K3 surfaces of Picard number four with finite automorphism group, whose N\'eron-Severi lattices have been classified by \`E.B. Vinberg. We provide projective models, we identify the degrees of a generating set of the Cox ring and in some cases we prove the unirationality of the associated moduli space.