On Cox rings of K3-surfaces

TL;DR

This paper investigates the structure of Cox rings of K3-surfaces, establishing conditions for finite generation and explicitly computing Cox rings for certain classes of K3-surfaces with specific geometric properties.

Contribution

It proves that a K3-surface's Cox ring is finitely generated if and only if its effective cone is polyhedral, and computes Cox rings for K3-surfaces with Picard number 2 to 5 with specific involutions.

Findings

Cox ring is finitely generated iff the effective cone is polyhedral.

Explicit Cox rings are computed for certain K3-surfaces with involutions.

Degrees of generators and relations are analyzed for Picard number two.

Abstract

We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3-surfaces of Picard number two, and explicitly compute the Cox rings of generic K3-surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.