Cox rings of K3 surfaces of Picard number three

TL;DR

This paper characterizes the generators of Cox rings for K3 surfaces with Picard number three, identifying specific classes and structures that generate the ring, and provides minimal generating sets and presentations in special cases.

Contribution

It introduces a detailed description of Cox ring generators for Picard number three K3 surfaces, including new classes and methods for minimal generating sets.

Findings

Cox ring generators are classes of smooth rational curves, sums of up to three Hilbert basis elements, or twice sums of elliptic fibration classes.

A minimal generating set for all Mori dream K3 surfaces of Picard number three is determined.

Explicit Cox ring presentations are provided for certain cases with few generators.

Abstract

Let $X$ be a projective K3 surface over $\mathbb C$. We prove that its Cox ring $R(X)$ has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form $2(f+f')$, where $f,f'$ are classes of elliptic fibrations with $f\cdot f'=2$. This result and techniques using Koszul's type exact sequences allow to determine a generating set for the Cox ring of all Mori dream K3 surfaces of Picard number three which is minimal in most cases. A presentation for the Cox ring is given in some special cases with few generators.