Cox rings of K3 surfaces of Picard number three

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 $2(f+f^{\prime })$, where $f,f^{\prime }$ are classes of elliptic fibrations with $f\cdot f^{\prime }=2$. 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.