Cox rings of nef anticanonical rational surfaces

Abstract

This paper deals with the problem of computing a generating set for the Cox ring $R(X)$ of a smooth projective rational surface $X$ with nef anticanonical class. In case $R(X)$ is finitely generated, we show that the degrees of its generators are either classes of negative curves, elements of the Hilbert basis of the nef cone or certain ample classes of anticanonical degree one, which only appear when $X$ is a rational elliptic surface of Halphen index $m>2$. Moreover, we partially characterize which elements of the Hilbert basis of the nef cone are irredundant for generating $R(X)$. We apply this result to compute explicit minimal generating sets for Cox rings of some rational elliptic surfaces of Halphen index $>1$.