Abstract In this paper we study smooth projective rational surfaces, defined over an algebraically closed field of any characteristic, with pseudo-effective anticanonical divisor. We provide a necessary and sufficient condition in order for any nef divisor to be semiample. We adopt our criterion to investigate Mori dream surfaces in the complex case.

Introduction

Let X be a smooth projective rational surface defined over an algebraically closed field K of any characteristic. A problem that has recently attracted attention consists in finding equivalent characterizations of Mori dream surfaces, that is surfaces whose Cox ring is finitely generated (see [ADHL] for basic definitions), for instance, in terms of the Iitaka dimension of the anticanonical divisor. Indeed, if the Iitaka dimension of the anticanonical divisor is 2, then X is always a Mori dream surface, as shown in [TVAV]. If the Iitaka dimension of the anticanonical divisor is 1, then X admits an elliptic fibration π : X → P1 and it is a Mori dream surface if and only if the relatively minimal elliptic fibration of π has a Jacobian fibration with finite Mordell-Weil group, as shown in [AL11]. The authors are not aware of any previously known example of a Mori dream surface with Iitaka dimension of the anticanonical divisor equal to 0.