In 1959, Nagata produced a counterexample to Hilbert’s 14th problem via a linear action of a non-reductive group. The resulting non-finitely generated invariant algebra can be interpreted as the Cox ring of a blow-up of P^2 at points, and in this framework he formulated his conjecture on plane curves. More recently, in joint work with Castravet, Tevelev and Ugaglia we show that if the blow-up of certain toric surfaces at a general point has a non-polyhedral pseudoeffective cone, then the pseudoeffective cone of M̄_{0,n} is not polyhedral for n ≥ 10. These results motivate a systematic study of Cox rings of blow-ups of toric varieties at a general point.

In this talk I focus on minimal toric surfaces, namely those without curves of negative self-intersection. Such surfaces are cyclic quotients of either P^2 or P^1 x P^1. For quotients of P^2, the blow-up at a general point may fail to be a Mori Dream Space, for instance when the semiample cone is not closed, or when the effective cone is not closed. In contrast, for quotients of P^1 x P^1, we prove that the blow-up at a general point is always a Mori Dream Space.

This is joint work with Luca Ugaglia.