The cone of curves and the Cox ring of rational surfaces given by divisorial valuations

We consider surfaces X defined by plane divisorial valuations v of the quotient field of the local ring R at a closed point p of the projective plane P-2 over an arbitrary algebraically closed field k and centered at R. We prove that the regularity of the cone of curves of X is equivalent to the fact that v is non-positive on Op(2) (P-2 \ L), where L is a certain line containing p. Under these conditions, we characterize when the characteristic cone of X is closed and its Cox ring finitely generated. Equivalent conditions to the fact that v is negative on Opt (P-2 \ L) k are also given. (C) 2015 Published by Elsevier Inc.