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

We consider surfaces X defined by plane divisorial valuations ν of the quotient field of the local ring R at a closed point p of the projective plane P2 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 ν is non-positive on OP2(P2∖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 ν is negative on OP2(P2∖L)∖k are also given.