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

Abstract

We consider surfaces X defined by plane divisorial valuations ν of the quo- tient 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 ν is non-positive on OP2 (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 ν is negative on OP2 (P

2

\ L) \ k are also given.