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

We consider surfaces $X$ defined by plane divisorial valuations $\nu$ of the quotient field of the local ring $R$ at a closed point $p$ of the projective plane $\mathbb{P}^2$ 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 $\nu$ is non positive on ${\mathcal O}_{\mathbb{P}^2}(\mathbb{P}^2\setminus L)$, 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 $\nu$ is negative on ${\mathcal O}_{\mathbb{P}^2}(\mathbb{P}^2\setminus L) \setminus k$ are also given.