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 $\nu$ 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.

Three references added with respect to the first version

References in corpus (3)

• Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves
• How to determine the sign of a valuation on C[x,y]?
• Mori dream surfaces associated with curves with one place at infinity

Cited by in corpus (5)

• Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces
• Discrete equivalence of non-positive at infinity plane valuations
• Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces
• The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces
• On the computation of Darboux first integrals of a class of planar polynomial vector fields