Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

Abstract

We consider rational surfaces $Z$ defined by divisorial valuations $\nu$ of Hirzebruch surfaces. We introduce the concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when $\nu$ is non-positive at infinity, the extremal rays of the cone of curves of $Z$ can be explicitly given.