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

Abstract

We consider rational surfaces Z defined by divisorial valuations ν of Hirzebruch surfaces. We introduce 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 ν is non-positive at infinity, the extremal rays of the cone of curves of Z can be explicitly given.