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.

References in corpus (1)

• How to determine the sign of a valuation on C[x,y]?

Cited by in corpus (3)

• On the degree of curves with prescribed multiplicities and bounded negativity
• 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