  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.
