We consider rational surfaces Z defined by divisorial valuations \(\nu \) 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 \(\nu \) is non-positive at infinity, the extremal rays of the cone of curves of Z can be explicitly given.

中文翻译：

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Hirzebruch曲面无穷大除数估值的正值和负值

我们考虑由Hirzebruch曲面的除数估值\（\ nu \）定义的有理曲面Z。我们为这些估值引入了无穷大的非正性和负性概念，并证明这些概念接受了良好的局部和全局等效条件。特别地，我们证明了，当\（\ nu \）在无穷大处为非正值时，可以明确给出Z曲线圆锥的极射线。