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If $X \rightarrow Y$ is a finite morphism of surfaces and $D$ is a nef divisor on $Y$, is the pullback of $D$ nef on X? I am interested in finite covers of Hirzebruch surfaces and need to know if some divisors coming from below are nef.

Thanks for any help!

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Yes, and you don't even need to know that $f$ is finite. If $C$ is an integral curve on $X$, we have the projection formula $f^*D \cdot C = D \cdot f_*C$; and $f_*C$ is a non-negative multiple of an integral curve on $Y$.

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  • $\begingroup$ Nice! Thank you very much for the answer. $\endgroup$
    – user9176
    Commented Sep 12, 2010 at 6:26
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Perhaps it is worth adding here that if you do assume that f is finite, then the pullback of an ample divisor remains ample.

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