Let $D$ be a big and nef divisor on a smooth complex projective minimal surface and let $\phi_D$ be the induced rational map. Is it true that $\phi_D$ is generically finite? Otherwise does someone know a counterexample?
Thank you
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4$\begingroup$ Even for surfaces of general type the canonical divisor need not have any non-zero section... $\endgroup$– M PCommented Sep 9, 2011 at 15:06
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No, you can only conclude that for some multiple $mD$, with $m$ large enough, the map is generically finite. In fact, "big" is actually equivalent to this condition.
For a counterexample to your question, take a minimal surface $S$ of general type with $p_g(S)=2$ (there are lots of them). Then $K$ is big and nef, but $p_g(S)=h^0(K)=2$ implies that $|K|$ is a pencil, so the map $$\phi_K \colon S \to \mathbb{P}^1$$ has $1$-dimensional fibres.
answered Sep 9, 2011 at 15:09