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



Rational map associated to a big and nef divisor
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.


