Let $f \colon X \to Y$ be a generically injective, finite morphism between projective varieties over $\mathbb{C}$, with $Y$ non-singular. Does $f$ have a section i.e., a morphism $g:Y \to X$ such that $f \circ g:Y \to Y$ is the identity?

If not true in general, is there any known condition under which $f$ has a section?

**EDIT.** The map $f$ is assumed to be dominant.

noin such a generality. Take the blow-up $\pi \colon X \to \mathbb{P}^2$ of the plane at at a point: of course it has no regular sections. $\endgroup$ – Francesco Polizzi Apr 2 '17 at 14:25