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.



Does a generically injective morphism of schemes have a section?
If $Y$ is normal, then every finite generically injective dominant morphism $f:X \to Y$ is an isomorphism. Indeed, take an open affine subset $U = Spec~ A$ of $Y$. Then $f^{-1}(Y) = Spec~B$ where $B$ is integral over $A$. Since $A$ is integrally closed and $A,B$ have the same field of fractions, $A = B$. 


So the answer to your question is: ``Yes, always!''


