In the book I'm reading ( Geometry of Algebraic Curves ), at some point (page $310$) they make the following claim:
One can use Nakai's criterion to establish the general fact that if $f:X\to Y$ is any finite surjective map and $L$ is a line bundle over $Y$, then $f^*L$ is ample if and only if $L$ is ample.
Do you know about any reference where I can find a proof for this statement?
Or could you give the sketch of a proof here?
Reference: Lazarsfeld, Positivity in Algebraic Geometry Volume 1, Proposition 1.2.13 and Corollary 1.2.28. I'll reproduce the arguments here for completeness.
To see that the finite pullback of an ample line bunde $L$ is still ample, use Serre's criterion: the projection formula shows that for any coherent sheaf $F$ on $Y$, the higher cohomology groups $H^i(Y, F \otimes f^*L^m)$ vanish for sufficiently large $m$, so Serre says that $f^*L$ is ample.
To show that if $f^*L$ is ample then $L$ must be, take an irreducible subvariety $V \subset Y$. We want to show that $L^{\operatorname{dim} \ V} \cdot V >0$. Take an irreducbile subvariety $W \subset X$ mapping finitely to $Y$: then it's not too hard to see that
$$ (f^*L)^{\operatorname{dim} \ V} \cdot W = \operatorname{deg}(W \rightarrow V) \left( L^{\operatorname{dim} \ V} \cdot V \right). $$
Now Nakai tells us that the left-hand side is strictly positive, therefore so too is the right, as required.