Let $X$ be a $n$-dimensional smooth projective variety over $\mathbb C$, by Noetherian normalization there will always be a finite surjective morphism from $X$ to $\mathbb P^n$, by miracle flatness this is a flat morphism. How do we classify those finite maps from $X$ to $\mathbb P^n$ which are surjective (equivalently flat)? In other words, what ample line bundles on $X$ will give finite flat maps from $X$ to $\mathbb P^n$ ?
I am mostly interested in the case $X$ is a complex surface, and wonder whether different maps will induce different maps on the rational singular cohomology groups.
If $X$ is a smooth projective surface and $L$ an ample line bundle, there exists a morphism $f_L:X\to\mathbb{P}^2$ such that $f_L^*(\mathcal{O}_{\mathbb{P}^2}(1))$ is a positive multiple of $L$. Since $H^2(X)$ has a basis (over $\mathbb{Q}$) of ample line bundles, we see that we have different maps to the plane so that the pull back of the generator of $H^2$ of the plane (it is one dimensional) maps to different elements of $H^2(X)$ and so $NS(X)$ has dimension greater than one, you get different maps on cohomologies.