If $p : \mathbb{C} \rightarrow \mathbb{C}$ is a polynomial, $B \subset \mathbb{C}$ is the set of its regular values and $E = p^{-1}(B)$, then $p_{|E} : E \rightarrow B$ is a covering map of finite degree $\deg(p)$. I would like to know about the generalisations of this fact.
I know that if $S$ and $S'$ are compact Riemann surfaces, then any non constant holomorphic map $f : S \rightarrow S'$ is a ramified covering space thus we obtain a covering map after removing a finite number of points in $S$ and $S'$.
And if $X$ and $Y$ are complex algebraic varieties of the same dimension (we maybe need more hypothesis like separated or irreducible, I am not sure this is why I am asking this question) and $f : X \rightarrow Y$ is a dominant morphism, then there is a dense Zariski open subset $V$ of $Y$ such that $f_{|f^{-1}(V)} : f^{-1}(V) \rightarrow V$ is a covering map for the analytic topology.
Do you have any reference for that please ? I can't find anything.