Let $f:X\to Y$ be a finite surjective morphism of normal varieties. We say that $f$ is a Galois covering if the extension $k(X)/k(Y)$ is Galois.
I have the following questions:
1.- Why we need to assume normality of the varieties for the definition of a Galois covering?
2.- Is it true that a finite surjective morphism between normal varieties is always flat?
3.- Is it true that $f$ is a Galois covering if and only if the group of automorphism $Aut(X/Y)$ acts transitively on all fibers of $f$?
Best Answer
The simplest counterexample to question 2 is the map $$ X := \mathbb{A}^2 \to \mathbb{A}^2/\{ \pm 1 \} =: Y. $$ Here $Y$ is a quadratic cone in $\mathbb{A}^3$. Both $X$ and $Y$ are normal, and the map is a Galois covering, but $X$ is not flat over $Y$.