Let $X$ be a smooth, quasi-projective variety, $G$ be a finite group which acts freely and properly on $X$. Denote by $\alpha:X \to X/G$ the quotient. Is $\alpha$ generically etale? 


Also, as I am new to this topic (of group action on varieties) can someone suggest a good reference for the topic? I am mainly interested in finite group actions.



Finite group action on quasi-projective varieties
Let $X$ be a normal variety over an algebraically closed field of characteristic 0 with a finite group $G$ acting effectively. Since $G$ is finite it is reductive and a geometric quotient $X/G$ exists. The quotient map $X\to X/G$ is étale at a point $x\in X$ if and only if the stabilizer at $x$ is trivial. Since the equations $gx-x$ are algebraic and define the locus where the action fails to be free, the complement is Zariski open and shows that the quotient map is always generically étale.


See Section 4.3 [here](https://webusers.imj-prg.fr/~jean-marc.drezet/papers/Wykno.pdf).


