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Disclaimer: I have asked this question on math exchange a week ago (here), but sadly to no avail. So I decided to escalate my question:

Let $G$ be a finite group acting faithfully on a smooth quasi-projective $k$-scheme $X$ via automorphisms, where $k$ is an algebraically closed field of any characteristic. Apparently, in characteristic $0$ it is then true that there is a dense, open and $G$-invariant subscheme $U \subseteq X$ on which $G$ acts freely, though I could not find a proof of this.

Now, I would like to know if this fact is also true in positive characteristic? Details or maybe a reference as to why this is true/false would be highly appreciated!

(I have been told that it fails for finite group schemes over $k$, because then some infinitesimal phenomena appear, but maybe it is still true for finite groups.)

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    $\begingroup$ This has nothing to do with characteristic. For each $g\neq 1$ in $G$, the fixed locus $X^{g}$ is a closed subvariety of $X$, with $X^g\neq X$ since $G$ acts faithfully. Then $G$ acts freely on $X\smallsetminus \bigcup X^g$. $\endgroup$
    – abx
    Nov 27 at 9:42
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    $\begingroup$ Oh, well now I feel stupid. Thank you very much for the clarification! $\endgroup$
    – OrdinaryAnon
    Nov 27 at 10:12

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