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I have a question about the statement of Remark 1.4 following on Thm 1.3 from Burt Totaro's paper "The Chow Ring of a Classifying Space" (p. 4):

Let $G$ be a reductive group over a field $k$.

Remark 1.4. If $G$ is finite, then the geometric quotient $V/G$ exists as an affine variety for all representations $V$ of $G$ [28: Mumford's GIT]. So $(V -S)/G$ is a quasi-projective variety for all closed subsets $S \subset V$ such that $G$ acts freely on $V - S$.

Question: Does anybody see to which explicit result from GIT Totaro refers to in his remark that for every closed subsets $S \subset V$ such that $G$ acts freely on $V - S$, the quotient $(V -S)/G$ exist as a quasi-projective variety? Clearly that's the same as to consider all open subsets $U \in V$ (which can be always canonically endowed with structure of an variety via open immersion; see https://stacks.math.columbia.edu/tag/01IM) with free $G$-action.

Nevertheless in GIT book I haven't found a result with these requirements implying quasi-projectivity of $U/G$. Does anybody know which GIT result Totaro has here in mind?

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    $\begingroup$ $(V-S)/G$ is an open subvariety of the affine variety $V/G$. $\endgroup$
    – abx
    Commented Nov 1, 2021 at 5:15
  • $\begingroup$ @abx: Is the premise that the $G$-action on $V-S$ is free crucial for this statement, or can it be weakened here? It seems to be quite strong, and in GIT I haven't found a result which uses this free action premise in essential way to garantee that $(V-S)/G$ carry open variety in $V/G$. The question remains, to which concrete GIT result exactly is Totaro referring to? $\endgroup$
    – user267839
    Commented Nov 1, 2021 at 23:21
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    $\begingroup$ You just need $S$ to be stable under $G$. The fact that the action on $V-S$ is free is crucial for the rest of the paper, so I think Totaro just kept it in the Remark. The "concrete GIT result" is the fact that $V/G$ is a geometric quotient and an affine variety. $\endgroup$
    – abx
    Commented Nov 2, 2021 at 4:44

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