Let $G$ be a finite group acting on a projective variety $X$. Then $G$ also acts on $X-X^G$, where $X^G$ is the fixed locus. The GIT quotient varieties $X/G$ and $(X-X^G)/G$ are projective and quasi-projective varieties respectively with respect to a suitable linearization. How are these two varieties related? Is it possible to construct $(X-X^G)/G$ from $X/G$? I guess $X/G$ can be constructed from $(X-X^G)/G$ by blowing up over some points but I am not sure.
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