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Let $X$ be a quasi-projective variety over an algebraicaly closed field and $G$ be a finite group acting on $X$ through automorphisms. Can you tell me how to prove that the quotient $X/G$ is also a quasi-projective variety. In particular: Why is this quotient a variety? Why is this variety quasi-projective?

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$X/G$ is quasi-projective: It is the Proj of $\bigoplus \Gamma(X,L^n)^G$ for some ample line bundle $L$ on $X$. – hilbert Aug 26 '11 at 14:58
Ah, I assumed $X$ projective. Maybe a GIT approach works for quasi-projective also. – hilbert Aug 26 '11 at 15:02
For the existence of the quotient you can see exercises 2.3.20, 2.3.21, 3.3.23 and proposition 3.3.36 of Qing Liu's book Algebraic Geometry and Arithmetic Curves. For quasi-projectiveness of the quotient you must read Mumford's Geometric Invariant Theory, I think. But I don't know if every finite quotient of a quasi-projective variety is a quasi-projective variety. – Andrea Aug 26 '11 at 15:15

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