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Let $V/G$ be the orbit space of a finite group $G$ of automorphisms of a complex projective variety $V$. Is $V/G$ a projective variety?

Example: $V/G$ is the space of sets in complex projective $n$-space $P$, of cardinality $\le k$. Here $V=P\times\dots\times P$ ($k$ factors) and $G$ is the permutation group on $k$ letters.

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Could you clarify what you mean by $V/G$? (I know its the quotient variety, but what conditions are you requiring it to satisfy?) –  jlk Jul 23 '11 at 3:44
    
The natural map $V\to V/G$ should be a morphism of projective varieties. –  Moe Hirsch Jul 24 '11 at 17:31

2 Answers 2

Let $A^\bullet$ be the homogeneous coordinate ring of $V$. Then $G$ acts on $A^\bullet$. Let $B^\bullet = (A^\bullet)^G$ be the ring of $G$ invariants. The projective spectrum $Proj(B^\bullet)$ is the required quotient. So, the answer is yes.

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This is true. There is a very thorough discussion of this result in Mumford's book on abelian varieties.

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