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.

  • $\begingroup$ Could you clarify what you mean by $V/G$? (I know its the quotient variety, but what conditions are you requiring it to satisfy?) $\endgroup$
    – jlk
    Jul 23, 2011 at 3:44
  • $\begingroup$ The natural map $V\to V/G$ should be a morphism of projective varieties. $\endgroup$
    – Moe Hirsch
    Jul 24, 2011 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.


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


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