Quotient varieties

If $V$ is a quasi-projective variety and $G$ is a finite group acting on $V$, then the quotient $V/G$ exists. The idea is the following: because $V$ is quasi-projective, any finite subset of $V$ lies in an affine open, hence every $G$-orbit lies in an affine open. Now a small argument shows that $V$ may be covered by $G$-invariant affine opens. For each $G$-invariant affine open $U$, we will construct the quotient $U/G$, and then glue them to form $V/G$.

If $U =$ Spec $A$, then the $G$-action on $U$ gives a $G$-action on $A$, and by definition $U/G =$ Spec $A^G$, where $A^G$ is the subring of $G$-invariant elements in $A$.

So to compute $V/G$, you have to (a) find a cover by $G$-invariant $U$; this shouldn't be too hard if your variety $V$, your group $G$, and your $G$-action on $V$ are explicit; (b) compute the various $G$-invariants $A^G$ --- this is probably the hardest part, although I imagine the right software can handle it in cases that aren't too complicated; (c) glue the various Spec $A^G$s together --- this is easy in principle, although it means that you end up with $V/G$ described in a somewhat abstract way; while $V/G$ will again be quasi-projective, you don't see this directly from this gluing procedure.

Incidentally, the dimension of $V/G$ will be the same as that of $V$ (because $G$ is finite). For things like singularities, the description of $V$ by gluing is perhaps not so bad, as you can check what is happening in one affine open at a time.