Quotient varieties I have a variety $V$ given by polynomial equations. These equations
admit a lot of symmetry. This means there are a lot of automorphisms
on $V$. I want to get rid of this symmetry. So I somewhat want to form a quotient variety.
However, googling for quotient variety does not really give me what I
am looking for. I found articles about geometric invariant theory and read that
quotient varieties need not exist in general and I am somewhat
wondering if it is even possible in my case.
To give a hint how my problem looks consider the projective curve(hyperbola)
$C:XY+Z$. There is an isomorphism $\phi:(x,y,z) \mapsto (y,x,z)$. Is it
possible to sort of mod out this isomorphism. I am thinking of this as
having a variety $V$  such that there is a surjective map $C \to V$ where two
points in $C$ have the same image if they they are mapped to each other
by $\phi$.
Is this possible? how can I compute equations for $V$. What is the general theory?
Furhtermore if this is not possible, can I find such a $V$ having this property
but maybe not being a variety but some other object(say maybe it is possble to
form quotients like these if we allow general schemes)?
EDIT: Ok, so I looked up example 11 and indeed it is what I want. But it only covers about half a page. The defintion is via the associated function fields. What about the computational aspect. Can someone point me to a reference where it is explained how to actually compute properties of the quotient variety. For example, dimension, defining equations etc..
 A: 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.
