10
$\begingroup$

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..

  • 0
    I do not really $k$now; but I have a feeling that you will find the answer in the boo$k$ "Ideals, Varieties and Algorithms"..2011-09-22

1 Answers 1

15

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.

  • 0
    Dear Matt, thanks!2013-04-02