# Singularities of a quotient of a smooth projective variety by a finite group

There is a nice smooth projective variety of high dimension, and a finite group G acting on it. Assume X/G exists. What can one say about the singularity of X/G? e.g. Are they always isolated? (I was wondering there may be some elementary facts from smooth manifolds that explain it.) Sometimes even when G has fixed points in X, the quotient X/G is smooth, just consider a double cover of $P^1(\mathbb{C})$ by itself. Are there some criterion that guarantee this happens?

I was wondering, is it always possible to find an ample divisor D on X which is invariant by G, and which is neither ramified nor in the inverse image of a singular point on X/G (does the former implies the latter?) Now can I look at the affine subset U=X-D, U is now invariant by G. Now I take the quotient U/G and take its projective closure $\overline{U/G}$, is it the isomorphic as X/G ? If so, then we can probably replace "projective" in the above question by "affine".

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The singularities are not always isolated: e.g. suppose that $G$ acts on $X$, and now choose any positive-dimensional smooth $Y$ on which $G$ acts trivially. Then $(X \times Y)/G = (X/G)\times Y$, and the singular locus is the product of the singular locus of $X/G$ and $Y$. – Matt E Sep 10 '11 at 12:49

First some good news: if a finite group $G$ acts on a projective variety $X$, nice or not, then the quotient always exists, so that you do not need to assume existence.

Another pleasant result: if $X$ is normal, so will be $X/G$. This explains your observation that a quotient of a smooth curve is smooth.

Even in higher dimensions, it can happen that the quotient $X/G$ of the smooth variety $X$ is smooth even though the action of $G$ has fixed points.
An interesting example is the action of the symmetric group $S_n$ on $X=(\mathbb P^1_k)^n$. The quotient variety is just $\mathbb P^n_k$. This is a projective geometric vision of the fundamental theorem on symmetric polynomials.

However you can also have situations where the quotient acquires singularities.
The simplest example is to divide out the (affine) variety $\mathbb A^2_k$ by the action of the two-element group $G=\lbrace I,s\rbrace$, where $s$ acts as $s\ast (x,y)=(-x,-y)$. The morphism $\mathbb A^2_k \to \mathbb A^3_k$ given by $u=x^2, v=xy, w=y^2$ descends to an embedding $X/G \stackrel {\sim} {\to} V \subset \mathbb A^3_k$, where the cone $V$ is given by the equation $v^2=uw$.
(Incidentally, this is a nice way to prove that $V$ is normal, by invoking the "pleasant result" above!)

The nature of the singularities (or absence thereof) is a whole industry in algebraic geometry: the key-worde are "rational singularities", "Du Val singularities",...

Two nice introductions to quotient varieties : Shafarevich's book Basic Algebraic Geometry and Mumford's Abelian Varieties.
The definitive reference is supposed to be Mumford-Fogarty's Geometric Invariant Theory, but I haven't read it.

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