7
$\begingroup$

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".

  • 1
    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 $Y$.2011-09-10

1 Answers 1