# What finite groups are stabilizers in Kirwan's desingularization construction?

Assume $X$ is a smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$ and let $M$ be the (singular) moduli space of semistable rank two vector bundles with trivial determinant on $X$. Then there is a desingularization of $M$ due to a more general construction of Kirwan.

As far as I understand the idea, one uses that $M=Q//G$ is a GIT quotient of a subscheme of some Quot-scheme $Quot(\mathcal{O}_X^r)$ and blows up smooth subschemes in $Q$ with nontrivial stabilizers to make the stabilizers "smaller". Finally one ends up with a scheme $Q'$ such that $G$ only has finite stabilzers.

What finite groups remain in this proces in the case of the moduli space of semistable rank two vector bundles on $X$?

For $M$ it is known that the singular set is the Kummer variety $K$ associated to $X$. Points in $Q$ above the singular points of $K$ have stabilizers $PGL(2)$ and points above smooth points have stabilizer $\mathbb{C}^{*}$. After three blow ups the scheme $Q'''//G$ is a desingularization of $M$. So what my question is: are all the stabilizers of $G$ in $Q'''$ in known?

I think the non-trivial ones are $\mathbb{Z}/2$ and $\mathbb{Z}/2 \oplus \mathbb{Z}/2$. This is proved on page 24 of O'Grady's paper : http://arxiv.org/pdf/alg-geom/9708009.pdf

Vorsicht : O'Grady deals in that paper with the moduli space of semis-stable sheaves (with respect to a generic polarization) of rank $2$ with $c_1 = 0$ and $c_2 = 2n$ on a K3 surface. Nevertheless, the analysis of singularities in your case is similar (if not easier) to the one he does and the finite stabilizers that you are looking for are the same as the ones that he describes.