Danilov's resolution and representations of the McKay quiver

Abstract

We construct a family of McKay quiver representations on the Danilov resolution of the $\frac{1}{r}(1,a,r-a)$ singularity. This allows us to show that the resolution is the normalization of the coherent component of the fine moduli space of $\theta$-stable McKay quiver representations for a suitable stability condition $\theta$. We describe explicitly the corresponding union of chambers of stability conditions for any coprime numbers $r,a$.