On $G$-Hilb $C^n$

Let $G$ be a finite subgroup of $SL_n(C)$. The scheme $G$-Hilb $C^n$ is the fine moduli space of $G$-clusters, the scheme-theoretic orbits of $G$ in $C^n$. It can also be thought of as a union of some of the connected components of the $G$-invariant part of the Hilbert scheme of $|G|$-tuple of points on $C^n$. When $n=2$ or $3$ it is a crepant resolution of the quotient singularity $C^n/G$. In this talk I give an overview of some of the cases where we have a fairly explicit understanding of $G$-Hilb $C^n$ and its geometry.