On G/N-Hilb of N-Hilb

Abstract

In this paper we consider the iterated $G$-equivariant Hilbert scheme $G/N$-Hilb($N$-Hilb) and prove that $G/N\text{-}\mathrm{Hilb}\left(N\text{-}\mathrm{Hilb}\right({\mathbb{C}}^3\left)\right)$ is a crepant resolution of ${\mathbb{C}}^3/G$ isomorphic to the moduli space ${\mathcal{M}}_{\theta }\left(Q\right)$ of $\theta$-stable representations of the McKay quiver $Q$ for certain stability condition $\theta$. We provide several explicit examples to illustrate this construction. We also consider the problem of when $G/N$-Hilb($N$-Hilb) is isomorphic to $G$-Hilb showing the fact that these spaces are most of the times different.