On G/N-Hilb of N-Hilb

TL;DR

This paper explores the structure of iterated G-equivariant Hilbert schemes, showing that G/N-Hilb(N-Hilb(C^3)) can serve as a crepant resolution of C^3/G and examining conditions for its isomorphism with G-Hilb.

Contribution

It establishes a connection between iterated G/N-Hilb(N-Hilb) and crepant resolutions, providing explicit examples and analyzing when these spaces are isomorphic.

Findings

G/N-Hilb(N-Hilb(C^3)) is a crepant resolution of C^3/G.

Explicit examples illustrating the construction.

Most of the time, G/N-Hilb(N-Hilb) and G-Hilb are not isomorphic.

Abstract

In this paper we consider the iterated G-equivariant Hilbert scheme G/N-Hilb(N-Hilb) and prove that G/N-Hilb(N-Hilb(C^3)) is a crepant resolution of C^3/G isomorphic to the moduli space of \theta-stable representations of the McKay quiver 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.