Wall-crossing for iterated Hilbert schemes (or 'Hilb of Hilb')

TL;DR

This paper explores wall-crossing phenomena in the McKay correspondence, focusing on how different crepant resolutions relate through stability conditions and conjectures, with some proven cases and connections to iterated Hilbert schemes.

Contribution

It introduces a conjecture about the placement of chambers for iterated Hilbert schemes within the McKay correspondence and proves it in specific cases.

Findings

Conjecture formulated on chamber placement for Hilb of Hilb resolutions.

Proof of the conjecture in certain examples and special cases.

Discussion of connections to other aspects of the McKay correspondence.

Abstract

We study wall-crossing phenomena in the McKay correspondence. Craw-Ishii show that every projective crepant resolution of a Gorenstein abelian quotient singularity arises as a moduli space of $\theta$-stable representations of the McKay quiver. The stability condition $\theta$ moves in a vector space with a chamber decomposition in which (some) wall-crossings capture flops between different crepant resolutions. We investigate where chambers for certain resolutions with Hilbert scheme-like moduli interpretations - iterated Hilbert schemes, or 'Hilb of Hilb' - sit relative to the principal chamber defining the usual $G$-Hilbert scheme. We survey relevant aspects of wall-crossing, pose our main conjecture, prove it for some examples and special cases, and discuss connections to other parts of the McKay correspondence.