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

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.

Written for proceedings of the conference 'The McKay Correspondence, Mutations, and Related Topics' hosted by IPMU in July 2020; 9 pages, 5 figures; comments welcome!

References in corpus (1)

• Combinatorial Reid's recipe for consistent dimer models