VOL. 88 | 2023 Wall-crossing for iterated Hilbert schemes (or ‘Hilb of Hilb’)
Ben Wormleighton

Editor(s) Yukari Ito, Akira Ishii, Osamu Iyama

Adv. Stud. Pure Math., 2023: 195-208 (2023) DOI: 10.2969/aspm/08810195


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.


Published: 1 January 2023
First available in Project Euclid: 8 May 2023

Digital Object Identifier: 10.2969/aspm/08810195

Primary: 14E16
Secondary: 14J17 , 14M25

Keywords: iterated Hilbert schemes , quiver representations , wall-crossing

Rights: Copyright © 2023 Mathematical Society of Japan


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