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All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

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Title: All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
Authors: Bellamy, G
Craw, A
Rayan, S
Schedler, T
Weiss, H
Item Type: Journal Article
Abstract: We demonstrate that the linear quotient singularity for the exceptional subgroup G in Sp(4, C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of C4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2n − 6; for example, we show that there are 1684 projective crepant resolutions when n = 6. We also prove that the resulting affine cones are not quotient singularities for n ≥ 6.
Issue Date: 1-Apr-2024
Date of Acceptance: 6-Nov-2023
URI: http://hdl.handle.net/10044/1/108094
DOI: 10.1090/jag/827
ISSN: 1056-3911
Publisher: American Mathematical Society
Start Page: 757
End Page: 793
Journal / Book Title: Journal of Algebraic Geometry
Volume: 33
Copyright Statement: © Copyright 2024 University Press, Inc. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (https://creativecommons.org/licenses/by-nc-nd/4.0/).
Publication Status: Published
Online Publication Date: 2024-04-01
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences
Mathematics



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