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

Bellamy, G. , Craw, A., Rayan, S., Schedler, T. and Weiss, H. (2024) All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces. Journal of Algebraic Geometry, (doi: 10.1090/jag/827) (Early Online Publication)

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.

• Item Type: Articles
• Additional Information: The first and second authors were partially supported by Research Project Grant RPG-2021-149 from the Leverhulme Trust. The third author was supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant. The fifth author was supported by the Deutsche Forschungsgemeinschaft (DFG) within SPP 2026 “Geometry at infinity”.
• Status: Early Online Publication
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Craw, Dr Alastair and Bellamy, Professor Gwyn
• Authors: Bellamy, G., Craw, A., Rayan, S., Schedler, T., and Weiss, H.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Algebraic Geometry
• Publisher: American Mathematical Society
• ISSN: 1056-3911
• ISSN (Online): 1534-7486
• Published Online: 01 April 2024
• Copyright Holders: Copyright © 2024 University Press, Inc.
• First Published: First published in Journal of Algebraic Geometry 2024
• Publisher Policy: Reproduced in accordance with the publisher copyright policy