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This article is available for purchase or by subscription. See below.
Abstract
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In this paper, we study and describe the universal Poisson deformation space of
hypertoric varieties concretely. In the first application, we show that affine
hypertoric varieties as conical symplectic varieties are classified by the associated
regular matroids (this is a partial generalization of the result by Arbo and
Proudfoot). As a corollary, we obtain a criterion when two quiver varieties whose
dimension vectors have all coordinates equal to one are isomorphic to each
other. Then we describe all 4- and 6-dimensional affine hypertoric varieties as
quiver varieties and give some examples of 8-dimensional hypertoric varieties
which cannot be raised as such quiver varieties. In the second application, we
compute explicitly the number of all projective crepant resolutions of some
4-dimensional hypertoric varieties by using the combinatorics of hyperplane
arrangements.
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Keywords
hypertoric variety, universal Poisson deformation,
classification, counting crepant resolutions, hyperplane
arrangement
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Mathematical Subject Classification 2010
Primary: 14B07, 14E15, 14M25, 52B40, 52C35
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Milestones
Received: 14 April 2019
Revised: 21 February 2021
Accepted: 22 May 2021
Published: 12 October 2021
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