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Geometric Reid’s recipe for dimer models

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A Correction to this article was published on 27 January 2021

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Abstract

Crepant resolutions of three-dimensional toric Gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models. By choosing a special stability parameter and hence a distinguished crepant resolution \(Y\), this derived equivalence generalises the Fourier-Mukai transform relating the \(G\)-Hilbert scheme and the skew group algebra \(\mathbb {C}[x,y,z]*G\) for a finite abelian subgroup of \(\mathrm{SL }(3,\mathbb {C})\). We show that this equivalence sends the vertex simples to pure sheaves, except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus. This generalises results of Cautis–Logvinenko (J Reine Angew Math 636:193–236, 2009) and Cautis–Craw–Logvinenko  (J Reine Angew Math arXiv:1205.3110, 2014) to the dimer setting, though our approach is different in each case. We also describe some of these pure sheaves explicitly and compute the support of the remainder, providing a dimer model analogue of results from Logvinenko (J Algebra 324:2064–2087, 2010).

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Acknowledgments

A. Craw thanks Akira Ishii and Kazushi Ueda for a preliminary draft of their preprint [20]. Thanks also to Alastair King for a useful remark, and to the anonymous referee for many helpful comments. A. Craw and A. Quintero Vélez were supported in part by EPSRC grant EP/G004048/1.

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Authors and Affiliations

  1. Korteweg de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, P.O. Box 94248, 1090 GE, Amsterdam, The Netherlands

    Raf Bocklandt

  2. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Alastair Craw

  3. Departamento de Matemáticas, Universidad del Valle, Ciudadela Universitaria Meléndez, Calle 13 No. 100-00, Edificio 320-3190, Cali, Colombia

    Alexander Quintero Vélez

Authors
  1. Raf Bocklandt
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  2. Alastair Craw
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  3. Alexander Quintero Vélez
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Correspondence to Alastair Craw.

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Bocklandt, R., Craw, A. & Quintero Vélez, A. Geometric Reid’s recipe for dimer models. Math. Ann. 361, 689–723 (2015). https://doi.org/10.1007/s00208-014-1085-8

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