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Mathematische Annalen

, Volume 361, Issue 3–4, pp 689–723 | Cite as

Geometric Reid’s recipe for dimer models

  • Raf Bocklandt
  • Alastair CrawEmail author
  • Alexander Quintero Vélez
Article

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).

Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Raf Bocklandt
    • 1
  • Alastair Craw
    • 2
    Email author
  • Alexander Quintero Vélez
    • 3
  1. 1.Korteweg de Vries Instituut voor WiskundeUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK
  3. 3.Departamento de MatemáticasUniversidad del Valle, Ciudadela Universitaria MeléndezCaliColombia

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