Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 083, 45 pages      arXiv:2309.16112      https://doi.org/10.3842/SIGMA.2026.083

Variations of GIT Quotients and Dimer Combinatorics for Toric Compound Du Val Singularities

Yusuke Nakajima
Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto, 603-8555, Japan

Received June 19, 2025, in final form August 18, 2026; Published online August 26, 2026

Abstract
A dimer model is a bipartite graph described on the real two-torus, from which we can define the associated quiver.It is known that for any three-dimensional Gorenstein toric singularity, there exists a dimer model such that a GIT quotient parametrizing stable representations of the associated quiver is a projective crepant resolution of this singularity for some stability parameter. It is also known that the space of stability parameters has a wall-and-chamber structure, and any projective crepant resolution of a three-dimensional Gorenstein toric singularity can be realized as the GIT quotient associated to a stability parameter contained in some chamber. In this paper, we consider dimer models giving rise to projective crepant resolutions of a toric compound Du Val singularity. We show that sequences of zigzag paths, which are special paths on a dimer model, determine the wall-and-chamber structure of the space of stability parameters. Moreover, we can track variations of stable representations under wall-crossing using the sequences of zigzag paths.

Key words: dimer models; toric compound Du Val singularities; variations of GIT quotients.

pdf (869 kb)   tex (65 kb)  

References

  1. Bellamy G., Craw A., Schedler T., Birational geometry of quiver varieties and other GIT quotients, Compos. Math. 162 (2026), 630-668, arXiv:2212.09623.
  2. Bender M., Mozgovoy S., Crepant resolutions and brane tilings II: Tilting bundles, arXiv:0909.2013.
  3. Bocklandt R., Consistency conditions for dimer models, Glasg. Math. J. 54 (2012), 429-447, arXiv:1104.1592.
  4. Bocklandt R., Generating toric noncommutative crepant resolutions, J. Algebra 364 (2012), 119-147, arXiv:1104.1597.
  5. Bocklandt R., Craw A., Quintero Vélez A., Geometric Reid's recipe for dimer models, Math. Ann. 361 (2015), 689-723, arXiv:1305.0156.
  6. Bondal A.I., Representations of associative algebras and coherent sheaves, Math. USSR-Izv. 34 (1990), 23-42.
  7. Bridgeland T., King A., Reid M., The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), 535-554, arXiv:math.AG/9908027.
  8. Broomhead N., Dimer models and Calabi-Yau algebras, Mem. Amer. Math. Soc. 215 (2012), viii+86 pages, arXiv:0901.4662.
  9. Cox D.A., Little J.B., Schenck H.K., Toric varieties, Grad. Stud. Math., Vol. 124, American Mathematical Society, Providence, RI, 2011.
  10. Craw A., The McKay correspondence and representations of the McKay quiver, Ph.D. Thesis, University of Warwick, 2001.
  11. Craw A., Heuberger L., Amador J.T., Combinatorial Reid's recipe for consistent dimer models, Épijournal Géom. Algébrique 5 (2021), 4, 29 pages, arXiv:2001.07506.
  12. Craw A., Ishii A., Flops of $G$-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124 (2004), 259-307, arXiv:math.AG/0211360.
  13. Dais D.I., Resolving 3-dimensional toric singularities, in Geometry of Toric Varieties, Sémin. Congr., Vol. 6, Société Mathématique de France, Paris, 2002, 155-186.
  14. Dais D.I., Haase C., Ziegler G.M., All toric local complete intersection singularities admit projective crepant resolutions, Tohoku Math. J. 53 (2001), 95-107.
  15. Derksen H., Weyman J., Zelevinsky A., Quivers with potentials and their representations I: Mutations, Selecta Math. (N.S.) 14 (2008), 59-119, arXiv:0704.0649.
  16. Dolgachev I.V., Hu Y., Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5-51.
  17. Donovan W., Segal E., Mixed braid group actions from deformations of surface singularities, Comm. Math. Phys. 335 (2015), 497-543, arXiv:1310.7877.
  18. Franco S., Hanany A., Martelli D., Sparks J., Vegh D., Wecht B., Gauge theories from toric geometry and brane tilings, J. High Energy Phys. 2006 (2006), no. 1, 128, 40 pages, arXiv:hep-th/0505211.
  19. Franco S., Hanany A., Vegh D., Wecht B., Kennaway K.D., Brane dimers and quiver gauge theories, J. High Energy Phys. 2006 (2006), no. 1, 096, 48 pages, arXiv:hep-th/0504110.
  20. Gulotta D.R., Properly ordered dimers, $R$-charges, and an efficient inverse algorithm, J. High Energy Phys. 2008 (2008), no. 10, 014, 31 pages, arXiv:0807.3012.
  21. Hanany A., Vegh D., Quivers, tilings, branes and rhombi, J. High Energy Phys. 2007 (2007), no. 10, 029, 35 pages, arXiv:hep-th/0511063.
  22. Ishii A., Ueda K., On moduli spaces of quiver representations associated with dimer models, in Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu, Vol. B9, Research Institute for Mathematical Sciences (RIMS), Kyoto, 2008, 127-141, arXiv:0710.1898.
  23. Ishii A., Ueda K., A note on consistency conditions on dimer models, in Higher dimensional algebraic geometry, RIMS Kôkyûroku Bessatsu, Vol. B24, Research Institute for Mathematical Sciences (RIMS), Kyoto, 2011, 143-164, arXiv:1012.5449.
  24. Ishii A., Ueda K., Dimer models and the special McKay correspondence, Geom. Topol. 19 (2015), 3405-3466, arXiv:0905.0059.
  25. Ishii A., Ueda K., Dimer models and crepant resolutions, Hokkaido Math. J. 45 (2016), 1-42, arXiv:1303.4028.
  26. Ito Y., Nakamura I., Hilbert schemes and simple singularities, in New Trends in Algebraic Geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge University Press, Cambridge, 1999, 151-233.
  27. Iyama O., Nakajima Y., On steady non-commutative crepant resolutions, J. Noncommut. Geom. 12 (2018), 457-471, arXiv:1509.09031.
  28. Iyama O., Reiten I., Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, Amer. J. Math. 130 (2008), 1087-1149, arXiv:math.RT/0605136.
  29. Iyama O., Wemyss M., Singular derived categories of $\mathbb{Q}$-factorial terminalizations and maximal modification algebras, Adv. Math. 261 (2014), 85-121, arXiv:1108.4518.
  30. Jung S.-J., On the Craw-Ishii conjecture, J. Pure Appl. Algebra 222 (2018), 1579-1605, arXiv:1601.05128.
  31. King A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515-530.
  32. Mozgovoy S., Crepant resolutions and brane tilings I: Toric realization, arXiv:0908.3475.
  33. Muto T., Tani T., Stability of quiver representations and topology change, J. High Energy Phys. 2001 (2001), no. 9, 008, 17 pages, arXiv:hep-th/0107217.
  34. Nagao K., Derived categories of small toric Calabi-Yau 3-folds and curve counting invariants, Q. J. Math. 63 (2012), 965-1007, arXiv:0809.2994.
  35. Nakajima Y., Mutations of splitting maximal modifying modules: The case of reflexive polygons, Int. Math. Res. Not. 2019 (2019), 470-550, arXiv:1601.05203.
  36. Nakajima Y., On 2-representation infinite algebras arising from dimer models, Q. J. Math. 73 (2022), 1517-1553, arXiv:1806.05331.
  37. Nakamura I., Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (2001), 757-779.
  38. Nolla de Celis Á., Sekiya Y., Flops and mutations for crepant resolutions of polyhedral singularities, Asian J. Math. 21 (2017), 1-45, arXiv:1108.2352.
  39. Reid M., Minimal models of canonical $3$-folds, in Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math., Vol. 1, North-Holland, Amsterdam, 1983, 131-180.
  40. Rickard J., Morita theory for derived categories, J. London Math. Soc. 39 (1989), 436-456.
  41. Thaddeus M., Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), 691-723, arXiv:alg-geom/9405004.
  42. Ueda K., Yamazaki M., A note on dimer models and McKay quivers, Comm. Math. Phys. 301 (2011), 723-747, arXiv:math.AG/0605780.
  43. Uranga A.M., Brane configurations for branes at conifolds, J. High Energy Phys. 1999 (1999), no. 1, 022, 27 pages, arXiv:hep-th/9811004.
  44. van den Bergh M., Non-commutative crepant resolutions, in The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, 749-770, arXiv:math.RA/0211064.
  45. Wemyss M., Flops and clusters in the homological minimal model programme, Invent. Math. 211 (2018), 435-521, arXiv:1411.7189.
  46. Wemyss M., A lockdown survey on cDV singularities, in McKay correspondence, mutation and related topics, Adv. Stud. Pure Math., Vol. 88, Mathematical Society of Japan, Tokyo, 2023, 47-93, arXiv:2103.16990.
  47. Wilson P.M.H., The Kähler cone on Calabi-Yau threefolds, Invent. Math. 107 (1992), 561-583.
  48. Wormleighton B., Walls for $G$-Hilb via Reid's recipe, SIGMA 16 (2020), 106, 38 pages, arXiv:1908.05748.
  49. Yamagishi R., Moduli of $G$-constellations and crepant resolutions II: The Craw-Ishii conjecture, Duke Math. J. 174 (2025), 229-285, arXiv:2209.11901.

Previous article  Next article  Contents of Volume 22 (2026)