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

SIGMA 16 (2020), 106, 38 pages      arXiv:1908.05748      https://doi.org/10.3842/SIGMA.2020.106

Walls for $G$-Hilb via Reid's Recipe

Ben Wormleighton
Department of Mathematics and Statistics, Washington University in St. Louis, MO 63130, USA

Received November 14, 2019, in final form October 14, 2020; Published online October 24, 2020

Abstract
The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein $3$-fold quotient singularities $\mathbb{A}^3/G$ with the representation theory of the group $G$. The first crepant resolution studied in depth was the $G$-Hilbert scheme $G\text{-Hilb}\,\mathbb{A}^3$, which is also a moduli space of $\theta$-stable representations of the McKay quiver associated to $G$. As the stability parameter $\theta$ varies, we obtain many other crepant resolutions. In this paper we focus on the case where $G$ is abelian, and compute explicit inequalities for the chamber of the stability space defining $G\text{-Hilb}\,\mathbb{A}^3$ in terms of a marking of exceptional subvarieties of $G\text{-Hilb}\,\mathbb{A}^3$ called Reid's recipe. We further show which of these inequalities define walls. This procedure depends only on the combinatorics of the exceptional fibre and has applications to the birational geometry of other crepant resolutions.

Key words:wall-crossing; McKay correspondence; Reid's recipe; quivers.

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References

  1. Artin M., Verdier J.L., Reflexive modules over rational double points, Math. Ann. 270 (1985), 79-82.
  2. Batyrev V.V., Dais D.I., Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology 35 (1996), 901-929, arXiv:alg-geom/9410001.
  3. Bocklandt R., Craw A., Quintero Vélez A., Geometric Reid's recipe for dimer models, Math. Ann. 361 (2015), 689-723, arXiv:1305.0156.
  4. Bridgeland T., Flops and derived categories, Invent. Math. 147 (2002), 613-632, arXiv:math.AG/0009053.
  5. 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.
  6. Cautis S., Craw A., Logvinenko T., Derived Reid's recipe for abelian subgroups of ${\rm SL}_3({\mathbb C})$, J. Reine Angew. Math. 727 (2017), 1-48, arXiv:1205.3110.
  7. Cautis S., Logvinenko T., A derived approach to geometric McKay correspondence in dimension three, J. Reine Angew. Math. 636 (2009), 193-236, arXiv:0803.2990.
  8. Craw A., The McKay correspondence and representations of the McKay quiver, Ph.D. Thesis, Warwick University, 2001.
  9. Craw A., An explicit construction of the McKay correspondence for $A$-Hilb ${\mathbb C}^3$, J. Algebra 285 (2005), 682-705, arXiv:math.AG/0010053.
  10. 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.
  11. Craw A., Ito Y., Karmazyn J., Multigraded linear series and recollement, Math. Z. 289 (2018), 535-565, arXiv:1701.01679.
  12. Craw A., Reid M., How to calculate $A$-Hilb ${\mathbb C}^3$, in Geometry of Toric Varieties, Sémin. Congr., Vol. 6, Soc. Math. France, Paris, 2002, 129-154, arXiv:math.AG/9909085.
  13. Denef J., Loeser F., Motivic integration, quotient singularities and the McKay correspondence, Compositio Math. 131 (2002), 267-290, arXiv:math.AG/9903187.
  14. Ishii A., Ito Y., Nolla de Celis A., On $G/N$-Hilb of $N$-Hilb, Kyoto J. Math. 53 (2013), 91-130, arXiv:1108.2310.
  15. Ishii A., Ueda K., Dimer models and the special McKay correspondence, Geom. Topol. 19 (2015), 3405-3466, arXiv:0905.0059.
  16. Ito Y., Nakajima H., McKay correspondence and Hilbert schemes in dimension three, Topology 39 (2000), 1155-1191, arXiv:math.AG/9803120.
  17. Ito Y., Nakamura I., McKay correspondence and Hilbert schemes, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 135-138.
  18. Ito Y., Wormleighton B., Wall-crossing for iterated $G$-Hilbert schemes, in preparation.
  19. King A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. 45 (1994), 515-530.
  20. Logvinenko T., Derived McKay correspondence via pure-sheaf transforms, Math. Ann. 341 (2008), 137-167, arXiv:math.AG/0606791.
  21. McKay J., Cartan matrices, finite groups of quaternions, and Kleinian singularities, Proc. Amer. Math. Soc. 81 (1981), 153-154.
  22. Nakamura I., Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (2001), 757-779.
  23. Reid M., La correspondance de McKay, arXiv:math.AG/9911165.
  24. Takahashi K., On essential representations in the McKay correpondence for ${\rm SL}_3({\mathbb C})$, Master's Thesis, Nagoya University, 2011.
  25. Wilson P.M.H., The Kähler cone on Calabi-Yau threefolds, Invent. Math. 107 (1992), 561-583.

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