Euler characteristics of Hilbert schemes of points on simple surface singularities

  • Ádám Gyenge
  • András Némethi
  • Balázs Szendrői
Research Article
  • 14 Downloads

Abstract

We study the geometry and topology of Hilbert schemes of points on the orbifold surface Open image in new window , respectively the singular quotient surface Open image in new window , where Open image in new window is a finite subgroup of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.

Keywords

Hilbert scheme Singularities Euler characteristic Generating series Young wall 

Mathematics Subject Classification

14N10 05E10 

Notes

Acknowledgements

The authors would like to thank Gwyn Bellamy, Alastair Craw, Eugene Gorsky, Ian Grojnowski, Kevin McGerty, Iain Gordon, Tomas Nevins and Tamás Szamuely for helpful comments and discussions.

References

  1. 1.
    Altman, A.B., Kleiman, S.L.: Joins of schemes, linear projections. Compositio Math. 31(3), 309–343 (1975)MathSciNetMATHGoogle Scholar
  2. 2.
    Andrews, G.E.: Generalized Frobenius Partitions. Memoirs of American Mathematical Society, vol. 49(301). American Mathematical Society, Providence (1984)Google Scholar
  3. 3.
    Białynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98(3), 480–497 (1973)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brion, M.: Invariant Hilbert schemes. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli. Vol. I. Advanced Lectures in Mathematics, vol. 24, pp. 64–117. International Press, Somerville (2013)Google Scholar
  5. 5.
    Cartan, H.: Quotient d’un espace analytique par un groupe d’automorphismes. In: Fox, R.H., Spencer, D.C., Tucker, A.W. (eds.) Algebraic Geometry and Topology, pp. 90–102. Princeton University Press, Princeton (1957)Google Scholar
  6. 6.
    Cheah, J.: On the cohomology of Hilbert schemes of points. J. Algebraic Geom. 5(3), 479–511 (1996)MathSciNetMATHGoogle Scholar
  7. 7.
    de Celis, Á.N.: Dihedral Groups and \(G\)-Hilbert Schemes. Ph.D. Thesis, University of Warwick (2008)Google Scholar
  8. 8.
    Dijkgraaf, R., Sułkowski, P.: Instantons on ALE spaces and orbifold partitions. J. High Energy Phys. 2008(3), Art. No. 013 (2008)Google Scholar
  9. 9.
    Eisenbud, D., Harris, J.: 3264 and All That. Cambridge University Press, Cambridge (2016)CrossRefMATHGoogle Scholar
  10. 10.
    Ellingsrud, G., Strømme, S.A.: On the homology of the Hilbert scheme of points in the plane. Invent. Math. 87(2), 343–352 (1987)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Etingof, P.: Symplectic reflection algebras and affine Lie algebras. Mosc. Math. J 12(3), 543–565 (2012)MathSciNetMATHGoogle Scholar
  12. 12.
    Frenkel, I.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62(1), 23–66 (1980/1981)Google Scholar
  13. 13.
    Fujii, Sh., Minabe, S.: A combinatorial study on quiver varieties. SIGMA Symmetry Integrability Geom. Methods Appl. 13, Art. No. 052 (2017)Google Scholar
  14. 14.
    Garvan, F., Kim, D., Stanton, D.: Cranks and \(t\)-cores. Invent. Math. 101(1), 1–17 (1990)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gordon, I.G.: Quiver varieties, category \(\fancyscript {O}\) for rational Cherednik algebras, and Hecke algebras. Int. Math. Res. Pap. IMRP 2008(3), Art. ID rpn006 (2008)Google Scholar
  16. 16.
    Göttsche, L.: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286(1–3), 193–207 (1990)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Grojnowski, I.: Instantons and affine algebras I: the Hilbert scheme and vertex operators. Math. Res. Lett. 3(2), 275–291 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gusein-Zade, S.M., Luengo, I., Melle-Hernández, A.: On generating series of classes of equivariant Hilbert schemes of fat points. Mosc. Math. J. 10(3), 593–602 (2010)MathSciNetMATHGoogle Scholar
  19. 19.
    Gyenge, Á.: Hilbert scheme of points on cyclic quotient singularities of type \((p, 1)\). Period. Math. Hungar. 73(1), 93–99 (2016)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gyenge, Á.: Hilbert Schemes of Points on Some Classes of Surface Singularities. Ph.D. Thesis, Eötvös Lóránd University (2016)Google Scholar
  21. 21.
    Gyenge, Á., Némethi, A., Szendrői, B.: Euler characteristics of Hilbert schemes of points on surfaces with simple singularities. Int. Math. Res. Not. IMRN 2017(13), 4152–4159 (2017)MathSciNetGoogle Scholar
  22. 22.
    Hartshorne, R.: Algebraic Geometry. Graduate Text in Mathematics, vol. 52. Springer, New York (1977)CrossRefGoogle Scholar
  23. 23.
    Ito, Y., Nakamura, I.: Hilbert schemes and simple singularities. In: Hulek, K., et al. (eds.) New Trends in Algebraic Geometry. London Mathematical Society Lecture Note Series, vol. 264, pp. 151–233. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  24. 24.
    Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls. Proc. London Math. Soc. 86(1), 29–69 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kang, S.-J., Kwon, J.-H.: Crystal bases of the Fock space representations and string functions. J. Algebra 280(1), 313–349 (2004)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316(3), 565–576 (2000)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kwon, J.-H.: Affine crystal graphs and two-colored partitions. Lett. Math. Phys. 75(2), 171–186 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211, 3rd edn. Springer, New York (2002)Google Scholar
  29. 29.
    Leclerc, B., Miyachi, H.: Some closed formulas for canonical bases of Fock spaces. Represent. Theory 6, 290–312 (2002)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Lorscheid, O., Weist, Th.: Quiver Grassmannians of type \(\widetilde{D}_n\), Part 1: Schubert systems and decompositions into affine spaces (2015). arXiv:1507.00392
  31. 31.
    Lorscheid, O., Weist, Th.: Quiver Grassmannians of type \(\widetilde{D}_n\), Part 2: Schubert decompositions and \(F\)-polynomials (2015). arXiv:1507.00395
  32. 32.
    Macdonald, I.G.: The Poincaré polynomial of a symmetric product. Proc. Cambridge Philos. Soc. 58(4), 563–568 (1962)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Maulik, D.: Stable pairs and the HOMFLY polynomial. Invent. Math. 204(3), 787–831 (2016)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Misra, K., Miwa, T.: Crystal base for the basic representation of \(U_q (\widehat{\mathfrak{sl}}(n))\). Comm. Math. Phys. 134(1), 79–88 (1990)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Nagao, K.: Quiver varieties and Frenkel–Kac construction. J. Algebra 321(12), 3764–3789 (2009)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Nakajima, H.: Quiver varieties and finite dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14(1), 145–238 (2001)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Nakajima, H.: Geometric construction of representations of affine algebras. In: Li, T. (ed.) Proceedings of the International Congress of Mathematicians (Beijing, 2002), vol. 1, pp. 423–438. Higher Education Press, Beijing (2002)Google Scholar
  38. 38.
    Nakajima, H., Yoshioka, K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Invent. Math. 162(2), 313–355 (2005)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Oblomkov, A., Shende, V.: The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link. Duke Math. J. 161(7), 1277–1303 (2012)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Savage, A., Tingley, P.: Quiver Grassmannians, quiver varieties and the preprojective algebra. Pacific J. Math. 251(2), 393–429 (2011)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Tingley, P.: Notes on Fock space (2011). http://webpages.math.luc.edu/~ptingley/lecturenotes/Fock_space-2010.pdf. Accessed 11 Mar 2018
  42. 42.
    Toda, Y.: S-duality for surfaces with \(A_n\)-type singularities. Math. Ann. 363, 679–699 (2015)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Young, B.: Generating functions for colored 3D Young diagrams and the Donaldson–Thomas invariants of orbifolds. With an appendix by Jim Bryan. Duke Math. J. 152(1), 115–153 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations