Mathematische Zeitschrift

, Volume 289, Issue 3–4, pp 1143–1168 | Cite as

Symmetric products of a semistable degeneration of surfaces

  • Yasunari Nagai


We explicitly construct a V-normal crossing Gorenstein canonical model of the relative symmetric products of a local semistable degeneration of surfaces without a triple point by means of toric geometry. Using this model, we calculate the stringy E-polynomial of the relative symmetric product. We also construct a minimal model of degeneration of Hilbert schemes explicitly.



The author would like to thank M. Brion, M. Lehn, and T. Yasuda for their interest and helpful comments. He also thanks the anonymous referee for giving him valuable comments. This work is partially supported by JSPS Grants-in-aid for young scientists (B) 26800025.


  1. 1.
    Batyrev, V.V.: Non-archimedean integrals and stringy euler numbers of log-terminal pairs. J. Eur. Math. Soc. (JEMS) 1(1), 5–33 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beauville, A.: Variétés kähleriennes dont la première classe de chern est nulle. J. Differ. Geom. 18(4), 755–782 (1984). (French)CrossRefzbMATHGoogle Scholar
  3. 3.
    Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Briançon, J.: Description de \({{\rm Hilb}}^{n}\mathbf{c}\{x, y\}\). Invent. Math. 41(1), 45–89 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties, Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)Google Scholar
  6. 6.
    Dolgachev, I.V.: Classical Algebraic Geometry. A Modern View. Cambridge University Press, Cambridge (2012)Google Scholar
  7. 7.
    Dolgachev, I., Lunts, V.: A character formula for the representation of a weyl group in the cohomology of the associated toric variety. J. Algebra 168(3), 741–772 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math 90, 511–521 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gulbrandsen, M.G., Halle, L. H., Hulek, K.: A GIT construction of degenerations of Hilbert schemes of points. Preprint arXiv:1604.00215 (2016)
  10. 10.
    Haiman, M.: Hilbert schemes, polygraphs and the macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kawamata, Y.: Francia’s Flip and Derived Categories. Algebraic Geometry, pp. 197–215. de Gruyter, Berlin (2002)zbMATHGoogle Scholar
  12. 12.
    Kollár, J., Mori, S.: Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese originalGoogle Scholar
  13. 13.
    Morrison, D.R., Stevens, G.: Terminal quotient singularities in dimensions three and four. Proc. Am. Math. Soc. 90(1), 15–20 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nagai, Y.: On monodromies of a degeneration of irreducible symplectic kähler manifolds. Math. Z. 258(2), 407–426 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nagai, Y.: Gulbrandsen–Halle–Hulek degeneration and Hilbert–Chow morphism. Preprint (2017)Google Scholar
  16. 16.
    Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, vol. 18. American Mathematical Society, Providence (1999)Google Scholar
  17. 17.
    Procesi, C.: The toric variety associated to Weyl chambers. Mots, Lang. Raison. Calc., pp. 153–161. Hermès, Paris (1990)Google Scholar
  18. 18.
    Ran, Z.: Cycle Map on Hilbert Schemes of Nodal Curves. Projective Varieties with Unexpected Properties, pp. 361–378. Walter de Gruyter GmbH & Co. KG, Berlin (2005)zbMATHGoogle Scholar
  19. 19.
    Reid, M.: Canonical 3-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/AlgebraicGeometry, Angers, 1979, pp. 273–310. Sijthoff & Noordhoff. Alphen aan den Rijn-Germantown, Md (1980)Google Scholar
  20. 20.
    Sagan, B.E.: The symmetric group, 2nd edn, Graduate Texts in Mathematics. Representations, Combinatorial Algorithms, and Symmetric Functions, vol. 203, Springer-Verlag, New York (2001)Google Scholar
  21. 21.
    Steenbrink, J.H.M.: Mixed Hodge Structure on the Vanishing Cohomology, Real and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 525–563. Sijthoff and Noordhoff, Alphen aan den Rijn (1977)Google Scholar
  22. 22.
    Stembridge, J.R.: Some permutation representations of Weyl groups associated with the cohomology of toric varieties. Adv. Math. 106(2), 244–301 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wang, J.: Degenerations of symmetric products of curves. Preprint Accessed 10 Aug 2017
  24. 24.
    Yasuda, T.: Motivic integration over deligne-mumford stacks. Adv. Math. 207(2), 707–761 (2006). MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and EngineeringWaseda UniversityTokyoJapan

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