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Computing top intersections in the tautological ring of \({\mathcal{M}_g}\)

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Abstract

We derive effective recursion formulae of top intersections in the tautological ring \({R^*(\mathcal{M}_g)}\) of the moduli space of curves of genus g ≥ 2. As an application, we prove a convolution-type tautological relation in \({R^{g-2}(\mathcal{M}_g)}\) and some interesting Bernoulli number identities.

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References

  1. Arbarello E., Cornalba M.: Combinatorial and algebro–geometric cohomology classes on the moduli spaces of curves. J. Alg. Geom. 5, 705–709 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Arbarello E., Cornalba M.: Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Pub. Math. IHES 88, 97–127 (1998)

    MathSciNet  MATH  Google Scholar 

  3. Buryak, A., Shadrin, S.: A new proof of Faber’s intersection number conjecture (2009). arXiv:0912.5115 [math.AG]

  4. Cavalieri R., Yang S.: Tautological pairings on moduli spaces of curves. Proc. Amer. Math. Soc. 139, 51–62 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheema, M.S., Motzkin, T.S.: Multipartitions and multipermutations. In: Combinatorics. Proc. Sympos. Pure Math., vol. XIX, University of California, Los Angeles 1968, pp. 39–70. American Mathematical Society, Rhode Island (1971)

  6. Faber, C.: A conjectural description of the tautological ring of the moduli space of curves. In: Moduli of curves and abelian varieties. Aspects Math., E33, pp. 109–129. Vieweg, Braunschweig (1999)

  7. Faber C.: Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians. In: Hulek, K., Catanese, F., Peters, C., Reid, M. (eds) New trends in algebraic geometry, pp. 93–109. Cambridge University Press, Cambridge (1999)

    Chapter  Google Scholar 

  8. Faber C., Pandharipande R.: Logarithmic series and Hodge integrals in the tautological ring (with an appendix by D. Zagier). Michigan Math. J. 48, 215–252 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Getzler E., Pandharipande R.: Virasoro constraints and the Chern classes of the Hodge bundle. Nuclear Phys. B 530, 701–714 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Givental A.: Gromov–Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1, 551–568 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Goulden, I.P., Jackson, D.M., Vakil R.: The moduli space of curves, double Hurwitz numbers and Faber’s intersection number conjecture. Ann. of Comb. (to appear)

  12. Ionel E.: Relations in the tautological ring of \({\mathcal{M}_g}\) . Duke Math. J. 129, 157–186 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kaneiwa R.: An asymptotic formula for Cayley’s double partition function p(2; n). Tokyo J. Math. 2, 137–158 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kaufmann R., Manin Yu., Zagier D.: Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves. Comm. Math. Phys. 181, 763–787 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147, 1–23 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu K., Xu H.: Recursion formulae of higher Weil-Petersson volumes. Int. Math. Res. Not. 2009, 835–859 (2009)

    MATH  Google Scholar 

  17. Liu K., Xu H.: A proof of the Faber intersection number conjecture. J. Differ. Geom. 83, 313–335 (2009)

    MATH  Google Scholar 

  18. Liu, K., Xu, H.: Descendent integrals and tautological rings of moduli spaces of curves. In: Proceedings of the conference “Geometric analysis: Present and Future” held at Harvard in August 27–September 1, 2008 (to appear)

  19. Looijenga E.: On the tautological ring of \({\mathcal M_g}\). Invent. Math. 121, 411–419 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Marian, A., Oprea, D., Pandharipande, R.: The moduli space of stable quotients. arXiv:0904.2992 [math.AG]

  21. Morita S.: Generators for the tautological algebra of the moduli space of curves. Topology 42, 787–819 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mumford D.: Towards an enumerative geometry of the moduli space of curves. In: Artin, M., Tate, J. (eds) Arithmetic and geometry part II, pp. 271–328. Birkhäuser, Switzerland (1983)

    Google Scholar 

  23. Pandharipande R.: Hodge integrals and degenerate contributions. Comm. Math. Phys. 208, 489–506 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Witten E.: Two-dimensional gravity and intersection theory on moduli space. Surv. Differ. Geom. 1, 243–310 (1991)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang, China

    Kefeng Liu & Hao Xu

  2. Department of Mathematics, University of California at Los Angeles, Los Angeles, CA, 90095-1555, USA

    Kefeng Liu

  3. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Hao Xu

Authors
  1. Kefeng Liu
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  2. Hao Xu
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Correspondence to Hao Xu.

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Liu, K., Xu, H. Computing top intersections in the tautological ring of \({\mathcal{M}_g}\) . Math. Z. 270, 819–837 (2012). https://doi.org/10.1007/s00209-010-0828-9

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  • DOI: https://doi.org/10.1007/s00209-010-0828-9

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