Geometry & Topology

Counting lattice points in compactified moduli spaces of curves

Norman Do and Paul Norbury

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We define and count lattice points in the moduli space ¯g,n of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space g,n. The enumeration produces polynomials whose top degree coefficients are tautological intersection numbers on ¯g,n and whose constant term is the orbifold Euler characteristic of ¯g,n. We prove a recursive formula which can be used to effectively calculate these polynomials. One consequence of these results is a simple recursion relation for the orbifold Euler characteristic of ¯g,n.

Article information

Geom. Topol., Volume 15, Number 4 (2011), 2321-2350.

Received: 12 May 2011
Revised: 26 August 2011
Accepted: 23 September 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 14N10: Enumerative problems (combinatorial problems) 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

moduli space stable maps Euler characteristic


Do, Norman; Norbury, Paul. Counting lattice points in compactified moduli spaces of curves. Geom. Topol. 15 (2011), no. 4, 2321--2350. doi:10.2140/gt.2011.15.2321.

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  • G Bini, J Harer, Euler characteristics of moduli spaces of curves, J. Eur. Math. Soc. $($JEMS$)$ 13 (2011) 487–512
  • B Eynard, N Orantin, Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007) 347–452
  • I P Goulden, S Litsyn, V Shevelev, On a sequence arising in algebraic geometry, J. Integer Seq. 8 (2005) 05.4.7
  • J Harer, D Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986) 457–485
  • M Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992) 1–23
  • Y I Manin, Generating functions in algebraic geometry and sums over trees, from: “The moduli space of curves (Texel Island, 1994)”, Progr. Math. 129, Birkhäuser, Boston (1995) 401–417
  • M Mulase, M Penkava, Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over $\wwbar{\mathbf{Q}}$, Asian J. Math. 2 (1998) 875–919
  • P Norbury, Counting lattice points in the moduli space of curves, Math. Res. Lett. 17 (2010) 467–481
  • P Norbury, Cell decompositions of moduli space, lattice points and Hurwitz problems, to appear in Handbook of Moduli
  • P Norbury, String and dilaton equations for counting lattice points in the moduli space of curves, to appear in Trans. Amer. Math. Soc.
  • R C Penner, Perturbative series and the moduli space of Riemann surfaces, J. Differential Geom. 27 (1988) 35–53
  • K Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 5, Springer, Berlin (1984)
  • E Witten, Two-dimensional gravity and intersection theory on moduli space, from: “Surveys in differential geometry (Cambridge, MA, 1990)”, Lehigh Univ., Bethlehem, PA (1991) 243–310
  • D Zvonkine, Strebel differentials on stable curves and Kontsevich's proof of Witten's conjecture