Counting lattice points in compactified moduli spaces of curves

Do, Norman¹ ; Norbury, Paul¹

¹ Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

Geometry & topology, Tome 15 (2011) no. 4, pp. 2321-2350.

Résumé

We define and count lattice points in the moduli space ${\overline{\mathcal{ℳ}}}_{g,n}$ of stable genus $g$ curves with $n$ labeled points. This extends a construction of the second author for the uncompactified moduli space ${\mathcal{ℳ}}_{g,n}$. The enumeration produces polynomials whose top degree coefficients are tautological intersection numbers on ${\overline{\mathcal{ℳ}}}_{g,n}$ and whose constant term is the orbifold Euler characteristic of ${\overline{\mathcal{ℳ}}}_{g,n}$. 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 ${\overline{\mathcal{ℳ}}}_{g,n}$.

Détail

DOI : 10.2140/gt.2011.15.2321

Keywords: moduli space, stable maps, Euler characteristic

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