#### Volume 15, issue 4 (2011)

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Counting lattice points in compactified moduli spaces of curves

### Norman Do and Paul Norbury

Geometry & Topology 15 (2011) 2321–2350
##### Abstract

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

##### Keywords
moduli space, stable maps, Euler characteristic
##### Mathematical Subject Classification 2000
Primary: 32G15
Secondary: 14N10, 05A15