Let Mg, n be the moduli space of n-pointed stable genus g curves, and let Mg, n be the moduli space of n-pointed smooth curves of genus g. In this paper, we obtain an asymptotic expansion for the characteristic of the free modular operad MV generated by a stable S-module V, allowing to effectively compute Sn-equivariant Euler characteristics of Mg, n in terms of Sn'-equivariant Euler characteristics of Mg', n' with 0 ≤ g' ≤ g and max[0; 3-2g'] ≤ n' ≤ 2(g-g') + n. This answers a question posed by Getzler and Kapranov by making their integral representation of the characteristic of the modular operad MV effective. To illustrate how the asymptotic expansion is used, we give formulas expressing the generating series of the Sn-equivariant Euler characteristics of Mg, n, for g = 0; 1 and 2, in terms of the corresponding generating series associated with Mg, n.
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I am grateful to Jonas Bergstrom and Dan Petersen for their comments and suggestions. I would also like to thank them for making available to me the formulas for the Sn-equivariant Euler characteristics of M2, n for small values of n. I am also grateful to the referee for pointing out an inaccuracy in the original version of the manuscript and for pointing out the reference [GK94].
- Asymptotic expansion
- Equivariant euler characteristics
- Free modular operad
- Moduli space
- Plethystic exponential