Equivariant Euler characteristics of Mg, n

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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.

Original languageEnglish (US)
Pages (from-to)523-543
Number of pages21
JournalAlgebraic Geometry
Issue number5
StatePublished - 2020

Bibliographical note

Publisher Copyright:
© Foundation Compositio Mathematica 2020.


  • Asymptotic expansion
  • Equivariant euler characteristics
  • Free modular operad
  • Moduli space
  • Plethystic exponential


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