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Stability in the homology of Deligne–Mumford compactifications

Part of: Curves Representation theory of rings and algebras

Published online by Cambridge University Press:  17 December 2021

Philip Tosteson
Philip Tosteson*
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL, USAptoste@math.uchicago.edu
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Abstract

Using the theory of ${\mathbf {FS}} {^\mathrm {op}}$ modules, we study the asymptotic behavior of the homology of ${\overline {\mathcal {M}}_{g,n}}$, the Deligne–Mumford compactification of the moduli space of curves, for $n\gg 0$. An ${\mathbf {FS}} {^\mathrm {op}}$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of ${\overline {\mathcal {M}}_{g,n}}$ the structure of an ${\mathbf {FS}} {^\mathrm {op}}$ module and bound its degree of generation. As a consequence, we prove that the generating function $\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$ is rational, and its denominator has roots in the set $\{1, 1/2, \ldots, 1/p(g,i)\},$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$. We also obtain restrictions on the decomposition of the homology of ${\overline {\mathcal {M}}_{g,n}}$ into irreducible $\mathbf {S}_n$ representations.

Keywords

moduli of curveshomological stabilityrepresentations of categories

MSC classification

Primary: 14H10: Families, moduli (algebraic) 16G20: Representations of quivers and partially ordered sets
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 12 , December 2021 , pp. 2635 - 2656
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The author is partially supported by NSF-Grant No. DMS-1903040.

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