Representation stability for moduli spaces of admissible covers

TL;DR

This paper establishes a representation stability result for the homology of moduli spaces of admissible covers, showing rationality of their generating functions and improving previous stability results for certain moduli spaces.

Contribution

It introduces a new stability framework for the homology of admissible cover moduli spaces, with explicit bounds and rational generating functions.

Findings

Homology groups form modules over a combinatorial category.

Modules are generated in degree at most g + 5i.

Generating functions for ranks are rational with specific poles.

Abstract

We prove a representation stability result for the sequence of spaces $\overline M_{g, n}^A$ of pointed admissible $A$-covers of stable $n$-pointed genus-$g$ curves, for an abelian group $A$. For fixed genus $g$ and homology degree $i$, we give the sequence of rational homology groups $H_i(\overline{M}_{g, n}^A;\mathbb Q)$ the structure of a module over a combinatorial category, a la Sam--Snowden, and prove that this module is generated in degree at most $g + 5 i$. This implies that the generating function for the ranks of the homology groups is rational, with poles in the set $\left\{-1, -\frac{1}{2}, \ldots, -\frac{1}{|A|^2\cdot(g + 5i)}\right\}$. In the case where $A$ is the trivial group, our work significantly improves on previous representation stability results on the Deligne--Mumford compactification $\overline M_{g, n}$.