Representation stability for moduli spaces of admissible covers
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913916820389888 |
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| author | Chang-Lee, Megan Kannan, Siddarth Tosteson, Philip |
| author_facet | Chang-Lee, Megan Kannan, Siddarth Tosteson, Philip |
| contents | 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 $\overline M_{g, n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22640 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representation stability for moduli spaces of admissible covers Chang-Lee, Megan Kannan, Siddarth Tosteson, Philip Algebraic Geometry Geometric Topology 14H10, 16G20 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 $\overline M_{g, n}$. |
| title | Representation stability for moduli spaces of admissible covers |
| topic | Algebraic Geometry Geometric Topology 14H10, 16G20 |
| url | https://arxiv.org/abs/2506.22640 |