标题： 可接受覆盖模空间的表示稳定性 显示英文标题

标题： Representation stability for moduli spaces of admissible covers

摘要： 我们证明了关于序列空间$\overline M_{g, n}^A$的表示稳定性结果，这些空间是稳定$n$-点亏格$g$曲线的带点可接受$A$-覆盖，对于一个阿贝尔群$A$。 对于固定的亏格$g$和同调次数$i$，我们给出有理同调群$H_i(\overline{M}_{g, n}^A;\mathbb Q)$的序列作为组合范畴上的模结构，类似于 Sam--Snowden 的方法，并证明该模在次数最多为$g + 5 i$的情况下生成。这表明同调群的秩的生成函数是有理的，极点位于集合$\left\{-1, -\frac{1}{2}, \ldots, -\frac{1}{|A|^2\cdot(g + 5i)}\right\}$中。当$A$是平凡群时，我们的工作显著改进了关于 Deligne--Mumford 紧化空间$\overline M_{g, n}$的先前表示稳定性结果。 显示英文摘要

摘要： 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 $\overline M_{g, n}$.