On the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915006500569088 |
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| author | Brandt, Madeline Chan, Melody Kannan, Siddarth |
| author_facet | Brandt, Madeline Chan, Melody Kannan, Siddarth |
| contents | For $g\ge 2$ and $n\ge 0$, let $\mathcal{H}_{g,n}\subset \mathcal{M}_{g,n}$ denote the complex moduli stack of $n$-marked smooth hyperelliptic curves of genus $g$. A normal crossings compactification of this space is provided by the theory of pointed admissible $\mathbb{Z}/2\mathbb{Z}$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$. Using this graph complex, we give a sum-over-graphs formula for the $S_n$-equivariant weight zero compactly supported Euler characteristic of $\mathcal{H}_{g, n}$. This formula allows for the computer-aided calculation, for each $g\le 7$, of the generating function $\mathsf{h}_g$ for these equivariant Euler characteristics for all $n$. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissible $G$-covers of genus zero curves, when $G$ is abelian, as a symmetric $Δ$-complex. We use these complexes to generalize our formula for $\mathsf{h}_g$ to moduli spaces of $n$-pointed smooth abelian covers of genus zero curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_01819 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$ Brandt, Madeline Chan, Melody Kannan, Siddarth Algebraic Geometry Combinatorics 05E14, 14H10 For $g\ge 2$ and $n\ge 0$, let $\mathcal{H}_{g,n}\subset \mathcal{M}_{g,n}$ denote the complex moduli stack of $n$-marked smooth hyperelliptic curves of genus $g$. A normal crossings compactification of this space is provided by the theory of pointed admissible $\mathbb{Z}/2\mathbb{Z}$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$. Using this graph complex, we give a sum-over-graphs formula for the $S_n$-equivariant weight zero compactly supported Euler characteristic of $\mathcal{H}_{g, n}$. This formula allows for the computer-aided calculation, for each $g\le 7$, of the generating function $\mathsf{h}_g$ for these equivariant Euler characteristics for all $n$. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissible $G$-covers of genus zero curves, when $G$ is abelian, as a symmetric $Δ$-complex. We use these complexes to generalize our formula for $\mathsf{h}_g$ to moduli spaces of $n$-pointed smooth abelian covers of genus zero curves. |
| title | On the weight zero compactly supported cohomology of $\mathcal{H}_{g, n}$ |
| topic | Algebraic Geometry Combinatorics 05E14, 14H10 |
| url | https://arxiv.org/abs/2307.01819 |