Vanishing $H^1$ for Hurwitz spaces of fully-marked admissible covers of degree 3
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918225845944320 |
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| author | Li, Amy Q. |
| author_facet | Li, Amy Q. |
| contents | We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers $H^1(\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ))$ vanishes for covers of degree $ d = 3$ and deduce the same result for the classical Hurwitz space of simply-branched covers. In degree 4, we compute examples where $H^1(\overline{\mathcal{H}}_{\underline{4},\underline{g}}(\underlineμ))$ is nonzero, which implies that $H^1(\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ))$ is nonvanishing for $d \geq 4$. We describe the stratification of the boundary of $\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ)$ by lower-dimensional $\overline{\mathcal{H}}_{\underline{d'},\underline{g'}}(\underline{μ'})$, and set up an inductive framework which may be used for future arguments involving the odd cohomology of $\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01553 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vanishing $H^1$ for Hurwitz spaces of fully-marked admissible covers of degree 3 Li, Amy Q. Algebraic Geometry 14H10, 14H30, 14C17 We show that the first cohomology group of the Hurwitz space of fully-marked admissible covers $H^1(\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ))$ vanishes for covers of degree $ d = 3$ and deduce the same result for the classical Hurwitz space of simply-branched covers. In degree 4, we compute examples where $H^1(\overline{\mathcal{H}}_{\underline{4},\underline{g}}(\underlineμ))$ is nonzero, which implies that $H^1(\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ))$ is nonvanishing for $d \geq 4$. We describe the stratification of the boundary of $\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ)$ by lower-dimensional $\overline{\mathcal{H}}_{\underline{d'},\underline{g'}}(\underline{μ'})$, and set up an inductive framework which may be used for future arguments involving the odd cohomology of $\overline{\mathcal{H}}_{\underline{d},\underline{g}}(\underlineμ)$. |
| title | Vanishing $H^1$ for Hurwitz spaces of fully-marked admissible covers of degree 3 |
| topic | Algebraic Geometry 14H10, 14H30, 14C17 |
| url | https://arxiv.org/abs/2512.01553 |