Chow Rings of Hurwitz Spaces with Marked Ramification
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918013141254144 |
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| author | Clader, Emily Hu, Zhengning Larson, Hannah Li, Amy Q. Lopez, Rose |
| author_facet | Clader, Emily Hu, Zhengning Larson, Hannah Li, Amy Q. Lopez, Rose |
| contents | The Hurwitz space $\overline{\mathscr{H}}_{k,g}$ is a compactification of the space of smooth genus-$g$ curves with a simply-branched degree-$k$ map to $\mathbb{P}^1$. In this paper, we initiate a study of the Chow rings of these spaces, proving in particular that when $k=3$ (which is the first case in which the Chow ring is not already known), the codimension-2 Chow group is generated by the fundamental classes of codimension-2 boundary strata. The key tool is to realize the codimension-1 boundary strata of $\overline{\mathscr{H}}_{3,g}$ as the images of gluing maps whose domains are products of Hurwitz spaces $\mathscr{H}_{k',g'}(μ)$ with a single marked fiber of prescribed (not necessarily simple) ramification profile $μ$, and to prove that the spaces $\mathscr{H}_{k',g'}(μ)$ with $k'=2,3$ have trivial Chow ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chow Rings of Hurwitz Spaces with Marked Ramification Clader, Emily Hu, Zhengning Larson, Hannah Li, Amy Q. Lopez, Rose Algebraic Geometry 14H10, 14H30, 14C15, 14C17 The Hurwitz space $\overline{\mathscr{H}}_{k,g}$ is a compactification of the space of smooth genus-$g$ curves with a simply-branched degree-$k$ map to $\mathbb{P}^1$. In this paper, we initiate a study of the Chow rings of these spaces, proving in particular that when $k=3$ (which is the first case in which the Chow ring is not already known), the codimension-2 Chow group is generated by the fundamental classes of codimension-2 boundary strata. The key tool is to realize the codimension-1 boundary strata of $\overline{\mathscr{H}}_{3,g}$ as the images of gluing maps whose domains are products of Hurwitz spaces $\mathscr{H}_{k',g'}(μ)$ with a single marked fiber of prescribed (not necessarily simple) ramification profile $μ$, and to prove that the spaces $\mathscr{H}_{k',g'}(μ)$ with $k'=2,3$ have trivial Chow ring. |
| title | Chow Rings of Hurwitz Spaces with Marked Ramification |
| topic | Algebraic Geometry 14H10, 14H30, 14C15, 14C17 |
| url | https://arxiv.org/abs/2505.04903 |