Boundary stratifications of Hurwitz spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908472094752768 |
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| author | Glynn, Darragh |
| author_facet | Glynn, Darragh |
| contents | Let $\mathcal{H}$ be a Hurwitz space that parametrises holomorphic maps to $\mathbb{P}^1$. Abramovich, Corti and Vistoli, building on work of Harris and Mumford, describe a compactification $\overline{\mathcal{H}}$ with a natural boundary stratification. We show that the irreducible strata of $\overline{\mathcal{H}}$ are in bijection with combinatorial objects called decorated trees (up to a suitable equivalence), and that containment of irreducible strata is given by edge contraction of decorated trees. This combinatorial description allows us to define a tropical Hurwitz space, which we identify with the skeleton of the Berkovich analytification of $\overline{\mathcal{H}}$. The tropical Hurwitz space that we obtain is a refinement of a version defined by Cavalieri, Markwig and Ranganathan. We also provide an implementation that computes the stratification of $\overline{\mathcal{H}}$, and discuss applications to complex dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05688 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary stratifications of Hurwitz spaces Glynn, Darragh Geometric Topology Algebraic Geometry Dynamical Systems 14H10 (Primary), 14T99, 37F34, 57K20 (Secondary) Let $\mathcal{H}$ be a Hurwitz space that parametrises holomorphic maps to $\mathbb{P}^1$. Abramovich, Corti and Vistoli, building on work of Harris and Mumford, describe a compactification $\overline{\mathcal{H}}$ with a natural boundary stratification. We show that the irreducible strata of $\overline{\mathcal{H}}$ are in bijection with combinatorial objects called decorated trees (up to a suitable equivalence), and that containment of irreducible strata is given by edge contraction of decorated trees. This combinatorial description allows us to define a tropical Hurwitz space, which we identify with the skeleton of the Berkovich analytification of $\overline{\mathcal{H}}$. The tropical Hurwitz space that we obtain is a refinement of a version defined by Cavalieri, Markwig and Ranganathan. We also provide an implementation that computes the stratification of $\overline{\mathcal{H}}$, and discuss applications to complex dynamics. |
| title | Boundary stratifications of Hurwitz spaces |
| topic | Geometric Topology Algebraic Geometry Dynamical Systems 14H10 (Primary), 14T99, 37F34, 57K20 (Secondary) |
| url | https://arxiv.org/abs/2503.05688 |