Finite isoresidual covers in strata of $k$-differentials
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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_ | 1866914071548264448 |
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| author | Chen, Dawei Gendron, Quentin Prado, Miguel Tahar, Guillaume |
| author_facet | Chen, Dawei Gendron, Quentin Prado, Miguel Tahar, Guillaume |
| contents | Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the $k$-factorial function, for the cardinality of each fiber. We prove this formula using intersection calculations on the multi-scale compactification of the strata of $k$-differentials. In special cases, we also give alternative proofs using flat geometry. Finally, we present an application to cone spherical metrics with dihedral monodromy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_01630 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite isoresidual covers in strata of $k$-differentials Chen, Dawei Gendron, Quentin Prado, Miguel Tahar, Guillaume Algebraic Geometry Combinatorics Geometric Topology Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the $k$-factorial function, for the cardinality of each fiber. We prove this formula using intersection calculations on the multi-scale compactification of the strata of $k$-differentials. In special cases, we also give alternative proofs using flat geometry. Finally, we present an application to cone spherical metrics with dihedral monodromy. |
| title | Finite isoresidual covers in strata of $k$-differentials |
| topic | Algebraic Geometry Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/2510.01630 |