K-differentials with prescribed singularities
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912542014570496 |
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| author | Gendron, Quentin Tahar, Guillaume |
| author_facet | Gendron, Quentin Tahar, Guillaume |
| contents | We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g \geq 0$ can have for $k \geq 3$. These local invariants include the orders of zeros and poles, as well as the $k$-residues at the poles. We show that for a given pattern of orders of zeros, there exists, with a few exceptions, a primitive holomorphic $k$-differential having zeros of these orders. In the meromorphic case, for genus $g \geq 1$, every expected tuple appears as a configuration of $k$-residues. On the other hand, for certain strata in genus zero, finitely many tuples (up to simultaneous scaling) do not occur as configurations of $k$-residues for a $k$-differential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_11654 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | K-differentials with prescribed singularities Gendron, Quentin Tahar, Guillaume Complex Variables Algebraic Geometry Geometric Topology We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g \geq 0$ can have for $k \geq 3$. These local invariants include the orders of zeros and poles, as well as the $k$-residues at the poles. We show that for a given pattern of orders of zeros, there exists, with a few exceptions, a primitive holomorphic $k$-differential having zeros of these orders. In the meromorphic case, for genus $g \geq 1$, every expected tuple appears as a configuration of $k$-residues. On the other hand, for certain strata in genus zero, finitely many tuples (up to simultaneous scaling) do not occur as configurations of $k$-residues for a $k$-differential. |
| title | K-differentials with prescribed singularities |
| topic | Complex Variables Algebraic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2208.11654 |