Holomorphic differentials of alternating four covers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909852899475456 |
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| author | Bleher, Frauke M. Gonzalez, Margarita Bustos |
| author_facet | Bleher, Frauke M. Gonzalez, Margarita Bustos |
| contents | Suppose $k$ is an algebraically closed field of characteristic two, let $A_4$ be an alternating group on four letters, and let $H$ be the unique Sylow two-subgroup of $A_4$. Let $X$ be a smooth projective irreducible curve over $k$ with a faithful $A_4$-action such that the quotient curve $X/H$ is a projective line and the $H$-cover $X\to X/H$ is totally ramified, in the sense that it is ramified and every branch point is totally ramified. Under these assumptions, we determine the precise $kA_4$-module structure of the space of holomorphic differentials of $X$ over $k$. We show that there are infinitely many different isomorphism classes of indecomposable $kA_4$-modules that can occur as direct summands, and we give precise formulas for the multiplicities with which they occur. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Holomorphic differentials of alternating four covers Bleher, Frauke M. Gonzalez, Margarita Bustos Algebraic Geometry Number Theory Primary 11G20, Secondary 20C20, 14H05, 14G17 Suppose $k$ is an algebraically closed field of characteristic two, let $A_4$ be an alternating group on four letters, and let $H$ be the unique Sylow two-subgroup of $A_4$. Let $X$ be a smooth projective irreducible curve over $k$ with a faithful $A_4$-action such that the quotient curve $X/H$ is a projective line and the $H$-cover $X\to X/H$ is totally ramified, in the sense that it is ramified and every branch point is totally ramified. Under these assumptions, we determine the precise $kA_4$-module structure of the space of holomorphic differentials of $X$ over $k$. We show that there are infinitely many different isomorphism classes of indecomposable $kA_4$-modules that can occur as direct summands, and we give precise formulas for the multiplicities with which they occur. |
| title | Holomorphic differentials of alternating four covers |
| topic | Algebraic Geometry Number Theory Primary 11G20, Secondary 20C20, 14H05, 14G17 |
| url | https://arxiv.org/abs/2510.15743 |