On gap sets in arbitrary Kummer extensions of $K(x)$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866918069016723456 |
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| author | Cotterill, Ethan Mendoza, Erik A. R. Speziali, Pietro |
| author_facet | Cotterill, Ethan Mendoza, Erik A. R. Speziali, Pietro |
| contents | Let $K$ be an algebraically closed field, and let $F/K(x)$ be a Kummer extension of function fields of genus $g$. We provide a compact and explicit description of the gap set $G(Q)$ at any totally ramified place $Q$ of the extension $F/K(x)$. As a consequence, we deduce structural properties of the Weierstrass semigroup $H(Q)$; in particular, we determine a generating set for $H(Q)$, and we characterize its symmetry in certain cases. We also generalize a formula due to Towse that describes the asymptotic behavior of the sum of the Weierstrass weights at all totally ramified places of the extension $F/K(x)$ relative to $g^3-g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On gap sets in arbitrary Kummer extensions of $K(x)$ Cotterill, Ethan Mendoza, Erik A. R. Speziali, Pietro Algebraic Geometry Let $K$ be an algebraically closed field, and let $F/K(x)$ be a Kummer extension of function fields of genus $g$. We provide a compact and explicit description of the gap set $G(Q)$ at any totally ramified place $Q$ of the extension $F/K(x)$. As a consequence, we deduce structural properties of the Weierstrass semigroup $H(Q)$; in particular, we determine a generating set for $H(Q)$, and we characterize its symmetry in certain cases. We also generalize a formula due to Towse that describes the asymptotic behavior of the sum of the Weierstrass weights at all totally ramified places of the extension $F/K(x)$ relative to $g^3-g$. |
| title | On gap sets in arbitrary Kummer extensions of $K(x)$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2506.19169 |