A study of H. Martens' Theorem on chains of cycles
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912353987067904 |
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| author | Coppens, Marc |
| author_facet | Coppens, Marc |
| contents | Let $Γ$ be a chain of cycles of genus $g$. Let $d$,$r$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-3+r$. Then $w^r_d(Γ)=d-2r$ implies $Γ$ is hyperelliptic. For each $g \geq 2r+3$ there exist non-hyperelliptic chains of cycles satisfying $w^r_{g-2+r}(Γ)=g-2-r$. In the case of algebraic curves such equality implies the curve is hyperelliptic. In particular we obtain the existence of chains of cycles $Γ$ such that $w^r_{g-2+r}(Γ) \neq w^1_{g-r}(Γ)$ in case $r \geq 2$. In the case of algebraic curves such numbers are equal because of the Riemann-Roch Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10804 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A study of H. Martens' Theorem on chains of cycles Coppens, Marc Combinatorics Algebraic Geometry 05C25, 14T15 Let $Γ$ be a chain of cycles of genus $g$. Let $d$,$r$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-3+r$. Then $w^r_d(Γ)=d-2r$ implies $Γ$ is hyperelliptic. For each $g \geq 2r+3$ there exist non-hyperelliptic chains of cycles satisfying $w^r_{g-2+r}(Γ)=g-2-r$. In the case of algebraic curves such equality implies the curve is hyperelliptic. In particular we obtain the existence of chains of cycles $Γ$ such that $w^r_{g-2+r}(Γ) \neq w^1_{g-r}(Γ)$ in case $r \geq 2$. In the case of algebraic curves such numbers are equal because of the Riemann-Roch Theorem. |
| title | A study of H. Martens' Theorem on chains of cycles |
| topic | Combinatorics Algebraic Geometry 05C25, 14T15 |
| url | https://arxiv.org/abs/2312.10804 |