${\mathbb Z}_{k}^{m}$-actions of signature $(0;k,\stackrel{n+1}{\ldots},k)$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914368416907264 |
|---|---|
| author | Hidalgo, Rubén A. Reyes-Carocca, Sebastián |
| author_facet | Hidalgo, Rubén A. Reyes-Carocca, Sebastián |
| contents | An action of a finite group $G$ is a pair $(S,\hat{G})$, where $S$ is a compact Riemann surface of genus $g \geqslant 2$ and $\hat{G} \leqslant {\rm Aut}(S)$ is isomorphic to $G$. To each action $(S,\hat{G})$ there is associated a signature $(γ;k_{1},\ldots,k_{r})$ that codifies the orbifold structure of $S/\hat{G}$. Two actions of $G$, say $(S_{1},G_{1})$ and $(S_{2},G_{2})$, are topologically equivalent if there is an orientation-preserving homeomorphism $φ:S_{1} \to S_{2}$ such that $φG_{1} φ^{-1}=G_{2}$. Topologically equivalent actions necessarily must have the same signature. The problem of determining the number of different topological actions of $G$ for a given signature is in general a difficult task. In this article, we describe, up to topological equivalence, those actions when $G$ is an abelian group and quotient genus $γ=0$. We are particularly interested in the case $G={\mathbb Z}_{k}^{m}$ and the quotient signature of the action to be of the form $(0;k,\stackrel{n+1}{\ldots},k)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | ${\mathbb Z}_{k}^{m}$-actions of signature $(0;k,\stackrel{n+1}{\ldots},k)$ Hidalgo, Rubén A. Reyes-Carocca, Sebastián Algebraic Geometry 30F10, 14H37, 30F35, 14H30 An action of a finite group $G$ is a pair $(S,\hat{G})$, where $S$ is a compact Riemann surface of genus $g \geqslant 2$ and $\hat{G} \leqslant {\rm Aut}(S)$ is isomorphic to $G$. To each action $(S,\hat{G})$ there is associated a signature $(γ;k_{1},\ldots,k_{r})$ that codifies the orbifold structure of $S/\hat{G}$. Two actions of $G$, say $(S_{1},G_{1})$ and $(S_{2},G_{2})$, are topologically equivalent if there is an orientation-preserving homeomorphism $φ:S_{1} \to S_{2}$ such that $φG_{1} φ^{-1}=G_{2}$. Topologically equivalent actions necessarily must have the same signature. The problem of determining the number of different topological actions of $G$ for a given signature is in general a difficult task. In this article, we describe, up to topological equivalence, those actions when $G$ is an abelian group and quotient genus $γ=0$. We are particularly interested in the case $G={\mathbb Z}_{k}^{m}$ and the quotient signature of the action to be of the form $(0;k,\stackrel{n+1}{\ldots},k)$. |
| title | ${\mathbb Z}_{k}^{m}$-actions of signature $(0;k,\stackrel{n+1}{\ldots},k)$ |
| topic | Algebraic Geometry 30F10, 14H37, 30F35, 14H30 |
| url | https://arxiv.org/abs/2510.14754 |