Generalized superelliptic Riemann surfaces
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2016
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910792600780800 |
|---|---|
| author | Hidalgo, Ruben A. Quispe, Saúl Shaska, Tony |
| author_facet | Hidalgo, Ruben A. Quispe, Saúl Shaska, Tony |
| contents | A closed Riemann surface $\mathcal X$, of genus $g \geq 2$, is called a generalized superelliptic curve of level $n \geq 2$ if it admits an order $n$ conformal automorphism $τ$ so that $\mathcal X/\langle τ\rangle$ has genus zero and $τ$ is central in ${\rm Aut}(\mathcal X)$; the cyclic group $H=\langle τ\rangle$ is called a generalized superelliptic group of level $n$ for $\mathcal X$. These Riemann surfaces are natural generalizations of hyperelliptic Riemann surfaces (when $n=2$). We provide an algebraic curve description of these Riemann surfaces in terms of their groups of automorphisms. Also, we observe that the generalized superelliptic group $H$ of level $n$ is unique, with the exception of a very particular family of exceptional generalized superelliptic Riemann surfaces for $n$ even. In particular, the uniqueness holds if either: (i) $n$ is odd or (ii) the quotient $\mathcal X/H$ has all its cone points of order $n$ (for instance, when $\mathcal X$ is a superelliptic curve of level $n$). In the non-exceptional case, we use this uniqueness property of its generalized superelliptic group $H$ to observe that the corresponding curves are definable over their fields of moduli if ${\rm Aut}(\mathcal X)/H$ is neither trivial or cyclic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1609_09576 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Generalized superelliptic Riemann surfaces Hidalgo, Ruben A. Quispe, Saúl Shaska, Tony Algebraic Geometry A closed Riemann surface $\mathcal X$, of genus $g \geq 2$, is called a generalized superelliptic curve of level $n \geq 2$ if it admits an order $n$ conformal automorphism $τ$ so that $\mathcal X/\langle τ\rangle$ has genus zero and $τ$ is central in ${\rm Aut}(\mathcal X)$; the cyclic group $H=\langle τ\rangle$ is called a generalized superelliptic group of level $n$ for $\mathcal X$. These Riemann surfaces are natural generalizations of hyperelliptic Riemann surfaces (when $n=2$). We provide an algebraic curve description of these Riemann surfaces in terms of their groups of automorphisms. Also, we observe that the generalized superelliptic group $H$ of level $n$ is unique, with the exception of a very particular family of exceptional generalized superelliptic Riemann surfaces for $n$ even. In particular, the uniqueness holds if either: (i) $n$ is odd or (ii) the quotient $\mathcal X/H$ has all its cone points of order $n$ (for instance, when $\mathcal X$ is a superelliptic curve of level $n$). In the non-exceptional case, we use this uniqueness property of its generalized superelliptic group $H$ to observe that the corresponding curves are definable over their fields of moduli if ${\rm Aut}(\mathcal X)/H$ is neither trivial or cyclic. |
| title | Generalized superelliptic Riemann surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/1609.09576 |