arXiv subjects

Ruben A. Hidalgo

Publications and source records attributed to Ruben A. Hidalgo.

At least 19 recordsLinked to original sources

arXiv2026

On $p$-gonal fields of definition

Let $S$ be a closed Riemann surface of genus $g \geq 2$ and $φ$ be a conformal automorphism of $S$, of prime order $p$ such that $S/\langle φ\rangle$ has genus zero. Let ${\mathbb K} \leq {\mathbb C}$ be a field of definition of $S$, that is, there is an irreducible curve $C$, defined over ${\mathbb K}$, whose Riemann surface structure is biholomorphic to $S$. We provide a simple argument for the existence of a field extension ${\mathbb F}$ of ${\mathbb K}$, of degree at most $2(p-1)$, for which $S$ is definable by a curve of the form $y^{p}=F(x) \in {\mathbb F}[x]$, in which case $φ$ corresponds to $(x,y) \mapsto (x,e^{2 πi/p} y)$. If, moreover, $φ$ is also definable over ${\mathbb K}$, then ${\mathbb F}$ can be chosen to be a quadratic extension of ${\mathbb K}$. For $p=2$, that is when $S$ is hyperelliptic and $φ$ is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and Lercier-Ritzenthaler-Sijslingit in the case that ${\rm Aut}(S)/φ\rangle$ is non-trivial.

math.AG↗

arXiv2026

Cyclic-Schottky strata of Schottky space

Schottky space ${\mathcal S}_{g}$, where $g \geq 2$ is an integer, is a connected complex orbifold of dimension $3(g-1)$; it provides a parametrization of the ${\rm PSL}_{2}({\mathbb C})$-conjugacy classes of Schottky groups $Γ$ of rank $g$. The branch locus ${\mathcal B}_{g} \subset {\mathcal S}_{g}$, consisting of those conjugacy classes of Schottky groups being a finite index proper normal subgroup of some Kleinian group, is known to be connected. If $[Γ] \in {\mathcal B}_{g}$, then there is a Kleinian group $K$ containing $Γ$ as a normal subgroup of index some prime integer $p \geq 2$. The structural description, in terms of Klein-Maskit Combination Theorems, of such a group $K$ is completely determined by a triple $(t,r,s)$, where $t,r,s \geq 0$ are integers such that $g=p(t+r+s-1)+1-r$. For each such a tuple $(g,p;t,r,s)$ there is a corresponding cyclic-Schottky stratum $F(g,p;t,r,s) \subset {\mathcal B}_{g}$. It is known that $F(g,2;t,r,s)$ is connected.In this paper, for $p \geq 3$, we study the connectivity of these $F(g,p;t,r,s)$.

math.GT↗

arXiv2025

Algebraic models of cyclic $k$-gonal curves

In this paper, we describe explicit algebraic equations of tame cyclic $k$-gonal curves, where $k \geq 2$ is an integer, reflecting the action of the normalizer of a tame cyclic $k$-gonal automorphism. For $k$ a prime integer, this was previously done by A. Wootton.

math.AG↗

arXiv2025

Generalized Fermat Riemann surfaces of infinite type

The Loch Ness monster (LNM) is, up to homeomorphisms, the unique orientable, connected, Hausdorff, second countable surface of infinite genus and with exactly one end. For each integer $k \geq 2$, we construct Riemann surface structures $S$ on the LNM admitting a group of conformal automorphisms $H \cong {\mathbb Z}_{k}^{\mathbb N}$ such that $S/H$ is planar. These structures can be described algebraically inside the projective space ${\mathbb P}^{\mathbb N}$ after deleting some limit points.

math.GT↗

arXiv2025

Generalized superelliptic Riemann surfaces

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.

math.AG↗

arXiv2024

Automorphisms of Generalized Fermat manifolds

Let $d \geq 1$, $k \geq 2$ and $n\geq d+1$ be integers. A $d$-dimensional smooth complex algebraic variety $M$ is called a generalized Fermat variety of type $(d;k,n)$ if there is a Galois holomorphic branched covering $π:M \to {\mathbb P}^{d}$, with deck group $H\cong {\mathbb Z}_{k}^{n}$, whose branch divisor consists of $n+1$ hyperplanes in general position, each one of branch order $k$. In this case, $H$ is called a generalized Fermat group of type $(d;k,n)$. In previous work, we proved that the generalized Fermat group $H$ is unique in the following cases: (i) $d=1$ and $(k-1)(n-1)>2$, or (ii) $d \geq 2$ and $(d;k,n) \notin \{(2;2,5), (2;4,3)\}$. To obtain this uniqueness fact, we used a differential method due to Kontogeorgis. This paper provides a different and shorter proof of the uniqueness of $H$. We also study the locus of fixed points of subgroups of $H$.

math.AG↗

arXiv2024

On the FOD/FOM parameter of rational maps

Let $χ$ be a (right) action of ${\rm PSL}_{2}({\mathbb L})$ on the space ${\mathbb L}(z)$ of rational maps defined over an algebraically closed field ${\mathbb L}$. If $R \in {\mathbb L}(z)$ and ${\mathcal M}_{R}^χ$ is its $χ$-field of moduli, then the parameter ${\rm FOD/FOM}_χ(R)$ is the smallest integer $n \geq 1$ such that there is a $χ$-field of definition of $R$ being a degree $n$ extension of ${\mathcal M}_{R}^χ$. When ${\mathbb L}$ has characteristic zero and $χ=χ_{\infty}$ is the conjugation action, then it is known that ${\rm FOD/FOM}_{χ_{\infty}}(R) \leq 2$. In this paper, we study the above parameter for general actions and any characteristic.

math.DS↗

arXiv2024

A simple remark on holomorphic maps on Torelli space of marked spheres

The configuration space of $k \geq 3$ ordered points in the Riemann sphere $\widehat{\mathbb C}$ is the Torelli space ${\mathcal U}_{0,k}$; a complex manifold of dimension $k-3$. If $m,n \geq 4$ and $F:{\mathcal U}_{0,m} \to {\mathcal U}_{0,n}$ is a non-constant holomorphic map, then we observe that (i) $n \leq m$ and (ii) each coordinate of $F$ is given by a cross-ratio.

math.CV↗

arXiv2024

Computing the Field of moduli of some non-hyperelliptic pseudo-real curves

The explicit computation of the field of moduli of a closed Riemann surface is, in general, a difficult task. In this paper, for each even integer $k \geq 2$, we consider a suitable $2$-real parameter family of non-hyperelliptic pseudo-real Riemann surfaces of genus $g=1+(2k-3)k^{4}$. For each of them, we compute its field of moduli and also a minimal field of definition.

math.AG↗

arXiv2023

$(g,k)$-Fermat curves: an embedding of moduli spaces

A group $H \cong {\mathbb Z}_{k}^{2g}$, where $g,k \geq 2$ are integers, of conformal automorphisms of a closed Riemann surface $S$ is called a $(g,k)$-Fermat group if it acts freely with quotient $S/H$ of genus $g$. We study some properties of these type of objects, in particular, we observe that $S$ is non-hyperelliptic and, if $k=p^{r}$, where $p>84(g-1)$ is a prime integer and $r \geq 1$, then $H$ is the unique $(g,k)$-Fermat group of $S$. Let $Γ$ be a co-compact torsion free Fuchsian group such that $S/H={\mathbb H}^{2}/Γ$. If $Γ_{k}$ is its normal subgroup generated by its commutators and the $k$-powers of its elements, then there is a biholomorphism between $S$ and ${\mathbb H}^{2}/Γ_{k}$ congugating $H$ to $Γ/Γ_{k}$. The inclusion $Γ_{k} < Γ$ induces a natural holomorphic embedding $Θ_{k}:{\mathcal T}(Γ) \hookrightarrow {\mathcal T}(Γ_{k})$ of the corresponding Teichmüller spaces. Such an embedding induces a holomorphic map, at the level of their moduli spaces, $Φ_{k}:{\mathcal M}(Γ) \to {\mathcal M}(Γ_{k})$. As a consequence of the results on $(g,k)$-Fermat groups, we provide sufficient conditions for the injectivity of $Φ_{k}$.

math.CV↗

arXiv2023

On quasiconformal equivalence of Schottky regions

In a recent paper, H. Shiga proved that the regions of discontinuity of any two Schottky groups of ranks at least two are quasiconformally equivalent. In this paper, we provide an alternative proof of such a fact. Our approach permits us to discuss quasiconformality equivalence of regions of discontinuity of Schottky type groups in terms of their signatures.

math.CV↗

arXiv2022

Quadrangular ${\mathbb Z}_{p}^{l}$-actions on Riemann surfaces

Let $p \geq 3$ be a prime integer and, for $l \geq 1$, let $G \cong {\mathbb Z}_{p}^{l}$ be a group of conformal automorphisms of some closed Riemann surface $S$ of genus $g \geq 2$. By the Riemann-Hurwitz formula, either $p \leq g+1$ or $p=2g+1$. If $l=1$ and $p=2g+1$, then $S/G$ is the sphere with exactly three cone points and, if moreover $p \geq 7$, then $G$ is the unique $p$-Sylow subgroup of ${\rm Aut}(S)$. If $l=1$ and $p=g+1$, then $S/G$ is the sphere with exactly four cone points and, if moreover $p \geq 13$, then $G$ is again the unique $p$-Sylow subgroup. The above unique facts permited many authors to obtain algebraic models and the corresponding groups ${\rm Aut}(S)$ in these situations. Now, let us assume $l \geq 2$. If $p \geq 5$, then either (i) $p^{l} \leq g-1$ or (ii) $S/G$ has genus zero, $p^{l-1}(p-3) \leq 2(g-1)$ and $2 \leq l \leq r-1$, where $r \geq 3$ is the number of cone points of $S/G$. Let us assume we are in case (ii). If $r=3$, then $l=2$ and $S$ happens to be the classical Fermat curve of degree $p$, whose group of automorphisms is well known. The next case, $r=4$, is studied in this paper. We provide an algebraic curve representation for $S$, a description of its group of conformal automorphisms, a discussion of its field of moduli and an isogenous decomposition of its jacobian variety.

math.AG↗

arXiv2022

Groups as automorphisms of dessins d'enfants

It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin d'enfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compact dessins. Moreover, we show that any tame action of a countable group is so realizable.

math.GR↗

arXiv2022

A structural description of extended ${\mathbb Z}_{2n}$-Schottky groups

Real points of Schottky space ${\mathcal S}_{g}$ are in correspondence with extended Kleinian groups $K$ containing, as a normal subgroup, a Schottky group $Γ$ of rank $g$ such that $K/Γ\cong {\mathbb Z}_{2n}$ for a suitable integer $n \geq 1$. These kind of groups are called extended ${\mathbb Z}_{2n}$-Schottky groups of rank $g$. In this paper, we provide a structural decomposition theorem, in terms of Klein-Maskit's combination theorems, of these kind of groups.

math.GT↗

arXiv2022

Structural description of dihedral extended Schottky groups and application in study of symmetries of handlebodies

Given a symmetry $τ$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $Γ$ uniformizing $S$, such that $K/Γ$ induces $\langle τ\rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of the Klein-Maskit combination theorems, of both Schottky and extended Schottky groups is well known. A dihedral extended Schottky group is a group generated by the elements of two different extended Schottky groups, both with the same orientation-preserving half. Such configuration of groups corresponds to closed Riemann surfaces together with two different symmetries and the aim of this paper is to provide a geometrical structure of them. This result can be used in study of three dimensional manifolds and as an illustration we give the sharp upper bounds for the total number of connected components of the locus of fixed points of two and three different symmetries of a handlebody with a Schottky structure.

math.GT↗

arXiv2022

On $p$-gonal fields of definition

Let $S$ be a closed Riemann surface of genus $g \geq 2$ and $φ$ be a conformal automorphism of $S$ of prime order $p$ such that $S/\langle φ\rangle$ has genus zero. Let ${\mathbb K} \leq {\mathbb C}$ be a field of definition of $S$. We provide an argument for the existence of a field extension ${\mathbb F}$ of ${\mathbb K}$, of degree at most $2(p-1)$, for which $S$ is definable by a curve of the form $y^{p}=F(x) \in {\mathbb F}[x]$, in which case $φ$ corresponds to $(x,y) \mapsto (x,e^{2 πi/p} y)$. If, moreover, $φ$ is also definable over ${\mathbb K}$, then ${\mathbb F}$ can be chosen to be at most a quadratic extension of ${\mathbb K}$. For $p=2$, that is when $S$ is hyperelliptic and $φ$ is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and Lercier-Ritzenthaler-Sijslingit in the case that ${\rm Aut}(S)/φ\rangle$ is non-trivial.

math.AG↗

arXiv2021

On Real and Pseudo-Real Rational Maps

The moduli space ${\rm M}_{d}$, of complex rational maps of degree $d \geq 2$, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps admitting antiholomorphic automorphisms. The locus of the real points ${\rm M}_{d}({\mathbb R})$ decomposes as a disjoint union of the loci ${\rm M}_{d}^{\mathbb R}$, consisting of the real rational maps, and ${\mathcal P}_{d}$, consisting of the pseudo-real ones. We obtain that, both ${\rm M}_{d}^{\mathbb R}$ and ${\rm M}_{d}({\mathbb R})$, are connected and that ${\mathcal P}_{d}$ is disconnected. We also observe that the group of holomorphic automorphisms of a pseudo-real rational map is either trivial or a cyclic group. For every $n \geq 1$, we construct pseudo-real rational maps whose group of holomorphic automorphisms is cyclic of order $n$. As the field of moduli of a pseudo-real rational map is contained in ${\mathbb R}$, these maps provide examples of rational maps which are not definable over their field of moduli. It seems that these are the only explicit examples in the literature (Silverman) of rational maps which cannot be defined over their field of moduli. We provide explicit examples of real rational maps which cannot be defined over their field of moduli. Finally, we also observe that every real rational map, which admits a model over the algebraic numbers, can be defined over the real algebraic numbers.

math.DS↗

arXiv2021

The structure of extended function groups

A function group is a finitely generated Kleinian group with an invariant connected component of its region of discontinuity. An extended function group is a finitely generated extended Kleinian group that contains orientation reversing elements and keep invariant a connected components of its region of discontinuity. An structural decomposition of function groups, in terms of the Klein-Maskit combination theorems, was provided by Maskit in the middle of the 70's. One should expect a similar decomposition structure for extended function groups, but it seems not to be stated in the existing literature. The aim of this paper is to state and procvide a proof of such a decomposition structural picture.

math.CV↗