arXiv subjects

Ruben A. Hidalgo

Publications and source records attributed to Ruben A. Hidalgo.

At least 37 records · Page 2Linked to original sources

arXiv2020

Sharp upper bounds for the number of fixed points components of two and three symmetries of handlebodies

An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodies. A group generated by two different extended Schottky groups, both with the same orientation-preserving half, is called a dihedral extended Schottky group. We provide a structural description of these type of groups and, as a consequence, we obtain sharp upper bounds for the sum of the cardinalities of the connected components of the locus of fixed points of two or three different symmetries of a handlebody.

math.GT↗

arXiv2020

Holomorphic differentials of Generalized Fermat curves

A non-singular complete irreducible algebraic curve $F_{k,n}$, defined over an algebraically closed field $K$, is called a generalized Fermat curve of type $(k,n)$, where $n, k \geq 2$ are integers and $k$ is relatively prime to the characteristic $p$ of $K$, if it admits a group $H \cong {\mathbb Z}_{k}^{n}$ of automorphisms such that $F_{k,n}/H$ is isomorphic to ${\mathbb P}_{K}^{1}$ and it has exactly $(n+1)$ cone points, each one of order $k$. By the Riemann-Hurwitz-Hasse formula, $F_{k,n}$ has genus at least one if and only if $(k-1)(n-1) >1$. In such a situation, we construct a basis, called an standard basis, of its space $H^{1,0}(F_{k,n})$ of regular forms, containing a subset of cardinality $n+1$ that provides an embedding of $F_{k,n}$ into ${\mathbb P}_{K}^{n}$ whose image is the fiber product of $(n-1)$ classical Fermat curves of degree $k$. For $p=2$, we obtain a lower bound (which is sharp for $n=2,3$) for the dimension of the space of the exact one-forms, that is, the kernel of the Cartier operator. Also, we done this for $p=3$, $k=2$ and $n=4$.

math.AG↗

arXiv2020

Geometric description of Virtual Schottky groups

A virtual Schottky group is a Kleinian group $K$ containing a Schottky group $G$ as a finite index normal subgroup. These groups correspond to those groups of automorphisms of closed Riemann surfaces which can be realized at the level of their Schottky uniformizations. In this paper we provides a geometrical structural decomposition of $K$ in the particular case when $K/G$ is an abelian group.

math.GT↗

arXiv2019

Hyperelliptic quotients of generalized Humbert curves

A group $H \cong {\mathbb Z}_{2}^{n}$, $n \geq 3$, of conformal automorphisms of a closed Riemann surface $S$ such that $S/H$ has genus zero and exactly $(n+1)$ cone points is called a generalized Humbert group of type $n$, in which case, $S$ is called a generalized Humbert curve of type $n$. It is known that a generalized Humbert curve $S$ of type $n \geq 4$ is non-hyperelliptic and that it admits a unique generalized Humbert group $H$ of type $n$. We describe those subgroups $K$ of $H$, acting freely on $S$, such that $S/K$ is hyperelliptic.

math.AG↗

arXiv2019

Jacobian varieties with many elliptic curves

In recent years there has been an interest in constructing examples of closed Riemann surfaces whose jacobian varieties are isogenous to a product of many elliptic factors and some other jacobian varieties. The first ones, provided by Ekedahl and Serre, are examples for which the isogenous decomposition has all factors being elliptic curves. It is well known that given two elliptic curves $E_{1}$ and $E_{2}$, there is a closed Riemann surface $X$ of genus two, with equations in terms of the elliptic curves, and whose jacobian variety $JX$ is isogenous to $E_{1} \times E_{2}$. In this paper, given $s \geq 3$ elliptic curves $E_{1},\ldots, E_{s}$, we provide an explicit construction of a closed Riemann surface $X$ of genus $g=1+2^{s-2}(s-2)$, with $JX$ isogenous to $E_{1} \times \cdots \times E_{s} \times A$, where $A$ is the product of some elliptic curves and jacobian varieties of hyperelliptic Riemann surfaces, all of them explicitly in terms of the given elliptic curves. In particular, for $s=3$, this provides explicit Riemann surface of genus three whose jacobian variety is isogenous to $E_{1} \times E_{2} \times E_{3}$, for given elliptic curves.

math.AG↗

arXiv2019

Automorphism groups of origami curves

A closed Riemann surface $S$ (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $β:S \to E$ with at most one branch value, where $E$ is a genus one Riemann surface. In this case, $(S,β)$ is called an origami pair and ${\rm Aut}(S,β)$ is the group of conformal automorphisms $ϕ$ of $S$ such that $β=β\circ ϕ$. Let $G$ be a finite group. It is a known fact that $G$ can be realized as a subgroup of ${\rm Aut}(S,β)$ for a suitable origami pair $(S,β)$. It is also known that $G$ can be realized as a group of conformal automorphisms of a Riemann surface $X$ of genus $g \geq 2$ and with quotient orbifold $X/G$ also of genus $γ\geq 2$. Given a conformal action of $G$ on a surface $X$ as before, we prove that there is an origami pair $(S,β)$, where $S$ has genus $g$ and $G \cong {\rm Aut}(S,β)$ such that the actions of ${\rm Aut}(S,β)$ on $S$ and that of $G$ on $X$ are topologically equivalent.

math.GT↗

arXiv2018

Automorphism groups of dessins d'enfants

Recently, Gareth Jones observed that every finite group $G$ can be realized as the group of automorphisms of some dessin d'enfant ${\mathcal D}$. In this paper, complementing Gareth's result, we prove that for every possible action of $G$ as a group of orientation-preserving homeomorphisms on a closed orientable surface of genus $g \geq 2$, there is a dessin d'enfant ${\mathcal D}$ admitting $G$ as its group of automorphisms and realizing the given topological action. In particular, this asserts that the strong symmetric genus of $G$ is also the minimum genus action for it to acts as the group of automorphisms of a dessin d'enfant of genus at least two.

math.CV↗

arXiv2018

On the fiber product of Riemann surfaces

Let $S_{0}, S_{1}$ and $S_{2}$ be connected Riemann surfaces and let $β_{1}:S_{1} \to S_{0}$ and $β_{2}:S_{2} \to S_{0}$ be surjective holomorphic maps. The associated fiber product $S_{1} \times_{(β_{1},β_{2})} S_{2}$ has the structure of a singular Riemann surface, endowed with a canonical map $β$ to $S_{0}$ satisfying that $β_{j} \circ π_{j}=β$, where $π_{j}$ is coordinate projection onto $S_{j}$. In this paper we provide a Fuchsian description of the fiber product and obtain that if one the maps $β_{j}$ is a regular branched cover, then all its irreducible components are isomorphic. In the case that both $β_{j}$ are of finite degree, we observe that the number of irreducible components is bounded above by the greatest common divisor of the two degrees; we study the irreducibility of the fiber product. In the case that $S_{0}=\widehat{\mathbb C}$, and $S_{1}$ and $S_{2}$ are compact, we define the strong field of moduli of the pair $(S_{1} \times_{(β_{1},β_{2})} S_{2},β)$ and observe that this field coincides with the minimal field containing the fields of moduli of both pairs $(S_{1},β_{1})$ and $(S_{2},β_{2})$. Finally, in the case that the fiber product is a connected Riemann surface, we provide an isogenous decomposition of its Jacobian variety.

math.AG↗

arXiv2018

On neoclassical Schottky groups

The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are "sufficiently complicated". We then show that every Schottky group in an appropriately defined relative conical neighborhood of any sufficiently complicated noded Schottky group is necessarily non-classical. Finally, we construct two examples; the first is a noded Riemann surface of genus $3$ that cannot be uniformized by any neoclassical Schottky group (i.e., classical noded Schottky group); the second is an explicit example of a sufficiently complicated noded Schottky group in genus $3$.

math.GT↗

arXiv2018

A sufficiently complicated noded Schottky group of rank three

The theoretical existence of non-classical Schottky groups is due to Marden. Explicit examples of such kind of groups are only known in rank two, the first one by by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to obtain examples of non-classical Schottky groups in any rank. The method assumes the knowledge of some algebraic limits of Schottky groups, called sufficiently complicated noded Schottky groups, whose existence was stated. In this paper we provide an explicit construction of a sufficiently complicated noded Schottky group of rank three and it is explained how to construct explicit non-classical Schottky groups of rank three.

math.GT↗

arXiv2017

On the connectedness of the set of Riemann surfaces with real moduli

The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann surfaces, is a complex orbifold of dimension $3(g-1)$ which carries a natural real structure i.e. it admits an anti-holomorphic involution $σ$. The involution $σ$ maps each point corresponding to a Riemann surface $S$ to its complex conjugate $\overline{S}$. The fixed point set of $σ$ consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside $\mathrm{Fix}(σ)$ is the locus ${\mathcal{M}}_{g}(\mathbb{R})$, the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Seppälä and R. Silhol. The complement $\mathrm{Fix}(σ)-{\mathcal{M}}_{g}(\mathbb{R})$ consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that $\mathrm{Fix}(σ)$ is connected.

math.CV↗

arXiv2017

Bipartite graphs and their dessins d'enfants

Each finite and connected bipartite graph induces a finite collection of non-isomorphic dessins d'enfants, that is, $2$-cell embeddings of it into some closed orientable surface. We describe an algorithm to compute all these dessins d'enfants, together their automorphims group, monodromy group and duality type.

math.CO↗

arXiv2017

About the Fricke-Macbeath curve

A Hurwitz curve is a closed Riemann surface of genus $g \geq 2$ whose group of conformal automorphisms has order $84(g-1)$. In 1895, Wiman proved that for $g=3$ there is, up to isomorphisms, a unique Hurwitz curve; this being Klein's plane quartic curve. Moreover, he also proved that there is no Hurwitz curve of genus $g=2,4,5,6$. Later, in 1965, Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus $g=7$; this known as the Fricke-Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a $4$-dimensional family of closed Riemann surfaces $S_μ$ admitting a group $G_μ \cong {\mathbb Z}_{2}^{3}$ of conformal automorphisms so that $S_μ/G_μ$ has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Wiman's description. We also observe that the jacobian variety of the surface $S_μ$ is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke-Macbeath curve, we obtain the well known fact that its jacobian variety is isogenous to $E^{7}$ for a suitable elliptic curve $E$.

math.AG↗

arXiv2017

Riemann surfaces defined over the reals

The known (explicit) examples of Riemann surfaces not definable over their field of moduli are not real whose field of moduli is a subfield of the reals. In this paper we provide explicit examples of real Riemann surfaces which cannot be defined over the field of moduli.

math.AG↗

arXiv2017

Real Structures on Marked Schottky Space

Schottky groups are exactly those Kleinian groups providing the regular lowest planar uniformizations of closed Riemann surfaces and also the ones providing to the interior of a handlebody of a complete hyperbolic structure with injectivity radius bounded away from zero. The space parametrizing quasiconformal deformations of Schottky groups of a fixed rank $g \geq 1$ is the marked Schottky space ${\mathcal M}{\mathcal S}_{g}$; this being a complex manifold of dimension $3(g-1)$ for $g \geq 2$ and being isomorphic to the punctured unit disc for $g=1$. In this paper we provide a complete description of the real structures of ${\mathcal M}{\mathcal S}_{g}$, up to holomorphic automorphisms, together their real part.

math.CV↗

arXiv2017

Uniformizations of stable $(γ,n)$-gonal Riemann surfaces

A $(γ,n)$-gonal pair is a pair $(S,f)$, where $S$ is a closed Riemann surface and $f:S \to R$ is a degree $n$ holomorphic map onto a closed Riemann surface $R$ of genus $γ$. If the signature of $(S,f)$ is of hyperbolic type, then there is pair $(Γ,G)$, called an uniformization of $(S,f)$, where $G$ is a Fuchsian group acting on the unit disc ${\mathbb D}$ containing $Γ$ as an index $n$ subgroup, so that $f$ is induced by the inclusion of $Γ<G$. The uniformization is uniquely determined by $(S,f)$, up to conjugation by holomorphic automorphisms of ${\mathbb D}$, and it permits to provide natural complex orbifold structures on the Hurwitz spaces parametrizing (twisted) isomorphic classes of pairs topologically equivalent to $(S,f)$. In order to produce certain compactifications of these Hurwitz spaces, one needs to consider the so called stable $(γ,n)$-gonal pairs, which are natural geometrical deformations of $(γ,n)$-gonal pairs. Due to the above, it seems interesting to search for uniformizations of stable $(γ,n)$-gonal pairs, in terms of certain class of Kleinian groups. In this paper we review such uniformizations by using noded Fuchsian groups, which are (geometric) limits of quasiconformal deformations of Fuchsian groups, and which provide uniformizations of stable Riemann orbifolds. These uniformizations permit to obtain a compactification of the Hurwitz spaces with a complex orbifold structure, these being quotients of the augmented Teichmüller space of $G$ by a suitable finite index subgroup of its modular group.

math.CV↗

arXiv2015

A remark on the decomposition of the Jacobian variety of Fermat curves of prime degree

Recently, Barraza-Rojas have described the action of the full automorphisms group on the Fermat curve of degree $p$, for $p$ a prime integer, and obtained the group algebra decomposition of the corresponding Jacobian variety. In this short note we observe that the factors in such a decomposition are given by the Jacobian varieties of certain $p$-gonal curves.

math.AG↗

arXiv2015

Connectivity of the branch locus of moduli space of rational maps

Milnor proved that the moduli space ${\rm M}_{d}$ of rational maps of degree $d \geq 2$ has a complex orbifold structure of dimension $2(d-1)$. Let us denote by ${\mathcal S}_{d}$ the singular locus of ${\rm M}_{d}$ and by ${\mathcal B}_{d}$ the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify ${\rm M}_2$ with ${\mathbb C}^2$ and, within that identification, that ${\mathcal B}_{2}$ is a cubic curve; so ${\mathcal B}_{2}$ is connected and ${\mathcal S}_{2}=\emptyset$. If $d \geq 3$, then ${\mathcal S}_{d}={\mathcal B}_{d}$. We use simple arguments to prove the connectivity of it.

math.DS↗