arXiv subjects

Milagros Izquierdo

Publications and source records attributed to Milagros Izquierdo.

9 recordsLinked to original sources

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

Modular companions in planar one-dimensional equisymmetric strata

Consider, in the moduli space of Riemann surfaces of a fixed genus, the subset of surfaces with non-trivial automorphisms. Of special interest are the numerous subsets of surfaces admitting an action of a given finite group, $G$, acting with a specific signature. In a previous study we declared two Riemann surfaces to be \emph{modular companions} if they have topologically equivalent $G$ actions, and that their $G$ quotients are conformally equivalent orbifolds. In this article we present a geometrically-inspired measure to decide whether two modular companions are conformally equivalent (or how different), respecting the $G$ action. Along the way, we construct a moduli space for surfaces with the specified $G$ action and associated equivariant tilings on these surfaces. We specifically apply the ideas to planar, finite group actions whose quotient orbifold is a sphere with four cone points.

math.GT↗

arXiv2021

On families of Riemann surfaces with automorphisms

In this article we determine the maximal possible order of the automorphism group of the form $ag + b$, where $a$ and $b$ are integers, of a complex three and four-dimensional family of compact Riemann surfaces of genus $g$, appearing for all genus. In addition, we construct and describe explicit complex three and four-dimensional families possessing these maximal number of automorphisms.

math.AG↗

arXiv2020

Groups of automorphisms of Riemann surfaces and maps of genus $p+1$ where $p$ is prime

We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $ρ(g-1)$ for some integer $ρ\ge 1$, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for $ρ>6$, and of the first and third authors for $ρ=3, 4, 5$ and $6$. As a corollary we classify the orientably regular hypermaps (including maps) of genus $p+1$, together with the non-orientable regular hypermaps of characteristic $-p$, with automorphism group of order divisible by the prime $p$; this extends results of Conder, \v Sirá\v n and Tucker for maps.

math.AG↗

arXiv2019

A note on large automorphism groups of compact Riemann surfaces

Belolipetsky and Jones classified those compact Riemann surfaces of genus $g$ admitting a large group of automorphisms of order $λ(g-1)$, for each $λ>6,$ under the assumption that $g-1$ is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting $5(g-1)$ and $6(g-1)$ automorphisms, with $g-1$ a prime number. As a consequence, we obtain the classification of Riemann surfaces admitting a group of automorphisms of order $3(g-1)$, with $g-1$ a prime number. We also provide isogeny decompositions of their Jacobian varieties.

math.AG↗

arXiv2015

Isometric point-circle configurations on surfaces from uniform maps

We embed neighborhood geometries of graphs on surfaces as point-circle configurations. We give examples coming from regular maps on surfaces with maximum number of automorphisms for their genus and survey geometric realization of pentagonal geometries coming from Moore graphs. An infinite family of point-circle $v_4$ configurations on $p$-gonal surfaces with two $p$-gonal morphisms is given. The image of these configuration on the sphere under the two $p$-gonal morphisms is also described.

math.AG↗

arXiv2015

Cyclic curves over the reals

In this paper we study the automorphism groups of real curves admitting a regular meromorphic function $f$ of degree $p$, so called real cyclic $p$-gonal curves. When $p=2$ the automorphism groups of real hyperelliptic curves where given by Bujalance et al. in \cite{BCGG}.

math.CV↗

arXiv2013

Connecting $p$-gonal loci in the compactification of moduli space

Consider the moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g\geq 2$ and its Deligne-Munford compactification $\bar{\mathcal{M}_{g}}$. We are interested in the branch locus ${\mathcal{B}_{g}}$ for $g>2$, i.e., the subset of $\mathcal{M}_{g}$ consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in $\mathcal{M}_{g}$ but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for $g\geq 5$ the set of (cyclic) trigonal surfaces is connected in $\bar{\mathcal{M}_{g}}$. To do so we exhibit an explicit nodal surface that lies in the completion of every equisymmetric set of 3-gonal Riemann surfaces. For $p>3$ the connectivity of the $p$-gonal loci becomes more involved. We show that for $p\geq 11$ prime and genus $g=p-1$ there are one-dimensional strata of cyclic $p$-gonal surfaces that are completely isolated in the completion $\bar{\mathcal{B}_{g}}$ of the branch locus in $\bar{\mathcal{M}_{g}}$.

math.GT↗

arXiv2012

On isolated strata of p-gonal Riemann surfaces in the branch locus of moduli spaces

The moduli space $\mathcal{M}_{g}$ of compact Riemann surfaces of genus $g$ has orbifold structure, and the set of singular points of such orbifold is the \textit{branch locus} $\mathcal{B}_{g}$. Given a prime number $p \ge 7$, $\mathcal{B}_{g}$ contains isolated strata consisting of $p$-gonal Riemann surfaces for genera $g \ge {3(p-1)\over 2}$,that are multiple of ${p-1 \over 2}$. This is a generalization of the results obtained in \cite{[BCI1]} for pentagonal Riemann surfaces, and the results of \cite{[K]} and \cite{[CI3]} for zero- and one-dimensional isolated strata in the branch locus.

math.GT↗