
Milagros IzquierdoLinköping University | LiU · Department of Mathematics (MAI)
Milagros Izquierdo
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66
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Introduction
Milagros Izquierdo currently works at the Department of Mathematics (MAI), Linköping University. Milagros does research in Algebra and Geometry and Topology.
Skills and Expertise
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January 2002 - present
Publications
Publications (66)
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} i...
Schottky space , where is an integer, is a connected complex orbifold of dimension ; it provides a parametrization of the ‐conjugacy classes of Schottky groups of rank . The branch locus , 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 , then...
The complex orbifold structure of the moduli space of Riemann surfaces of genus g (\(g\ge 2\)) produces a stratification into complex subvarieties named equisymmetric strata. Each equisymmetric stratum is formed by the surfaces where the group of automorphisms acts in a topologically equivalent way. The Riemann surfaces in the equisymmetric strata...
Schottky space S g , where g ≥ 2 is an integer, is a connected complex orbifold of dimension 3(g − 1); it provides a parametrization of the PSL 2 (C)-conjugacy classes of Schottky groups Γ of rank g. The branch locus B g ⊂ S g , consisting of those conjugacy classes of Schottky groups being a finite index proper normal subgroup of some Kleinian gro...
The moduli space M g \mathcal {M}_{g} of surfaces of genus g ≥ 2 g\geq 2 is the space of conformal equivalence classes of closed Riemann surfaces of genus g g . This space is a complex, quasi-projective variety of dimension 3 g − 3 3g-3 . The singularity set of the moduli space, which is roughly the same as the branch locus , becomes increasingly c...
We classify compact Riemann surfaces of genus \(g\), where \(g-1\) is a prime \(p\), which have a group of automorphisms of order \(\rho(g-1)\) for some integer \(\rho\ge 1\), and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for \(\rho>6\), and of the first and...
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 m...
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 possessi...
We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $\rho(g-1)$ for some integer $\rho\ge 1$, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for $\rho>6$, and of the first and third author...
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 numb...
Belolipetsky and Jones classified those compact Riemann surfaces of genus $g$ admitting a large group of automorphisms of order $\lambda (g-1)$, for each $\lambda >6,$ 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 $6(g-1)$ automorphi...
Unfortunatelly an erratum appears in the foreword of this volume.
We prove that the maximal number \(ag+b\) of automorphisms of equisymmetric and complex-uniparametric families of Riemann surfaces appearing in all genera is \(4g+4\). For each integer \(g\ge 2\) we find an equisymmetric complex-uniparametric family \({\mathcal {A}}_{g}\) of Riemann surfaces of genus g having automorphism group of order \(4g+4\). F...
In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus (Formula presented.) with one boundary component is connected and in the case of non-orientable Klein surfaces it has (Formula presented.) components, if (Formula presented.) is odd, and (Formula presented.) components for even (Formula presented.). We...
Given a compact Riemann surface $X$ with an action of a finite group $G$, the group algebra $\QQ[G]$ provides an isogenous decomposition of its Jacobian variety $JX$, known as the group algebra decomposition of $JX$. We consider the set of equisymmetric Riemann surfaces $\mathcal{M}(2n-1, D_{2n}, \theta)$ for all $n\geq 2$. We study the group algeb...
Given a compact Riemann surface $X$ with an action of a finite group $G$, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety $JX$, known as the group algebra decomposition of $JX$. We consider the set of equisymmetric Riemann surfaces $\mathcal{M}(2n-1, D_{2n}, \theta)$ for all $n\geq 2$. We study the group algebra d...
We determine, for all genus the Riemann surfaces of genus g with exactly 4g automorphisms. For g≠ or 30, these surfaces form a real Riemann surface in the moduli space : the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the family. Furthermore we determine the topological...
We determine, for all genus $g\geq2$ the Riemann surfaces of genus $g$ with $4g$ automorphisms. For $g\neq$ $3,6,12,15$ or $30$, this surfaces form a real Riemann surface $\mathcal{F}_{g}$ in the moduli space $\mathcal{M}_{g}$: the Riemann sphere with three punctures. The set of real Riemann surfaces in $\mathcal{F}_{g}$ consists of three intervals...
We embed neighborhood geometries of graphs on surfaces as point-circle configurations. We give examples coming from regular maps on surfaces with a 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...
We construct isometric point-circle configurations on surfaces from uniform maps. This gives one geometric realisation in terms of points and circles of the Desargues configuration in the real projective plane, and three distinct geometric realisations of the pentagonal geometry with seven points on each line and seven lines through each point on t...
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 sur...
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}.
The moduli space ℳ g , of compact Riemann surfaces of genus g has orbifold structure since ℳ g is the quotient space of the Teichmüller space by the action of the mapping class group. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space we find the orbifold structur...
Let M be a handlebody of genus g >= 2. The space T(M), that parametrizes marked Kleinian structures on M up to isomorphisms, can be identified with the space MSg, of marked Schottky groups of rank g, so it carries a structure of complex manifold of finite dimension 3(g - 1). The space M(M) parametrizing Kleinian structures on M up to isomorphisms,...
Let (Formula presented.) be the moduli space of orientable Klein surfaces of genus (Formula presented.) boundary components (see Alling and Greenleaf in Lecture notes in mathematics, vol 219. Springer, Berlin, 1971; Natanzon in Russ Math Surv 45(6):53–108, 1990). The space (Formula presented.) has a natural orbifold structure with singular locus (F...
In this work we obtain the group of conformal and anticonformal automorphisms of real cyclic p-gonal Riemann surfaces, where p>=3 is a prime integer and the genus of the surfaces is at least (p-1)^2+1. We use Fuchsian and NEC groups, and cohomology of finite groups.
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 surf...
The moduli space Mg of compact Riemann surfaces of genus g has orbifold structure and the set of singular points of the orbifold is the branch locus Bg. In this article we show that Bg is connected for genera three, four, thirteen, seventeen, nineteen and fiftynine, and disconnected for any other genus. In order to prove this we use Fuchsian groups...
Let g be an integer ≥3 and let ℬ g ={X∈ℳ g :Aut(X)≠Id} be the branch locus of ℳ g , where ℳ g denotes the moduli space of compact Riemann surfaces of genus g. The structure of ℬ g is of substantial interest because ℬ g corresponds to the singularities of the action of the modular group on the Teichmüller space of surfaces of genus g (see [W. J. Har...
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 $g \ge {3(p-1)\over 2}$,tha...
Let g be an integer ≥ 3 and let Bg = {X ∈ MgAut(X) ≠ 1d}, where Mg denotes the moduli space of compact Riemann surfaces of genus g. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci corresponding to Riemann surfaces with automorphism gr...
Let p be a prime number, p>2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is great...
Let g be an integer >= 3 and let B(g) = {X is an element of M(g)vertical bar Aut(X) not equal 1(d)}, where M(g) denotes the moduli space of compact Riemann surfaces of genus g. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci correspon...
The moduli space \(
\mathcal{M}_g
\)
of compact Riemann surfaces of genus g has the structure of an orbifold and the set of singular points of such orbifold is the branch locus
\(
\mathcal{B}_g
\). In this article we present some results related with the topology of \(
\mathcal{B}_g
\). We study the connectedness of \(
\mathcal{B}_g
\) for g ≤ 8, t...
Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, complex analysis, fractal geometry, conformal dynamics, discrete groups, geometric group theory, algebraic curves and their moduli, various kinds of deformation theory, coding, thermodynamic formalism, and topology of three-dimensional manifolds. This colle...
In this paper we find the maximal order of an automorphism of a trigonal Riemann surface of genus g, g⩾5. We find that this order is smaller for generic than for cyclic trigonal Riemann surfaces, showing that generic trigonal surfaces have “less symmetry” than cyclic trigonal surfaces. Finally we prove that the maximal order is attained for infinit...
A closed Riemann surface which can be realized as a three-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann surface. Using the characterization of cyclic trigonality by Fuchsian g...
Hurwitz spaces are spaces of pairs where is a Riemann surface and a meromorphic
function. In this work, we study -dimensional Hurwitz spaces of meromorphic -fold functions with four branched points, three
of them fixed; the corresponding monodromy representation over each branched
point is a product of transpositions and the monodromy group
is the...
'Groups St Andrews 2005' was held in the University of St Andrews in August 2005 and this second volume of a two-volume book contains selected papers from the international conference. Four main lecture courses were given at the conference, and articles based on their lectures form a substantial part of the Proceedings. This volume contains the con...
Una superficie de Riemann que es una cubierta regular de 3 hojas de la esfera se llama cíclica trigonal, y la cubierta un morfismo trigonal. Accola probó que el morfismo trigonal es único si el género de la superficie es mayor o igual que 5. Costa-Izquierdo-Ying encontraron una familia de superficies de Riemann de género 4 cíclicas trigonales con v...
A closed Riemann surface X which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface X is called real trigonal if there is an anticonformal involution (symmetry) σ of X commuting with the trigonal morphism. If the trigonal morphism is a...
Let Γ be a non-Euclidean crystallographic group. Γ is said to be non-maximal if there exists a non-Euclidean crystallographic
group Γ′ such that Γ ≤ Γ′ and the dimension of the Teichmüller space of Γ equals the dimension of the Teichmüller space of
Γ′. The full list of such pairs of groups is computed in the case when Γ is non-normal in Γ′. The cor...
A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such
a covering will be called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface
is called cyclic trigonal Riemann surface. Accola showed that the trigonal morphism is unique for Riemann...
A closed Riemann surface X which can be realised as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. A p-gonal Riemann surface is called real p-gonal if there is an anticon- formal involution (symmetry) σ of X commuting with the p-gonal morphism. If the p-gonal morphism is a cyclic regu...
Let X be a Riemann surface. Two coverings and are said to be equivalent if p2=ϕp1 for some conformal homeomorphism . In this paper we determine, for each integer g⩾2, the maximum number ρR(g) of inequivalent ramified coverings between compact Riemann surfaces X→Y of degree 2, where X has genus g. Moreover, for infinitely many values of g, we comput...
It is well known that the functorial equivalence between pairs (X, σ) , where X is a Riemann surface which admits an antiholomorphic involution (symmetry) σ: X → X , and real algebraic curves. We shall refer to such Riemann surfaces as real Riemann surfaces, following Klein's terminology. We consider the sets M R g and M 2R g of real curves and rea...
Two projective nonsingular complex algebraic curves X and Y defined over the field R of real numbers can be isomorphic while their sets X(R) and Y(R) of R-rational points could be even non homeomorphic. This leads to the count of the number of real forms of a complex algebraic curve X, that is, those nonisomorphic real algebraic curves whose comple...
Let X be a Riemann surface of genus gÖg\sqrt g
+1) nonconjugate symmetries and, again, this bound is attained for infinitely many of g. Recently we have showed that a Riemann surface of even genus g admits at most four symmetries. Our aim here is to show, using NEC groups and combinatorial methods, that three nonconjugate symmetries of a surface of...
Natanzon proved that a Riemann surface X of genus g 2 has at most 2( p g + 1) conjugacy classes of symmetries, and this bound is attained for innitely many genera g. The aim of this note is to prove that a Riemann surface of even genus g has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even g as well. An...
Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of...
Let g be an integer ≥ 3 and let B g = {X ∈ M g | Aut X = e} , where M g denotes the moduli space of a compact Riemann surface. The geometric structure of B g is of substantial interest because B g corresponds to the singularities of the action of the modular group on the Teichmüller space of surfaces of genus g (see [H]). Surprisingly R.S. Kulkarni...
Let Fg be a compact Riemann surface of genus g. A symmetry S of Fg is an anticonformal involution acting on Fg . The fixed-point set of a symmetry is a collection of disjoint simple closed curves, called the mirrors of the symmetry. The number of mirrors |S| of a symmetry of a surface of genus g can be any integer k with 0 ≤ k ≤ g+1. However, if a...
We introduce a theory of hypermaps on surfaces with boundary. A topological hypermap can be associated to an algebraic hypermap which is a quintuple (G,Ω,c 1 ,c 2 ,c 3 ), where G is a group, generated by three involutions c 1 , c 2 and c 3 , that acts transitively on the set Ω. Conversely, the topological hypermap can be reconstructed from the alge...
Izquierdo är professor vid Matematiska institutionen, Linköpings universitet och ledamot av nationalkommitté för matematik och svenska kommitté för matematikutbildning. Hon forskar inom geometri och topologi. Hon har medverkat flera gånger som föreläsare i kurser om konst och matematik. Möbiusband Möbiusbandet är, topologiskt, en icke-orienterbar y...