
Sebastián Reyes-CaroccaUniversity of Chile · Departamento de Matemáticas
Sebastián Reyes-Carocca
Ph. D.
My webpage: https://sites.google.com/site/sreyescarocca/
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54
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Introduction
Geometry: Riemann surfaces, algebraic curves, Jacobians and abelian varieties. Moduli, automorphisms and related aspects.
Skills and Expertise
Publications
Publications (54)
This article deals with dihedral group actions on compact Riemann surfaces and the interplay between different geometric data associated to them. First, a bijective correspondence between geometric signatures and analytic representations is obtained. Second, a refinement of a result of Bujalance, Cirre, Gamboa and Gromadzki about signature realizat...
In this article, we consider certain irreducible subvarieties of the moduli space of compact Riemann surfaces determined by the specification of actions of finite groups. We address the general problem of determining which among them are non-normal subvarieties of the moduli space. We obtain several new examples of subvarieties with this property.
In this short note, we prove the existence of infinitely many pairwise nonisomorphic, non‐hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also find all compact Riemann surfaces with automorphism group acting transitively on the Weierstrass points, under the assumption that they are simple.
In this short note, we study the Jacobian variety of the Accola-Maclachlan curve of genus four and obtain explicitly its Poincaré isogeny decomposition. More precisely, we show that its Jacobian variety is isomor-phic to the product of two abelian surfaces that are simple, and provide explicitly a Riemann matrix for each one of the involved abelian...
In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is 3g and 3g + 3, where g is the genus. We prove that if g is odd (respecti...
In this article, we consider certain irreducible subvarieties of the moduli space of compact Riemann surfaces determined by the specification of actions of finite groups. We address the general problem of determining which among them are non-normal subvarieties of the moduli space.
In this short note, we prove the existence of infinitely many pairwise non-isomorphic non-hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also found all those compact Riemann surfaces with automorphism group acting transitively on the Weierstrass points, under the assumption that they are sim...
If S 0 , S 1 , S 2 S_{0}, S_{1}, S_{2} are connected Riemann surfaces, β 1 : S 1 → S 0 \beta _{1}:S_{1} \to S_{0} and β 2 : S 2 → S 0 \beta _{2}:S_{2} \to S_{0} are surjective holomorphic maps, then the associated fiber product S 1 × ( β 1 , β 2 ) S 2 S_{1} \times _{(\beta _{1},\beta _{2})} S_{2} has the structure of a one-dimensional complex analy...
In this article we classify compact Riemann surfaces of genus $1+q^2$ with a group of automorphisms of order $3q^2,$ where $q$ is a prime number. We also study decompositions of the corresponding Jacobian varieties.
In this article we classify compact Riemann surfaces of genus 1+q2 with a group of automorphisms of order 3q2, where q is a prime number. We also study decompositions of the corresponding Jacobian varieties.
In this article we consider Riemann surfaces and abelian varieties endowed with a group of automorphisms isomorphic to a generalized quaternion group. We provide isogeny decompositions of each abelian variety with this action, compute dimensions of the corresponding factors and provide conditions under which this decomposition is nontrivial. We the...
Let m≥6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 6$$\end{document} be an even integer. In this short note we prove that the Jacobian variety of a quasiplato...
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 extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex d-dimensional family of compact Riemann surfaces, where \(d \geqslant 1.\) We provide conditions under which these bounds are sharp. In addition, for the one-dimensional case, we construct and describe an expl...
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...
Let p and q be odd prime numbers. In this paper we study non-abelian pq-fold regular covers of the projective line, determine algebraic models for some special cases and provide a general isogeny decomposition of the corresponding Jacobian varieties. We also give a classification and description of the one-dimensional families of compact Riemann su...
Let \(p \geqslant 3\) be a prime number and let \(n \geqslant 0\) be an integer such that \(p-1\) divides n. In this short note, we construct a family of (p, n)-gonal Riemann surfaces of maximal genus \(2np+(p-1)^2\) with more than one (p, n)-gonal group.
Let $m \geqslant 6$ be an even integer. In this short note we prove that the Jacobian variety of a quasiplatonic Riemann surface with associated group of automorphisms isomorphic to $C_2^2 \rtimes_2 C_m$ admits complex multiplication. We then extend this result to provide a criterion under which the Jacobian variety of a quasiplatonic Riemann surfa...
In this article we study compact Riemann surfaces of genus g with an automorphism of prime order g + 1. The main result provides a classification of such surfaces. In addition, we give a description of them as algebraic curves, determine and realise their full automorphism groups and compute their fields of moduli. We also study some aspects of the...
Let p and q be odd prime numbers. In this paper we study non-abelian pq-fold regular covers of the projective line, determine algebraic models for some special cases and provide a general isogeny decomposition of the corresponding Ja-cobian varieties. We also give a classification and description of the one-dimensional families of compact Riemann s...
Let $p \geqslant 3$ be a prime number and let $n \geqslant 0$ be an integer such that $p-1$ divides $n.$ In this short note we construct a family of $(p,n)$-gonal Riemann surfaces of maximal genus $2np+(p-1)^2$ with more than one $(p,n)$-gonal group.
In this article we study compact Riemann surfaces with a non-large group of automorphisms of maximal order, namely, compact Riemann surfaces of genus g with a group of automorphisms of order 4g–4. Under the assumption that g–1 is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian v...
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...
In this article we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex $d$-dimensional family of compact Riemann surfaces, where $d \geqslant 1.$ We provide conditions under which these bounds are sharp and, in addition, for the one-dimensional case we construct and describe an ex...
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...
Let $S$ be a compact Riemann surface and let $H$ be a finite group. It is known that if $H$ acts on $S$ then there is a $H$-equivariant isogeny decomposition of the Jacobian variety $JS$ of $S,$ called the group algebra decomposition of $JS$ with respect to $H.$ If $S_1 \to S_2$ is a regular covering map, then it is also known that the group algebr...
It is known that the universal cover of compact Riemann surface is either the projective line, the complex plane or the unit disk. In this article we construct a very explicit family of complex surfaces that gives rise to uncountably many mutually non-biholomorphic universal covers. The slope of these surfaces, which are going to be total spaces of...
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...
In this article we study compact Riemann surfaces with a non-large group of automorphisms of maximal order; namely, compact Riemann surfaces of genus $g$ with a group of automorphisms of order $4g-4.$ Under the assumption that $g-1$ is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jaco...
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...
Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus $g \ge 2$ different from $3, 6, 12, 15$ and 30, with exactly $4g$ automorphisms form an equisymmetric one-dimensional family, denoted by $\mathcal{F}_g.$ In this paper, for every prime number $q \ge 5,$ we explore further properties of each Riemann surface...
Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus $g \ge 2$ different from $3, 6, 12, 15$ and 30, with exactly $4g$ automorphisms form an equisymmetric one-dimensional family, denoted by $\mathcal{F}_g.$ In this paper, for every prime number $q \ge 5,$ we explore further properties of each Riemann surface...
Let $S$ be a compact Riemann surface and let $H$ be a finite group. It is known that if $H$ acts on $S$ then there is a $H$-equivariant isogeny decomposition of the Jacobian variety $JS$ of $S,$ called the group algebra decomposition of $JS$ with respect to $H.$ If $S_1 \to S_2$ is a regular covering map, then it is also known that the group algebr...
Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g \ge 3.$ In this paper we prove the connectedness of the singular sublocus of $\mathcal{A}_g$ consisting of those abelian varieties which possess an involution different from $-id$.
Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g \ge 3.$ In this paper we prove the connectedness of the singular sublocus of $\mathcal{A}_g$ consisting of those abelian varieties which possess an involution different from $-id$.
It is well known that every closed Riemann surface S of genus g≥2, admitting a group G of conformal automorphisms so that S/G has triangular signature, can be defined over a finite extension of ℚ. It is interesting to know, in terms of the algebraic structure of G, if S can in fact be defined over ℚ. This is the situation if G is either abelian or...
In this short paper we generalise a theorem due to Kani and Rosen on decomposition of Jacobian varieties of Riemann surfaces with group action. This generalisation extends the set of Jacobians for which it is possible to obtain an isogeny decomposition where all the factors are Jacobians.
In this short paper we generalise a theorem due to Kani and Rosen on decomposition of Jacobian varieties of Riemann surfaces with group action. This generalisation extends the set of Jacobians for which it is possible to obtain an isogeny decomposition where all the factors are Jacobians.
Given non-constant holomorphic maps $\beta_{j}:S_{j} \to S_{0}$, $j=1,2$, between closed Riemann surfaces, there is associated its fiber product (in the set theoretical sense), which may or not be connected and when it is connected it may or not be irreducible. A Fuchsian group description of the irreducible components of the fiber product is given...
Let $S \to C$ be a Kodaira fibration. Here we show that whether or not the algebraic surface $S$ is defined over a number field depends only on the biholomorphic class of its universal cover.
As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$ can be described by a non-singular complex projective algebraic curve $C$. A field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may choose $C$ to be defined by polynomials in $D[x_0, \ldots, x_n]$. The field of moduli of $S$ is ${\mathbb R}$ if and on...
The computation of the field of moduli of a closed Riemann surface seems to be a very difficult problem and even more difficult is to determine if the field of moduli is a field of definition. In this paper we consider the family of closed Riemann surfaces of genus five admitting a group of conformal automorphisms isomorphic to ${\mathbb Z}_{2}^{4}...
It is well known that every closed Riemann surface $S$ of genus $g \geq 2$, admitting a group $G$ of conformal automorphisms so that $S/G$ has triangular signature, can be defined over a finite extension of ${\mathbb Q}$. It is interesting to know, in terms of the algebraic structure of $G$, if $S$ can in fact be defined over ${\mathbb Q}$. This is...
A consequence of the results of Bers and Griffiths on the uniformization of
complex algebraic varieties is that the universal cover of a family of Riemann
surfaces, with base and fibers of finite hyperbolic type, is a contractible
2-dimensional domain that can be realized as the graph of a holomorphic motion
of the unit disk.
In this paper we deter...
A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2-dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we deter...
Let S → C be a Kodaira fibration. Here we show that whether or not the algebraic surface S is defined over a number field depends only on the biholomorphic class of its universal cover.
The computation of the field of moduli of a closed Riemann surface seems to be a very difficult problem and even more difficult
is to determine whether the field of moduli is a field of definition. In this paper we consider the family of closed Riemann
surfaces S of genus five admitting a group H of conformal automorphisms isomorphic to ℤ42. It tur...
As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$ can be described by a non-singular complex projective algebraic curve $C$. A field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may choose $C$ to be defined by polynomials in $D[x_0, \ldots, x_n]$. The field of moduli of $S$ is ${\mathbb R}$ if and on...
Let ${\mathcal K}<{\mathcal L}$ be a finite Galois extension and let $X$ be
an algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem
provides sufficient conditions for $X$ to be definable over ${\mathcal K}$,
that is, the existence of a Galois decent datum for $X$ asserts the existence
of an algebraic variety $Y$, defined ove...