On the Moduli Space of Cyclic Trigonal Riemann Surfaces of Genus 4 (original) (raw)

Equisymmetric Strata of the Moduli Space of Cyclic Trigonal Riemann Surfaces of Genus 4

Glasgow Mathematical Journal, 2008

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 groups, we find the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

Cyclic Trigonal Riemann Surfaces of Genus 4

2004

A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus g ≥ 5. This thesis will characterize the Riemann surfaces of genus 4 with non-unique trigonal morphism. We will describe the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

On the Connectedness of the Branch Locus of the Moduli Space of Riemann Surfaces of Genus 4

Glasgow Mathematical Journal, 2010

Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

On the moduli space of singular euclidean surfaces

Handbook of Teichmüller Theory, Volume I

The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.

Analytic computation of some automorphism groups of Riemann surfaces

Kodai Mathematical Journal, 2007

Equations for the locus of Riemann Surfaces of genus three with a nonabelian automorphism group generated by involutions are determined from vanishings of Riemann's theta function. Torelli's Theorem implies that all of the properties of a non-hyperelliptic compact Riemann Surface (complex algebraic curve) X are determined by its period matrix W. This paper shows how to compute the group Aut X of conformal automorphisms of a surface X of genus three using W, in the case when the group is nonabelian and generated by its involutions. The connection between W and X is Riemann's theta function yðz; WÞ. Accola ([1], [2], [3]), building on classical results about hyperelliptic surfaces, found relationships between the theta divisor Y ¼ fz A JacðX Þ : yðz; WÞ ¼ 0g and Aut X. In the case of genus three, certain vanishings of y at quarter-periods of JacðX Þ imply that X has an automorphism s of degree two (or involution) such that X =hsi has genus one (making s an elliptic-hyperelliptic involution). This work derives equations in the moduli space of surfaces of genus three for many of the loci consisting of surfaces with a given automorphism group. It is a two-step process. First, topological arguments determine the order of the dihedral group generated by two non-commuting involutions. Then, combinatorial arguments about larger groups generated by involutions determine the theta vanishings corresponding to each. Much of the work here is based on the author's 1981 PhD dissertation [7] at Brown University. It appears now because of renewed interest in these questions, some of which is inspired by questions in coding theory: See [3], [5]. The research was directed by R. D. M. Accola, and Joe Harris was also a valuable resource. The author extends his (belated) thanks to them. 1. Preliminaries and notation In all that follows, X is a compact Riemann Surface (or complex algebraic curve) of genus three with automorphism group Aut X , period matrix W, jacobian 394

Riemann Surfaces, Plane Algebraic Curves and Their Period Matrices

Journal of Symbolic Computation, 1998

The aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program Cars ) that can be used to de ne Riemann surfaces for computations. Cars allows one also to perform the Fenchel{Nielsen twist and other deformations on Riemann surfaces.

On the orbifold structure of the moduli space of Riemann surfaces of genera four and five

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2013

The moduli space M g , of compact Riemann surfaces of genus g has orbifold structure since M g is the quotient space of the Tiechmüller 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 structure of the moduli spaces of Riemann surfaces of genera 4 and 5.