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

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 real trigonal Riemann surfaces

MATHEMATICA SCANDINAVICA, 2006

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 cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry σ is the number of connected components of the fixed point set Fix(σ) and the orientability of the Klein surface X/ σ. We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.

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.

Basis of homology adapted to the trigonal automorphism of a Riemann surface

2007

A closed (compact without boundary) Riemann surface S of genus g is said to be trigonal if there is a three sheeted covering (a trigonal morphism) from S to the Riemann sphere, f : S −→Ĉ. If there is an automorphism of period three, φ, on S permuting the sheets of the covering, we shall call S cyclic trigonal and φ will be called trigonal automorphism.

Maximal order of automorphisms of trigonal Riemann surfaces

Journal of Algebra, 2010

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 infinitely many genera in both the cyclic and the generic case.

On compact Riemann surfaces with dihedral groups of automorphisms

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

We study compact Riemann surfaces of genus g 2 having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group D N. The question of extendability of the action of D N is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

Riemann surface with cyclic automorphisms group

Proyecciones (Antofagasta), 1997

In t his paper. we present tllC' uniformization of y 2 = .rP-l, with p > 5 aurl prime. i. e .. the only hyperelliptic Riemann surface of gt'nus (/-7. \\"hich admit Z j2pZ as automorphism group. This 1111ifonnization is fouud by using a fuc:hsian group which rcflects the actiou of Z/2pZ aud is coustructed starting of a triangle group of !YJW (0:¡>.p.p). I\loreover. we describe completely the action of the automorphism group in hmnology. so that we can describe the invariant subvariety for Z /2pZ in A 9 (principally polarized abelian varieties of dimension y). which is detPrmiued bv the real Abe! aplication from M 9 in A 9 .

Triangulations and moduli spaces of Riemann surfaces with group actions

Manuscripta Mathematica, 1995

We study that subset of the moduli space Ma of stable genus g, g > 1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite group F acts. We show first that this subset is compact. It turns out that, for general finite groups F, the above subset is not connected. We show, however, that for Z2 actions this subset is connected. Finally, we show that even in the moduli space of smooth genus g Riemann surfaces, the subset of those Riemann surfaces on which Z2 acts is connected, ha view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certain regular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in apl)roximating eigenfunctions of tim Laplace operator numerically.