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.

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.

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.

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.

One-dimensional families of Riemann surfaces of genus g with 4\text {g}+4$$ 4 g + 4 automorphims

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2017

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 ≥ 2 we find an equisymmetric complex-uniparametric family A g of Riemann surfaces of genus g having automorphism group of order 4g + 4. For g ≡ −1mod4 we present another uniparametric family K g with automorphism group of order 4g + 4. The family A g contains the Accola-Maclachlan surface and the family K g contains the Kulkarni surface.

On Riemann surfaces of genus g with 4g–4 automorphisms

Israel Journal of Mathematics

In this article we study compact Riemann surfaces with a nonlarge 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 varieties.

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 .

On cyclic p-gonal Riemann surfaces with several p-gonal morphisms

Geometriae Dedicata, 2009

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 greater than (p − 1) 2 the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p−1) 2 which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.

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.