On the Moduli Space of Cyclic Trigonal Riemann Surfaces of Genus 4 (original) (raw)
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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.
The Moduli Space of Surfaces with K
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Introduction 1 1. Surfaces with 2-divisible canonical divisor 4 2. Surfaces of type II and III b. 6 3. On rings associated to curves of genus 3 9 4. The family of deformations 12 The research of the authors was performed in the realm of the DFG SCHWER-PUNKT "Globale Methode in der komplexen Geometrie", and of the EAGER EEC Project. The third author was supported by the Schwerpunkt and by P.R.I.N. 2002 "Geometria delle variet algebriche" of M.I.U.R. and is a member of G.N.S.A.G.A. of I.N.d.A.M. .
Journal of the American Mathematical Society, 1990
Dedicated to Lipman Bers on the occasion of his seventy-fifth birthday TABLE OF CONTENTS O. Introduction and statement of main results 1. Horocyclic coordinates 2. The zw = t plumbing construction 3. The plumbing construction for an admissible graph 4. Deformation (TeichmiiUer) and moduli (Riemann) spaces 5. Torsion free terminal b-groups 6. One-dimensional deformation spaces 7. Deformation spaces for torsion free terminal b-groups 8. One-dimensional moduli spaces 9. Moduli spaces for torsion free terminal b-groups 10. Forgetful maps 11. Metrics on surfaces and their Teichmiiller spaces 12. Appendix I: Calculations in PSL(2, C) and SL(2, C) 13. Appendix II: A computer program for computing torsion free terminal bgroups 14. Appendix III: Independence of gluing on choice of annuli O. INTRODUCTION AND STATEMENT OF MAIN RESULTS This paper is concerned with the general problem of explicitly describing intrinsic parameters for Teichmiiller and Riemann spaces. Ideally, we want to be able to read off from a given Riemann surface its position in moduli space. Further, we want to attach various geometric and analytic objeCts such
A note on isolated points in the branch locus of the moduli space of compact Riemann surfaces
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 [K] has found isolated points in B g. He showed that they appear if and only if 2g + 1 is an odd prime distinct from 7. The aim of this paper is to find a geometrical explanation of this phenomenon using the fact that the isolated points are given by surfaces admitting anticonformal involutions (symmetries). The points in the Teichmüller space, corresponding to groups uniformizing surfaces with a symmetry, is a (non disjoint) union of submanifolds. We shall obtain that the isolated intersections of such submanifolds give us the isolated points in the branch loci. Also we prove that there are no isolated points in the moduli space of Klein surfaces which are not Riemann surfaces.
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.