# Transformations hyperboliques et courbes algebriques en genre 2 et 3

Abstract : The uniformization theorem of Poincaré-Koebe states that any smooth compact Riemann surface of genus $g>1$
is a quotient of the upper half-plane by a Fuchsian
group. On the other hand, a Riemann surface is also a complex
algebraic curve. In genus 2 and 3, these curves can always be
realized as plane curves, i.e as the set of zeros of a homogeneous
polynomial equation $P(x,y,z)=0$ with complex coefficients.

In this thesis we deal with the explicit link between these two
descriptions for surfaces of genus 2 and 3 with non-trivial automorphisms.

In genus 2, we first deal with surfaces having a non-trivial
involution. We describe the correspondence between the actions of two
groups, the first acting on the algebraic structures, and the second on
the hyperbolic structures of these surfaces. The relation between
these two groups enables us to interpret in terms of Dehn twists and
half-twists the links between the covers branched over the
same five distinct points of $\mathbb{P}^1(\mathbb{C})$. In particular, the
action of some Dehn twists can be read on the equations.
A similar study is done for surfaces having an order 3 automorphism.
We then study special algebraic families, in which the surfaces are
defined by a smaller number of parameters than those of the ambient spaces (but
not having necessarily more automorphisms).
We then deal with real families. We show in particular that the
various groups enable us to describe the algebraic and geometric links
between surfaces whose real components have different topological types.

In genus 3, we study the relations between the equations of
the four genus 3 double
covers of a genus 1 curve branched over four given points. We also
describe the relations between their hyperbolic structure when they
are tiled by two right-angled hyperbolic hexagons.
Mots-clés :
Document type :
Theses
Domain :
Complete list of metadata

https://tel.archives-ouvertes.fr/tel-00001154
Contributor : Aline Aigon <>
Submitted on : Wednesday, February 27, 2002 - 2:27:15 PM
Last modification on : Thursday, January 11, 2018 - 6:12:22 AM
Long-term archiving on: : Tuesday, September 11, 2012 - 4:55:11 PM

### Identifiers

• HAL Id : tel-00001154, version 1

### Citation

Aline Aigon. Transformations hyperboliques et courbes algebriques en genre 2 et 3. Mathématiques [math]. Université Montpellier II - Sciences et Techniques du Languedoc, 2001. Français. ⟨tel-00001154⟩

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