Families of Klein surfaces
and real-etale rational fonctions

phD defended the 16 December 2004, abstract.
Author: Mathilde Lahaye-Hitier

The main topic of this thesis is the classification -- up to isotopy -- of real-etale rational functions of $\mathbb{P}^{\smash{1}}_{\mathbb{R}}=\mathbb{P}^{\smash{1}}$.

A real rational function is a fraction of two polynomes with real coefficients, or, equivalently, an endomorphism of $\mathbb{P}^{\smash{1}}$. We say that such a function is real-etale if it is unramified over the real points of $\mathbb{P}^{\smash{1}}$. As we will see later, these functions are interesting because of their link with M-surfaces. Our study is in relation with the article [EG02]of A. Eremenko and A. Gabrielov. They solve there a B. and M. Shapiro conjecture in dimension 1. . Therefore, they study the rational functions of $\mathbb{P}^{\smash{1}}$ with only real ramification points.

Looking at the real-etale rational functions up to homotopy, we may pass through rational functions that have ramification over real points. This is too rough a classification. That's why we rather study the real-etale rational functions up to isotopy. Two such functions are isotopes if it is possible to pass from one to the other by continuous deformations in the set of real-etale rational functions with same degree.
To give a more precise definition of the notion of isotopy, a first part of the thesis develops the theory of continuous families of Klein surfaces. Therefore, I take the view point of locally ringed spaces. In particular, it allows a more natural definition of morphisms of Klein surfaces than the one given in the classical theory. Moreover, it makes the work with families easier.In this study, I also prove a Riemann Existence Theorem for this families.

The main objects of the classification are the signed trees associated to a real-etale rational function. Topologically, an endomorphism of $\mathbb{P}^{\smash{1}}$ est un revêtement ramifié du disque fermé par lui-même. Une fonction rationnelle $f$ sur $\mathbb{P}^{\smash{1}}$ s real-etale if and only if the inverse image $f^{-1}\bigl(\mathbb{P}^{\smash{1}}(\mathbb{R})\bigr)$ of the set of real points is a disjoint union of topological circles in $\mathbb{C}$.
This circles are the vertices of the tree. The vertices are the connected components of $f^{-1}\bigl(\mathbb{P}^{\smash{1}}\setminus\mathbb{P}^{\smash{1}}(\mathbb{R})\bigr)$. An edge e is the extremity of a vertex v if the topological circle e is included in the closure of v dans $\mathbb{P}^{\smash{1}}$. Moreover, to each edge e is associated a weight which correspond to the degree of the restriction of $f$ restreint à e . An orientation on $\mathbb{P}^{\smash{1}}$ induces an orientation on its set of real points. We then add at the foot of the weighted tree of $f$ a "+" or "-" sign depending on whether $f$ respects or inverses respectively the orientation on $\mathbb{P}^{\smash{1}}(\mathbb{R})$. This way, we get the signed tree of $f$.
Conversely, to each signed tree may be associated a real-etale rational function.

Bibliographie

BEGG90

Emilio Bujalance, José J. Etayo, José M. Gamboa, and Grzegorz Gromadzki.
Automorphism Groups of Compact Bordered Klein Surfaces, volume 1439 of Lecture Notes in Mathematics.
Springer-Verlag, 1990.

EG02

A. Eremenko and A. Gabrielov.
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry.
Ann. of Math. (2), 155(1):105-129, 2002.

SS89

M. Seppälä and R. Silhol. Moduli spaces for real algebraic curves and real abelian varieties. Math. Z., 201(2):151-165, 1989.