×

Real geometry of dessins d’enfant. (Géométrie réelle des dessins d’enfant.) (French) Zbl 1078.14089

A known construction is the following: to each “dessin d’enfant” (in the sense of Grothendieck), one can assign a ramified covering (topological or algebraic) of the complex projective line \(\mathbb {P}^1 (\mathbb{C})\), non ramified over \(\mathbb{P}^1 (\mathbb{C})\setminus\{0,1,\infty\}\). Then one can take the preimage of the real projective line \(\mathbb{P}^1 (\mathbb{R})\) by this covering. In this paper, the author defines an algebraic curve \(\Gamma\) over \(\mathbb{ R}\) such that the set of real points of \(\Gamma\) is precisely the preimage considered above. Then, the author proceeds to study some properties of the curve \(\Gamma\), such as, what are the irreducible components of \(\Gamma\) like (absolutely irreducible or not) and, whether these components do have real points or not. The paper contains two appendices: one describes the Weil restriction of \(\mathbb{C}\) to \(\mathbb{R}\) and the second one presents the notion and properties of “dessins d’enfant”. Some examples are given.

MSC:

14P25 Topology of real algebraic varieties
14P05 Real algebraic sets
14H30 Coverings of curves, fundamental group
57M12 Low-dimensional topology of special (e.g., branched) coverings
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21, Springer-Verlag, (1990). · Zbl 0705.14001
[2] J.-M. Couveignes, Calcul et rationalité de fonctions de Belyi en genre 0. Annales de l’Institut Fourier, 44 (1) (1994), p. 1-38. · Zbl 0791.11059
[3] P. Dèbes, J.-C. Douai, Algebraic covers : field of moduli versus field of definition. Ann. scient. Éc. Norm. Sup, tome 30 (1997), p. 303-338. · Zbl 0906.12001
[4] A. Grothendieck, Éléments de Géométrie Algébrique : I, Le langage des schémas. Institut des Hautes Études Scientifiques, Publications Mathématiques 4 (1960). · Zbl 0118.36206
[5] A. Grothendieck, Éléments de Géométrie Algébrique : II, Étude globale élémentaire de quelques classes de morphismes. Institut des Hautes Études Scientifiques, Publications Mathématiques 8 (1961). · Zbl 0118.36206
[6] A. Grothendieck, Éléments de Géométrie Algébrique : IV, Étude locale des schémas et morphismes de schémas (Première Partie). Institut des Hautes Études Scientifiques, Publications Mathématiques 20 (1964). · Zbl 0136.15901
[7] A. Grothendieck, Éléments de Géométrie Algébrique : IV, Étude locale des schémas et morphismes de schémas (Seconde Partie). Institut des Hautes Études Scientifiques, Publications Mathématiques 24 (1965). · Zbl 0135.39701
[8] A. Grothendieck, Éléments de Géométrie Algébrique : IV, Étude locale des schémas et morphismes de schémas (Quatrième Partie). Institut des Hautes Études Scientifiques, Publications Mathématiques 32 (1967). · Zbl 0153.22301
[9] J. Oesterlé, Revêtements de \({\mathbf{P}}_1\moinsgras\lbrace 0,1,∞ \rbrace \). Notes du cours de DEA prises par A. Kraus, année 1993-1994, non publiées.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.