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COURBES ALGÉBRIQUES RÉELLES ET COURBES FLEXIBLES SUR LES SURFACES RÉGLÉES DE BASE [Copf ]P1

Published online by Cambridge University Press:  23 July 2002

JEAN-YVES WELSCHINGER
JEAN-YVES WELSCHINGER
Affiliation:
Institut de Recherche Mathématique Avancée, 7, rue René Descartes, 67084 Strasbourg Cedex, France. Adresse actuelle : École Normale Supérieure de Lyon, UMPA, 46, allée d'Italie, 69364 Lyon Cedex 07, France. [email protected]
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Abstract

A first aim of this paper is to answer, in the case of real ruled surfaces $X_l$ of base $\mathbb{C}P^1$, with $l \geq 2$, a question of V. A. Rokhlin: is it true that the equivariant isotopy class does not suffice to distinguish the connected components of the space of smooth real algebraic curves of $X_l$? A second aim is to prove that there exist in these surfaces some real schemes realized by real flexible curves but not by smooth real algebraic curves. These two results of real algebraic geometry are deduced from the following comparison theorem: when $m = l + 2k$, with $k > 0$, the discriminants of the surface $X_m$ are deduced from those of the surface $X_l$ via weighted homotheties. All these results are obtained from a study of a deformation of ruled surfaces.

The paper is written in French.

2000 Mathematical Subject Classification: 14H10, 14J26, 14P25.

Keywords

real algebraic curvediscriminantHirzebruch's surface
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 85 , Issue 2 , September 2002 , pp. 367 - 392
Copyright
2002 London Mathematical Society

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