Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base C P 1

Jean-Yves Welschinger (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

A first aim of this paper is to answer, in the case of real ruled surfaces $X_l$ of base $\C P^1$, $l \geq 2$, to a question of V.A. Rokhlin : 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 exists 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 $m = l+ 2k$, $k>0$, the discriminants of the surface $X_m$ are deduced from those of the surface $X_l$ via weighted homotheties. All these results are obtained thanks to a study of a deformation of ruled surfaces.

Domaines: Géométrie algébrique [math.AG]

Jean-Yves Welschinger. Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base C P 1. Proceedings of the London Mathematical Society, 2002, 85 (2), pp.367-392. ⟨10.1112/S0024611502013606⟩. ⟨hal-02081358⟩