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Real algebraic curves and flexible curves on ruled surfaces of base $$\mathbb{C} \mathbb{P}^1$$. (Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base $$\mathbb{C} \mathbb{P}^1$$.) (French) Zbl 1016.14019
A first aim of this paper is to answer, in the case of real ruled surfaces $$X_l$$ of base $$\mathbb{C}\mathbb{P}^1$$, $$l \geq 2$$, a question of V. A. Rokhlin [Usp. Mat. Nauk 33, No. 5(203), 77-89 (1978; Zbl 0437.14013)]: 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.

##### MSC:
 14J26 Rational and ruled surfaces 14H10 Families, moduli of curves (algebraic) 14P25 Topology of real algebraic varieties
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