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Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base C P 1
Author(s) -
Welschinger JeanYves
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013606
Subject(s) - mathematics , isotopy , surface (topology) , base (topology) , algebraic curve , algebraic number , combinatorics , space (punctuation) , pure mathematics , equivariant map , geometry , mathematical analysis , computer science , operating system
A first aim of this paper is to answer, in the case of real ruled surfaces X l of base C P 1 , with l ⩾ 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 provethat 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 + 2 k , 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.