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Real algebraic curves and real pseudoholomorphic curves in ruled surfaces

Abstract : This thesis is motivated by the study of real algebraic curves in the real projective plane and in rational geometrically ruled surfaces equipped with their standard real structure. We were especially interested in two particular problems. The ovals of a nonsingular curves in the projective plane of even degree are naturally divided in two disjoint sets : even ovals, contained in an even number of ovals, and odd ovals. Combining the Harnack and Petrovsky inequalities, one obtains an upper bound on the number of even ovals, and on the number of odd ovals with respect to the degree of the curve. We generalize here a previous construction of I. Itenberg, and show that this upper bound is asymptotically sharp. Almost all known prohibitions on the topology of real algebraic curves are still valid for a wider class of objects, real pseudoholomorphic curves. An open problem is the existence of a real scheme realizable by a nonsingular real pseudoholomorphic curve but not realizable by a nonsingular real algebraic curve of the same degree. In this thesis, we study real nonsingular algebraic and pseudoholomorphic symmetric curves of degree 7 in the real projective plane. In particular, we give several classifications and exhibit two real schemes realizable by dividing nonsingular real symmetric pseudoholomorphic curves of degree 7, but not realizable by such algebraic curves. Some results of this thesis are based on the techniques of dessins d'enfants. In real algebraic geometry these objects were first used by S. Yu. Orevkov. In particular, they allow one to answer the following question : does there exist two real polynomials P and Q of degree n such that the real roots of P, Q, and P+Q realize a given arrangement? Following Orevkov, we give a necessary and sufficient condition for the existence of two such polynomials, formulated in terms of dessins d'enfants. We also give an algorithm which gives a possibility to check whether a given L-scheme is realizable by a real trigonal algebraic curve.
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Contributor : Erwan Brugallé <>
Submitted on : Thursday, March 3, 2005 - 3:47:23 PM
Last modification on : Thursday, January 7, 2021 - 4:24:44 PM
Long-term archiving on: : Friday, April 2, 2010 - 9:37:52 PM


  • HAL Id : tel-00008652, version 1


Erwan Brugallé. Real algebraic curves and real pseudoholomorphic curves in ruled surfaces. Mathematics [math]. Université Rennes 1, 2004. English. ⟨tel-00008652⟩



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