Displaying similar documents to “The polar curve of a foliation on ℙ 2 ”

Multiplicity of a foliation on projective spaces along an integral curve.

Julio García (1993)

Revista Matemática de la Universidad Complutense de Madrid

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We compute the global multiplicity of a 1-dimensional foliation along an integral curve in projective spaces. We give a bound in the way of Poincaré problem for a complete intersection curves. In the projective plane, this bound give us a bound of the degree of non irreducible integral curves in function of the degree of the foliation.

Holomorphic foliations by curves on ℙ 3 with non-isolated singularities

Gilcione Nonato Costa (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let ℱ be a holomorphic foliation by curves on ℙ 3 . We treat the case where the set Sing ( ℱ ) consists of disjoint regular curves and some isolated points outside of them. In this situation, using Baum-Bott’s formula and Porteuos’theorem, we determine the number of isolated singularities, counted with multiplicities, in terms of the degree of ℱ , the multiplicity of ℱ along the curves and the degree and genus of the curves.

Foliations in algebraic surfaces having a rational first integral.

Alexis García Zamora (1997)

Publicacions Matemàtiques

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Given a foliation in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case S = P some new counter-examples to the classic formulation of the Poincaré problem are presented. If S is a rational surface and has singularities of type (1, 1) or (1,-1) we prove that the general solution is a non-singular curve.

On dicritical foliations and Halphen pencils

Luís Gustavo Mendes, Paulo Sad (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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The aim of this article is to provide information on the number and on the geometrical position of singularities of holomorphic foliations of the projective plane. As an application it is shown that certain foliations are in fact Halphen pencils of elliptic curves. The results follow from Miyaoka’s semipositivity theorem, combined with recent developments on the birational geometry of foliations.