Displaying similar documents to “Holomorphic foliations in ℂ ℙ ( 2 ) having an invariant algebraic curve”

The polar curve of a foliation on ℙ 2

Rogério S. Mol (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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We study some properties of the polar curve P l ℱ associated to a singular holomorphic foliation ℱ on the complex projective plane ℙ 2 . We prove that, for a generic center l ∈ ℙ 2 , the curve P l ℱ is irreducible and its singular points are exactly the singular points of ℱ with vanishing linear part. We also obtain upper bounds for the algebraic multiplicities of the singularities of ℱ and for its number of radial singularities.

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.

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.