The polar curve of a foliation on $\mathbb{P}^2$

Rogério S. Mol

doi:10.5802/afst.1268

The polar curve of a foliation on

ℙ^{2}

Rogério S. Mol ¹

¹ Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627 C.P. 702, 30123-970 - Belo Horizonte - MG, BRAZIL

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 849-863.

Abstract
Abstract

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 $l \in ℙ^{2}$ , 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 étudie dans cet article quelques propriétés de la courbe polaire $P_{l}^{ℱ}$ associée à un feuilletage holomorphe singulier $ℱ$ dans le plan projectif complexe $ℙ^{2}$ . On démontre que, pour un centre $l \in ℙ^{2}$ générique, la courbe $P_{l}^{ℱ}$ est irréductible et ses points singuliers sont précisément les points singuliers de $ℱ$ avec partie linéaire nulle. On obtient aussi des bornes supérieurs pour la multiplicité algébrique des singularités de $ℱ$ et pour son nombre de singularités radiales.

MR Zbl

DOI: 10.5802/afst.1268

Author's affiliations:

Rogério S. Mol ¹

¹ Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627 C.P. 702, 30123-970 - Belo Horizonte - MG, BRAZIL

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     author = {Rog\'erio S. Mol},
     title = {The polar curve of a foliation on $\mathbb{P}^2$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {849--863},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
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     year = {2010},
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Rogério S. Mol. The polar curve of a foliation on $\mathbb{P}^2$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 849-863. doi : 10.5802/afst.1268. https://afst.centre-mersenne.org/articles/10.5802/afst.1268/

References
Cited by

[1] Bodin (A.), Débes (P.), and Najib (S.).— Irreducibility of hypersurfaces. Comm. Algebra, 37(6):1884-1900, (2009). | MR | Zbl

[2] Brieskorn (E.) and Knörrer (H.).— Plane algebraic curves. Birkhäuser Verlag, Basel, (1986). | MR | Zbl

[3] Camacho (C.), Lins Neto (A.), and Sad (P.).— Topological invariants and equidesingularization for holomorphic vector fields. J. Differential Geom., 20(1):143-174, (1984). | MR | Zbl

[4] Campillo (A.) and Olivares (J.).— Polarity with respect to a foliation and Cayley- Bacharach theorems. J. Reine Angew. Math., 534:95-118, (2001). | MR | Zbl

[5] Corral (N.).— Sur la topologie des courbes polaires de certains feuilletages singuliers. Ann. Inst. Fourier (Grenoble), 53(3):787-814, (2003). | Numdam | MR | Zbl

[6] Corral (N.).— Infinitesimal adjunction and polar curves. Bull. Braz. Math. Soc. (N.S.), 40(2):181-224, (2009). | MR | Zbl

[7] Corral (N.).— Polar pencil of curves and foliations. Astérisque, (323):161-179, (2009). | MR

[8] Gómez-Mont (X.), Seade (J.), and Verjovsky (A.).— The index of a holomorphic ow with an isolated singularity. Math. Ann., 291(4):737-751, (1991). | MR | Zbl

[9] Griffiths (P.) and Harris (J.).— Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons Inc., New York, (1994). | MR | Zbl

[10] Mol (R.S.).— Classes polaires associées aux distributions holomorphes de sous-espaces tangents. Bull. Braz. Math. Soc. (N.S.), 37(1):29-48, (2006). | MR | Zbl

[11] Schinzel (A.).— Polynomials with special regard to reducibility, volume 77 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, (2000). | MR | Zbl

[12] Wall (C.T.C.).— Singular points of plane curves, volume 63 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, (2004). | MR | Zbl

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