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

Mol, Rogério S.

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

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 ${\mathbb{P}}^2$. On démontre que, pour un centre $l\in {\mathbb{P}}^2$ 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.

We study some properties of the polar curve $P_l^{ℱ}$ associated to a singular holomorphic foliation $ℱ$ on the complex projective plane ${\mathbb{P}}^2$. We prove that, for a generic center $l\in {\mathbb{P}}^2$, 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.

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