Foliations by curves with curves as singularities

[Feuilletages par des courbes ayant des courbes comme singularités]

Corrêa Jr, M. ; Fernández-Pérez, A. ; Nonato Costa, G. ; Vidal Martins, R.

Annales de l'Institut Fourier, Tome 64 (2014), p. 1781-1805 / Harvested from Numdam

Résumé

Soit $\mathcal{F}$ un feuilletage holomorphe unidimensionnel sur ${\mathbb{P}^n}$, dont les composantes du lieu singulier $\Sigma $ sont des courbes $C_i$ et des points $p_j$. On exprime le nombre de tels points $p_j$, comptés avec leurs multiplicités, en termes des invariants de $\mathcal{F}$ et $C_i$, en supposant que $\mathcal{F}$ est spécial le long des courbes $C_i$. En supposant qu’il n’y a qu’une seule composante de $\Sigma $ de dimension non nulle, on obtient aussi des résultats lorsque le feuilletage est déterminé par ses lieux singuliers.

Abstract

Let $\mathcal{F}$ be a holomorphic one-dimensional foliation on ${\mathbb{P}^n}$ such that the components of its singular locus $\Sigma $ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $\mathcal{F}$ and $C_i$, assuming that $\mathcal{F}$ is special along the $C_i$. Allowing just one nonzero dimensional component on $\Sigma $, we also prove results on when the foliation happens to be determined by its singular locus.

Détail

Publié le : 2014-01-01

MR 3329679 | Zbl 06387323

DOI : https://doi.org/10.5802/aif.2896
Classification:  32S65,  58K45
Mots clés: feuilletages holomorphes, singularités non-isolées

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