Generic singularities of holomorphic foliations by curves on ${\mathbb{P}}^n$

Abstract

Let ${\mathcal{F}}_d({\mathbb{P}}^n)$ be the space of all singular holomorphic foliations by curves on ${\mathbb{P}}^n$ ($n\ge 2$) with degree $d\ge 1.$ We show that there is subset ${\mathcal{S}}_d({\mathbb{P}}^n)$ of ${\mathcal{F}}_d({\mathbb{P}}^n)$ with full Lebesgue measure with the following properties: 1. for every $\mathcal{F}\in {\mathcal{S}}_d({\mathbb{P}}^n),$ all singular points of $\mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d\ge 2,$ then every $\mathcal{F}$ does not possess any invariant algebraic curve.

20 pages