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

TL;DR

This paper proves that for most holomorphic foliations by curves on projective space, all singularities are linearizable hyperbolic, and for degree at least 2, these foliations lack invariant algebraic curves.

Contribution

It establishes that a full measure subset of such foliations has hyperbolic linearizable singularities and no invariant algebraic curves for degree ≥ 2.

Findings

Most singular points are linearizable hyperbolic.

Foliations of degree ≥ 2 have no invariant algebraic curves.

Full measure subset with these properties exists.

Abstract

Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ 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, $d \geq 2,$ then every $\mathcal{F}$ does not possess any invariant algebraic curve.