A structural theorem for codimension-one foliations on $P^n, n ≥ 3$, with an application to degree-three foliations
Résumé
Let $\mathcal F$ be a codimension-one foliation on $\mathbb P^{n}$ : for each point $p\in \mathbb P^{n}$ we define $\mathcal J (\mathcal F,p)$ as the order of the first non-zero jet $j^{k}_{p}(\omega)$ of a holomorphic 1-form $\omega$ defining $\mathcal F$ at $p$. The singular set of $\mathcal F$ is $sing (\mathcal F)=\{p\in \mathbb P^{n} | \mathcal J (\mathcal F,p)\leq 1\}$. We prove (main Theorem 1.2) that a foliation $\mathcal F$ satisfying $\mathcal J (\mathcal F,p)\leq 1$ for all $p\in \mathbb P^{n}$ has a non-constant rational first integral. Using this fact we are able to prove that any foliation of degree-three on $\mathbb P^{n}$, with $n\geq 3$, is either the pull-back of a foliation on $\mathbb P^{2}$, or has a transverse affine structure with poles. This extends previous results for foliations of degree at most two.