A structural theorem for codimension-one foliations on ${\mathbb{P}}^n$, $n\ge 3$, with an application to degree-three foliations

Let $ℱ$ be a codimension-one foliation on ${\mathbb{P}}^n$: for each point $p\in {\mathbb{P}}^n$ we define $𝒥(ℱ,p)$ as the order of the first non-zero jet $j_p^k\left(\omega \right)$ of a holomorphic 1-form $\omega$ defining $ℱ$ at $p$. The singular set of $ℱ$ is $sing\left(ℱ\right)=\{p\in {\mathbb{P}}^n|𝒥(ℱ,p)\ge 1\}$. We prove (main Theorem 1.2) that a foliation $ℱ$ satisfying $𝒥(ℱ,p)\le 1$ 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 $n\ge 3$, 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.