Preprint A635/2009
A structural theorem for codimension one foliations on $\mathbb{P}^n$, $n\ge3$, with an application to degree three foliations.

Dominique Cerveau | Lins Neto, Alcides

**Keywords: **
Foliation | degree three

Let $\fa$ be a codimension one foliation on $\p^n$ : to each point $p\in\p^n$ we set
$\jj(\fa,p)=$ the order of the first non-zero jet $j^k_p(\om)$ of a holomorphic 1-form $\om$ defining $\fa$ at $p$. The singular set of $\fa$ is $sing(\fa)=\{p\in\p^n\,|\,\jj(\fa,p)\ge1\}$. We prove (main theorem \ref{t:ip}) that a foliation $\fa$ satisfying $\jj(\fa,p)\le1$ for all $p\in\p^n$ has a non-constant rational first integral. Using this fact, we are able to prove that any foliation of degree three on $\p^n$, $n\ge3$, is either the pull-back of a foliation on $\p^2$, or has a transverse affine structure with poles.
This extends previous results for foliations of degree $\le2$.