Codimension one foliations of degree three on projective spaces

TL;DR

This paper classifies degree three codimension one foliations on projective spaces of dimension three or higher, identifying the number of irreducible components with and without rational first integrals.

Contribution

It extends previous results to higher dimensions and precisely counts the irreducible components of the foliation space.

Findings

18 irreducible components without rational first integrals

At least 6 irreducible components with rational first integrals

Structure theorem for degree three foliations on projective spaces

Abstract

We establish a structure theorem for degree three codimension one foliations on projective spaces of dimension $n\ge 3$, extending a result by Loray, Pereira, and Touzet for degree three foliations on $\mathbb P^3$. We show that the space of codimension one foliations of degree three on $\mathbb{P}^n$, $n\ge 3$, has exactly $18$ distinct irreducible components parameterizing foliations without rational first integrals, and at least $6$ distinct irreducible components parameterizing foliations with rational first integrals.