Codimension one foliations of degree three 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.