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.

Maple code uploaded as ancillary files. Final version. To appear in Bulletin des Sciences Mathématiques