Codimension one foliations of degree three on projective spaces

We establish a structure theorem for degree three codimension one foliations on projective spaces of dimension n >= 3, extending a result by Loray, Pereira, and Touzet for degree three foliations on P-3. We show that the space of codimension one foliations of degree three on P-n, n >= 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. (C) 2021 Elsevier Masson SAS. All rights reserved.