Convex foliations of degree 4 on the complex projective plane

Abstract

We show that up to automorphisms of ${\mathbb{P}}_{\mathbb{C}}^2$ there are $5$ homogeneous convex foliations of degree four on ${\mathbb{P}}_{\mathbb{C}}^2.$ Using this result, we give a partial answer to a question posed in $2013$ by D. {Marín} and J. {Pereira} about the classification of reduced convex foliations on~${\mathbb{P}}_{\mathbb{C}}^2.$