In this paper, we study homogeneous convex foliations on the complex projective plane \(\mathbb {P}^2\) . A foliation is called convex if all of its leaves, except straight lines, have no inflection points, and such foliations form a Zariski closed subset in the space of degree d foliations on \(\mathbb {P}^2\) . Using projective duality, every foliation can be associated with a d-web on the dual plane via its Legendre transform, and it is known that the Legendre transform of a homogeneous convex foliation is flat. Our first main result provides a classification of homogeneous convex foliations admitting exactly three radial singularities on the line at infinity. As a second result, we complete the classification of convex homogeneous foliations of degree 6, extending previous classifications in degrees 4 and 5.
