Homogeneous Convex Foliations of degree 6

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 $5$.