Homogeneous Convex Foliations of degree 6

TL;DR

This paper classifies homogeneous convex foliations of degree 6 on the complex projective plane, focusing on those with specific singularities, and extends previous classifications for degrees 4 and 5.

Contribution

It provides a complete classification of convex homogeneous foliations of degree 6, including those with three radial singularities, building on earlier work for degrees 4 and 5.

Findings

Classified homogeneous convex foliations with three radial singularities.

Extended the classification of convex homogeneous foliations to degree 6.

Connected foliations to flat webs via Legendre transform.

Abstract

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 $5$.