Homogeneous Convex Foliations of degree 6
Carla Pracias, Maycol Falla Luza
TL;DR
We address the classification of homogeneous convex foliations on the complex projective plane $\mathbb{P}^2$, and in particular provide a complete degree-$6$ census. The approach combines projective duality via the Legendre transform, which yields flat $d$-webs, with a detailed analysis of the inflection divisor and Gauss map for homogeneous foliations. The results include a classification of foliations with exactly three radial singularities on the line at infinity, a closed formula for the number $N(d)$ of foliations with $\deg\mathcal{T}_{\mathcal{H}}=3$, and explicit normal forms for all degree-$6$ cases organized by $\deg\mathcal{T}_{\mathcal{H}}=3,4,5,6$, together with an appendix detailing a SageMath computation. These contributions advance the understanding of convex foliations and their flat Legendre transforms in complex projective geometry, providing concrete models and computational tools for further study.
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 $5$.