Classification of foliations of degree three on $\mathbb{P}^{2}_{\mathbb{C}}$ with a flat Legendre transform

TL;DR

This paper classifies degree three foliations on the complex projective plane with flat Legendre transforms, identifying 16 distinct types and analyzing their geometric structure.

Contribution

It provides a complete classification of degree three foliations with flat Legendre transforms, revealing 16 types and 12 irreducible components.

Findings

16 foliations with flat Legendre transform up to automorphism

12 irreducible components of the set of such foliations

4 convex foliations of degree three identified

Abstract

The set $\mathbf{F}(3)$ of foliations of degree three on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $23$ on which acts $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$. The subset $\mathbf{FP}(3)$ of $\mathbf{F}(3)$ consisting of foliations of $\mathbf{F}(3)$ with a flat Legendre transform (dual web) is a Zariski closed subset of $\mathbf{F}(3)$. We classify up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ the elements of $\mathbf{FP}(3)$. More precisely, we show that up to automorphism there are $16$ foliations of degree three with a flat Legendre transform. From this classification we deduce that $\mathbf{FP}(3)$ has exactly $12$ irreducible components. We also deduce that up to automorphism there are $4$ convex foliations of degree three on $\mathbb{P}^{2}_{\mathbb{C}}.$