Tissus plats et feuilletages homog\`enes sur le plan projectif

TL;DR

This paper investigates flat Legendre transforms of foliations on the complex projective plane, providing criteria for flatness, explicit examples, and classification results for foliations with flat dual webs.

Contribution

It establishes effective criteria for flatness of dual webs of homogeneous foliations and classifies certain foliations with flat Legendre transforms.

Findings

Identified 11 homogeneous degree 3 foliations with flat dual webs.

Developed criteria for flatness of dual webs of homogeneous foliations.

Connected flatness of dual webs to the homogeneous framework under certain conditions.

Abstract

The aim of this work is to study the foliations on the complex projective plane with flat \textsc{Legendre} transform (dual web). We establish some effective criteria for the flatness of the dual $d$-web of a homogeneous foliation of degree $d$ and we describe some explicit examples. These results allow us to show that up to automorphism of $\mathbb{P}^2$ there are $11$ homogeneous foliations of degree $3$ with flat dual web. We will see also that it is possible, under certain assumptions, to bring the study of flatness of the dual web of a general foliation to the homogeneous framework. We get some classification results about foliations with non-degenerate singularities and flat \textsc{Legendre} transform.