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

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.