Une nouvelle d\'emonstration de la classification des feuilletages convexes de degr\'{e} deux sur $\mathbb P^2_{\mathbb C}$

TL;DR

This paper provides a new, simpler proof for classifying convex degree 2 holomorphic foliations on the complex projective plane, avoiding previous real polynomial vector field results and remaining within the holomorphic context.

Contribution

It introduces a novel proof method based on properties of convex foliation models and the discriminant of the dual web, simplifying prior classification approaches.

Findings

New proof of classification of convex degree 2 foliations

Avoids reliance on real polynomial vector field results

Utilizes properties of dual web discriminants

Abstract

A holomorphic foliation on $\mathbb P^2_{\mathbb C}$, or a real analytic foliation on $\mathbb{P}^{2}_{\mathbb{R}},$ is said to be convex if its leaves other than straight lines have no inflection points. The classification of the convex foliations of degree $2$ on $\mathbb P^2_{\mathbb C}$ has been established in $2015$ by C.~\textsc{Favre} and J.~\textsc{Pereira}. The main argument of this classification was a result obtained in~$2004$ by~D.~\textsc{Schlomiuk} and N.~\textsc{Vulpe} concerning the real polynomial vector fields of degree $2$ whose associated foliation on $\mathbb{P}^{2}_{\mathbb{R}}$ is convex. We present here a new proof of this classification, that is simpler, does not use this result and does not leave the holomorphic framework. It is based on the properties of certain models of convex foliations of $\mathbb P^2_{\mathbb C}$ of arbitrary degree and of the discriminant of the dual web of a foliation of $\mathbb P^2_{\mathbb C}$.