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

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 $\mathbb P^2_{\mathbb C}$.