A new proof of the classification of convex foliations of degree two on the complex projective plane [Une nouvelle démonstration de la classification des feuilletages convexes de degré deux sur P2C]

Abstract

A holomorphic foliation on P2C, or a real analytic foliation on P2R, 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 P2C has been established in 2015 by C. Favre and J. Pereira. The main argument of this classification was a result obtained in 2004 by D. Schlomiuk and N. Vulpe concerning the real polynomial vector fields of degree 2 whose associated foliation on P2R 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 P2C of arbitrary degree and of the discriminant of the dual web of a foliation of P2C.