Feuilletages de degr\'e trois du plan projectif complexe ayant une transform\'ee de Legendre plate

Abstract

The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $(d+2)^2-2$ on which acts $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$. The subset $\mathbf{FP}(d)$ of $\mathbf{F}(d)$ consisting of foliations of $\mathbf{F}(d)$ with a flat Legendre transform (dual web) is a Zariski closed subset of $\mathbf{F}(d)$. In this dissertation we study foliations of $\mathbf{FP}(d)$ and we try to better understand the topological structure of $\mathbf{FP}(3)$. First, 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. 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. Second, we classify up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ the elements of $\mathbf{FP}(3)$. More precisely, we show that up to automorphism there are $16$ foliations of degree $3$ with a flat Legendre transform. From this classification we deduce that $\mathbf{FP}(3)$ has exactly $12$ irreducible components.