Le tissu dual d'un pr\'e-feuilletage convexe r\'eduit sur $\mathbb{P}^{2}_{\mathbb{C}}$ est plat

Abstract

A holomorphic pre-foliation $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a reduced complex projective curve $\mathcal{C}$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation $\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$. When the foliation $\mathcal{F}$ is convex (resp. reduced convex) and the curve $\mathcal{C}$ is invariant by $\mathcal{F}$, we say that the pre-foliation $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}$ is convex (resp. reduced convex). We prove that the dual web of a reduced convex pre-foliation on $\mathbb{P}^{2}_{\mathbb{C}}$ is flat. This generalizes our previous result obtained in the case where the associated curve consists only of invariant lines.