Platitude des tissus duaux de certains pr\'e-feuilletages convexes du plan projectif complexe

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 and the curve $\mathcal{C}$ is invariant by $\mathcal{F}$, we speak of convex pre-foliation. In a previous paper, we showed that if a foliation $\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is reduced convex or homogeneous convex and if $\mathcal{C}$ is an invariant line of $\mathcal{F}$, then the dual web of the convex pre-foliation $\mathcal{C}\boxtimes\mathcal{F}$ is flat. In this paper, we propose to extend this result to the case of a curve $\mathcal{C}$ consisting of several invariant lines.