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.
Categories: math.DS