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

TL;DR

This paper extends previous results on the flatness of dual webs associated with convex pre-foliations on the complex projective plane, specifically when the invariant curve is composed of multiple lines.

Contribution

It generalizes earlier findings by proving the flatness of dual webs for convex pre-foliations with invariant curves made up of several lines.

Findings

Dual webs are flat when the invariant curve has multiple lines.

Extension of flatness results from single invariant lines to multiple lines.

Supports broader understanding of convex pre-foliations in complex geometry.

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 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.