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

TL;DR

This paper proves that the dual web of a reduced convex pre-foliation on the complex projective plane is flat, extending previous results from invariant lines to more general curves.

Contribution

It establishes the flatness of the dual web for reduced convex pre-foliations on a2a2a2, generalizing earlier work limited to invariant lines.

Findings

Dual web of reduced convex pre-foliation is flat.

Generalization from invariant lines to arbitrary invariant curves.

Extends previous flatness results to broader class of pre-foliations.

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.