Pre-foliations of co-degree one on $\mathbb{P}^{2}_{\mathbb{C}}$ with a flat Legendre transform

TL;DR

This paper classifies and analyzes pre-foliations of co-degree 1 on the complex projective plane with flat Legendre transforms, identifying explicit examples and establishing flatness conditions for their dual webs.

Contribution

It provides a classification of homogeneous pre-foliations of co-degree 1 with flat dual webs, including explicit examples and flatness criteria, extending known results for foliations.

Findings

Identified two families and six examples of degree 3 pre-foliations with flat dual webs.

Proved flatness of dual webs for reduced convex pre-foliations of co-degree 1.

Extended results on foliations with flat Legendre transforms to pre-foliations.

Abstract

A holomorphic pre-foliation $\mathscr{F}=\ell\boxtimes\mathcal{F}$ of co-degree $1$ and degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a line $\ell$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation $\mathcal{F}$ on $\mathbb{P }^{2}_{\mathbb{C}}$ of degree $d-1.$ We study pre-foliations of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{ C}}$ with a flat Legendre transform (dual web). After having established some general results on the flatness of the dual $d$-web of a homogeneous pre-foliation of co-degree $1$ and degree $d$, we describe some explicit examples and we show that up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ there are two families and six examples of homogeneous pre-foliations of co-degree $1$ and degree $3$ on $\mathbb {P}^{2}_{\mathbb{C}}$ with a flat dual web. This allows us to prove an analogue for pre-foliations of co-degree $1$ and degree~$3$ of a result, obtained in collaboration with D. Mar\'{\i}n, on foliations of degree $3$ with non-degenerate singularities and a flat Legendre transform. We also show that the dual web of a reduced convex pre-foliation of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is flat. This is an analogue of a result on foliations of $\mathbb{P}^{2}_{\mathbb{C}}$ due to D. Mar\'{\i}n and J. V. Pereira.