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

Abstract

A holomorphic pre-foliation $\mathcal{F}=\ell ⊠\mathcal{F}$ of co-degree $1$ and degree $d$ on ${\mathbb{P}}_{\mathbb{C}}^2$ is the data of a line $\ell$ of ${\mathbb{P}}_{\mathbb{C}}^2$ and a holomorphic foliation $\mathcal{F}$ on ${\mathbb{P}}_{\mathbb{C}}^2$ of degree $d-1.$ We study pre-foliations of co-degree $1$ on ${\mathbb{P}}_{\mathbb{C}}^2$ 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 ${\mathbb{P}}_{\mathbb{C}}^2$ there are two families and six examples of homogeneous pre-foliations of co-degree $1$ and degree $3$ on ${\mathbb{P}}_{\mathbb{C}}^2$ 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\'ın, 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 ${\mathbb{P}}_{\mathbb{C}}^2$ is flat. This is an analogue of a result on foliations of ${\mathbb{P}}_{\mathbb{C}}^2$ due to D. Mar\'ın and J. V. Pereira.

34 pages