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

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