Homogeneous pre-foliations of co-degree one and degree four on the projective plane

TL;DR

This paper classifies all homogeneous pre-foliations of co-degree one and degree four on the complex projective plane with a flat Legendre transform, completing the cases based on the degree of the underlying foliation.

Contribution

It provides a complete classification of these pre-foliations by combining curvature criteria, explicit normal forms, and symbolic computation.

Findings

Finite list of explicit one-forms parametrized by ramification data.

Complete classification for cases where the underlying foliation degree is 3 or 4.

Extended previous results to include new cases with detailed normal forms.

Abstract

We classify, up to projective automorphism, all homogeneous pre-foliations of co-degree one and degree four on the complex projective plane $\Ptwo$ whose Legendre transform defines a flat $4$-web. The classification is organized according to the type of the underlying homogeneous foliation $\Hcal$ of degree~$3$, distinguishing the cases $\deg(\Tcal_{\Hcal})=2$, $3$, and~$4$. The case $\deg(\Tcal_{\Hcal})=2$ was treated by Bedrouni, while the cases $\deg(\Tcal_{\Hcal})=3$ and $\deg(\Tcal_{\Hcal})=4$ are completed here. The proof combines Bedrouni's curvature-holomorphy criteria with explicit normal forms and symbolic computation; the result yields a finite list of explicit one-forms, parametrised by the ramification data of the Gauss map of~$\Hcal$.