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

Abstract

We classify, up to projective automorphism, all homogeneous pre-foliations of co-degree $1$ and degree $4$ 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, parametrized by the ramification data of the Gauss map of~$\Hcal$, all displayed explicitly in the article (the few cases whose parameters are roots of higher-degree polynomials being written out in full in Appendix A).