G\'eom\'etrie classique de certains feuilletages quadratiques

TL;DR

This paper classifies quadratic foliations on the complex projective plane with a single singularity, revealing only four such types and analyzing the action of automorphisms on their moduli space.

Contribution

It provides a complete classification of quadratic foliations with one singularity and describes the automorphism group action on their moduli space.

Findings

Only four quadratic foliations with one singularity exist up to conjugacy.

The dynamics of three are well-understood, while the fourth remains mysterious.

The automorphism group acts with orbits of dimension more than 6, with specific orbit dimension counts.

Abstract

The set $\mathscr{F}(2;2)$ of quadratic foliations on the complex projective plane can be identified with a \textsc{Zariski}'s open set of a projective space of dimension 14 on which acts $\mathrm{Aut}(\mathbb{P}^2(\mathbb{C})).$ We classify, up to automorphisms of $\mathbb{P}^2(\mathbb{C}),$ quadratic foliations with only one singularity. There are only four such foliations up to conjugacy; whereas three of them have a dynamic which can be easily described the dynamic of the fourth is still mysterious. This classification also allows us to describe the action of $\mathrm{Aut}(\mathbb{P}^2(\mathbb{C}))$ on $\mathscr{F}(2;2).$ On the one hand we show that the dimension of the orbits is more than 6 and that there are exactly two orbits of dimension $6;$ on the other hand we obtain that the closure of the generic orbit in $\mathscr{F} (2;2)$ contains at least seven orbits of dimension~7 and exactly one orbit of dimension $6.$