Classification of foliations of degree three on ${\mathbb{P}}_{ℂ}^2$ with a flat Legendre transform

Bedrouni, Samir; Marín, David

Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1757-1790.

L’ensemble $\mathbf{F}\left(3\right)$ des feuilletages de degré trois du plan projectif complexe s’identifie à un ouvert de Zariski dans un espace projectif de dimension $23$ sur lequel agit le groupe $\mathrm{Aut}\left({\mathbb{P}}_{ℂ}^2\right)$. Le sous-ensemble $\mathbf{FP}\left(3\right)$ de $\mathbf{F}\left(3\right)$ formé des feuilletages de $\mathbf{F}\left(3\right)$ ayant une transformée de Legendre (tissu dual) plate est un fermé de Zariski de $\mathbf{F}\left(3\right)$. Nous classifions à automorphisme de ${\mathbb{P}}_{ℂ}^2$ près les éléments de $\mathbf{F}\left(3\right)$ ; plus précisément, nous montrons qu’à automorphisme près il y a $16$ feuilletages de degré $3$ ayant une transformée de Legendre plate. De cette classification nous obtenons la décomposition de $\mathbf{F}\left(3\right)$ en ses composantes irréductibles. Nous en déduisons aussi la classification à automorphisme près des feuilletages convexes de degré $3$ de ${\mathbb{P}}_{ℂ}^2$.

The set $\mathbf{F}\left(3\right)$ of foliations of degree three on the complex projective plane can be identified with a Zariski’s open set of a projective space of dimension $23$ on which acts $\mathrm{Aut}\left({\mathbb{P}}_{ℂ}^2\right)$. The subset $\mathbf{FP}\left(3\right)$ of $\mathbf{F}\left(3\right)$ consisting of foliations of $\mathbf{F}\left(3\right)$ with a flat Legendre transform (dual web) is a Zariski closed subset of $\mathbf{F}\left(3\right)$. We classify up to automorphism of ${\mathbb{P}}_{ℂ}^2$ the elements of $\mathbf{FP}\left(3\right)$. More precisely, we show that up to an automorphism there are $16$ foliations of degree three with a flat Legendre transform. From this classification we deduce that $\mathbf{FP}\left(3\right)$ has exactly $12$ irreducible components. We also deduce that up to an automorphism there are $4$ convex foliations of degree three on ${\mathbb{P}}^2.$

Classification : 14C21, 32S65, 53A60
Keywords: web, flatness, Legendre transformation, homogeneous foliation
Mot clés : tissu, platitude, transformation de Legendre, feuilletage homogène

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