Classification of foliations of degree three on 2 with a flat Legendre transform
[Classification des feuilletages de degré trois sur 2 ayant une transformée de Legendre plate]
Annales de l'Institut Fourier, Online first, 34 p.

L’ensemble F(3) 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 Aut( 2 ). Le sous-ensemble FP(3) de F(3) formé des feuilletages de F(3) ayant une transformée de Legendre (tissu dual) plate est un fermé de Zariski de F(3). Nous classifions à automorphisme de 2 près les éléments de F(3) ; 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 F(3) en ses composantes irréductibles. Nous en déduisons aussi la classification à automorphisme près des feuilletages convexes de degré 3 de 2 .

The set F(3) 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 Aut( 2 ). The subset FP(3) of F(3) consisting of foliations of F(3) with a flat Legendre transform (dual web) is a Zariski closed subset of F(3). We classify up to automorphism of 2 the elements of FP(3). 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 FP(3) has exactly 12 irreducible components. We also deduce that up to an automorphism there are 4 convex foliations of degree three on  2 .

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DOI : https://doi.org/10.5802/aif.3431
Classification : 14C21,  32S65,  53A60
Mots clés : tissu, platitude, transformation de Legendre, feuilletage homogène
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Bedrouni, Samir; Marín, David. Classification of foliations of degree three on $\protect \mathbb{P}^{2}_{\protect \mathbb{C}}$ with a flat Legendre transform. Annales de l'Institut Fourier, Online first, 34 p.

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