Classification of foliations of degree three on ℙ²_ℂ with a flat Legendre transform

2018/03/22 by Samir Bedrouni, Bedrouni, Samir, David Marín +1
Mathematics · #14C21 #32S65 #53A60 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1803.08526

openalex publication_date 2018/03/22 · openalex created_date 2018/04/06 · openalex updated_date 2026/07/28

Abstract

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(ℙ²_ℂ). 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 ℙ²_ℂ the elements of FP(3). More precisely, we show that up to 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 automorphism there are 4 convex foliations of degree three on ℙ²_ℂ.

Citations

• Feuilletages de degr 'e trois du plan projectif complexe ayant une\n transform 'ee de Legendre plate

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