2017/12/11 by Samir Bedrouni, Bedrouni, Samir · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.1712.03895
openalex publication_date 2017/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The set \F(d) of foliations of degree d on the complex projective\nplane can be identified with a Zariski's open set of a projective space of\ndimension (d+2)2-2 on which acts\n\Aut(\ℙ2\ℂ). The subset \FP(d) of\n\F(d) consisting of foliations of \F(d) with a flat\nLegendre transform (dual web) is a Zariski closed subset of \F(d). In\nthis dissertation we study foliations of \FP(d) and we try to better\nunderstand the topological structure of \FP(3). First, we establish\nsome effective criteria for the flatness of the dual d-web of a homogeneous\nfoliation of degree d and we describe some explicit examples. We will see\nalso that it is possible, under certain assumptions, to bring the study of\nflatness of the dual web of a general foliation to the homogeneous framework.\nSecond, we classify up to automorphism of \ℙ2\ℂ the\nelements of \FP(3). More precisely, we show that up to automorphism\nthere are 16 foliations of degree 3 with a flat Legendre transform. From\nthis classification we deduce that \FP(3) has exactly 12\nirreducible components.\n