2023/09/22 by Samir Bedrouni, Bedrouni, Samir
Mathematics · #14C21 #32S65 #53A60 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2309.12837
openalex publication_date 2023/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A holomorphic pre-foliation \mathscrF=ℓ\boxtimesF of co-degree 1 and degree d on ℙ2ℂ is the data of a line ℓ of ℙ2ℂ and a holomorphic foliation F on ℙ2ℂ of degree d-1. We study pre-foliations of co-degree 1 on ℙ2ℂ with a flat Legendre transform (dual web). After having established some general results on the flatness of the dual d-web of a homogeneous pre-foliation of co-degree 1 and degree d, we describe some explicit examples and we show that up to automorphism of ℙ2ℂ there are two families and six examples of homogeneous pre-foliations of co-degree 1 and degree 3 on \mathbb P2ℂ with a flat dual web. This allows us to prove an analogue for pre-foliations of co-degree 1 and degree~3 of a result, obtained in collaboration with D. Mar'ın, on foliations of degree 3 with non-degenerate singularities and a flat Legendre transform. We also show that the dual web of a reduced convex pre-foliation of co-degree 1 on ℙ2ℂ is flat. This is an analogue of a result on foliations of ℙ2ℂ due to D. Mar'ın and J. V. Pereira.