Convex foliations of degree 5 on the complex projective plane

2018/12/22 by Bedrouni, Samir, Marín, David · 1 citation
#32M25 #32S65 #37F75 #Complex Variables (math.CV) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.03174

Abstract

We show that up to automorphisms of \mathbb P²_{\mathbb C} there are 14 homogeneous convex foliations of degree 5 on \mathbb P²_{\mathbb C}. We establish some properties of the Fermat foliation \mathcal F₀^d of degree d≥2 and of the Hilbert modular foliation F_H⁵ of degree 5. As a consequence, we obtain that every reduced convex foliation of degree 5 on \mathbb P²_{\mathbb C} is linearly conjugated to one of the two foliations \mathcal F₀⁵ or F_H⁵, which is a partial answer to a question posed in 2013 by D. Mar'ın and J.V. Pereira. We end with two conjectures about the Camacho-Sad indices along the line at infinity at the non radial singularities of the homogeneous convex foliations of degree d≥2 on \mathbb P²_{\mathbb C}.

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