A Family of Foliations with One Singularity
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A Family of Foliations with One Singularity S. C. Coutinho1
· Filipe Ramos Ferreira1
Received: 28 May 2019 / Accepted: 9 November 2019 © Sociedade Brasileira de Matemática 2019
Abstract For every integer k ≥ 3 we describe a new family of foliations of degree k with one singularity. We show that a very generic member of this family has trivial isotropy group and a line as its unique Darboux polynomial. Keywords Holomorphic foliation · Algebraic solution · Singularity Mathematics Subject Classification Primary 17B35 · 16S32; Secondary 37F75
1 Introduction In a well-known paper Darboux (1878), published in 1878, G. Darboux introduced a new method to find first integrals of differential equations of the first order and the first degree with polynomial coefficients. Following on the footsteps of A. Clebsch, Darboux interpreted these equations as differential 1-forms with homogeneous coefficients. This allowed him to use results of projective geometry to analyse these differential equations and to build first integrals from special solutions, defined by the vanishing of polynomials. Although Darboux’s method made its way into classic texts like Ince’s Ordinary differential equations (Ince 1956, section 2.21, p. 29–32), it disappeared from both elementary and advanced texts until 1979, when it was reworked in the language of modern algebraic geometry and vastly generalised by J.-P. Jouanolou in his monograph (Jouanolou 1979). In this language, what Darboux called a differential equation, corresponds to a foliation over the complex projective plane P2 . These are defined by 1-forms = Ad x + Bdy + Cdz, where A, B, C ∈ C[x, y, z] are
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S. C. Coutinho [email protected] Filipe Ramos Ferreira [email protected]
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Departamento de Ciência da Computação, Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, Rio de Janeiro, RJ 21945-970, Brazil
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S. C. Coutinho, F. R. Ferreira
homogeneous polynomials of the same degree ν > 0 that satisfy x A + y B + zC = 0. Under these hypothesis, the degree of the foliation defined by is ν − 1. The key players in Darboux’s method of finding first integrals are the homogeneous non-constant polynomials F ∈ C[x, y, z] that satisfy ∧ d F = Fη, for some 2-form η; which are now often called Darboux polynomials. The set of zeros of a Darboux polynomial is an algebraic curve in P2 that is, locally, almost everywhere tangent to the vector field dual to . In the first chapter of his monograph, Jouanolou classified the foliations of degree 1. These foliations had already appeared as examples in Darboux’s paper (Darboux 1878, p. 72), where they are called Jacobi equations, because they had been studied by C.G.J. Jacobi in Jacobi (1842). Jacobi equations always admit Darboux polynomials, and Jouanolou determines all of them as part of his classification. More recently, Cerveau et al. (2010) have classified the foliations of degree 2 with only one singularity, and investigated their properties. In Cerveau et al. (2010, Théorème 1, p. 163), they show that, up
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