Degree of the First Integral of a Foliation in the Pencil ${\mathcal{P}}_4$

Abstract

Let ${\mathcal{P}}_4$ be the linear family of foliations of degree 4 in ${\mathbb{P}}^2$ given by A. Lins Neto, whose set of parameters with first integral $I_p({\mathcal{P}}_4)$ is dense and countable. In this work, we will calculate explicitly the degree of the rational first integral of the foliations in this linear family, as a function of the parameter.