Degree of the first integral of a pencil in $\mathbb{P}^2$ defined by Lins Neto

Abstract

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