Some families of pencils with a unique base point and their associated foliations (Q6562443)

Consider a pencil of curves \(\Lambda\) in the complex projective plane \(\mathbb{P}^{2}\) given by the closure of affine curves \(f_{\lambda}(x, y) = f_{0}(x, y) + \lambda f_{\infty}(x, y)=0\), with \(\lambda\in \mathbb{C}\cup\{\infty\}\), where \(f_{0}(x, y)\) and \(f_{\infty}(x, y)\) are polynomials of the same degree and \(f_{\lambda}(x,y)=0\) is (generically) an irreducible curve. After removing codimension \(1\) singularities of \(f_{0}(x, y)df_{\infty}(x, y)-f_{\infty}df_{0}(x,y)\) it is obtained a 1-form \(\omega\) which defines a foliation of \(\mathbb{P}^{2}\) with isolated singularities whose leaves are the curves \(f_{\lambda}(x, y)=0\).\N\NIf such a \(\Lambda\) exists for a \(1\)-form \(\omega\), it is called a first integral for \(\omega\), and \(\omega\) is said to be algebraically integrable.\N\NThe work under review studies if it is possible to bound the degree of \(\Lambda\) in terms of information that only depends on \(\omega\) (Poincaré's Problem) for the case where \(\omega\), an thus the foliation, have just one singular point. The main result of the article under review provide examples of families of integrable foliations with a unique singular point, such that the degree and genus of \(\Lambda\) are not bounded for each fixed degree of \(\omega\) (Theorem 1). Moreover, the authors also prove rationality of the variety defined by these families of pencils.\N\NThis nice results cover the study of Poincaré's Problem for foliations that have a unique singular point, in contrast with the examples previously given by other authors for foliations that have the maximum number of singularities.\N\NThis work contains interesting results concerning algebraically integrable foliations on \(\mathbb{P}^{2}\) and Poincaré's Problem.