# Lecture 3: Ambients for the Betti map and the question of its rank

In this last lecture we shall further consider some of the mentioned contexts involving the Betti map. We shall also discuss in short some recent work with Yves André and Pietro Corvaja, where we obtain some general criteria ensuring that on appropriate assumptions the Betti map has a differential of maximal rank. The results extend what comes from Manin's theorem and for instance imply topological density of points in the base which yield torsion for the relevant section.