Brieskorn module and Center conditions: pull-back of differential equations in projective space

Abstract

The moduli space of algebraic foliations on P2 of a fixed degree and with a center singularity has many irreducible components. We find a basis of the Brieskorn module defined for a rational function and prove that set of pull-back foliations forms an irreducible component of the moduli space. The main tools are the Picard-Lefschetz theory of a rational function in two variables, period integrals, and the Brieskorn module.