Brieskorn Module and Center Conditions: Pull-Back of Differential Equations in Projective Space

Abstract

The moduli space of algebraic foliations on \(\mathbb {P}^{2}\) 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 pull-back foliations forms an irreducible component of the moduli space. The main tools are Picard-Lefschetz theory of a rational function in two variables, period integrals and Brieskorn module.