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

TL;DR

This paper investigates the structure of algebraic foliations with center singularities on the projective plane, demonstrating that pull-back foliations form an irreducible component of the moduli space using Brieskorn modules and Picard-Lefschetz theory.

Contribution

It introduces a basis for the Brieskorn module for rational functions and proves the irreducibility of the pull-back foliation component in the moduli space.

Findings

Pull-back foliations form an irreducible component of the moduli space.

A basis for the Brieskorn module for rational functions is established.

Application of Picard-Lefschetz theory and period integrals to foliation analysis.

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.