Monodromy problem and Tangential center-focus problem for product of generic lines in $\mathbb{P}^2$

TL;DR

This paper investigates the monodromy and tangential center-focus problems for a rational map defined by products of lines in projective space, analyzing the homology and Abelian integrals related to singularities.

Contribution

It characterizes the monodromy action on homology and identifies conditions for vanishing Abelian integrals in the context of product of lines in projective space.

Findings

Characterization of monodromy action on homology groups.

Conditions for vanishing Abelian integrals around singularities.

Analysis of the fibration associated with the rational map.

Abstract

We consider the rational map $F$ defined by the quotient of products of lines in general position and we study the monodromy problem and tangential center-focus problem for the fibration associated with $F$. Thus, we study the submodule of the 1-homology group of a regular fiber of $F$ generated by the orbit of the monodromy action on a vanishing cycle. Moreover, we characterize the meromorphic 1-forms $\omega$ in $\mathbb{P}^2$ such that the Abelian integral $\int_{\delta_t}\omega$ vanishes on a family of cycles $\delta_t$ around a center singularity.