Resonance varieties, admissible line combinatorics and combinatorial pencils

TL;DR

This paper introduces a combinatorial concept analogous to pencils in line arrangements, establishing a link between resonance varieties and combinatorial pencils to study fundamental group isomorphisms.

Contribution

It defines a new combinatorial structure called a combinatorial pencil, generalizing existing notions, and proves its correspondence with resonance variety components.

Findings

Establishes a correspondence between resonance variety components and combinatorial pencils.

Provides a new combinatorial framework to analyze fundamental groups of line arrangement complements.

Generalizes the concept of nets in line arrangements.

Abstract

In this paper we define the combinatorial analogous of a pencil, and show its relationship with the concept of admissibility. Such an object is usefull to study the isomorphisms between fundamental groups of the complements of line arrangement with the same combinatorial type. This definition generalizes the idea of net given by Yuzvinsky and others. The main theorem in this paper states that there is a correspondence between components of the resonance variety and combinatorial pencils.