Characteristic varieties for a class of line arrangements

TL;DR

This paper characterizes the non-local irreducible components of the first resonance variety for a specific class of line arrangements in the complex projective plane, linking geometric parallelograms to algebraic properties.

Contribution

It identifies the structure of non-local components of the resonance variety for arrangements with points of multiplicity ≥3 on two lines, connecting geometry and algebra.

Findings

Non-local irreducible components are 2-dimensional.

These components correspond to parallelograms with sides in the arrangement.

The parallelograms have a specific diagonal relation with the lines.

Abstract

Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the non-local irreducible components of the first resonance variety $\mathcal{R}_1(\mathcal{A})$ are 2-dimensional and correspond to parallelograms $\mathcal{P}$ in $\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides are in $\mathcal{A}$ and for which $H_0$ is a diagonal.