Double star arrangement and the pointed multinet

TL;DR

This paper investigates the relationship between hyperplane arrangement combinatorics and cohomology jump loci, showing that certain arrangements exhibit translated components not explained by existing multinet structures.

Contribution

It demonstrates that the double star arrangement provides a counterexample to the conjecture that all translated components are explained by pointed multinets.

Findings

Double star arrangement has translated positive-dimensional components not explained by pointed multinets.

Counterexample to the conjecture on the origin of all translated components in cohomology jump loci.

Shows limitations of multinet-based combinatorial descriptions in hyperplane arrangements.

Abstract

Let $\mathcal{A}$ be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of $\mathcal{A}$. By the work of Falk and Yuzvinsky \cite{FY}, all the irreducible components passing through the origin are determined by the multinet structure, which are combinatorially determined. Denham and Suciu introduced the pointed multinet structure to obtain examples of arrangements with translated positive-dimensional components in the degree one cohomology jump loci \cite{DS}. Suciu asked the question if all translated positive-dimensional components appear in this manner \cite{Suc14}. In this paper, we show that the double star arrangement introduced by Ishibashi, Sugawara and Yoshinaga \cite[Example 3.2]{ISY22} gives a negative answer to this question.