Double star arrangement and the pointed multinet

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.