The Characteristic polynomials of semigeneric graphical arrangements

Abstract

The purpose of this paper is twofold. Firstly, we generalize the notion of semigeneric braid arrangement to semigeneric graphical arrangement. Secondly, we give a formula for the characteristic polynomial of the semigeneric graphical arrangement $ \mathcal{S}_G: = \{x_i- x_j = a_i: ij\in E(G), 1\leq i < j\leq n \} $, where $ a_i{\rm{'s}} $ are generic. The formula is obtained by characterizing central subarrangements of $ \mathcal{S}_{G} $ via the corresponding graph $ {G} $.

Keywords

• hyperplane arrangement
• characteristic polynomial
• semigeneric graphical arrangement
• semigeneric braid arrangement
• generic element