On the connection between coordinate and diagonal arrangement complements

Abstract

We study diagonal arrangement complements $D(K)$ in $\mathbb{C}^m$. We consider the class of simplicial complexes $K$ in which any two missing faces have a common vertex, and prove that the coordinate arrangement complement $U(K)$ is the double suspension of the diagonal arrangement complement $D(K)$. In the case of subspace arrangements in $\mathbb{R}^m$ the coordinate arrangement complement $U_{\mathbb{R}}(K)$ is the single suspension of $D_{\mathbb{R}}(K)$.