Vector fields of graphic arrangements and face rings of simplicial posets

Abstract

A graphic arrangement $\A_G$ associated with a simple graph $G$ is a classical and well-studied object in the theory of hyperplane arrangements. In this note, we show that, for a connected graph $G$, a slight modification of the logarithmic vector field $D(\A_G)$ of $\A_G$ is isomorphic to the face ring of a certain simplicial poset. This allows us to give formulas for several algebraic invariants of $D(\A_G)$, such as its Hilbert series, local cohomology, projective dimension, and Castelnuovo--Mumford regularity, in terms of combinatorial and topological information about the corresponding simplicial poset. As a by-product, we also give an explicit vector space basis of $D(\A_G)$.