On the projective dimension of weakly chordal graphic arrangements

A hyperplane arrangement is called free if its module of derivations is free. A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices.

In this talk, we first present the concept of freeness in the graphic setting and extend it to the case of graphic arrangements of projective dimension at most 1, whose underlying graphs form the class of weakly chordal graphs (a graph is weakly chordal if the graph and its complement have no chordless cycle with five or more vertices).