On the Gorensteinness of broken circuit complexes and Orlik--Terao ideals

TL;DR

This paper characterizes when the broken circuit complex of an ordered matroid and the Orlik--Terao algebra of a hyperplane arrangement are Gorenstein, linking this property to being a complete intersection and analyzing their $h$-vectors.

Contribution

It establishes that Gorensteinness of these complexes and algebras is equivalent to being a complete intersection, providing new characterizations and a method to determine this from the $h$-vector.

Findings

Gorenstein broken circuit complex iff it is a complete intersection

Gorenstein Orlik--Terao algebra iff it is a complete intersection

Gorensteinness determined by last two nonzero $h$-vector entries

Abstract

It is proved that the broken circuit complex of an ordered matroid is Gorenstein if and only if it is a complete intersection. Several characterizations for a matroid that admits such an order are then given, with particular interest in the $h$-vector of broken circuit complexes of the matroid. As an application, we prove that the Orlik--Terao algebra of a hyperplane arrangement is Gorenstein if and only if it is a complete intersection. Interestingly, our result shows that the complete intersection property (and hence the Gorensteinness as well) of the Orlik--Terao algebra can be determined from the last two nonzero entries of its $h$-vector.