A Broken Circuit Ring

TL;DR

This paper introduces a graded ring associated with a matroid that generalizes the Stanley-Reisner ring of the broken circuit complex, providing new algebraic and geometric insights into matroid theory.

Contribution

It defines a new Cohen-Macaulay graded ring R(L) that degenerates to the broken circuit complex's Stanley-Reisner ring and offers a geometric interpretation of its spectrum.

Findings

R(L) is Cohen-Macaulay.

R(L) degenerates to the Stanley-Reisner ring of the broken circuit complex.

Provides a stratification of Spec R(L) indexed by flats of the matroid.

Abstract

Given a matroid M represented by a linear subspace L in n-space (equivalently by an arrangement of n hyperplanes in L), we define a graded ring R(L) which degenerates to the Stanley-Reisner ring of the broken circuit complex for any choice of ordering of the ground set. In particular, R(L) is Cohen-Macaulay, and may be used to compute the h-vector of the broken circuit complex of M. We give a geometric interpretation of Spec R(L), as well as a stratification indexed by the flats of M.