A broken circuit ring. (English) Zbl 1095.13024

Proudfoot, Nicholas; Speyer, David

Beitr. Algebra Geom. 47, No. 1, 161-166 (2006).

Summary: Given a matroid \(M\) represented by a linear subspace \(L\subset{\mathbb C}^n\) (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 \(\text{Spec} R(L)\), as well as a stratification indexed by the flats of \(M\).

MSC:

13F55	Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13D40	Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series