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A broken circuit ring. (English) Zbl 1095.13024
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
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