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Semi-inverted linear spaces and an analogue of the broken circuit complex. (English) Zbl 1417.05254
Summary: The image of a linear space under inversion of some coordinates is an affine variety whose structure is governed by an underlying hyperplane arrangement. In this paper, we generalize work by N. Proudfoot and D. Speyer [Beitr. Algebra Geom. 47, No. 1, 161–166 (2006; Zbl 1095.13024)] to show that circuit polynomials form a universal Gröbner basis for the ideal of polynomials vanishing on this variety. The proof relies on degenerations to the Stanley-Reisner ideal of a simplicial complex determined by the underlying matroid, which is closely related to the external activity complex defined by F. Ardila and A. Boocher [J. Algebr. Comb. 43, No. 1, 199–235 (2016; Zbl 1331.05051)]. If the linear space is real, then the semi-inverted linear space is also an example of a hyperbolic variety, meaning that all of its intersection points with a large family of linear spaces are real.
##### MSC:
 05E45 Combinatorial aspects of simplicial complexes 05B35 Combinatorial aspects of matroids and geometric lattices 52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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