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 Proudfoot and Speyer 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 Ardila and Boocher. 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.

Revised : 2019-02-01

Accepted : 2019-02-12

Published online : 2019-08-01

DOI : https://doi.org/10.5802/alco.65

Keywords: Matroid, hyperplane arrangement, simplicial complex, reciprocal linear space

@article{ALCO_2019__2_4_645_0, author = {Scholten, Georgy and Vinzant, Cynthia}, title = {Semi-inverted linear spaces and an analogue of the broken circuit complex}, journal = {Algebraic Combinatorics}, publisher = {MathOA foundation}, volume = {2}, number = {4}, year = {2019}, pages = {645-661}, doi = {10.5802/alco.65}, language = {en}, url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_4_645_0} }

Semi-inverted linear spaces and an analogue of the broken circuit complex. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 645-661. doi : 10.5802/alco.65. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_645_0/

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