Semi-inverted linear spaces and an analogue of the broken circuit complex

Abstract

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 Groebner 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. 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.