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

TL;DR

This paper explores the algebraic and geometric properties of semi-inverted linear spaces, showing they have a universal Groebner basis and are hyperbolic varieties, extending prior work on hyperplane arrangements and matroids.

Contribution

It generalizes previous results by proving circuit polynomials form a universal Groebner basis for semi-inverted linear spaces and establishes their hyperbolic nature in the real case.

Findings

Circuit polynomials form a universal Groebner basis.

Semi-inverted linear spaces are hyperbolic varieties in the real case.

Degenerations relate these spaces to Stanley-Reisner ideals.

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.