New examples of oriented matroids with disconnected realization spaces

TL;DR

This paper constructs new examples of oriented matroids with disconnected realization spaces, including the smallest known case, and explores their geometric properties.

Contribution

It introduces the first known oriented matroids of rank 3 on 13 points with disconnected realization spaces, improving on previous examples with larger point sets.

Findings

Constructed oriented matroids of rank 3 on 13 points with disconnected realization spaces.

Presented the first example with a connected, non-irreducible realization space on 13 points.

Abstract

We construct oriented matroids of rank 3 on 13 points whose realization spaces are disconnected. They are defined on smaller points than the known examples with this property. Moreover, we construct the one on 13 points whose realization space is a connected and non-irreducible semialgebraic variety.