New Examples of Oriented Matroids with Disconnected Realization Spaces

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. 1 Oriented Matroids and Matrices Throughout this section, we fix positive integers r and n. Let X = (x1, . . . , xn) ∈ R be a real (r, n) matrix of rank r, and E = {1, . . . , n} be the set of labels of the columns of X. For such matrix X, a map χX can be defined as χX : E r → {−1, 0,+1}, χX (i1, . . . , ir) := sgn det(xi1 , . . . , xir ). The map χX is called the chirotope of X. The chirotope χX encodes the information on the combinatorial type which is called the oriented matroid of X. In this case, the oriented matroid determined by χX is of rank r on E. We note for some properties which the chirotope χX of a matrix X satisfies. 1. χX is not identically zero. 2. χX is alternating, i.e. χX (iσ(1), . . . , iσ(r)) = sgn(σ)χX (i1, . . . , ir) for all i1, . . . , ir ∈ E and all permutation σ. 3. For all i1, . . . , ir, j1, . . . , jr ∈ E such that χX (jk, i2, . . . , ir) · χX (j1, . . . , jk−1, i1, jk+1, . . . , jr) ≥ 0 for k = 1, . . . , r, we have χX (i1, . . . , ir) · χX (j1, . . . , jr) ≥ 0. ∗Department of Human Coexistence, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan [email protected]