Biased Graphs. VI. Synthetic Geometry

Abstract

A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called balanced, such that no theta subgraph contains exactly two balanced circles. A biased graph $\Omega$ has two natural matroids, the frame matroid $G(\Omega )$, and the lift matroid $L(\Omega )$, and their extensions the full frame matroid $G^{{}^{{}_{{}_{\bullet }}}}(\Omega )$ and the extended (or complete) lift matroid $L_0(\Omega )$. In Part IV we used algebra to study the representations of these matroids by vectors over a skew field and the corresponding embeddings in Desarguesian projective spaces. Here we redevelop those representations, independently of Part IV and in greater generality, by using synthetic geometry.

arXiv comment48 pp. V2=V3 is the first half of V1; 21 pp. The second half of V1 is now arXiv:1708.00095. V4 28 pp., many minor improvements