Biased Graphs. VI. Synthetic Geometry

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.
Comment: 48 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