Lines in 3-space. (English) Zbl 1175.51002

Blunck, Andrea; Pianta, Silvia

Mitt. Math. Ges. Hamb. 27, 189-202 (2008).

The authors identify lines in the three-dimensional projective space \({\mathbb P}_3(F)\) over a (possibly non-commutative) field \(F\) with elements of the projective line over the ring \(M_2(F)\) of \(2 \times 2\)-matrices with entries in \(F\). Within this framework, they study reguli and regular spreads in \({\mathbb P}_3(F)\). Most result refer to the relation of regular spreads and quadratic field extensions of \(F\).
The article makes some efforts to be self-contained: the list of references is detailed, all fundamental concepts and notions are explained, and proofs for known theorems are provided.

Reviewer: Hans-Peter Schröcker (Innsbruck)

MSC:

51A45	Incidence structures embeddable into projective geometries
51A30	Desarguesian and Pappian geometries
51B05	General theory of nonlinear incidence geometry
51C05	Ring geometry (Hjelmslev, Barbilian, etc.)

Keywords:

projective line over a ring; regulus; spread; Baer subspace