Non-commutative projective lines

Question

There have been many approaches to the notion of projective line:

• combinatorial approaches (e.g. as certain permutation groups, such as $\mathrm{PGL}_2(k)$ in its natural action on $\mathbb{P}^1(k)$, $k$ a field);
• algebro-geometric approaches (e.g. the Proj-construction applied on graded commutative rings such as $k[X,Y]$, $k$ a field);
• (...)

My first question is: what other approaches have been shown to be fundamental ?

Secondly, and that is my main question: is there a theory about projective lines over non-commutative rings (so as to be "coordinatized over a non-commutative ring") ?

Answer 1

Marshall Hall showed that Hilbert's axioms for projective plane geometry give rise to ternary rings, which are generally noncommutative and nonassociative, and conversely every ternary ring arises in this way.

MR0008892 (5,72c) Reviewed Hall, Marshall Projective planes. Trans. Amer. Math. Soc. 54 (1943), 229–277. 48.0X

It is natural then to ask about topological projective geometry, which is studied in the elegant book: MR1384300 (97b:51009) Reviewed Salzmann, Helmut(D-TBNG-MI); Betten, Dieter(D-KIEL); Grundhöfer, Theo(D-WRZB-IM); Hähl, Hermann(D-KIEL); Löwen, Rainer(D-BRNS-AN); Stroppel, Markus(D-DARM) Compact projective planes. With an introduction to octonion geometry. De Gruyter Expositions in Mathematics, 21. Walter de Gruyter & Co., Berlin, 1995. which summarizes the state of the art at that time; little has been done since.

If you really want algebraic geometry, Beniamino Segre constructed examples where the lines of a projective geometry are actually real algebraic curves in the usual real projective plane: MR0099988 (20 #6424) Reviewed Segre, Beniamino Plans graphiques algébriques réels non desarguésiens et correspondances crémoniennes topologiques. (French) Rev. Math. Pures Appl. 1 (1956), no. 3, 35–50. 14.00 (50.00)

These are known to be Nash projective planes, i.e. the map taking two points to the projective line connecting them is a Nash map.