Requirements:
- Scalar fields other than $\mathbb{R}$ and $\mathbb{C}$.
- Precise. Visual explanations are good, but they must complement definitions and proofs, not replace them.
- No repetition of text. It may reference other textbooks for linear algebra, order theory, etc.
- Modern. Semilinear transformations, category theory.
- Well-known prerequisites. Phrases “it is a well-known fact” and “it is evident” without references must occur as rarely as possible.
There was a similar request, but concentrated on problem-solving, not on slick theory. Some examples I found so far:
- Stubbe, Steirteghem (2007). “Propositional systems, Hilbert lattices and generalized Hilbert spaces”, chapter in: “Handbook of Quantum Logic and Quantum Structures: Quantum Structures (edited by K. Engesser, D. M. Gabbay and D. Lehmann), Elsevier”, pp. 477-524. The chapter “2. Projective geometries, projective lattices” is the best. But it is short and dense, it is not a textbook after all, just a chapter in the scientific article.
- Joseph J. Rotman (1999). “Introduction to the theory of groups.” In “Chapter 9. Permutations and the Mathieu Groups”, subchapters “Affine Geometry”, “Projective Geometry.”
- Beutelspacher, Rosenbaum. “Projective Geometry, From Foundations to Applications.” It seems promising, I have just started reading it.
- Baer R. “Linear algebra and projective geometry.” It is somewhat old. Its style IMO is sloppy, imprecise.
- Dieudonné, Jean (1955). “La géométrie des groupes classiques.” It is somewhat old. It requires unknown prerequisites, IMO Bourbaki's “Algebra” is not enough.
