Projective geometry axioms

In mathematics, an axiomatic projective space S is a set P (the set of points), together with a set of subsets of P (the set of lines), all of which have at least three elements, satisfying these axioms :

• Each two distinct points p and q are in exactly one line.
• Veblen's axiom : When a line L intersects the line through p and q (different from p and q), and intersects the line through q and r (different from q and r), it also intersects the line through p and r.
• There is a point p and a line L that are disjoint.

The last axiom is there to prevent degenerations.

A subspace of the projective space is a subset X, such that any line containing two points of X is a subset of X. The full space and the empty space are also considered subspaces.

The geometric dimension of the space is said to be n if that is the largest number for which there is a strictly ascending chain of subspaces of this form : $\emptyset =X_{-1}\subset X_{0}\subset \cdots X_{n}=P$ As a consequence of the third axiom : $n\geq 2$.

Classification

Oswald Veblen and John Wesley Young (1879-1932) proved that, if $n\geq 3$, every axiomatic projective space is isomorphic with a PG(n,K), the n-dimensional projective space over some division ring K. This is now known as the Veblen-Young theorem, and appeared in the first volume, from 1910, of their two-volume book Projective Geometry.

The two-dimensional case

One can check that the definition for n = 2 is completely equivalent with that of an projective plane (identifying lines with the set of points incident with it). However it turns out that these are much harder to classify, as not all of them are isomorphic with a PG(d,K). Tits generalized this phenomenon with his generalized n-gons : the generalized 3-gons with at least three points on every line are precisely the axiomatic projective planes.

External links

http://mathworld.wolfram.com/ProjectiveSpace.html
http://eom.springer.de/P/p075350.htm
http://planetmath.org/encyclopedia/ProjectiveSpace.html

References

Beutelspacher A./Rosenbaum U. : Projective Geometry. From Foundations to Applications: Cambridge University Press (1998)
Coxeter,H.S.M. : Projective Geometry: University of Toronto Press (1974)
Dembowski,P. : Finite Geometries. Springer (1968)