Chain space

An incidence system to axiomatically describe chain geometries (cf. Chain geometry).

A weak chain space is an incidence system satisfying the three conditions below. Here, the elements of are called chains and two different points (i.e., elements of ) are called distant if they are incident with a common chain.

i) any three pairwise distant points are contained in exactly one chain;

ii) any chain contains at least three points;

iii) any point lies in at least one chain.

For a point , let be the set of all points distant to and let . Then the incidence system is called the residual space of at .

A partial parallel structure is an incidence system together with an equivalence relation on satisfying the two conditions below. Here, the elements of are called lines.

a) two different points are incident with at most one line;

b) for a line and point , there is exactly one line, , incident with and such that . Condition b) is the Euclid parallel axiom.

A partial parallel structure is called a partial affine space if there is an affine space such that is the set of points of , is the set of straight lines of and is the natural parallelism on .

A weak chain space is called a chain space if all residual spaces of it are partial affine spaces.

Every proper chain geometry is a chain space. Conversely, the chain spaces that are proper chain geometries can be characterized by suitable automorphism groups [a1].

A contact space is a weak chain space together with a family , where is an equivalence relation on with the following properties:

1) if , then is the only point common to and ;

2) if and is a point distant to , then there is a unique chain incident with and for which .

Clearly, for a contact space any residual space of the incidence system gives rise to a partial parallel structure . Conversely, any chain space is a contact space (taking for the natural parallelism of the affine space underlying ). One can characterize the contact spaces that are chain spaces by certain configurations together with richness conditions [a3].

An affine chain space is a contact space, where is the point set of an affine space . The elements of are called affine chains and are normal rational curves in , i.e., affine parts of curves which are a Veronese variety (cf. Veronese mapping). For the set of all affine chains that are straight lines, the structure is a partial affine space. Affine chain spaces can be constructed by means of Jordan algebras [a2] (cf. also Jordan algebra).

A classical example is the quadric model of a chain space, constructed on a quadric by means of plane sections. Moreover, the stereographic projection from a simple point of (to a hyperplane different from the tangent plane of at ) then gives rise to an affine chain space (cf. also Benz plane).

References

[a1]	A. Herzer, "Chain geometries" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995)
[a2]	A. Herzer, "Affine Kettengeometrien über Jordan-Algebren" Geom. Dedicata , 59 (1996) pp. 181–195
[a3]	S. Meuren, A. Herzer, "Ein Axiomsystem für partielle affine Räume" J. Geom. , 50 (1994) pp. 124–142