Chain space

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

A weak chain space is an incidence system $ {\mathcal I} = ( P, \mathfrak C,I ) $ satisfying the three conditions below. Here, the elements of $ \mathfrak C $ are called chains and two different points (i.e., elements of $ P $) 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 $ p $, let $ D _ {p} $ be the set of all points distant to $ p $ and let $ ( p ) = \{ {C \in \mathfrak C } : {pIC } \} $. Then the incidence system $ {\mathcal I} _ {p} = ( D _ {p} , ( p ) ,I ) $ is called the residual space of $ {\mathcal I} $ at $ p $.

A partial parallel structure $ ( P, \mathfrak B,I, \| ) $ is an incidence system $ ( P, \mathfrak B,I ) $ together with an equivalence relation $ \| $ on $ \mathfrak B $ satisfying the two conditions below. Here, the elements of $ \mathfrak B $ are called lines.

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

b) for a line $ L $ and point $ p $, there is exactly one line, $ L ^ \prime $, incident with $ p $ and such that $ L \| L ^ \prime $. Condition b) is the Euclid parallel axiom.

A partial parallel structure $ ( P, \mathfrak B,I, \| ) $ is called a partial affine space if there is an affine space $ {\mathcal A} $ such that $ P $ is the set of points of $ {\mathcal A} $, $ \mathfrak B $ is the set of straight lines of $ {\mathcal A} $ and $ \| $ is the natural parallelism on $ {\mathcal A} $.

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 $ {\mathcal C} = ( P, \mathfrak B,I, ( \rho _ {p} ) _ {p \in P } ) $ is a weak chain space $ ( P, \mathfrak B,I ) $ together with a family $ ( \rho _ {p} ) _ {p \in P } $, where $ \rho _ {p} $ is an equivalence relation on $ ( p ) $ with the following properties:

1) if $ C \rho _ {p} C ^ \prime $, then $ p $ is the only point common to $ C $ and $ C ^ \prime $;

2) if $ pIC $ and $ q $ is a point distant to $ p $, then there is a unique chain $ C ^ \prime $ incident with $ p $ and $ q $ for which $ C \rho _ {p} C ^ \prime $.

Clearly, for a contact space $ ( P, \mathfrak B,I, ( \rho _ {p} ) _ {p \in P } ) $ any residual space of the incidence system $ ( P, \mathfrak B,I ) $ gives rise to a partial parallel structure $ ( D _ {p} , ( p ) ,I, \rho _ {p} ) $. Conversely, any chain space $ {\mathcal I} $ is a contact space (taking for $ \rho _ {p} $ the natural parallelism of the affine space underlying $ {\mathcal I} $). One can characterize the contact spaces that are chain spaces by certain configurations together with richness conditions [a3].

An affine chain space $ \mathfrak A = ( P, \mathfrak C,I ) $ is a contact space, where $ P $ is the point set of an affine space $ {\mathcal A} $. The elements of $ \mathfrak C $ are called affine chains and are normal rational curves in $ {\mathcal A} $, i.e., affine parts of curves which are a Veronese variety (cf. Veronese mapping). For the set $ \mathfrak L $ of all affine chains that are straight lines, the structure $ ( P, \mathfrak L,I, \| ) $ 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 $ Q $ by means of plane sections. Moreover, the stereographic projection from a simple point $ p $ of $ Q $( to a hyperplane different from the tangent plane of $ Q $ at $ p $) 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