Geometry on the lines of polar spine spaces
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Aequationes Mathematicae
Geometry on the lines of polar spine spaces Krzysztof Petelczyc , Krzysztof Praz˙ mowski , and ˙ Mariusz ZYNEL
Abstract. The concept of spine geometry over a polar Grassmann space is introduced. We determine conditions under which the structure of lines together with a binary coplanarity relation is a sufficient system of primitive notions for polar spine spaces. These assumptions also allow to characterize polar spine spaces in terms of the binary relation of being in one pencil of lines. Mathematics Subject Classification. 51A15, 51A45. Keywords. Grassmann space, Projective space, Polar space, Spine space, Coplanarity, Pencil of lines.
Introduction In [7,8] we proved that lines with coplanarity relation π suffice to define polar Grassmann spaces (cf. [5,6,9]) and spine spaces (cf. [10,12,13,15]). A natural question arises: can we do the same for a geometry that possesses both ‘polar’ and ‘spine’ properties? Although, regardless of this question, a polar spine geometry M is worth considering in details, this is not addressed in the paper. We determine only those properties that are necessary for the proof of our main theorem. Firstly, we have to establish parameters for which M is nondegenerate. Following [7,8] we would like to identify points of our geometry with vertices of bundles of lines. Every bundle is covered by semibundles of lines, each contained in some maximal strong subspace of M. So, maximal strong subspaces are essential here and we need to know what they and their intersections look like. Generally, as in all ‘Grassmann-type’ spaces, in M we have stars and tops. Stars and tops can be either projective or semiaffine (cf. [16]) spaces, so actually the family of all maximal strong subspaces splits into four classes. In this M resembles a
K. Petelczyc et al.
AEM
spine space. However, in spine spaces and polar spaces all maximal subspaces belonging to the same class have equal dimensions, which is not true for M: the dimension of projective stars is not constant. The next and main difference between M and spine or polar spaces is that M can be disconnected. We do not solve the problem of connectedness of M entirely, but we shed some light on this. Let us return to the main question. The maximal cliques of coplanarity in M are the same as in spine spaces and quite a bit of the reasoning from [8] can be repeated. There are some new difficulties though, mainly caused by multidimensional stars. It is hard to use techniques from [8] to characterize the family of all maximal π-cliques in terms of lines and the relation π. Nevertheless, all the proper top semibundles are just what we need in the rest of our reasoning. So, it suffices to distinguish them in the set of (‘uniformly’) definable maximal π-cliques. Since the dimension of top semibundles might be the same as the dimension of some star semibundles a couple of additional assumptions must be imposed on M (cf. Theorem 4.5). Pencils of lines are an important tool to recognize and compare the dimensions of maximal π-cliques. We consider
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