Abstract.
In this paper a generalisation of the notion of polarity is exhibited which allows to completely describe, in an incidence-geometric way, the linear complexes of h-subspaces. A generalised polarity is defined to be a partial map which maps (h−1)-subspaces to hyperplanes, satisfying suitable linearity and reciprocity properties. Generalised polarities with the null property give rise to linear complexes and vice versa. Given that there exists for h > 1 a linear complex of h-subspaces which contains no star – this seems to be an open problem over an arbitrary ground field – the combinatorial structure of a partition of the line set of the projective space into non-geometric spreads of its hyperplanes can be obtained. This line partition has an additional linearity property which turns out to be characteristic.
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Dedicated to Helmut Karzel on the occasion of his 80th birthday
Received: December 3, 2007. Revised: December 13, 2007.
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Havlicek, H., Zanella, C. Incidence and Combinatorial Properties of Linear Complexes. Result. Math. 51, 261–274 (2008). https://doi.org/10.1007/s00025-007-0275-z
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DOI: https://doi.org/10.1007/s00025-007-0275-z