Abstract
Let ℘ be a projective space. In this paper we consider sets ℰ of planes of ℘ such that any two planes of ℰ intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:
- If ℰ is a set of planes of a projective space intersecting mutually in one point, then the set of intersection points spans a subspace of dimension ≤6. There are up to isomorphism only three sets ℰ where this dimension is 6. These sets are related to the Fano plane.
- If ℰ is a set of planes of PG(d,q) intersecting mutually in one point, and if q≥3, ∣ℰ∣≥3(q2+q+1), then ℰ is either contained in a Klein quadric in PG(5,q), or ℰ is a dual partial spread in PG(4,q), or all elements of ℰ pass through a common point.
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Beutelspacher, A., Eisfeld, J. & Müller, J. On Sets of Planes in Projective Spaces Intersecting Mutually in One Point. Geometriae Dedicata 78, 143–159 (1999). https://doi.org/10.1023/A:1005294416997
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DOI: https://doi.org/10.1023/A:1005294416997