On sets of planes in projective spaces intersecting mutually in one point
type: Item
instance of: scholarly article
title: On sets of planes in projective spaces intersecting mutually in one point
zbMATH Open document ID: 0945.51001
DOI: 10.1023/A:1005294416997
author: Albrecht Beutelspacher
author: J. Eisfeld
author: Jörg Müller
published in: Geometriae Dedicata
publication date: 2000-03-27T00:00:00Z
review text: Let \({\mathcal P}\) be a projective space. A set \({\mathcal K}\) of projective subplanes of \(\mathcal P\) is called a Klein set, if any two of these subplanes intersect in exactly one point. The projective subspace that is spanned by all of the intersection points of any two distinct planes of \(\mathcal K\) is called the base space of \(\mathcal K\) and its dimension is called the base dimension of \(\mathcal K\). The authors give a complete description of Klein sets with base dimension 3. Determining Klein sets of base dimension 4 is shown to be equivalent to classifying partial spreads in a four-dimensional projective space. Klein sets of base dimension 5 are classified in the case when \(\mathcal K\) is not contained in its base space. If \(\mathcal K\) is contained in its base space, the authors prove for \({\mathcal P} = \text{ PG}(5,q)\), \(q \geq 3\), and \(|\mathcal K|\geq 3(q^2+q+1)\) that \(\mathcal K\) is contained in a hyperbolic Klein set, i.e. a Klein set that arises from a hyperbolic quadric in \({\mathcal P}\). Finally, the authors classify all Klein sets with a base dimension greater than 5. It turns out that in this case there are, up to isomorphism, only three Klein sets, all of them being related to the Fano plane.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1384068
zbMATH Keywords: projective space
zbMATH Keywords: PG
zbMATH Keywords: Klein quadric
zbMATH Keywords: Fano plane
zbMATH Keywords: subplane
MaRDI profile type: Publication
full work available at URL: a:1005294416997
OpenAlex ID: W105886966
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description: scientific article; zbMATH DE number 1384068