A class of topological spreads with unisymplecticly complemented regulization
type: Item
instance of: scholarly article
title: A class of topological spreads with unisymplecticly complemented regulization
zbMATH Open document ID: 0978.51006
DOI: 10.1007/BF03322014
author: Rolf Riesinger
published in: Results in Mathematics
publication date: 2002-01-25T00:00:00Z
review text: The author constructs a \(3\)-parameter family of Thas-Walker line sets \(c_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) with respect to a particular Lie quadric \(L_4\) in \(\text{PG}(4,\mathbb R)\) (i.e. \(c_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) is a set of lines such that the intersections of \(L_4\) and the images of these lines under the polarity associated with \(L_4\) form a so-called flockoid of \(L_4\)). By the extended Thas-Walker construction [the author, Bull. Belg. Math. Soc. 6, No. 2, 237-247 (1999; Zbl 0940.51006)], each set \(c_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) defines a spread \({\mathcal A}_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) in \(\text{PG}(3,\mathbb R)\). It turns out that all \({\mathcal A}_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) are topological spreads with unisymplecticly complemented regulizations. In particular, the corresponding translation planes are \(4\)-dimensional compact projective planes. Moreover, if \(\varepsilon_1\varepsilon_2\varepsilon_3 \not= 0\), then \({\mathcal A}_{\varepsilon_1,\varepsilon_2,\varepsilon_3}\) is algebraic and rigid (its automorphims group contains only the identity), and hence every collineation of the corresponding translation plane fixes the translation axis pointwise. For \(\varepsilon_1\varepsilon_2 \not= 0\), the automorphism group of \({\mathcal A}_{\varepsilon_1,\varepsilon_2,0}\) consists of precisely two elements.
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 1563154
zbMATH Keywords: algebraic spread
zbMATH Keywords: rigid algebraic spread
zbMATH Keywords: flockoid
zbMATH Keywords: rigid 4-dimensional translation plane
reviewed by: Harald Löwe
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322014
OpenAlex ID: W2000169914
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description: scientific article; zbMATH DE number 1563154