Bruck's vision of regular spreads or what is the use of a Baer superspace?
type: Item
instance of: scholarly article
title: Bruck's vision of regular spreads or what is the use of a Baer superspace?
zbMATH Open document ID: 0799.51001
DOI: 10.1007/BF02941330
author: Albrecht Beutelspacher
author: Johannes Ueberberg
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1994-11-09T00:00:00Z
review text: Let \(\mathbb{P}^* = \text{PG} (d,F^*)\) be a \(d\)-dimensional pappian projective space such that the field \(F^*\) admits a subfield \(F\) of index 2. Assuming \(d\) at least 3, \(\mathbb{P}\) is called a Baer subspace of \(\mathbb{P}^*\) if every point of \(\mathbb{P}^*\) is incident with at least one line of \(\mathbb{P}\), it follows that \(\mathbb{P} = \text{PG} (d,F)\). For every point \(x\) of \(\mathbb{P}^* \setminus \mathbb{P}\), there is exactly one line through \(x\) that intersects \(\mathbb{P}\) in a line \(I\). One says that the line \(I\) is induced by the point \(x\). R. H. Bruck constructed regular spreads in Baer subspaces of 3-dimensional finite projective spaces. He proved that if \(I^*\) is a line of \(\mathbb{P}^* = \text{PG} (3,q^ 2)\) disjoint from \(\mathbb{P} = \text{PG} (3,q)\), then the family of lines of \(\mathbb{P}\) induced by the point of \(I^*\) is a regular spread of \(\mathbb{P}\). Moreover he proved that for every regular spread \(\varphi\) of \(\mathbb{P}\), there are exactly two lines \(I_ 1^*\) and \(I_ 2^*\) of \(\mathbb{P}^*\) disjoint to \(\mathbb{P}\) such that \(\varphi\) is induced by the points of \(I_ 1^*\) and by the points of \(I_ 2^*\). In this paper the authors generalize this result of Bruck for infinite pappian projective spaces of odd dimension \(2t+1\). Their main result reads as follows. Let \(\mathbb{P}^* = \text{PG} (2t + 1,F^*)\) be a \(2t+1\)-dimensional pappian projective space such that the field \(F^*\) admits a subfield of index 2. If \(U^*\) is a \(t\)-dimensional subspace of \(\mathbb{P}^*\) that is disjoint to a Baer subspace \(\mathbb{P} = \text{PG} (2t + 1,F)\), then the family \(\varphi\) of lines of \(\mathbb{P}\) induced by points of \(U^*\) is a regular line spread of \(\mathbb{P}\). Conversely if \(\mathbb{P} = \text{PG} (2t + 1,F)\) admits a regular line spread \(\varphi\), then there exists a quadratic extension \(F^*\) of \(F\) such that \(\mathbb{P}\) is a Baer subspace of \(\mathbb{P}^*…
Mathematics Subject Classification ID: 51A30
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 492581
zbMATH Keywords: spreads
zbMATH Keywords: Baer subspaces
zbMATH Keywords: pappian projective spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 492581