Hyper-reguli and non-André quasi-subgeometry partitions of projective spaces
type: Item
instance of: scholarly article
title: Hyper-reguli and non-André quasi-subgeometry partitions of projective spaces
zbMATH Open document ID: 1056.51010
author: Norman L. Johnson
published in: Journal of Geometry
publication date: 2004-05-27T00:00:00Z
review text: Suppose that \(\mathcal A\neq \mathcal B\) are partial spreads in PG\((2ds-1,q)\), each of \(N=(q^{ds}-1)/(q-1)\) mutually skew \((ds-1)\)-dimensional (projective) subspaces. If \(| A\cap B |=1\), for all \(A\in\mathcal A\), \(B\in\mathcal B\) then \(\mathcal A\) is called a hyper-regulus in PG\((2d-1,q)\) [cf. \textit{T. G. Ostrom}, J. Geom. 48, No. 1--2, 157--166 (1993; Zbl 0792.51001)]. (When \(ds=2\) then \(\mathcal A\) and \(\mathcal B\) are a pair of mutually opposite reguli.) It is well known that André planes of order \(q^{ds}\) can be obtained by replacing a certain family of hyper-reguli in a regular spread \(\mathcal S\) of PG\((2ds-1,q)\) [see \textit{J. André}, Math. Z. 60, 156--186 (1954; Zbl 0056.38503)]. One of the main results of the present paper says that `non-André hyper-reguli' exist whenever \(d,s>1\). Replacement yields then a tremendous variety of generalized André planes and, under an extra assumption, also a wealth of interesting new subgeometry partitions.
Mathematics Subject Classification ID: 51E23
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 2067297
zbMATH Keywords: hyper-regulus
zbMATH Keywords: subgeometry
zbMATH Keywords: quasi-subgeometry
zbMATH Keywords: partition
zbMATH Keywords: fan
reviewed by: Hans Havlicek
MaRDI profile type: Publication
full work available at URL: s00022-003-1671-5
OpenAlex ID: W2076913607
DOI: 10.1007/S00022-003-1671-5
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description: scientific article; zbMATH DE number 2067297