Full embeddings of (\(\alpha\),\(\beta\))-geometries in projective spaces. II.
type: Item
instance of: scholarly article
title: Full embeddings of (\(\alpha\),\(\beta\))-geometries in projective spaces. II.
zbMATH Open document ID: 1039.51002
DOI: 10.1016/S0012-365X(02)00805-1
author: Sara Cauchie
published in: Discrete Mathematics
publication date: 2003-06-09T00:00:00Z
review text: The author continues the classification of proper \((\alpha,\beta)\)-geometries fully embedded into PG\((n,q)\) [Part I [Eur. J. Comb. 23, No. 6, 635--646 (2002; Zbl 1026.51003)] . The present paper deals with the case \(\beta=q\), odd, and \(\alpha>1\). It turns out that only two cases occur, namely \(\alpha=q-1\) and \(\alpha=q-\sqrt q\). In the case \(\alpha=q-1\) there are two types of geometries, which show up whether or not a certain type of planes containing an antiflag exists. In the case \(\alpha=q-\sqrt q\) only non-degenerate geometries are considered. One type of examples has \(n=3,\) or \(4\). For the other two possibilities (they must have \(n=3\)) it is not known whether examples exist. The section ``Conclusion'' wraps up the results of this and the previous paper [Part I, loc. cit.] to give a nice overview of what is known so far. (One of the rare instances where such a section makes sense.) Some partial results for the case \(q\) even are given, too.
reviewed by: Hubert Kiechle
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 1924763
zbMATH Keywords: \((\alpha
zbMATH Keywords: \beta)\)-geometry
zbMATH Keywords: full embedding
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1924763