Lax embeddings of the Hermitian unital
type: Item
instance of: scholarly article
title: Lax embeddings of the Hermitian unital
zbMATH Open document ID: 1292.51004
author: Valentina Pepe
author: Hendrik Van Maldeghem
published in: Designs, Codes and Cryptography
publication date: 2013-05-30T00:00:00Z
review text: A lax Veronesian embedding of a linear space \(({\mathcal P},{\mathcal L})\) in a projective space \(\mathrm{PG}(d,{\mathbb F})\) is a map \(\theta\) such that the image of \({\mathcal P}\) generates \(\mathrm{PG}(d,{\mathbb F})\) and the image of the points of any line is a plane oval. If \({\mathcal U}=H(2,{\mathbb L}),\) (\({\mathbb L}\) is a quadratic extension of \({\mathbb K})\) is an Hermitian unital of \(\mathrm{PG}(2,{\mathbb L}),\) then it has an Veronesian embedding in \(\mathrm{PG}(7,{\mathbb F})\) (\({\mathbb F} \simeq {\mathbb K})\) in which the plane ovals are conics (standard Veronesian embedding). In this paper, the authors prove that if \(\theta\) is a lax generalized Veronesian embedding of the Hermitian unital of \(\mathrm{PG}(2,{\mathbb L}),\) where \({\mathbb L}\) is a quadratic extension of \({\mathbb K},\) \(|{\mathbb K}|>2,\) in a \(\mathrm{PG}(d,{\mathbb F})\) \((d\geq 7),\) such that disjoint blocks span disjoint subspaces of \(\mathrm{PG}(d,{\mathbb F}),\) then either (a) \(d=7\) and the embedding is the standard Veronesian embedding in a canonical subgeometry \(\mathrm{PG}(7,{\mathbb K}')\) of \(\mathrm{PG}(7,{\mathbb F})\), or (b) the embedding is the projection from a point \(p \in {\mathcal U}\) of \({\mathcal U} \setminus \{p\}\) from a subgeometry \(\mathrm{PG}(7, {\mathbb K}')\) into a hyperplane \(\mathrm{PG}(6,{\mathbb K}')\) together with an arbitrary point \(p'\) lying outside the \({\mathbb F}-\)span of \(\mathrm{PG}(6, {\mathbb K}')\) where \({\mathbb K}'\simeq{\mathbb K}\).
reviewed by: Guglielmo Lunardon
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E05
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 6168452
zbMATH Keywords: lax embedding
zbMATH Keywords: Hermitian unital
zbMATH Keywords: standard Veronesean embedding
MaRDI profile type: Publication
full work available at URL: s10623-011-9571-4
OpenAlex ID: W2062580157
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DOI: 10.1007/S10623-011-9571-4
description: scientific article; zbMATH DE number 6168452