Veronesean embeddings of Hermitian unitals
type: Item
instance of: scholarly article
title: Veronesean embeddings of Hermitian unitals
zbMATH Open document ID: 1202.51004
author: J. Martínez
published in: European Journal of Combinatorics
publication date: 2010-09-10T00:00:00Z
review text: Let \({\mathcal S}=({\mathcal P},{\mathcal L})\) be a point-line geometry. A \textit{Veronesean (projective) embedding} of \({\mathcal S}\) in the projective space \(PG(d,{\mathbf K})\) over a field \(\mathbf K\) is an injective mapping \(\alpha:{\mathcal P}\to\,PG_{0}(d,{\mathbf K})\), where \(PG_{0}(d,{\mathbf K})\) denotes the point set of \(PG(d,{\mathbf K})\), such that the \(\alpha\)-image of any line is a plane oval, and such that the \(\alpha\)-image of \(\mathcal P\) generates \(PG(d,{\mathbf K})\). (It suffices to assume only commutative fields because of the lack of ``classical'' ovals in projective planes over non-commutative skew fields.) Let \(\mathbf L\) be a quadratic Galois extension of \(\mathbf K\), the involution of the corresponding Galois group maps \(x\in{\mathbf L}\) to \(\overline {x}\), and let \(\eta\in{\mathbf L}\setminus{\mathbf K}\) be an arbitrarily chosen element. We consider the mapping \(\beta:PG_0(2,{\mathbf L})\to\,PG_0(8,{\mathbf K})\) with \[ \begin{multlined}\beta(x_0,x_1,x_2)\mapsto (x_0\overline{x}_0,x_1\overline{x}_1,x_2\overline{x}_2; x_0\overline{x}_1+\overline{x}_{0}x_1,x_1\overline{x}_2+\overline{x}_{1}x_2,x_2\overline{x}_0+\overline{x}_{2}x_0;\\ \eta\,x_0\overline{x}_1+\overline{\eta\,x}_{0}x_1, \eta\,x_1\overline{x}_2+\overline{\eta\,x}_{1}x_2, \eta\,x_2\overline{x}_0+\overline{\eta\,x}_{2}x_0).\end{multlined} \] The direct image of \(\beta\) is called the \textit{Hermitian Veronesean} \(\mathcal H\) of \(PG(2,\mathbf L)\) with respect to \(\mathbf L\). Any non-degenerate Hermitian curve \(c\) of \(PG(2,\mathbf L)\) in an obvious way defines a point-line geometry, say \(\mathcal C=(\mathcal P, \mathcal L)\), named the \textit{Hermitian unital over \(\mathbf L/\mathbf K\)} (usually the elements of \(\mathcal L\) are called blocks). The \(\beta\)-image of \(c\) generates a hyperplane \(H\) of \(PG(8,\mathbf K)\) and \({\mathcal H}\cap\,H\) is a Veronesean embedding of \(\mathcal C\) in \(PG(7,\mathbf K)\), one speaks of …
reviewed by: Rolf Riesinger
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51M35
zbMATH DE Number: 5782186
zbMATH Keywords: Veronesean embedding
zbMATH Keywords: Hermitian unital
zbMATH Keywords: Hermitian Veronesean
zbMATH Keywords: generalized polygon
zbMATH Keywords: Hermitian ovoid
zbMATH Keywords: Hermitian spread
zbMATH Keywords: triality
zbMATH Keywords: split Cayley hexagon
zbMATH Keywords: Grassmannian
zbMATH Keywords: Segre variety
zbMATH Keywords: Baer subline
zbMATH Keywords: Baer subpencil
MaRDI profile type: Publication
full work available at URL: j.ejc.2009.08.004
OpenAlex ID: W1987793340
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DOI: 10.1016/J.EJC.2009.08.004
description: scientific article; zbMATH DE number 5782186