Symplectic semifield spreads of \(\mathrm{PG}(5,q)\) and the Veronese surface
type: Item
instance of: scholarly article
title: Symplectic semifield spreads of \(\mathrm{PG}(5,q)\) and the Veronese surface
zbMATH Open document ID: 1228.51003
author: Guglielmo Lunardon
author: Olga Polverino
author: Rocco Trombetti
author: Giuseppe Marino
published in: Ricerche di Matematica
publication date: 2011-07-01T00:00:00Z
review text: A spread of \(\mathrm{PG}(2n-1, q)\) is a partition \({\mathcal S}\) of the point set of \(\mathrm{PG}(2n-1, q)\) into \((n-1)\)-dimensional subspaces. With any spread \({\mathcal S}\) is associated a translation plane \(A({\mathcal S})\). The spread \({\mathcal S}\) is called Desarguesian if \(A({\mathcal S})\) is isomorphic to \(\mathrm{AG}(2,q^n)\). The spread \({\mathcal S}\) is a semifield spread with respect to an element \(X\in {\mathcal S}\) if there exists an elementary abelian subgroup \(G\) of \(\mathrm{PGL}(2n,q)\) of order \(q^n\) which fixes \(X\) pointwise and acts regularly on \({\mathcal S}\setminus \{X\}\). The spread \({\mathcal S}\) is a symplectic spread with respect to a nonsingular symplectic polarity \(\perp\) of \(\mathrm{PG}(2n-1, q)\) when all subspaces of \({\mathcal S}\) are totally isotropic with respect to \(\perp\). The authors show that starting from a symplectic semifield spread \({\mathcal S}\) of \(\mathrm{PG}(5, q)\), \(q\) odd, another symplectic semifield spread of \(\mathrm{PG}(5,q)\) can be obtained, called the symplectic dual of \({\mathcal S}\), and prove that the symplectic dual of a Desarguesian spread of \(\mathrm{PG}(5,q)\) is the symplectic semifield spread arising from a generalized twisted field. They construct a new symplectic semifield spread of \(\mathrm{PG}(5,q)\) (\(q = s^{2}\), \(s\) odd), describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, they determine the nuclei of the commutative pre-semifields constructed by \textit{Z. Zha, G. M. Kyureghyan} and \textit{X. Wang} [Finite Fields Appl 15, No. 2, 125--133 (2009; Zbl 1194.12003)].
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 12K10
Mathematics Subject Classification ID: 51E99
zbMATH DE Number: 5916729
zbMATH Keywords: symplectic semifield
zbMATH Keywords: spread set
zbMATH Keywords: isotopy
MaRDI profile type: Publication
full work available at URL: s11587-010-0098-1
OpenAlex ID: W2012617510
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DOI: 10.1007/S11587-010-0098-1
description: scientific article; zbMATH DE number 5916729