On the rigidity of the Figueroa replacement in \(\mathrm{PG}(2,q^3)\)
type: Item
instance of: scholarly article
title: On the rigidity of the Figueroa replacement in \(\mathrm{PG}(2,q^3)\)
zbMATH Open document ID: 1389.51001
author: Mauro Biliotti
author: Alessandro Montinaro
published in: Combinatorica
publication date: 2017-11-10T00:00:00Z
review text: A part from the Desarguesian plane, the unique known example of a finite projective plane of order \(q^3\) admitting a collineation group \(G\) isomorphic to \(\mathrm{PGL}(3,q)\) is the Figueroa plane. Its construction can be obtained in the following manner: let \(\pi_0\) be any subplane of order \(q\) of a Desarguesian plane \(D\) of order \(q^3\). The sets \(L_j(D)\), \(j = 1,2,3\), consisting of the lines of \(D\) that are secant, tangent or external to \(\pi_0\), respectively, are orbits under the copy \(G\) of \(\mathrm{PGL}(3;q)\) preserving \(\pi_0\). The sets \(P_j(D)\), \(j = 1,2,3\), consisting of the points of \(D\) that are on \(q+1\), \(1\) or \(0\) secants to \(\pi_0\), respectively, are \(G\)-orbits. Each external line to \(\pi_0\) consists of \(2\) fixed points and \(q-1\) orbits of length \(q^2+q+1\) under a cyclic subgroup of \(G\) of order \(q^2+q+1\) inducing a Singer cyclic group on \(\pi_0\). The Figueroa plane of order \(q^3\) is constructed by replacing in each external line to \(\pi_0\) specific \(q-2\) orbits of length \(q^2+q+1\) by \(q-2\) suitable subplanes of order \(q\) of \(D\). The obtained Figueroa plane inherits the collineation group \(G\) from \(D\). The authors show that there is only one \(G\)-admissible replacement of the external lines by such orbits which defines a projective plane, namely the replacement leading to the Figueroa planes. More precisely, they prove the following: let \(\pi\) be any projective plane of order \(q^3\) admitting \(G\cong \mathrm{PGL}(3,q)\) as collineation group (and then the existence of a \(G\)-invariant subplane isomorphic to \(\mathrm{PG}(2,q\)) is guaranteed). For each \(j = 1,2,3\), the sets \(P_j(\pi)\) and \(L_j(\pi)\) have the same meaning referred to \(\pi\) and \(\mathcal I_{ij}(\pi)\) is the incidence relation in \(\pi\) between \(P_i(\pi)\) and \(L_j(\pi)\). Then, \(\pi\) is either Desarguesian or the Figueroa plane if the following three conditions are satisfied: {\parindent=0.5cm …
reviewed by: Gloria Rinaldi
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 6806145
zbMATH Keywords: Figueroa replacement
MaRDI profile type: Publication
full work available at URL: s00493-015-3337-y
OpenAlex ID: W2269350331
cites work: Circle geometry in higher dimensions. II
cites work: The Jamison method in Galois geometries
cites work: Circle geometry and three-dimensional subregular translation planes
cites work: PSL(3,q) on projective planes of order \(q^ 3\)
cites work: Some big sets of mutually disjoint subgeometries of \(PG(d,q^ t)\)
cites work: A fimily of not \((V,\ell)\)-transitive projective planes of order \(q^3\), \(q \not\equiv 1 (\mod 3)\) and \(q>2\)
cites work: A synthetic construction of the Figueroa planes
cites work: Q3963629
cites work: Translation ovoids of orthogonal polar spaces
cites work: On linear sets on a projective line
DOI: 10.1007/S00493-015-3337-Y
description: scientific article; zbMATH DE number 6806145