On the rigidity of the Figueroa replacement in \(\mathrm{PG}(2,q^3)\)
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 \begin{itemize}\item[{\(\bullet\)}] \(P_i(\pi)=P_i(D)\), \(L_j(\pi)=L_j(D)\), \(\mathcal I_{ij}(\pi)=\mathcal I_{ij}(D)\) for each \(i=1,2,3\) and \(j=1,2\); \item[{\(\bullet\)}] The actions of \(G\) on the set of points of \(\pi\) and on \(D\) are the same; \item[{\(\bullet\)}] The actions of \(G\) on the sets of lines \(L_j(\pi)\) and \(L_j(D)\), \(j=1,2\) are the same. \end{itemize}}
scientific article; zbMATH DE number 6806145