Classical mixed partitions
A mixed partition \(\mathcal P\) of \(\Sigma= \text{PG}(2n-1,q^2)\) is a partition of the points of \(\Sigma\) into \((n-1)\)-dimensional subspaces and in Baer subgeometries isomorphic to \(\text{PG}(2n-1,q).\) Embed \(\Sigma\) into \(\Sigma'= \text{PG}(2n,q^2)\) as a hyperplane and define a new point-line incidence structure \(\pi\) in the following way. The points are the points of \(\Sigma'\) not incident with \(\Sigma.\) The lines are either the \(n\)-dimensional subspaces \(X\) of \(\Sigma'\) such that \(X \cap \Sigma\) belongs to \(\mathcal P\) or the Baer subgeometries \(\Omega\simeq \text{PG}(2n,q)\) of \(\Sigma'\) such that \(\Omega \cap \Sigma\) belongs to \(\mathcal P.\) The incidence is the natural one. It is easy to prove that \(\pi\) is a translation plane of order \(q^4\) with kernel containing \(\text{GF}(q).\) Hence there is a spread \(\mathcal S\) of \(\text{PG}(4n-1,q)\) such that the translation plane \(A({\mathcal S})\) associated with \(\mathcal S\) is isomorphic to \(\pi.\) In this paper the author proves that if \(\pi\) is a desarguesian plane, then either \(\mathcal P\) is a spread of \(\text{PG}(2n-1,q^2)\) or \(\mathcal P\) is defined by a Singer group and contains two subspaces and \((q-1)(q^{2n-2}+q^{2n-4}+\cdots+q^2+1)\) Baer subgeometries.
scientific article; zbMATH DE number 2083739
scientific article