Linear geometries of Baer subspaces (Q1971025)

In a finite desarguesian projective space \(PG(d,q^2)\) of square order a subspace of the same dimension \(d\) and of order \(q\) is called a Baer subspace. Each Baer subspace consists of the fixed elements in \(PG(d,q^2)\) of a unique so-called Baer involution. Now the main results of the paper can be stated as follows. Theorem 1.1. Let \(B\) and \(B'\) be two Baer subplanes of \(PG(2,q^2)\) with involutions \(\tau\) and \(\tau'\), respectively; moreover, let \(\delta = \tau\tau'\), \(s := \text{ord}(\delta)\), \(D := \langle\tau, \tau' \rangle\) and \({\mathbf B}(B,B') := \{ \delta^i(B), \delta^i(B') \mid i = 0,1,\dots, s-1 \}\). If \(B\) and \(B'\) have a triangle \(PQR\) in common, then the following hold: (a) \(s\) is a divisor of \(q+1\). (b) \(D\) is a dihedral group of order \(2s\) such that the reflections of \(D\) are the involutions of the Baer subplanes of \({\mathbf B}(B,B')\). (c) There exists a Baer subplane \(\overline{B}\) through the triangle \(PQR\) such that \({\mathbf B}(B,B') \subseteq {\mathbf B}(B,\overline{B})\) and \(|{\mathbf B}(B,\overline{B})|= q + 1\). Theorem 1.3. Let \(P,Q\) and \(R\) be three non collinear points of \(PG(2,q^2)\) and let \(T\) be the set of Baer subplanes through \(P,Q\) and \(R\) . Define a geometry \(A_{q+1}\) as follows: (1) The points of \(A_{q+1}\) are the Baer subplanes of \(T\) . (2) The lines are the sets \({\mathbf B}(B,B')\) with \(|{\mathbf B}(B,B')|= q + 1\) and \(B,B' \in T\). (3) Incidence is inclusion. If \(q+1\) is a prime number, then \(A_{q+1}\) is a desarguesian affine plane of order \(q+1\). Furthermore, the authors generalize the construction of \(AG(d,q+1)\) out of \(PG(d,q^2)\) to dimensions \(d \geq 2\) constructing the corresponding vector space.