Affine planes and flag linear spaces (Q1431613)

Rajola and Scafati-Tallini introduce in the paper under review a very interesting familiy of linear spaces, called the authors flag linear spaces by. The construction is as follows: Let \(P\) be a (finite or infinite) projective plane, let \(l_{\infty}\) be a line of \(P\), and let \(A := P \setminus l_{\infty}\) be the corresponding affine plane. Let \(\Gamma = \Gamma(P, l_{\infty})\) be the following point-line-geometry: The points of \(\Gamma\) are the flags of \(A\) and the points on the line \(l_{\infty}\). The lines of \(\Gamma\) are as follows: (i) the parallel classes \(\pi\) of \(A\); (ii) the sets \(L(g, \pi) := \{ (x, l) \mid x \in g, l \in \pi, x \in l\}\), where \(\pi\) is a parallel class of \(A\) and \(g\) a line of \(A\) not contained in \(\pi\); (iii) the sets \(L(x) := \{(x, l) \mid (x, l) \text{ is a flag of } A\}\) for all points \(x\) of \(A\); (iv) the sets \(L(x, g) := \{ (y, l) \mid y \in g, l = xy\} \cup \{(x, \pi_l(x)\}\) (where \(\pi_l(x)\) denotes the line of \(A\) through \(x\) parallel to \(l\)) for all non-incident point-line-pairs \((x, g)\) of \(A\); (v) the line at infinity \(l_{\infty}\) The incidence is defined in the natural way. The main results of Rajola and Scafati-Tallini are as follows: Theorem 1. The linear space \(\Gamma = \Gamma(P, l_{\infty})\) is a linear space containing a familiy of subspaces through the line \(l_{\infty}\) isomorphic to the projective plane \(P\). Theorem 2. Let \(P\) be a finite projective plane of order \(q\). (a) Then \(\Gamma = \Gamma(P, l_{\infty})\) is a 2-\((q^3+q^2+q+1,q+1,1)\)-design, hence admitting the parameters of the projective space \(PG(3, q)\). (b) \(\Gamma\) admits exactly \(q+1\) proper subspaces which are no lines. All of them are isomorphic to \(P\). As an corollary one deduces that every finite projective plane of order \(q\) is embedded in a 2-\((q^3+q^2+q+1,q+1,1)\)-design.