On the Grassmann space representing the lines of an affine space (Q658126)

In the spirit of \textit{G. Tallini} [London Math. Soc. Lect. Notes Ser. 49, 354--358 (1981; Zbl 0469.51006)] the authors provide axioms on the family \(\Sigma\) of maximal singular subspaces for the Grassmann spaces of the lines of an affine space. Among others the following concepts are used by the authors. A point-line incidence geometry \(\Gamma=({\mathcal P},{\mathcal L)}\) is called a \textit{partial linear space}, if any point belongs to at least one line, if any two distinct points belong to at most one line, and if any line contains at least two points. One speaks of an \textit{irreducible} partial linear space, if any line contains at least three points. A partial linear space is said to be \textit{proper}, if it contains two non-collinear points. An example of a proper partial linear space is the \textit{Grassmann space \(\mathrm{Gr}(1,\mathbf A)\) of lines of an affine space} \(\mathbf A\) of dimension at least \(3\); the points of \(\mathrm{Gr}(1,\mathbf A)\) are the lines of \(\mathbf A\), the lines of \(\mathrm{Gr}(1,\mathbf A)\) are the pencils of lines of \(\mathbf A\), a proper pencil being the set of all lines passing through a fixed point and contained in a fixed plane, and an improper pencil being the set of pairwise parallel lines contained in a fixed plane. The authors prove the subsequent characterization of \(\mathrm{Gr}(1,\mathbf A)\): Let \(\Gamma=({\mathcal P},{\mathcal L)}\) be a proper irreducible partial linear space whose lines are not maximal singular subspaces and let \(\Sigma_1\) and \(\Sigma_2\) be two non-empty families of maximal singular subspaces of \(\Gamma\), \(\Sigma:=\Sigma_1\cup\Sigma_2\), such that the following axioms hold: (P1) Every line \(L\in\mathcal L\) is contained in at most two maximal singular subspaces of \(\Gamma\). (P2) For \(i=1,2\), for every \(S\in\Sigma_i\) and every point \(p\not\in\,S\) there exist exactly \(i-1\) elements of \(\Sigma\) passing through \(p\) and disjoint from \(S\). Furthermore, every \(S'\in\Sigma\) passing through \(p\) and not disjoint from \(S\) intersects \(S\) at a unique point, and these points trace out a line coinciding with \(p^{\perp}\cap\,S\) where \(p^{\perp}\) denotes the set of points collinear with \(p\). (P3) There exists a point \(q\in\mathcal P\) which is contained in at most one element of \(\Sigma_2\). Then there exists an affine space \(\mathbf A\) of dimension at least \(3\) such that \(\Gamma\) is isomorphic to the Grassmann space \(\mathrm{Gr}(1,\mathbf A)\) of lines of \(\mathbf A\).