On the incidence geometry of Grassmann spaces (Q1290868)

The main result is a characterization of the Grassmann space \({\mathbf G}\) of a projective space \(\mathcal P\). By definition, the point set \(P\) of \({\mathbf G}\) is the set of lines of \(\mathcal P\), the line set \(\mathcal L\) of \({\mathbf G}\) consists of all plane line pencils in \(\mathcal P\). An arbitrary pair \((P,\mathcal L)\) is called a semilinear space, if any two distinct points in \(P\) are incident with at most one line in \(\mathcal L\). If \(L\in \mathcal L\), then \(L^{\perp}\) denotes the set of all points in \(P\) which are collinear with each point of \(L\). Theorem. A semilinear space \((P,\mathcal L)\) is isomorphic to the Grassmann space of some projective space \(\mathcal P\) if and only if the following two conditions hold: (A) if \(L\in\mathcal L\), the \(L^{\perp}\) contains \(2\), but not \(3\) pairwise non-collinear points, (B) for every pair \(K,L\) of non-collinear and non-intersecting lines, either (a) \(1 \leq |K^{\perp} \cap L^{\perp}|\leq 2\) and \(K^{\perp} \cap L^{\perp}\) is not contained in \(K \cup L\), or (b) \(K^{\perp} \cap L^{\perp}\) is a line intersecting \(K \cup L\).