Grassmann space associated with a planar space (Q1568169)

A linear space is called planar if there is a family of proper subspaces (called ``planes'') such that any three non-collinear points lie on exactly one plane, every plane contains three non-collinear points, and there are at least two planes. Each planar space gives rise to an associated Grassmann space as follows: Its ``points'' are the lines of the planar space, its ``lines'' are the pencils of lines of the planar space. The authors extend a geometric characterization due to \textit{G. Tallini} [Lond. Math. Soc. Lect. Note Ser. 49, 354-358 (1981; Zbl 0469.51006)] of the Grassmann space of a projective space to the more general situation of planar spaces. Further, they show that every finite planar spaces of order \(n\) in which any two distinct planes intersect in a line, and satisfying the bundle theorem, is embeddable in \(\text{PG}(3,q)\).