In this paper we prove that every collineation of the Segre product of strongly connected partial line spaces is (up to permutation of indices) the product of collineations of its components (Thm. 1.10). Spaces of pencils are strongly connected, so the claim holds for Segre products of them (Thm. 1.14). In the second part we study the extendability of collineations of Segre products of spaces of pencils under some natural embeddings. (2000): 51A45, 51M35. Mathematics Subject Classification

In this paper we introduce the notion of spine space, generalizing the notion of affine grassmannians. We describe the set of strong subspaces of a spine space (Thm. 2.6, 2.7), construct the horizon of a spine space, and study the automorphisms of a spine space. (2000): 51A45, 51D25. Mathematics Subject Classification

Abstract. The Grassmann space of fc-subspaces of a polar space is defined and its geometry is examined. In particular, its cliques, subspaces and automorphisms are characterized. An analogue of Chow's theorem for the Grassmann space of fc-subspaces of a polar spaces is proved.