Affine Tallini Sets and Grassmannians

In 1982/1983 A. Bichara and F. Mazzocca characterized the Grassmann space Gr(h, ) of index h of an affine space of dimension at least 3 over a skew-field K by means of the intersection properties of the three disjoint families of maximal singular subspaces of Gr(h, ) and, till now, their result represents the only known characterization of Gr(h, ). If K is a commutative field and has finite dimension m, then the image under the well known Plücker morphism is a proper subset of PG(M, K), , called the affine Grassmannian of the h-subspaces of . The aim of this paper is to introduce the notion of Affine Tallini Set and provide a natural and intrinsic characterization of from the point-line geometry point of view. More precisely, we prove that if a projective space over a skew-field K contains an Affine Tallini Set ? satisfying suitable axioms on “perp” of lines, then the skew-field K is forced to be a commutative field and ? is an affine Grassmannian, up to projections. Furthermore, several results concerning Affine Tallini Sets are stated and proved.