Affine Tallini sets and Grassmannians

Abstract. In 1982/1983 A. Bichara and F. Mazzocca characterized the Grassmann space Gr(h;A) of index h of an affine space A 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;A) and, till now, their result represents the only known characterization of Gr(h;A). If K is a commutative field and A has finite dimension m, then the image Gr(h;A)} under the well known Pl¨ucker morphism } is a proper subset Am;h;K of PG(M;K), called the affine Grassmannian of the h-subspaces of A. The aim of this paper is to introduce the notion of Affine Tallini Set and provide a natural and intrinsic characterization of Am;h;K 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 Δ satisfying suitable axioms on “perp” of lines, then the skew-field K is forced to be a commutative field and Δ is an affine Grassmannian, up to projections. Furthermore, several results concerning Affine Tallini Sets are stated and proved.