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 under the well known Plücker morphism is a proper subset 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 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.