In this paper we prove that if a Grassmann space Δ = Gr A (m, h, K) of the h-subspaces of an affine space A = AG(m, K) has an embedding e into a projective space P G(n, K ) over a skew-field K , and e satisfies two suitable conditions (α) and (β), then K and K are isomorphic fields and Δ e is, up to projections, an affine Grassmannian.
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 Plücker morphism ℘ is a proper subset A m,h,K of P G(M, K), 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 A m,h,K from the point-line geometry point of view. More precisely, we prove that if a desarguesian 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 the incidence structure is an affine Grassmannian, up to projections. Furthermore, several result concerning Affine Tallini Sets are stated and proved.2000 Mathematics Subject Classification: 51A45, 51A30 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.