A Characterization of Grassmann Spaces of Index h of a Projective Space

Abstract

In 1982/83 Bichara and Tallini characterized Grassmann spaces Gr(h, P) of a projective space P involving intersection properties of the two disjoint families of maximal subspaces of Gr(h, P). In 1984 Melone and Olanda characterized Gr(1, P) using only one family of maximal subspaces. In this paper the generalization of the result of Melone and Olanda to general index is given. More precisely, I prove that the natural extension of the axiom of Melone and Olanda is not sufficient to characterize Gr(h, P) when h > 1, since the affine Grassmann space Gr(1, A) of the lines of an affine space A satisfies the axioms, too. Thus, an additional axiom is given and the characterization follows.