On tangent spaces and external flats to Grassmannians of lines over finite fields

Abstract

Let G 1,n,K be the Grassmannian of lines of the projective space PG( n, K), K any field. For any point P of G 1,n,K , let T P( G 1,n,K) denote the tangent space to G 1,n,K at P. We give a description of T P( G 1,n,K) in terms of incident lines of PG( n, K). Also, by using geometric tools and looking at the orbits of cyclic collineation groups, we construct subspaces of maximal dimension missing the Grassmannian G 1,n,q of lines of the finite projective space PG( n, q).