Tangential Tallini sets in finite Grassmannians of lines

A Tallini set in a semilinear space is a set B of points, such that each line not contained in B intersects B in at most two points. In this paper, the following notion of a tangential Tallini set in the Grassmannian G( n , 1) , q , q odd, is investigated: a Tallini set is called tangential when it meets every ruled plane (i.e. the set of lines contained in a plane of PG ( n , q ) ) in either q + 1 or q^2 + q + 1 elements. A Tallini set Q_B in PG ( n , q ) can be associated with each tangential Tallini set B in G(n,1,q) . Each l in B is a line of PG ( n , q ) intersecting Q_B in either 0, or 1, or q + 1 points; when n <> 4 and B is covered by ( n - 2 ) -dimensional projective subspaces of G(n,1,q) the first case does not occur. If B is a tangential Tallini set in G(n ,1,q) covered by ( n - 2 ) -dimensional subspaces, any of which is in PG ( n , q ) the set of all lines through a point and in a hyperplane, then either Q_B is a quadric, and B is the set of all lines contained in, or tangent to, Q_B , or B is a linear complex.