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 Γ 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 Γ n , 1 , q . Each ℓ ∈ 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 Γ n , 1 , q the first case does not occur. If B is a tangential Tallini set in Γ 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.