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 Γn,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 q2+q+1 elements. A Tallini set QB in PG(n,q) can be associated with each tangential Tallini set B in Γn,1,q. Each ℓ∈B is a line of PG(n,q) intersecting QB in either 0, or 1, or q+1 points; when n≠4 and B is covered by (n-2)-dimensional projective subspaces of Γn,1,q the first case does not occur. If B is a tangential Tallini set in Γn,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 QB is a quadric, and B is the set of all lines contained in, or tangent to, QB, or B is a linear complex.