On line covers of finite projective and polar spaces

An m-cover of lines of a finite projective space PG (r, q) (of a finite polar space P) is a set of lines L of PG (r, q) (of P) such that every point of PG (r, q) (of P) contains m lines of L, for some m. Embed PG (r, q) in PG (r, q2). Let L¯ denote the set of points of PG (r, q2) lying on the extended lines of L. An m-cover L of PG (r, q) is an (r- 2) -dual m-cover if there are two possibilities for the number of lines of L contained in an (r- 2) -space of PG (r, q). Basing on this notion, we characterize m-covers L of PG (r, q) such that L¯ is a two-character set of PG (r, q2). In particular, we show that if L is invariant under a Singer cyclic group of PG (r, q) then it is an (r- 2) -dual m-cover. Assuming that the lines of L are lines of a symplectic polar space W(r, q) (of an orthogonal polar space Q(r, q) of parabolic type), similarly to the projective case we introduce the notion of an (r- 2) -dual m-cover of symplectic type (of parabolic type). We prove that an m-cover L of W(r, q) (of Q(r, q)) has this dual property if and only if L¯ is a tight set of an Hermitian variety H(r, q2) or of W(r, q2) (of H(r, q2) or of Q(r, q2)). We also provide some interesting examples of (4 n- 3) -dual m-covers of symplectic type of W(4 n- 1 , q).