Self-dual sets of type (m,n) in projective space

In this paper we introduce and analyze the notion of self-dual k-sets of type (m, n). We show that in a non-square order projective space such sets exist only if the dimension is odd. We prove that, in a projective space of odd dimension r=2s+1(s>=1) and order q, self-dual k-sets of type (m, n), with k ϵ{ϑ2s−q^s,ϑ2s+q^s}, are of elliptic and hyperbolic type, respectively. As a corollary we obtain a new characterization of the non-singular elliptic and hyperbolic quadrics.