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

Abstract

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 \( r = 2 s + 1 (s \geq 1) \) and order q, self-dual k-sets of type (m, n), with \( k \in \{\vartheta_{2s}-q^s, \vartheta_{2s}+q^s\} \), are of elliptic and hyperbolic type, respectively. As a corollary we obtain a new characterization of the non-singular elliptic and hyperbolic quadrics.