An extension of a theorem of G. Tallini

Abstract

A Tallini set in a projective space P is a set Q of points of P such that each line not contained in Q intersects Q in at most two points. We prove that if P is a finite projective space with odd order q > 3 and dimension d > 2 and if | Q| > q d − 1 + 2 q d − 3 + q d − 4 + … + 1, then Q is essentially an orthogonal quadric. The proof of this theorem is based on a characterization of the orthogonal quadrics in every finite dimensional projective space (with possibly infinite order).