Embedding pseudo-complements of quadrics in PG( n, q), n⩾3, q⩾2

Abstract

Let S be a finite, planar, linear space of dimension n⩾3 such that (1) each line has q − 1, q, or q + 1 points; (2) in any subspace R, the number of lines on any point of R is (q dimR − 1) (q − 1) , where q⩾2. We prove that S embeds in a unique way in PG( n, q). If in addition S has at most q n points, it follows, using a result of Tallini, that S is the complement in PG( n, q) of a parabolic or hyperbolic quadric, a parabolic quadric plus a subspace of its nucleus space, a cone projecting from a PG( n − 3, q) a plane ( q + 1)-arc plus a subspace of the PG( n − 2, q) joining the knot of the arc with the PG( n − 3, q), or a hyperplane along with a subspace of PG( n, q).