Lax embeddings of polar spaces in finite projective spaces

Abstract

A polar space with point set P is laxly embedded in the projective space PG(d, q), d ≥ 2, if the following conditions are satisfied: (i) P is a point set of PG(d, q) which generates PG(d, q), and (ii) each line L of is a subset of a line L′ of (d, q), and distinct lines L1, L2 of define distinct lines L′1, L′2 of PG(d, q). In this paper we determine all polar spaces of rank at least three which are laxly embedded in PG(d, q), where d ≥ 4 if is isomorphic to the polar space W(2m + 1, s), m ≥ 2 and s odd, arising from a symplcctic polarity in PG(2m + 1, s), and where d ≥ 3 in all other cases. Laxly embedded generalized quadrangles were considered in a foregoing paper.