Nearly all subspaces of a classical polar space arise from its universal embedding

Abstract

Let Γ be an embeddable non-degenerate polar space of finite rank n≥2. Assuming that Γ admits the universal embedding (which is true for all embeddable polar spaces except grids of order at least 5 and certain generalized quadrangles defined over quaternion division rings), let ε:Γ→PG(V) be the universal embedding of Γ. Let S be a subspace of Γ and suppose that S, regarded as a polar space, has non-degenerate rank at least 2. We shall prove that S is the ε-preimage of a projective subspace of PG(V).