Lax Projective Embeddings of Polar Spaces

Abstract

Let Γ be a non-degenerate polar space of rank n ≥ 3 where all of its lines have at least three points. We prove that, if Γ admits a lax embedding e : Γ → Σ in a projective space Σ defined over a skewfield K, then Γ is a classical and defined over a sub-skewfield K0 of K. Accordingly, Γ admits a full embedding e0 in a K0-projective space Σ0. We also prove that, under suitable hypotheses on e and e0, there exists an embedding \(\hat e:\Sigma _0 \to \Sigma \) such that \(\hat ee_0 = e\) and \({\hat e}\) preserves dimensions.