On the simple connectedness of hyperplane complements in dual polar spaces

Let Δ be a dual polar space of rank n ≥ 4 , H be a hyperplane of Δ and Γ ≔ Δ ∖ H be the complement of H in Δ . We shall prove that, if all lines of Δ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.