Dominant lax embeddings of polar spaces (Q2471478)

It is known that every lax projective embedding \(e : \Gamma\to PG(V)\) of a point-line geometry \(\Gamma\) admits a hull, namely a projective embedding \(\widetilde e :\Gamma\to PG(\widetilde V)\) uniquely determined up to isomorphisms by the following property: \(V\) and \(\widetilde V\) are defined over the same skewfield, say \(K\), there is a morphism of embeddings \(\widetilde f:\widetilde e\to e\) and, for every embedding \(e' : \Gamma\to PG(V')\) with \(V'\) defined over \(K\), if there is a morphism \(g : e'\to e\) then a morphism \(f :\widetilde e\to e'\) also exists such that \(\widetilde f = gf\). If \(e = \widetilde e\) then we say that \(e\) is dominant. Clearly, hulls are dominant. Let now \(\Gamma\) be a non-degenerate polar space of rank \(n \geq 3\). We prove the following: A lax embedding \(e :\Gamma\to PG(V)\) is dominant if and only if, for every geometric hyperplane \(H\) of \(\Gamma\), \(e(H)\) spans a hyperplane of \(PG(V)\). We also give some applications of the above result.