Dominant lax embeddings of polar spaces

Abstract

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