Embedded polar spaces revisited

Abstract

In this paper we introduce generalized pseudo-quadratic forms and develope some theory for them. Recall that the codomain of a $(\sigma,\varepsilon)$-quadratic form is the group $\overline{K} := K/K_{\sigma,\varepsilon}$, where $K$ is the underlying division ring of the vector space on which the form is defined and $K_{\sigma,\varepsilon} := \{t-t^\sigma\varepsilon\}_{t\in K}$. Generalized pseudo-quadratic forms are defined in the same way as $(\sigma,\varepsilon)$-quadratic forms but for replacing $\overline{K}$ with a quotient $\overline{K}/\overline{R}$ for a subgroup $\overline{R}$ of $\overline{K}$ such that $\lambda^\sigma\overline{R}\lambda = \overline{R}$ for any $\lambda\in K$. In particular, every non-trivial generalized pseudo-quadratic form admits a unique sesquilinearization, characterized by the same property as the sesquilinearization of a pseudo-quadratic form. Moreover, if $q:V\rightarrow \overline{K}/\overline{R}$ is a non-trivial generalized pseudo-quadratic form and $f:V\times V\rightarrow K$ is its sesquilinarization, the points and the lines of $\mathrm{PG}(V)$ where $q$ vanishes form a subspace $S_q$ of the polar space $S_f$ associated to $f$. After a discussion of quotients and covers of generalized pseudo-quadratic forms we prove the following: let $e:S\rightarrow \mathrm{PG}(V)$ be a projective embedding of a non-degenerate polar space $S$ of rank at least $2$; then $e(S)$ is either the polar space $S_q$ associated to a generalized pseudo-quadratic form $q$ or the polar space $S_f$ associated to an alternating form $f$. By exploiting this theorem we also obtain an elementary proof of the following well known fact: an embedding $e$ as above is dominant if and only if either $e(S) = S_q$ for a pseudo-quadratic form $q$ or $\mathrm{char}(K)\neq 2$ and $e(S) = S_f$ for an alternating form $f$.