Absolutely universal projective embeddings of long-root geometries of orthogonal type

Abstract

Let $G$ be a split Chevalley group of type $B_n$ ($n\ge 3$) or $D_n$ ($n\ge 4$), let $L$ be its Lie algebra and $\Gamma$ its long-root geometry, namely $\Gamma$ is the line-grassmannian of the polar space associated to $G$. It is well known that $L$ hosts a projective embedding of $\Gamma$, called the adjoint embedding, which realizes $\Gamma$ as a projective variety in the projective geometry of the underlying vector space of $L$. It is also known that $\Gamma$ admits the absolutely universal projective embedding, which is such that all projective embeddings of $\Gamma$ can be obtained from it as projections. When the field $F$ over which $G$ is defined is finite, the universal embedding of $\Gamma$ is just the same as the adjoint embedding. The same is not necessarily true when $F$ is infinite. Indeed, as we shall prove in this paper, the universal projective embedding of $\Gamma$ lives in a direct sum $K\oplus L$ for an $F$-vector space $K$, the dual of which is isomorphic to the derivation space $\mathrm{Der}(F)$ of $F$, which in general is non-trivial. Accordingly, the adjoint embedding of $\Gamma$ is universal if and only if $\mathrm{Der}(F)$ is trivial, namely $F$ is either perfect of positive characteristic or algebraic over its minimum subfield.