Geometric hyperplanes of the Lie geometry $A_{n,\{1,n\}}(\mathbb{F})$

In this paper we investigate hyperplanes of the point-line geometry $A_{n,\{1,n\}}(\mathbb{F})$ of point-hyerplane flags of the projective geometry $\mathrm{P}\mathrm{G}(n,\mathbb{F})$. Renouncing a complete classification, which is not yet within our reach, we describe the hyperplanes which arise from the natural embedding of $A_{n,\{1,n\}}(\mathbb{F})$, that is the embedding which yields the adjoint representation of $\mathrm{S}\mathrm{L}(n+1,\mathbb{F})$. The information we shall collect on these hyperplanes will allow us to prove that all hyperplanes of $A_{n,\{1,n\}}(\mathbb{F})$ are maximal subspaces of $A_{n,\{1,n\}}(\mathbb{F})$. Hyperplanes of $A_{n,\{1,n\}}(\mathbb{F})$ can also be contructed starting from suitable line-spreads of $\mathrm{P}\mathrm{G}(n,\mathbb{F})$ (provided that $\mathrm{P}\mathrm{G}(n,\mathbb{F})$ admits line-spreads, of course). Explicitly, let $\mathfrak{S}$ be a line-spread of $\mathrm{P}\mathrm{G}(n,\mathbb{K})$ satisfying certain conditions to be stated in this paper (which hold for all line-spreads obtained via the most popular constructions). The set of point-hyperplane flags $(p,H)$ of $\mathrm{P}\mathrm{G}(n,\mathbb{F})$ such that $H$ contains the member of $\mathfrak{S}$ through the point $p$ is a hyperplane of $A_{n,\{1,n\}}(\mathbb{F})$. We call these hyperplanes {\em hyperplanes of spread type}. Many of them arise from the natural embedding. We don’t know if this is the case for all of them.