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

Abstract

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 $\textrm{PG}(n,\mathbb {F})$. 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 $\textrm{SL}(n+1,\mathbb {F})$. By exploiting properties of a particular sub-class of these hyerplanes, namely the singular hyperplanes, we shall 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 $\textrm{PG}(n,\mathbb {F})$ (provided that $\textrm{PG}(n,\mathbb {F})$ admits line-spreads, of course). Explicitly, let $\mathfrak {S}$ be a composition line-spread of $\textrm{PG}(n,\mathbb {F})$ such that every hyperplane of $\textrm{PG}(n,\mathbb {F})$ contains a sub-hyperplane of $\textrm{PG}(n,\mathbb {F})$ spanned by lines of $\mathfrak {S}$. Then the set of points (p, H) of $A_{n,\{1,n\}}(\mathbb {F})$ such that H contains the member of $\mathfrak {S}$ through p is a hyperplane of $A_{n,\{1,n\}}(\mathbb {F})$. We call these hyperplanes hyperplanes of spread type. Many but not all of them arise from the natural embedding.