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

In this paper we investigate hyperplanes of the point-line geometry \(A_{n,\{1,n\}}(\mathbb {F})\) $A_{n,\{1,n\}}(\mathbb{F})$ of point-hyerplane flags of the projective geometry \(\textrm{PG}(n,\mathbb {F})\) $\text{PG}(n,\mathbb{F})$ . Renouncing a complete classification, which is not yet within our reach, we describe the hyperplanes which arise from the natural embedding of \(A_{n,\{1,n\}}(\mathbb {F})\) $A_{n,\{1,n\}}(\mathbb{F})$ , that is the embedding which yields the adjoint representation of \(\textrm{SL}(n+1,\mathbb {F})\) $\text{SL}(n+1,\mathbb{F})$ . By exploiting properties of a particular sub-class of these hyerplanes, namely the singular hyperplanes, we shall prove that all hyperplanes of \(A_{n,\{1,n\}}(\mathbb {F})\) $A_{n,\{1,n\}}(\mathbb{F})$ are maximal subspaces of \(A_{n,\{1,n\}}(\mathbb {F})\) $A_{n,\{1,n\}}(\mathbb{F})$ . Hyperplanes of \(A_{n,\{1,n\}}(\mathbb {F})\) $A_{n,\{1,n\}}(\mathbb{F})$ can also be contructed starting from suitable line-spreads of \(\textrm{PG}(n,\mathbb {F})\) $\text{PG}(n,\mathbb{F})$ (provided that \(\textrm{PG}(n,\mathbb {F})\) $\text{PG}(n,\mathbb{F})$ admits line-spreads, of course). Explicitly, let \(\mathfrak {S}\) $\mathfrak{S}$ be a composition line-spread of \(\textrm{PG}(n,\mathbb {F})\) $\text{PG}(n,\mathbb{F})$ such that every hyperplane of \(\textrm{PG}(n,\mathbb {F})\) $\text{PG}(n,\mathbb{F})$ contains a sub-hyperplane of \(\textrm{PG}(n,\mathbb {F})\) $\text{PG}(n,\mathbb{F})$ spanned by lines of \(\mathfrak {S}\) $\mathfrak{S}$ . Then the set of points (p, H) of \(A_{n,\{1,n\}}(\mathbb {F})\) $A_{n,\{1,n\}}(\mathbb{F})$ such that H contains the member of \(\mathfrak {S}\) $\mathfrak{S}$ through p is a hyperplane of \(A_{n,\{1,n\}}(\mathbb {F})\) $A_{n,\{1,n\}}(\mathbb{F})$ . We call these hyperplanes hyperplanes of spread type. Many but not all of them arise from the natural embedding.