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 \(\textrm{PG}(n,\mathbb {F})\). 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.