Geometric hyperplanes of Lie incidence geometries (Q674790)

Let \(\Gamma=({\mathcal P},{\mathcal L})\) be a geometry of points and lines. A subspace of \(\Gamma\) is a set of points which contains every line that meets it in at least two points. An embedding \(\mu\) of \(\Gamma\) in a finite-dimensional vector space \(V\) consists of a map \(\mu_1\) of \({\mathcal P}\) into the set of 1-subspaces of \(V\) and a map \(\mu_2\) of \({\mathcal L}\) into the set of 2-subspaces such that a) incidence is preserved, b) every 1-subspace in the image of a line is the image of a point, and c) \(V\) is spanned by the image of \({\mathcal P}\). A geometric hyperplane of \(\Gamma\) is a proper subspace which meets every line. If \(\mu\) is an embedding of \(\Gamma\) in \(V\), and \(W\) is a hyperplane of \(V\), the set \(H_W\) of points of \(\Gamma\) whose \(\mu_1\)-images lie in \(W\) is a geometric hyperplane. The question is whether all geometric hyperplanes of \(\Gamma\) arise from an embedding in this way. Using results from a fundamental paper by \textit{M. Ronan} [Eur. J. Comb. 8, 179-185 (1987; Zbl 0624.51007)], the authors give an affirmative answer for a number of Lie incidence systems: (1) the Grassmann spaces of type \(A_{n,2}\); (2) the half-spin geometry \(D_{5,5}\); (3) \(E_{6,1}\).