On existence of embeddings for point-line geometries (Q1932150)

Many important point-line geometries can be found as subgeometries of projective spaces. Any embedding of a point-line geometry into projective space over a division ring \(\mathbb{D}\) defines a presheaf over \(\mathbb{D}\). Vice versa, \textit{M. A. Ronan} [Eur. J. Comb. 8, 179--185 (1987; Zbl 0624.51007)] established that under certain conditions \(\mathbb{D}\) -presheaves give rise to projective embeddings. In this paper, the author considers a point-line geometry \(\Gamma\), a family \(\mathcal{S}\) of full subgeometries of \(\Gamma\) such that every point and every line of \(\Gamma\) lies in at least one member of \(\mathcal{S}\), a graph on \(\mathcal{S}\) with arc set \(\mathcal{A}\), a set \(\{\mathcal{F}_S | S\in\mathcal{S}\}\) of \(\mathbb{D}\)-presheaves on the geometries in \(\mathcal{S}\), and a set of presheaf isomorphisms \(\psi_{S,T}:\mathcal{F}_S|(S\cap T)\to \mathcal{F}_T|(S\cap T)\) such that \(\psi_{T,S}=\psi_{S,T}^{-1}\) for each \((S,T)\in\mathcal{A}\). She shows that under a number of additional conditions there exists a presheaf \(\mathcal{F}\) on \(\Gamma\) such that \(\mathcal{F}|S\) is isomorphic to \(\mathcal{F}_S\) for every \(S\in \mathcal{S}\). The additional conditions made are further investigated and several other assumptions are presented that imply the original conditions. As an application of her construction, the author gives new proofs that the following point-line geometries with thick lines are embeddable in projective spaces: nondegenerate polar spaces of finite rank at least four and Grassmannians of spherical buildings of type \(F_{4,1}\) with embeddable symplecta, \(D_{5,5}\), \(E_{n,n}\) where \(n\in\{6, 7, 8\}\), \(E_{6,2}\), \(E_{7,1}\).