Complements of Grassmann substructures in projective Grassmannians (Q744033)

Let \(V\) be a vector space of dimension \(n\), take two subspaces \(Z\subseteq W\) and define the set \({\mathcal D}\) of outer subspaces \(U\) by \(Z\not\subseteq U\) or \(U\not\subseteq W\). We assume that \({\mathcal D}\) is not empty. For \(1\leq k\leq n-2\), the \(k\)-th Grassmannian \(G_{k}\) is an incidence structure consisting of the \(k\)-dimensional elements of \({\mathcal D}\) and of the \(k+1\)-dimensional elements of \({\mathcal D}\) with \(\subset\) as incidence relation. The author shows that the underlying projective space as well as the maximal possible \(Z\) and the minimal possible \(W\) can be recovered from any of the \(G_{k}\)'s. This is also rephrased into a theorem about the extension of automorphisms.