Affine Grassmannians
Let \({\mathbf P \mathbf G}(V)\) be an \((n + 1)\)-dimensional projective space arising from a right \((n + 2)\)-dimensional vector space over some skewfield \(D\). Let \(0 \leq k \leq n\) and let \({\mathcal H}\) be a collection of \(k\)-dimensional subspaces of \({\mathbf P \mathbf G} (V)\) with the property that for every \((k-1)\)-dimensional subspace \(H_ 0\) and every \((k+1)\)- dimensional subspace \(H_ 1\) with \(H_ 0 \subseteq H_ 1\), the set of all \(k\)-dimensional subspaces containing \(H_ 0\) and contained in \(H_ 1\) either contains a unique element of \(\mathcal H\) or is entirely contained in \(\mathcal H\). Thus \(\mathcal H\) is a geometric hyperplane of the \(k\)-Grassmann geometry of \({\mathbf P \mathbf G} (V)\). The author characterizes by the diagram the subgeometry of \({\mathbf P \mathbf G} (V)\) obtained from \({\mathbf P \mathbf G} (V)\) by deleting all subspaces of dimension \(\geq k + 2\) and all those of dimension \(\leq k + 1\) which do not contain and are not contained in any \(k\)-dimensional subspace not belonging to \(\mathcal H\). The diagram is an \(A_{k + 2}\)-diagram in which the \(k\)th edge is replaced by an edge representing a dual affine plane, and the \((k+1)\)th edge (= last edge) is replaced by an edge representing the class of linear spaces. One additional assumption is hypothesized: if \(k > 1\), then lines must have at least 5 points.
scientific article; zbMATH DE number 753619