Geometric hyperplanes of non-embeddable Grassmannians
The author gives a complete classification of geometric hyperplanes for Grassmann spaces arising from the \(k\)-dimensional subspaces of a vector space \(V\) over a proper skew field. (Such a Grassmann space is not embeddable in a projective space.) Every geometric hyperplane is formed by all \(k\)-subspaces that meet a subspace of codimension \(k\) nontrivially. Hence to within projective transformations there is just a unique type of geometric hyperplane. This is in sharp contrast to geometric hyperplanes of Grassmann spaces over a commutative field. They arise as hyperplane sections of the associated Grassmann variety [the reviewer, J. Geom. 16, 168-180 (1982; Zbl 0463.51004), the second author, J. Algebra 145, No. 1, 55-82 (1992; Zbl 0751.51002)] and are not in one orbit with respect to the projective group.
scientific article; zbMATH DE number 167693