Root shadow spaces
A root shadow space is the point-line truncation of the \(J\)-Grassmannian of a spherical building, where \(J\) is the type of the set of nodes corresponding to the fundamental roots not perpendicular to the longest root. In the paper under review, the authors characterize these geometries (except when the root shadow space is a polar space) using the notion of a root filtration space. The latter is a point-line space in which the pairs of points are divided up in five ``relations'', describing the mutual positions of points (among which the simplest are ``being identical'' and ``being collinear''). The main result states that a geometry is a non-degenerate root filtration space if and only if it is a root shadow space (with finite singular rank). The proof is purely geometric. This is rather surprising, as the theorem extends some of Timmesfelds' very group theoretical results on abstract root subgroups (the five relations above express the different possibilities for the commutator of two rank 1 groups --- usually SL\(_2\)'s). This shows how beautiful geometry can be.
scientific article; zbMATH DE number 5154714