Higher dimensional analogues of Klein's quadric (Q1184500)

A spread can be described by a class of \(d\)-dimensional subspaces of a \(2d\)-dimensional vector space over a field \(K\). Therefore this class can be seen as a subset \(S\) of the grassmannian \(G_{2d-1,d-1}\) of the \(({2d\choose d}-1)\)-dimensional projective space \(P\). Without saying it the author develops properties of such manifolds and the belonging quadrics. He shows that no three points of \(S\) are collinear in \(G_{2d- 1,d-1}\). For the dimensions \(d=3,4\) he defines a set \(Q\supset G_{2d-1,d-1}\) of quasi-special points of \(P\) such that the line through two different points of \(S\) doesn't meet \(Q\) either.