Some subspaces of the projective space PG(Λ^kV) related to regular spreads of PG(V)

Abstract

Let V be a 2m-dimensional vector space over a field F (m ≥ 2) and let k ∈ {1, .. .,2m − 1}. Let A2m−1,k denote the Grassmannian of the (k − 1)-dimensional subspaces of PG(V ) and let egr denote the Grassmann embedding of A2m−1,k into PG(Vk V ). Let S be a regular spread of PG(V ) and let XS denote the set of all (k − 1)-dimensional subspaces of PG(V ) which contain at least one line of S. Then we show that there exists a subspace Σ of PG(Vk V ) for which the following holds: (1) the projective dimension of Σ is equal to 2m k − 2 · m k − 1; (2) a (k−1)-dimensional subspace α of PG(V ) belongs to XS if and only if egr(α) ∈ Σ; (3) Σ is generated by all points egr(p), where p is some point of XS.