ELA
SOME SUBSPACES OF THE PROJECTIVE SPACE
PG(
VK
V
)
RELATED TO REGULAR SPREADS OF
PG(
V
)
∗BART DE BRUYN†
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 letegrdenote the Grassmann embedding ofA2m−1,k into PG(VkV). LetSbe a reg-ular 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 ofS. Then we show that there exists a subspace Σ of PG(VkV) for which the following holds: (1) the projective dimension of Σ is equal to 2m
k
−2· mk
−1; (2) a (k−1)-dimensional subspaceαof PG(V) belongs toXSif and only ifegr(α)∈Σ; (3) Σ is generated by all pointsegr(p), wherepis some point ofXS.
Key words. Regular spread, Grassmannian, Grassmann embedding, Klein correspondence.
AMS subject classifications.15A75, 51A45.
∗Received by the editors February 18, 2010. Accepted for publication July 13, 2010. Handling
Editor: Raphael Loewy.
†Department of Mathematics, Ghent University, Gent, Belgium (bdb@cage.ugent.be). Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society Volume 20, pp. 354-366, July 2010
