Some subspaces of the projective space $PG(\Lambda ^{k}$V) related to regular spreads of $PG(V)$.
ELA. The Electronic Journal of Linear Algebra, Tome 20 (2010), pp. 354-366.

Voir la notice de l'article dans Electronic Library of Mathematics

Summary: Let V be a 2m-dimensional vector space over a field F (m $\geq 2$) and let k $\in {1, . . . , 2m - 1}$. Let A 2m - 1,k denote the Grassmannian of the (k - 1)-dimensional subspaces of $PG(V )$ and let e gr denote the Grassmann embedding of A 2m - 1,k into $PG( k V)$. Let S be a regular spread of $PG(V )$ and let X Sdenote 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 $\Sigma $of $PG( k V)$ for which the following holds: (1) the projective dimension of $\Sigma $is equal to 2m k - $2 \cdot m$ k - 1; (2) a (k - 1)-dimensional subspace $\alpha $of $PG(V )$ belongs to X Sif and only if e gr $(\alpha ) \in \Sigma $; (3) $\Sigma $is generated by all points e gr (p), where p is some point of X S.
Classification : 15A75, 51A45
Mots-clés : regular spread, Grassmannian, Grassmann embedding, Klein correspondence 
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     author = {De Bruyn, Bart},
     title = {Some subspaces of the projective space $PG(\Lambda ^{k}${V)} related to regular spreads of $PG(V)$.},
     journal = {ELA. The Electronic Journal of Linear Algebra},
     pages = {354--366},
     publisher = {mathdoc},
     volume = {20},
     year = {2010},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/EEJLA_2010__20__a27/}
}
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De Bruyn, Bart. Some subspaces of the projective space $PG(\Lambda ^{k}$V) related to regular spreads of $PG(V)$.. ELA. The Electronic Journal of Linear Algebra, Tome 20 (2010), pp. 354-366. https://geodesic-test.mathdoc.fr/item/EEJLA_2010__20__a27/