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Some subspaces of the projective space PG(Lambda(K) V) related to regular spreads of PG(V)

Bart De Bruyn (UGent)
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Abstract
Let V be a 2m-dimensional vector space over a field F (m >= 2) and let k is an element of {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(Lambda(k) V). Let S be a regular spread of PG(V) and let X-S 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 Sigma of PG(Lambda(k) V) for which the following holds: (1) the projective dimension of Sigma is equal to ((2m)(k)) - 2 . ((m)(k)) - 1; (2) a (k - 1)-dimensional subspace alpha of PG(V) belongs to X-S if and only if e(gr)(alpha) is an element of Sigma; (3) Sigma is generated by all points e(gr)(p), where p is some point of X-S.
Keywords
Klein correspondence, Grassmann embedding, Regular spread, Grassmannian

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Citation

Please use this url to cite or link to this publication:

Chicago
De Bruyn, Bart. 2010. “Some Subspaces of the Projective Space PG(Lambda(K) V) Related to Regular Spreads of PG(V).” Electronic Journal of Linear Algebra 20: 354–366.
APA
De Bruyn, B. (2010). Some subspaces of the projective space PG(Lambda(K) V) related to regular spreads of PG(V). ELECTRONIC JOURNAL OF LINEAR ALGEBRA, 20, 354–366.
Vancouver
1.
De Bruyn B. Some subspaces of the projective space PG(Lambda(K) V) related to regular spreads of PG(V). ELECTRONIC JOURNAL OF LINEAR ALGEBRA. 2010;20:354–66.
MLA
De Bruyn, Bart. “Some Subspaces of the Projective Space PG(Lambda(K) V) Related to Regular Spreads of PG(V).” ELECTRONIC JOURNAL OF LINEAR ALGEBRA 20 (2010): 354–366. Print.
@article{1105471,
  abstract     = {Let V be a 2m-dimensional vector space over a field F (m >= 2) and let k is an element of {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(Lambda(k) V). Let S be a regular spread of PG(V) and let X-S 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 Sigma of PG(Lambda(k) V) for which the following holds: (1) the projective dimension of Sigma is equal to ((2m)(k)) - 2 . ((m)(k)) - 1; (2) a (k - 1)-dimensional subspace alpha of PG(V) belongs to X-S if and only if e(gr)(alpha) is an element of Sigma; (3) Sigma is generated by all points e(gr)(p), where p is some point of X-S.},
  author       = {De Bruyn, Bart},
  issn         = {1081-3810},
  journal      = {ELECTRONIC JOURNAL OF LINEAR ALGEBRA},
  keywords     = {Klein correspondence,Grassmann embedding,Regular spread,Grassmannian},
  language     = {eng},
  pages        = {354--366},
  title        = {Some subspaces of the projective space PG(Lambda(K) V) related to regular spreads of PG(V)},
  volume       = {20},
  year         = {2010},
}

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