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Tetrads of lines spanning PG(7,2)

Shaw, Ron; Gordon, Neil; Havlicek, Hans

Authors

Ron Shaw

Hans Havlicek



Abstract

Our starting point is a very simple one, namely that of a set L₄ of four mutually skew lines in PG(7,2). Under the natural action of the stabilizer group G(L₄)<GL(8,2) the 255 points of PG(7,2) fall into four orbits ω₁,ω₂,ω₃,ω4, of respective lengths 12,54,108,81. We show that the 135 points ∈ω₂∪ω₄ are the internal points of a hyperbolic quadric H7 determined by L₄, and that the 81-set ω₄ (which is shown to have a sextic equation) is an orbit of a normal subgroup G₈₁≅(Z₃)4 of G(L4). There are 40 subgroups ≅(Z₃)3 of G₈₁, and each such subgroup H<G₈₁ gives rise to a decomposition of ω4 into a triplet {RH,R′H,R′′H} of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S₃(2) in PG(7,2). This ties in with the recent finding 225-239 --- that each S=S₃(2) in PG(7,2) determines a distinguished Z₃ subgroup of GL(8,2) which generates two sibling copies S′,S′′ of S.

Citation

Shaw, R., Gordon, N., & Havlicek, H. (2013). Tetrads of lines spanning PG(7,2). Bulletin of the Belgian Mathematical Society, Simon Stevin, 20(4), 735-752. https://doi.org/10.36045/bbms/1382448192

Journal Article Type Article
Acceptance Date Feb 1, 2012
Publication Date Jan 1, 2013
Deposit Date Sep 30, 2015
Publicly Available Date Nov 23, 2017
Journal Bulletin of the Belgian Mathematical Society - Simon Stevin
Print ISSN 1370-1444
Peer Reviewed Peer Reviewed
Volume 20
Issue 4
Pages 735-752
DOI https://doi.org/10.36045/bbms/1382448192
Keywords Segre variety S₃(2), Line-spread, Invariant polynomials
Public URL https://hull-repository.worktribe.com/output/379399
Publisher URL https://projecteuclid.org/euclid.bbms/1382448192#abstract
Additional Information Authors' accepted manuscript of article published in: Bulletin of the Belgian Mathematical Society - Simon Stevin, 2013, v.20, issue 4

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