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Tight sets in finite classical polar spaces

(2017) ADVANCES IN GEOMETRY. 17(1). p.109-129
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Abstract
We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG(2r + 1, root q) and generators of H(2r + 1, q), if q >= 81 is an odd square and i < (q(2/3) - 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Delta,Delta(perpendicular to)}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Delta,Delta(perpendicular to)} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity perpendicular to of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under perpendicular to. This improves previous results where i < q(5/8)/root 2+ 1 was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q(2/3) - 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).
Keywords
Tight sets, finite classical polar spaces, minihypers, blocking sets, Hermitian varieties, symplectic polar spaces, MULTIPLE BLOCKING SETS, BAER SUBPLANES, MINIHYPERS, ARCS

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

Chicago
Nakić, Anamari, and Leo Storme. 2017. “Tight Sets in Finite Classical Polar Spaces.” Advances in Geometry 17 (1): 109–129.
APA
Nakić, A., & Storme, L. (2017). Tight sets in finite classical polar spaces. ADVANCES IN GEOMETRY, 17(1), 109–129.
Vancouver
1.
Nakić A, Storme L. Tight sets in finite classical polar spaces. ADVANCES IN GEOMETRY. 2017;17(1):109–29.
MLA
Nakić, Anamari, and Leo Storme. “Tight Sets in Finite Classical Polar Spaces.” ADVANCES IN GEOMETRY 17.1 (2017): 109–129. Print.
@article{8517621,
  abstract     = {We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG(2r + 1, root q) and generators of H(2r + 1, q), if q >= 81 is an odd square and i < (q(2/3) - 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Delta,Delta(perpendicular to)}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Delta,Delta(perpendicular to)} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity perpendicular to of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under perpendicular to. This improves previous results where i < q(5/8)/root 2+ 1 was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q(2/3) - 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).},
  author       = {Nakić, Anamari and Storme, Leo},
  issn         = {1615-715X},
  journal      = {ADVANCES IN GEOMETRY},
  keywords     = {Tight sets,finite classical polar spaces,minihypers,blocking sets,Hermitian varieties,symplectic polar spaces,MULTIPLE BLOCKING SETS,BAER SUBPLANES,MINIHYPERS,ARCS},
  language     = {eng},
  number       = {1},
  pages        = {109--129},
  title        = {Tight sets in finite classical polar spaces},
  url          = {http://dx.doi.org/10.1515/advgeom-2016-0034},
  volume       = {17},
  year         = {2017},
}

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