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- Title
Tight sets in finite classical polar spaces.
- Authors
Nakić, Anamari; Storme, Leo
- 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, √q) and generators of H(2r +1, q), if q ≥ 81 is an odd square and i <(q2/3 -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 {Δ, Δ⊥}, and (2r +1)-dimensional Baer subgeometries. For W(2r +1, q) with r even, pairs of disjoint r-spaces {Δ, Δ⊥} cannot occur. The (2r +1)-dimensional Baer subgeometries in the i-tight set of W(2r +1, q) are invariant under the symplectic polarity ⊥ of W(2r +1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under ⊥. This improves previous results where i < q5/8/√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 (q2/3 -1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r +1, q).
- Subjects
HERMITIAN structures; PARTIAL algebras; UNIVERSAL algebra; SYMPLECTIC geometry; DIFFERENTIAL geometry
- Publication
Advances in Geometry, 2017, Vol 17, Issue 1, p109
- ISSN
1615-715X
- Publication type
Article
- DOI
10.1515/advgeom-2016-0034