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Tight sets in finite classical polar spaces

  • Anamari Nakić and Leo Storme EMAIL logo
Published/Copyright: February 16, 2017
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Advances in Geometry
From the journal Advances in Geometry Volume 17 Issue 1

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).

Keywords: Tight sets; finite classical polar spaces; minihypers; blocking sets; Hermitian varieties; symplectic polar spaces
MSC 2010: 05B25; 51E20

Communicated by: G. Korchmáros

Funding statement: Anamari Nakić is supported in part by an STSM grant from the COST project Random network coding and designs over GF(q) (COST IC-1104) and in part by the Croatian Science Foundation under the project 1637

Acknowledgements

The authors thank the referee for the many suggestions which improved the article. We especially wish to thank the referee for the proof of Lemma 2.16 and for giving us the example of two disjoint Baer subgeometries PG(3, 3) in PG(3, 9) which correspond to each other under a symplectic polarity W(3, 9) (Theorem 2.26), thus providing us with an example of a (2q+2) -tight set in W(2r + 1, q) = W(3, 9) which is the union of two disjoint Baer subgeometries PG(2r+1,q)=PG(3,3).

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Received: 2015-5-8
Accepted: 2015-12-15
Published Online: 2017-2-16
Published in Print: 2017-1-1

© 2017 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  3. In memoriam Karl Strambach
  5. Dominance order on signed integer partitions
  7. Connectedness Bertini Theorem via numerical equivalence
  9. Invariants of the Brill–Noether curve
  11. Rigidity of entire graphs in weighted product spaces with nonnegative Bakry–Émery–Ricci tensor
  13. A note on surfaces with pg = q = 2 and an irrational fibration
  15. Mean width of random perturbations of random polytopes
  17. The automorphism groups of doubly transitive bilinear dual hyperovals
  19. Tight sets in finite classical polar spaces
https://doi.org/10.1515/advgeom-2016-0034
Keywords for this article
Tight sets; finite classical polar spaces; minihypers; blocking sets; Hermitian varieties; symplectic polar spaces

Articles in the same Issue

  1. Frontmatter
  3. In memoriam Karl Strambach
  5. Dominance order on signed integer partitions
  7. Connectedness Bertini Theorem via numerical equivalence
  9. Invariants of the Brill–Noether curve
  11. Rigidity of entire graphs in weighted product spaces with nonnegative Bakry–Émery–Ricci tensor
  13. A note on surfaces with pg = q = 2 and an irrational fibration
  15. Mean width of random perturbations of random polytopes
  17. The automorphism groups of doubly transitive bilinear dual hyperovals
  19. Tight sets in finite classical polar spaces
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