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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 17, Issue 1


Tight sets in finite classical polar spaces

Anamari Nakić / Leo Storme
Published Online: 2017-02-16 | DOI: https://doi.org/10.1515/advgeom-2016-0034


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


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About the article

Received: 2015-05-08

Accepted: 2015-12-15

Published Online: 2017-02-16

Published in Print: 2017-01-01

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

Citation Information: Advances in Geometry, Volume 17, Issue 1, Pages 109–129, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2016-0034.

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