scholarly journals On the Smallest Non-Trivial Tight Sets in Hermitian Polar Spaces

10.37236/6461 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Jan De Beule ◽  
Klaus Metsch
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Tight Set ◽  
Tight Sets

We show that an $x$-tight set of the Hermitian polar spaces $\mathrm{H}(4,q^2)$ and $\mathrm{H}(6,q^2)$ respectively, is the union of $x$ disjoint generators of the polar space provided that $x$ is small compared to $q$. For $\mathrm{H}(4,q^2)$ we need the bound $x<q+1$ and we can show that this bound is sharp.

scholarly journals Tight sets in finite classical polar spaces

2017 ◽  
Vol 17 (1) ◽  
pp. 109-129 ◽  
Author(s):  
Anamari Nakić ◽  
Leo Storme
Keyword(s):  
Pairwise Disjoint ◽  
Polar Space ◽  
Alternative Proof ◽  
Blocking Sets ◽  
Hermitian Variety ◽  
Tight Set ◽  
Tight Sets ◽  
Symplectic Polarity ◽  
Classical Polar Spaces ◽  
Even Pairs

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 $\text{PG}(2r+1,\,\sqrt{q})$ 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 $i \lt q^{5/8} / \sqrt{2} +1$ 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).

scholarly journals On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces

10.37236/90 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Hermitian Variety ◽  
Projective Embedding ◽  
Dual Polar Space ◽  
Transfinite Recursion ◽  
Symplectic Dual Polar Space ◽  
Dual Polar Spaces

Let $\Delta$ be a symplectic dual polar space $DW(2n-1,{\Bbb K})$ or a Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$, $n \geq 2$. We define a class of hyperplanes of $\Delta$ arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$ arising from its Grassmann-embedding if and only if there exists an empty $\theta$-Hermitian variety in ${\rm PG}(n-1,{\Bbb K})$. Using this result we are able to give the first examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the first examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transfinite recursion.

scholarly journals Generating Symplectic and Hermitian Dual Polar Spaces over Arbitrary Fields Nonisomorphic to ${\Bbb F}_2$

10.37236/972 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Bart De Bruyn ◽  
Antonio Pasini
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Dual Polar Space ◽  
Symplectic Dual Polar Space ◽  
Dual Polar Spaces

Cooperstein proved that every finite symplectic dual polar space $DW(2n-1,q)$, $q \neq 2$, can be generated by ${2n \choose n} - {2n \choose n-2}$ points and that every finite Hermitian dual polar space $DH(2n-1,q^2)$, $q \neq 2$, can be generated by ${2n \choose n}$ points. In the present paper, we show that these conclusions remain valid for symplectic and Hermitian dual polar spaces over infinite fields. A consequence of this is that every Grassmann-embedding of a symplectic or Hermitian dual polar space is absolutely universal if the (possibly infinite) underlying field has size at least 3.

scholarly journals The Valuations of the Near Octagon ${\Bbb I}_4$

10.37236/1102 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Bart De Bruyn ◽  
Pieter Vandecasteele
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Near Polygon ◽  
Hyperbolic Quadric ◽  
Dual Polar Space ◽  
Full Embedding ◽  
Parabolic Quadric ◽  
Dual Polar Spaces

The maximal and next-to-maximal subspaces of a nonsingular parabolic quadric $Q(2n,2)$, $n \geq 2$, which are not contained in a given hyperbolic quadric $Q^+(2n-1,2) \subset Q(2n,2)$ define a sub near polygon ${\Bbb I}_n$ of the dual polar space $DQ(2n,2)$. It is known that every valuation of $DQ(2n,2)$ induces a valuation of ${\Bbb I}_n$. In this paper, we classify all valuations of the near octagon ${\Bbb I}_4$ and show that they are all induced by a valuation of $DQ(8,2)$. We use this classification to show that there exists up to isomorphism a unique isometric full embedding of ${\Bbb I}_n$ into each of the dual polar spaces $DQ(2n,2)$ and $DH(2n-1,4)$.

scholarly journals Simple connectivity in polar spaces

2016 ◽  
Vol 28 (3) ◽  
Author(s):  
Max Horn ◽  
Reed Nessler ◽  
Hendrik Van Maldeghem
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Simple Connectivity ◽  
Small Parameters

AbstractWe settle the simple connectivity of the geometry opposite a chamber in a polar space of rank 3 by completing the job for the non-embeddable polar spaces and some polar spaces with small parameters.

scholarly journals Tight sets and m-ovoids of finite polar spaces

2007 ◽  
Vol 114 (7) ◽  
pp. 1293-1314 ◽  
Author(s):  
John Bamberg ◽  
Shane Kelly ◽  
Maska Law ◽  
Tim Penttila
Keyword(s):  
Polar Spaces ◽  
Tight Sets

scholarly journals Cross-Intersecting Erdős-Ko-Rado Sets in Finite Classical Polar Spaces

10.37236/4734 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ferdinand Ihringer
Keyword(s):  
Maximum Size ◽  
Upper Bounds ◽  
Polar Space ◽  
Polar Spaces ◽  
The Cross ◽  
Classical Polar Spaces

A cross-intersecting Erdős-Ko-Rado set of generators of a finite classical polar space is a pair $(Y, Z)$ of sets of generators such that all $y \in Y$ and $z \in Z$ intersect in at least a point. We provide upper bounds on $|Y| \cdot |Z|$ and classify the cross-intersecting Erdős-Ko-Rado sets of maximum size with respect to $|Y| \cdot |Z|$ for all polar spaces except some Hermitian polar spaces.

An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces

2019 ◽  
Vol 31 (2) ◽  
pp. 491-502 ◽  
Author(s):  
Klaus Metsch
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Classical Polar Spaces

AbstractIn this paper, we call a set of lines of a finite classical polar space an Erdős–Ko–Rado set of lines if no two lines of the polar space are opposite, which means that for any two lines l and h in such a set there exists a point on l that is collinear with all points of h. We classify all largest such sets provided the order of the underlying field of the polar space is not too small compared to the rank of the polar space. The motivation for studying these sets comes from [7], where a general Erdős–Ko–Rado problem was formulated for finite buildings. The presented result provides one solution in finite classical polar spaces.

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