Blocking sets of tangent and external lines to a hyperbolic quadric in \(\mathrm{PG}(3, q)\)
type: Item
instance of: scholarly article
title: Blocking sets of tangent and external lines to a hyperbolic quadric in \(\mathrm{PG}(3, q)\)
zbMATH Open document ID: 1398.51006
author: Bart De Bruyn
author: Binod K. Sahoo
author: Bikramaditya Sahu
published in: Discrete Mathematics
publication date: 2018-08-15T00:00:00Z
review text: In this paper, the authors characterise the smallest blocking sets with respect to the set of tangent and external lines to a hyperbolic quadric in \(\mathrm{PG}(3,q)\). Their main result is the following: Let \(B\) be a point set in \(\mathrm{PG}(3,q)\), \(q\leq 3\), blocking all tangent and external lines to a hyperbolic quadric \(\mathcal{H}\). Then \(|B|\geq q^2+q\) and equality holds if and only if \(B=\pi_x\setminus\{x\}\) for some point \(x\in \mathcal{H}\). Here, \(\pi_x\) denotes the tangent plane to \(\mathcal{H}\) at the point \(x\). This result was shown for \(q\) even in [the second and third author, ``Blocking sets of tangent and external lines to a hyperbolic quadric in \(\mathrm{PG}(3,q)\), \(q\) even'', Proc. Indian Acad. Sci., Math. Sci. (to appear)]. A similar result was proven in [\textit{P. Biondi} and \textit{P. M. Lo Re}, J. Geom. 92, No. 1--2, 23--27 (2009; Zbl 1170.51004)] and [\textit{P. Biondi} et al., Beitr. Algebra Geom. 48, No. 1, 209--215 (2007; Zbl 1121.51006)] for blocking sets with respect to external lines (where some small values of \(q\) had to be excluded). The proof of the theorem is combinatorial in nature and makes use of earlier results in the planar case, i.e., results on blocking sets with respect to tangent and/or external lines to a conic in \(\mathrm{PG}(2,q)\).
reviewed by: Geertrui Van de Voorde
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 6918738
zbMATH Keywords: projective space
zbMATH Keywords: blocking set
zbMATH Keywords: irreducible conic
zbMATH Keywords: hyperbolic quadric
MaRDI profile type: Publication
full work available at URL: j.disc.2018.06.027
OpenAlex ID: W2884915842
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DOI: 10.1016/J.DISC.2018.06.027
description: scientific article; zbMATH DE number 6918738