Blocking sets of tangent and external lines to a hyperbolic quadric in \(\mathrm{PG}(3, q)\)
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 \(\mathrm{PG}(2,q)\).
scientific article; zbMATH DE number 6918738