Blocking sets of tangent and external lines to an elliptic quadric in \(\mathrm{PG}(3,q)\)
For a given nonempty set \(\mathcal L\) of lines of \(\mathrm{PG}(d, q),\) a set \(B\) of points of \(\mathrm{PG}(d, q)\) is called an \(\mathcal L\)-blocking set if each line in \(\mathcal L\) contains at least one point of \(B.\) Let \(Q^-(3,q)\) be an ellipic quadric of \(\mathrm{PG}(3,q),\) \(q\) odd. The lines of \(\mathrm{PG}(3, q)\) are divided into three collections: (i) the set \(\mathcal E\) of external lines which are disjoint from \(Q^-(3, q),\) (ii) the set \(\mathcal T\) of tangent lines which meet \(Q^-(3, q)\) exaclty in a point, (iii) the set \(\mathcal S\) of secant lines which meet \(Q^-(3, q)\) in two points. The authors study the \(\mathcal L\)-blocking set when either \(\mathcal L=\mathcal T\cup{\mathcal E},\) or \(\mathcal L=\mathcal E.\) They prove the following characterizations: \begin{itemize} \item[(a)] If \(\mathcal L=\mathcal T\cup\mathcal E,\) an \(\mathcal L\)-blocking set \(B\) in \(\mathrm{PG}(3, q)\) has size \(|B| \geq q^2 + q.\) Equality holds if and only if \(B=x^{\perp}\setminus \{x\}\) for some point \(x\) of \(Q^-(3, q).\) \item[(b)]If \(B\) is an \(\mathcal E\)-blocking set in \(\mathrm{PG}(3, q),\) then \(|B| \geq q^2.\) Equality holds if and only if \(B = \pi \setminus Q^-(3, q)\) for some secant plane \(\pi\) of \(\mathrm{PG}(3, q).\) \end{itemize}
scientific article; zbMATH DE number 7434836
scientific article