Blocking sets of external lines to a conic in \(\text{PG}(2,q)\), \(q\) odd
A point set \(B\subseteq \text{PG}(2,q)\) is a blocking set for a family \(\mathcal L\) of subsets of \( \text{PG}(2,q)\) if it meets any element of \({\mathcal L}\) in at least one point. Blocking sets for the set \(\mathcal F\) of all the lines of the plane have been widely investigated. An interesting problem is to determine minimal blocking sets for suitable subsets of \(\mathcal F\). In this paper, the authors investigate minimal blocking sets for the set of all the external lines to a conic \(\mathcal C\) in \( \text{PG}(2,q)\), \(q\) odd. They show that such blocking sets \(B\) always contain at least \(q-1\) points. For \(q\geq 9\) and \(q=3\), equality holds if only if \(B\) is linear, that is \(B\) consists of the \(q-1\) points of a chord \(r\) of \(\mathcal C\) not on \(\mathcal C\). For \(q=5\) or \(q=7\) they prove that, up to projectivities, there is just one more sporadic example of blocking set of size \(q-1\). The proof of these results is based on the investigation of the linear system of all polynomial vanishing at every internal point of \(\mathcal C\) and the study of the involutions of the group \( \text{PGL}(2,q)\).
scientific article; zbMATH DE number 5082371