Semiarcs with a long secant in \(\mathrm{PG}(2, q)\) (Q2634811)

A non-empty point set \({\mathcal S}_t\) in a projective plane \(\Pi_q\) of order \(q\) is called a \textit{\(t\)-semiarc} if for every point \(P\) in \({\mathcal S}_t,\) there exist exactly \(t\geq 1\) (tangent) lines \(\ell_1,\ell_2,\dots,\ell_t\) such that \({\mathcal S}_t\cap \ell_i=\{P\}\) for \(i=1,2,\dots,t.\) If a line \(\ell\) meets \({\mathcal S}_t\) in \(k\) points, then \(\ell\) is called a \textit{\(k\)-secant}. A \textit{blocking set} \(B\) of \(\Pi_q\) is a set of points such that every line contains at least a point of \(B.\) If there is a line \(\ell\) such that \(|B|=q+|B \cap \ell|,\) then \(B\) is called a Redéi blocking set. In this paper, the authors embed a \(t\)-semiarc with a \(k\)-secant where \(2\leq k\leq q\) and \(1\leq t\leq q-3,\) in a small Redéi blocking set. Using strong results on Redéi blocking sets, they characterise such semiarcs.