\(2\)-Semiarcs in \(PG(2,q)\), \(q \leq 13\)

Abstract

A 2-semiarc is a pointset \(\mathcal{S}_2\) with the property that the number of tangent lines to \(\mathcal{S}_2\) at each of its points is two. Using theoretical results and computer-aided search, we provide the complete classification of 2-semiarcs in \(PG(2, q)\) for \(q \leq 7\), determine the spectrum of their sizes for \(q \leq 9\), and prove existence results for \(q = 11\) and \(q = 13\). Additionally, for several sizes of 2-semiarcs in \(PG(2, q)\) with \(q \leq 7\), classification results have been obtained through theoretical proofs.