Semiovals with large collinear subsets (Q1841861)

A semioval in a projective plane \(\pi\) of order \(q-1\) is a set \(S\) of points with the property that every point \(P \in S\) is contained in a unique tangent line. The author considers semiovals for which \(|L \setminus S|\leq 2\) for some line \(L\) of \(\pi\). He proves that for any \(q\) no line exists for which \(|L \setminus S|=0\). If \(q>3\), then also \(|L \setminus S|=1\) is impossible. If \(q >5\) and if there is more than one \((q-1)\)-secant, then \(S\) is obtained from a vertexless triangle by removing some subset of points from one side.