Blocking semiovals of type \((1,m+1,n+1)\) (Q2753536)

A set \(X\) of points in a projective plane \(\Pi\) of order \(q\) is called a blocking semioval if the following conditions are satisfied: (a) each point \(P\) of \(X\) is contained in a unique tangent line, (b) every line meets \(X\) in at least one point. A set \(X\) of points in a projective plane \(\Pi\) is of type \((m_1,m_2,\ldots,m_k)\) if each line of \(\Pi\) meets \(X\) in \(m_i\) points for some \(i \in \{ 1,\ldots,k \}\), and for each \(i \in \{ 1,\ldots,k \}\) some line of \(\Pi\) meets \(X\) in exactly \(m_i\) points. NEWLINENEWLINENEWLINEThe paper studies blocking semiovals of type \((1,m+1,n+1)\) with \(1 \leq m < n\). An example of such a bocking semioval consists of the \(3(q-1)\) points on three lines in general position, the three intersection points being omitted. The authors prove that this is the only example if \(q^2+q+1\) is a prime or three times a prime. This is also proved to be correct for almost all prime powers \(q \leq 1024\), and the parameter sets of the remaining unresolved cases are listed.