On blocking semiovals with an 8-secant in projective planes of order 9

Abstract

Let $S$ be a blocking semioval in an arbitrary projective plane $\Pi$ of order 9 which meets some line in 8 points. According to Dover in $[2]$, $20\leq \vert S\vert\leq 24$. In $[8]$ one of the authors showed that if $\Pi$ is desarguesian, then $22\leq\vert S\vert\leq 24$. In this note all blocking semiovals with this property in all non-desarguesian projective planes of order 9 are completely determined. In any non-desarguesian plane $\Pi$ it is shown that $21\leq \vert S\vert \leq 24$ and for each $i\in \{ 21,22,23,24\}$ there exist blocking semiovals of size $i$ which meet some line in 8 points. Therefore, the Dover's bound is not sharp.