Transitive hyperovals in finite projective planes.
Angelo Sonnino · CINECA IRIS Institutional Research Information System (University of Basilicata) · 2005
Let $\\Omega$ be a hyperoval in a projective plane $\\pi$ of even order $n$, and $G$ the collineation group of $\\pi$ preserving $\\Omega$. If $G$ acts transitively on the points of $\\Omega$, then $\\Omega$ is a transitive hyperoval. By a deep result due to M. Biliotti and G. Korchm\\'{a}ros (1987), if $\\Omega$ is transitive and $|G|$ is divisible by $4$, then either $n=2,4$ and $\\Omega$ is a hyperconic, or $n=16$ and $|G|\\leq 144$. In this paper, it is shown that the case $n=16$ with $|G|=144$ only occurs when $\\pi\\cong\\mathrm{PG}(2,16)$ and $\\Omega$ is the Lunelli-Sce-Hall hyperoval.