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.

Transitive hyperovals in finite projective planes

SONNINO, Angelo
2005-01-01

Abstract

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.
2005
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11563/1540
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