The determination of ovoids of PG(3,\(q\)) containing a pointed conic
A pointed conic in a finite desarguesian plane of even order is a \((q+1)\)-arc obtained by interchanging a point of a conic with its nucleus. For \(q=4\) a pointed conic is still a conic. B. Segre [see, e.g., \textit{A. Barlotti}: Some topics in finite geometrical structures. Institute of Statistics Mimeo Series No. 439, University ot North Carolina, North Carolina, 1965] constructed the first non-classical example of ovoid in \(PG(3,8)\) using pointed conics and G. Fellegara in [\textit{G. Fellegara}, Atti Accad. Naz. Lincei., Rend., Cl. Sci. Fis. Mat. Nat., VIII Ser. 32, 170-176 (1962; Zbl 0109.39403)] proved, with the aid of a computer, that there is a unique non-classical ovoid in \(PG(3,8)\). Hence, Segre gived a different construction of the Tits-ovoid of \(PG(3,8)\) [see \textit{J. Tits}, Sem. Bourbaki 13, No. 210 (1961; Zbl 0267.20041)]. Using the theory of generalized quadrangles, the author proves that, if an ovoid contains a pointed conic as a plane section, then either \(q=4\) and it is an elliptic quadric, or \(q=8\) and it is a Tits ovoid. Therefore the old Segre's original idea for constructing non-classical ovoids using pointed conics only works for \(q=8\).
scientific article; zbMATH DE number 1443423
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