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Ovoids and translation ovals. (English) Zbl 0915.51003

The only known classes of ovoids in 3-dimensional projective spaces over fields of even order are the elliptic quadrics and the Tits ovoids. The authors present a common characterization of these two classes as follows. An ovoid \(\mathcal O\) of PG\((3,q)\), where \(q > 2\) is even, is an elliptic quadric or a Tits ovoid if and only if \(\mathcal O\) admits a tangent line \(L\) such that all secant planes of \(\mathcal O\) containing \(L\) intersect \(\mathcal O\) in a translation oval and \(L\) is an axis of at least one of these ovals.
This result generalizes known characterizations of elliptic quadrics, but it seems to be the first one of its type which includes the Tits ovoids.

MSC:

51E20 Combinatorial structures in finite projective spaces
51B10 Möbius geometries
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