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Translation ovoids. (English) Zbl 1042.51008
The aim of the paper is to present some recent results on translation ovoids in a homogeneous way. The author uses the relationship between spreads of the projective space PG\((3,q)\) and ovoids of the Klein quadric \(Q^+(5,q)\) as a guideline for the approach to semifield flocks.
Having recalled that translation ovoids of an orthogonal polar space \(P\) exist if and only if \(P\) is one of \(Q^+(3,q), Q(4,q)\) or \(Q^+(5,q)\), the author describes the relationship between translation ovoids of \(Q^+(5,q)\) and semifield spreads of PG\((3,q)\). Furthermore, he discusses a bound for the existence of sporadic semifield flocks due to S. Ball, A. Blockhuis and M. Lavrauw. Using a particular model \(H\) of \(Q^+(5,q)\) he also gives a characterization of the ovoid associated with a flock. Finally, he presents a different construction of the ovoid associated with a semifield flock working for all values of \(q\) and he gives a characterization of the sporadic flock of order \(3^5\) due to I. Cardinali, O. Polverino and R. Trombetti.

MSC:
51E20 Combinatorial structures in finite projective spaces
51A50 Polar geometry, symplectic spaces, orthogonal spaces
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