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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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