Ovoids in \(\text{PG}(3,q)\): A survey
Let \(q\) be a prime power, \(q > 2\). An ovoid in \(PG(3,q)\) may be defined to be a set \(\mathcal O\) of \(1 + q^2\) points of \(PG(3,q)\) with no three on a line. The lines tangent to \(\mathcal O\) at a given point of \(\mathcal O\) form a plane tangent to \(\mathcal O\) at that point. Each plane \(\pi\) not tangent to \(\mathcal O\) is a secant plane, i.e., \(\pi \cap {\mathcal O}\) is an oval in \(\pi\). An elliptic quadric in \(PG(3,q)\) provides an ovoid for all \(q > 2\), and these were shown by A. Barlotti and G. Panella (independently) in 1955 to be the only ovoids when \(q \) is odd. For \(q = 8\) (B. Segre) and \(q = 2^e \geq 8\), \(e\) odd, (J. Tits) one other ovoid is known. It is conjectured by some that these two families are the only ovoids that exist, and a great deal of work has been done characterizing these ovoids by their plane sections. When the paper under review was written in 1991, it was a remarkably complete and well-written survey of the results obtained (but in several cases not yet published) up to that time. In the meantime, the author along with several other colleagues mostly from Perth and Adelaide have pushed their techniques further. For example, it is now known that for \(2 < q \leq 32\), each ovoid of \(PG(3,q)\) is either an elliptic quadric or a Tits ovoid [cf. the author, \textit{T. Penttila} and \textit{G. F. Royle}, J. Geom. 50, No. 1-2, 143-150 (1994; Zbl 0805.51004)]. Nevertheless, we recommend this paper as a pleasant introduction to the subject, except for one sentence. The claim in Section 3 that ovoids and spreads of \(PG(3,q)\) are ovoids and spreads, respectively, of \(W(q)\) is not correct. For example, when \(q\) is odd, \(PG(3,q)\) has an ovoid but \(W(q)\) does not. For \(q\) even, the claim is true ``in spirit'' [cf., \textit{J. A. Thas}, Chapters 7 and 9 of `Handbook of incidence geometry', ed. \textit{F. Buekenhout}, North-Holland (1995; Zbl 0821.00012): see Handbook of incidence geometry, 295-347 (1995; Zbl 0822.51007) and ibid., 383-431 (1995; Zbl 0823.51009)].
scientific article; zbMATH DE number 896540
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