Desarguesian and unitary complete partial ovoids (Q1947190)

An ovoid of a finite classical polar space \({\mathcal P}\) is a set of points intersecting every generator of \({\mathcal P}\) in exactly one point. A partial ovoid of a finite classical polar space \({\mathcal P}\) is a set of points intersecting every generator of \({\mathcal P}\) in at most one point. A partial ovoid of a finite classical polar space \({\mathcal P}\) is called complete when it is not contained in a larger partial ovoid of \({\mathcal P}\). The hyperbolic quadric \(Q^+(7,q)\) contains different types of ovoids, among which the Desarguesian and the unitary ovoid. These two ovoids are contained in a subvariety of the Grassmannian. This article focusses on these two families of ovoids. The author presents for the polar spaces \(Q^+(2^t-1,q)\), with \(t\) even, or, \(t\) odd and \(q\) even, and for the symplectic polar space of \(\mathrm{PG}(2^t-1,q)\), with both \(t\) and \(q\) odd, complete partial ovoids of size \(q^t+1\), which for \(t=3\) reduce to the Desarguesian ovoid of \(Q^+(7,q)\), \(q\) even. In a second construction, the author applies the twisted tensor embedding to the translation ovoid \({\mathcal O}\) of \(H(3,q^2)\), \(q\) odd, obtained by a commuting orthogonal polarity of \(\mathrm{PG}(3,q)\), which leads to a complete partial ovoid of the elliptic quadric \(Q^-(9,q)\), for \(q=p^{2h+1}\), \(p\) prime with \(p\equiv -1,-3 \pmod{8}\), and of the hyperbolic quadric \(Q^+(9,q)\) for all the other odd prime powers \(q\).