Ovoids of the quadric Q\((2n,q)\) (Q1345531)

Let \(Q(2n,q)\) be a nonsingular quadric in the projective space \(PG (2n,q)\) for \(n \geq 2\). An avoid of \(Q(2n,q)\) is a set of points of \(Q(2n,q)\) which has exactly one point in common with each subspace of maximal dimension on \(Q(2n,q)\). The authors prove that if \(q\) is odd, \(q \neq 3\), and every ovoid of the nonsingular quadric \(Q(4,q)\) in \(PG(4,q)\) is an elliptic quadric, then \(Q(6,q)\), and hence also \(Q(2n,q)\) with \(n \geq 3\), has no ovoid. As a corollary, it follows that \(Q(2n,5)\) and \(Q(2n,7)\), \(n \geq 3\), have no ovoid. Also, a survey of the known results about existence and non-existence of ovoids in \(Q(2n,q)\) is given.