On the classification of low-degree ovoids of \(Q^+ (5,q)\) (Q6639854)

Let \(\mathrm{Q}^+(5,q)\) denote a hyperbolic quadric embedded in the projective space \(\mathrm{PG}(5,q)\), it is the quadric determined by a non-singular quadratic form of Witt index \(3\). Hence the quadric \(\mathrm{Q}^+(5,q)\) contains points, lines and planes of \(\mathrm{PG}(5,q)\), and this quadric is also known as the quadric of Klein. The so-called Klein correspondence is a bijection between the lines of the projective space \(\mathrm{PG}(3,q)\) and the points of \(\mathrm{Q}^+(5,q)\). Notably, two lines of \(\mathrm{PG}(3,q)\) correspond to two collinear points on \(\mathrm{Q}^+(5,q)\) if and only if the two lines intersect.\N\NAn ovoid of \(\mathrm{Q}^+(5,q)\) is a set \(\mathcal{O}\) of points of \(\mathrm{Q}^+(5,q)\) such that every plane of \(\mathrm{Q}^+(5,q)\) meets \(\mathcal{O}\) in exactly one point. Under the Klein correspondence, a spread of \(\mathrm{PG}(3,q)\), i.e. a set of lines partitioning the point set, becomes an ovoid of \(\mathrm{Q}^+(5,q)\) (and vice versa). The regular spread of \(\mathrm{PG}(3,q)\) becomes an elliptic quadric embedded in a \(3\)-dimensional subspaces section of \(\mathrm{Q}^+(5,q)\). Hence ovoids of \(\mathrm{Q}^+(5,q)\) exist for all \(q\), but apart from the elliptic quadric, many other (infinite families) of ovoids of \(\mathrm{Q}^+(5,q)\) are known.\N\NThe nice paper under review contributes to the classification of particular ovoids. It is known (and recalled in the paper) that to any ovoid of \(\mathrm{Q}^+(5,q)\), \(q\) a prime power, two polynomials \(f_1(X,Y)\) and \(f_2(X,Y)\) can be associated. The classification is then obtained for ovoids determined by the polynomials \(f_1(X,Y)\) and \(f_2(X,Y)\) under the assumptions that \(f_1 = Y + g(X)\) and \(q > 6.31 (d+1)^{(13/3)}\) with \(d = \max \{\mathrm{deg}(f_1),\mathrm{def}(f_2)\}\)