On semiclassical translation ovoids of \(H(3,q^2)\) (Q2051431)

Let \(\mathbb{F}_q\) be the finite field with \(q\) elements, where \(q = p^h\), \(p\) a prime. An \textit{I-permutation} of \(\mathbb{F}_{q^2}\) is a map \(f: \mathbb{F}_{q^2} \rightarrow \mathbb{F}_{q^2}\) such that for all \(x, y \in \mathbb{F}_{q^2}\), \(x \ne y\), it holds that \(\frac{f(x) - f(y)}{x - y} \notin \mathbb{F}_q\). Let \(\mathrm{PG}(2,q^2)\) be the Desarguesian projective plane of order \(q^2\), \(\ell\) a line of \(\mathrm{PG}(2,q^2)\) and \(\mathcal B\) be a Baer subline of \(\ell\). A set \(\mathcal X\) of \(q^2\) points of \(\mathrm{PG}(2,q^2) \setminus \ell\) is called an \textit{indicator set} if no line of \(\mathrm{PG}(2,q^2)\) through two distinct points of \(\mathcal X\) has a point in common with \(\mathcal B\). The set \(\{(1, x, f(x)) \mid x \in \mathbb{F}_{q^2}\}\) is an indicator set of \(\mathrm{PG}(2, q^2) \setminus \ell\) if and only if \(f\) is an \(I\)-permutation, [\textit{A. Cossidente} et al., Adv. Geom. 6, No. 4, 523--542 (2006; Zbl 1136.51006)]. Let \(\mathcal{H}(3, q^2)\) be a non-degenerate Hermitian surface of \(\mathrm{PG}(3, q^2)\). An \textit{ovoid} of \(\mathcal{H}(3, q^2)\) is a set of points of \(\mathcal{H}(3, q^2)\) meeting each line of \(\mathcal{H}(3, q^2)\) in one point. An ovoid of \(\mathcal{H}(3, q^2)\) is said to be \textit{locally Hermitian} with respect to a point \(P\) of \(\mathcal{H}(3, q^2)\) if it is the union of \(q^2\) secant lines of \(\mathcal{H}(3, q^2)\) through \(P\). Indicator sets are in one-to-one correspondence with locally Hermitian ovoids of \(\mathcal{H}(3, q^2)\). Moreover, two indicator sets are equivalent if and only if their associated locally Hermitian ovoids are isomorphic [\textit{A. Cossidente} et al., Adv. Geom. 7, No. 3, 357--373 (2007; Zbl 1130.51003)]. There are two standard examples of indicator sets: \begin{itemize} \item the \textit{classical} indicator set consisting of the \(q^2\) points of \(\mathrm{PG}(2, q^2) \setminus \ell\) on a line \(r\), where \(r \ne \ell\) and \(r \cap \ell \notin \mathcal{B}\). \item the \textit{semiclassical} indicator set formed by the \(q^2\) points of \(\mathrm{PG}(2, q^2) \setminus \ell\) of a Baer subplane \(\pi\), where \(|\pi \cap \ell| = q+1\) and \(|\pi \cap \mathcal{B}| = 0\). \end{itemize} To these indicator sets there correspond the so called \textit{classical} and \textit{semiclassical} ovoids of \(\mathcal{H}(3, q^2)\), respectively. In [Ann. Comb. 25, No. 2, 495--514 (2021; Zbl 1475.51003)], the author studied the equivalence issue of two indicator sets in terms of equivalence of the two \(I\)-permutations defining them. Based on this result, in the paper under review the author determines the number of inequivalent semiclassical ovoids of \(\mathcal{H}(3, q^2)\) extending a result established in [\textit{A. Cossidente} et al., Adv. Geom. 7, No. 3, 357--373 (2007; Zbl 1130.51003)] for \(\mathcal{H}(3, p^2)\).