Ovoids of \(\mathcal{H}(3, q^2)\), \(q\) odd, admitting a group of order \(\frac{(q + 1)^3}{2}\) (Q1752688)

A \textit{generalised quadrangle} of order \((s, t )\) is a partial linear space with the properties that every point is incident with \(t + 1\) lines, every line is incident with \(s + 1\) points, and through any point \(P\), not on a line \(l\), there is a unique line intersecting \(l\). An \textit{ovoid} of a generalised quadrangle is a subset of its points which meets every line of the generalised quadrangle in exactly one point. The generalised quadrangle \(\mathcal{H}(3,q^2)\) consists of the points and lines of a non-degenerate Hermitian surface of \(\mathrm{PG}(3,q^2)\), it has order \((q^2,q)\). In this paper, a new construction of an ovoid \(\mathcal{O}\) of the Hermitian surface \(\mathcal{H}(3,q^2)\), \(q > 3\) odd, is provided. The stabiliser group of the ovoid is calculated; it depends on the particular elements \(a,b\) in \(\mathrm{GF}(q)\) that are used in the construction. But in all cases, the stabiliser of \(\mathcal{O}\) has a subgroup of order \((q+1)^3\). Finally, it is also shown that \(\mathcal{O}\) cannot be obtained from a Hermitian curve by means of multiple derivation and it is not locally Hermitian.