On Singer action on Hermitian varieties (Q2352467)

Let \(\mathrm{PG}(n,q)\), \(q=p^h\), \(p\) prime, be the projective space of dimension \(n\) over the field \(\mathrm{GF}(q)\). Consider the Hermitian variety \(\mathcal{H}(2n,q^2)\) contained in \(\mathrm{PG}(2n,q^2)\). When \(n\) equals \(1\) a known result shows that the Hermitian curve \(\mathcal{H}(2,q^2)\) can be partitioned into Baer subconics \(Q(2,q)\), if \(q\) is odd, or admits a cyclic spread if \(q\) is even; see [\textit{R. D. Baker} et al., Lond. Math. Soc. Lect. Note Ser. 191, 17--30 (1993; Zbl 0804.51013)] and [\textit{S. Barwick} and \textit{G. Ebert}, Unitals in projective planes. New York, NY: Springer (2008; Zbl 1156.51006)]. In this paper, the author generalizes the previous results to the following: { Theorem.} In \(\mathrm{PG}(2n,q^2)\) a non-degenerate Hermitian variety \(\mathcal{H}(2n,q^2)\) can be partitioned into Baer parabolic quadrics \(\mathcal{Q}(2n,q)\), if \(q\) is odd, or in Baer symplectic spaces \(\mathcal{W}(2n-1,q)\), if \(q\) is even. Also, a partial spread of \(\mathcal{H}(4,q^2)\), that is a collection of mutually disjoint lines of the generalized quadrangle \(\mathcal{H}(4,q^2)\), of size \((q^5+1)/(q+1)\) is provided. Finally, a hyperoval of \(\mathcal{DH}(4,q^2)\) (the dual of the generalized quadrangle \(\mathcal{H}(4,q^2)\)), that is a set of points intersecting every line in \(0\) or \(2\) points, of size \(2(q^5+1)/(q+1)\) is presented. The partial spread and the hyperoval admit a group of size \((q^5+1)/(q+1)\) and \(2(q^5+1)/(q+1)\) respectively.