On quasi-Hermitian varieties in \(\operatorname{PG}(3, q^2)\) (Q898101)

The author gives two combinatorial characterizations of the Hermitian surface of the finite 3-dimensional projective space in terms of quasi-Hermitian varieties. Specifically, he proves the following results. If \(s\) and \(t\) are integers with \(0 \leq s <t \leq q^2-1\) then a quasi-Hermitian variety of \(\mathrm{PG}(3,q^2)\) of line class \([s+1,t+1,q^2+1]_1\) is a Hermitian variety of \(\mathrm{PG}(3,q^2)\). In \(\mathrm{PG}(3,q^2)\) a quasi-Hermitian variety of line class \([1,q+1,\dots, q^2+1]_1\) is a Hermitian variety of \(\mathrm{PG}(3,q^2)\).