On Singer action on Hermitian varieties

Abstract

We show that a non-degenerate Hermitian variety ${\mathcal{H}(2n, q^2)}$ in PG(2n, q ²) can be partitioned in Baer parabolic quadrics Q(2n, q), if q is odd, or in Baer symplectic spaces ${\mathcal{W}(2n-1, q)}$ , if q is even. We construct a partial spread of ${\mathcal{H}(4, q^2)}$ of size (q ⁵ + 1)/(q + 1), admitting a group of order (q ⁵ + 1)/(q + 1) and a hyperoval of size 2(q ⁵ + 1)/(q + 1) on ${\mathcal{DH}(4, q^2)}$ , the point line dual generalized quadrangle of ${\mathcal{H}(4, q^2)}$ , admitting a dihedral group of order 2(q ⁵ + 1)/(q + 1).