A relative <Emphasis Type="Italic">m</Emphasis>-cover of a Hermitian surface is a relative hemisystem

An m-cover of the Hermitian surface \(\mathrm {H}(3,q^2)\) of \(\mathrm {PG}(3,q^2)\) is a set \(\mathcal {S}\) of lines of \(\mathrm {H}(3,q^2)\) such that every point of \(\mathrm {H}(3,q^2)\) lies on exactly m lines of \(\mathcal {S}\), and \(0<m<q+1\). Segre (Annali di Matematica Pura ed Applicata Serie Quarta 70:1–201, 1965) proved that if q is odd, then \(m=(q+1)/2\), and called such a set \(\mathcal {S}\) of lines a hemisystem. Penttila and Williford (J Comb Theory Ser A 118(2):502–509, 2011) introduced the notion of a relative hemisystem of a generalised quadrangle \(\varGamma \) with respect to a subquadrangle \(\varGamma '\): a set of lines \(\mathcal {R}\) of \(\varGamma \) disjoint from \(\varGamma '\) such that every point P of \(\varGamma \setminus \varGamma '\) has half of its lines (disjoint from \(\varGamma '\)) lying in \(\mathcal {R}\). In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of generalised quadrangles of order \((q^2,q)\) with respect to a subquadrangle and proving that m must be q / 2 when the subquadrangle is doubly subtended. In particular, a relative m-cover of \(\mathrm {H}(3,q^2)\) with respect to a symplectic subgeometry \(\mathrm {W}(3,q)\) is a relative hemisystem.