A RELATIVE m-COVER OF A HERMITIAN SURFACE IS A RELATIVE HEMISYSTEM

An m-cover of the Hermitian surface H(3, q2) of PG(3, q2) is a set S of lines of H(3, q2) such that every point of H(3, q2) lies on exactly m lines of S, and 0 < m < q + 1. Segre (1965) proved that if q is odd, then m = (q + 1)/2, and called such a set S of lines a hemisystem. Penttila and Williford (2011) introduced the notion of a relative hemisystem: a set of lines R of H(3, q2), q even, disjoint from a symplectic subgeometry W(3, q) such that every point of H(3, q2)\W(3, q) lies on exactly q/2 elements of R. In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of H(3, q2) with respect to a symplectic subgeometry and proving that m must necessarily be q/2.