A relative $m$-cover of a Hermitian surface is a relative hemisystem

TL;DR

This paper introduces the concept of relative m-covers of Hermitian surfaces with respect to symplectic subgeometries and proves that the parameter m must be q/2, extending Segre's classical result.

Contribution

It generalizes the notion of hemisystems to relative m-covers in Hermitian surfaces and establishes a necessary condition for m in this context.

Findings

m must be q/2 for relative m-covers

Extension of Segre's hemisystem result to relative covers

Provides a new framework for studying line sets in Hermitian surfaces

Abstract

An $m$-cover of the Hermitian surface $H(3,q^2)$ of $PG(3,q^2)$ is a set $\mathcal{S}$ of lines of $H(3,q^2)$ such that every point of $H(3,q^2)$ lies on exactly $m$ lines of $\mathcal{S}$, and $0<m<q+1$. Segre (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 (2011) introduced the notion of a relative hemisystem: a set of lines $\mathcal{R}$ of $H(3,q^2)$, $q$ even, disjoint from a symplectic subgeometry $W(3,q)$ such that every point of $H(3,q^2)\setminus W(3,q)$ lies on exactly $q/2$ elements of $\mathcal{R}$. In this paper, we provide an analogue of Segre's result by introducing relative $m$-covers of $H(3,q^2)$ with respect to a symplectic subgeometry and proving that $m$ must necessarily be $q/2$.