A note on relative hemisystems of Hermitian generalised quadrangles

TL;DR

This paper introduces criteria for constructing relative hemisystems of Hermitian spaces, unifies known families, and discovers new related relative hemisystems, advancing the understanding of their structure and existence.

Contribution

It provides a unified framework for known relative hemisystems and introduces new techniques for constructing and identifying them in Hermitian geometries.

Findings

Unified criteria for relative hemisystems of $ ext{H}(3,q^2)$

New proofs of existing infinite families

Discovery of new relative hemisystems related to Cossidente's family

Abstract

In this paper we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space $\mathrm{H}(3,q^2)$, unifying all known infinite families. We use these conditions to provide new proofs of the existence of the known infinite families of relative hemisystems. Reproving these results has allowed us to find new relative hemisystems closely related to an infinite family of Cossidente's, and develop techniques that are likely to be useful in finding relative hemisystems in future.