A family of hemisystems on the parabolic quadrics

TL;DR

This paper introduces a new family of hemisystems on parabolic quadrics for all ranks and odd prime powers, expanding the known constructions and revealing symmetries related to the group a3_3(q).

Contribution

It provides the first known construction of hemisystems on a3(2d, q) for ranks d e; 2, broadening the understanding of geometric structures in finite projective spaces.

Findings

Constructed hemisystems for all ranks d e; 2 and odd prime powers q.

Identified symmetries related to a3_3(q) group.

First known constructions for d e; 4.

Abstract

We constuct a family of hemisystems of the parabolic quadric $\mathcal{Q}(2d, q)$, for all ranks $d \ge 2$ and all odd prime powers $q$, that admit $\Omega_3(q) \cong \mathrm{PSL}_2(q)$. This yields the first known construction for $d \ge 4$.