Every special set of the Hermitian surface $\mathsf{H}(3,q^2)$ is classical

John Bamberg; Ethan Kelley

Every special set of the Hermitian surface $\mathsf{H}(3,q^2)$ is classical

Combinatorics 2026-07-09 v1

Authors: John Bamberg , Ethan Kelley

Abstract

Special sets of the Hermitian surface $\mathsf{H}(3,q^2)$ , $q$ odd, were introduced by Shult and Thas (1995) in order to construct new finite generalised quadrangles, yet only one example is known to exist and it gives rise to a classical generalised quadrangle. We show that there can be no other special sets of the Hermitian surface.

Keywords

Hermitian surface special set generalised quadrangle finite geometry classical

Cite

@article{arxiv.2607.08135,
  title  = {Every special set of the Hermitian surface $\mathsf{H}(3,q^2)$ is classical},
  author = {John Bamberg and Ethan Kelley},
  journal= {arXiv preprint arXiv:2607.08135},
  year   = {2026}
}