Quasi-Hermitian Varieties and Their Barlotti--Cofman Representation

TL;DR

This paper explores the geometric structure of quasi-Hermitian varieties, specifically Buekenhout--Metz and Buekenhout--Tits types, within the Barlotti--Cofman representation, revealing their quadratic and non-quadratic cone configurations.

Contribution

It explicitly determines the BC representation of BM and BT quasi-Hermitian varieties in projective spaces, clarifying their geometric and structural differences.

Findings

BM varieties correspond to quadratic cones with hyperbolic bases

BT varieties correspond to non-quadratic cones

The configuration of spread elements at infinity is described

Abstract

Quasi-Hermitian varieties arise as higher-dimensional generalizations of non-classical unitals, including the Buekenhout--Metz (BM) and Buekenhout--Tits (BT) families. After reviewing known constructions and structural properties, we determine explicitly the BC representation of BM and BT quasi-Hermitian varieties in $\mathrm{PG}(3,q^2)$ inside $\mathrm{PG}(6,q)$. We show that BM varieties correspond to quadratic cones with hyperbolic base, whereas BT varieties give rise to non-quadratic cones, and we describe the associated configuration of spread elements in the section at infinity. These results provide a geometric interpretation of the non-classical nature of BM and BT varieties within the BC framework.