Quasi-Hermitian Varieties and Their Barlotti--Cofman Representation

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.