On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaces

The complete classification of the orbits on subspaces under the action of the projective stabiliser of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of $\text{PGL}(2,q^2)$ (and $\text{PSL}(2,q^2)$) arising from the Hermitian Veronese curve in $\text{PG}(3,q^2)$, a maximal rational curve embedded on a smooth Hermitian surface with some fascinating properties. The study of its orbits leads to a new construction of quasi-Hermitian surfaces: sets of points with the same combinatorial and geometric properties as a non-degenerate Hermitian surface.