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 stabilizer of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of PGL(2, q2) (and PSL(2, q2)) arising from the Hermitian Veronese curve in PG(3, q2), 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.& COPY; 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).