A generalization of the quadratic cone of PG(3,qn) and its relation with the affine set of the Lüneburg spread

Using a variation of Seydewitz’s method of projective generation of quadratic cones, we define an algebraic surface of PG (3 , q n ) , called σ-cone, whose Fqn-rational points are the union of a line with a set A of q 2 n points. If q n = 2 2 h + 1 , h≥ 1 , and σ is the automorphism of Fqn given by x↦x2h, then the set A is the affine set of the Lüneburg spread of PG (3 , q n ). If n= 2 and σ is the involutory automorphism of Fq2, then a σ-cone is a subset of a Hermitian cone and the set A is the union of q non-degenerate Hermitian curves.