A generalization of the quadratic cone of $$\mathop {\mathrm{PG}}(3,q^n)$$ and its relation with the affine set of the Lüneburg spread

Abstract

Using a variation of Seydewitz’s method of projective generation of quadratic cones, we define an algebraic surface of $$\mathop {\mathrm{PG}}(3,q^n)$$ , called $$\sigma $$ -cone, whose $$\mathbb {F}_{q^n}$$ -rational points are the union of a line with a set $$\mathcal{A}$$ of $$q^{2n}$$ points. If $$q^n=2^{2h+1}, h\ge 1$$ , and $$\sigma $$ is the automorphism of $$\mathbb {F}_{q^n}$$ given by $$x \mapsto x^{2^h} $$ , then the set $$\mathcal{A}$$ is the affine set of the Luneburg spread of $$\mathop {\mathrm{PG}}(3,q^n)$$ . If $$n=2$$ and $$\sigma $$ is the involutory automorphism of $$\mathbb {F}_{q^2}$$ , then a $$\sigma $$ -cone is a subset of a Hermitian cone and the set $$\mathcal{A}$$ is the union of q non-degenerate Hermitian curves.