Two projectively generated subsets of the Hermitian surface

Abstract

Using a variation of Seydewitz’s method of projective generation of quadrics we define two algebraic surfaces of $PG\left(3,q^2\right)$, called elliptic ${\mathcal{Q}}_F$-sets and semi-hyperbolic ${\mathcal{Q}}_F$-sets, and we show that these surfaces are contained in the Hermitian surface of $PG\left(3,q^2\right)$. Also, we characterize a semi-hyperbolic ${\mathcal{Q}}_F$-set as the intersection of two Hermitian surfaces. Finally we describe all possible configurations of the absolute set of an $\alpha$-correlation in $PG\left(2,q^2\right)$, where $\alpha$ is the involutory automorphism of $GF\left(q^2\right)$.