On absolute points of correlations in PG$(2,q^n)$

Abstract

Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the $d$-dimensional projective space PG$(V)$. Everything is known in this case for both degenerate and non-degenerate reflexive forms if $\mathbb{F}$ is either ${\mathbb{R}}$, ${\mathbb{C}}$ or a finite field ${\mathbb{F}}_q$. In this paper we consider degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of PG$(2,q^n)$ whose set of absolute points are, besides cones, the (possibly degenerate) $C_F^m$-sets. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in PG$(2,q^n)$ induced by a non-degenerate, non-reflexive sesquilinear form of $V=\mathbb{F}_{q^n}^3$.