On Absolute Points of Correlations of $\mathrm{PG}(2,q^n)$

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 $\mathrm{PG}(V)$. 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 $\mathrm{PG}(2,q^n)$ whose set of absolute points are, besides cones, the (possibly degenerate) $C_F^m$-sets studied by Donati and Durante in 2014. 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 $\mathrm{PG}(2,q^n)$ induced by a non-degenerate, non-reflexive sesquilinear form of $V=\mathbb{F}_{q^n}^3$.