On absolute points of correlations of \(\mathrm{PG}(2,q^{n})\)

Jozefien D'haeseleer¹ ; Nicola Durante²

¹PhD fellow of FWO (Research foundation-Flanders, Belgium).\\ Department of Mathematics: Analysis, Logic and Discrete Mathematics.\\ Krijgslaan 281, 9000 Ghent, Campus De Sterre, Building S8.
²Università di Napoli "Federico II"

The electronic journal of combinatorics, Tome 27 (2020) no. 2

Résumé

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 $V=\mathbb{F}_{q^n}^3$.

Détail

DOI : 10.37236/8920

Classification : 51E21, 05B25
Mots-clés : sesquilinear forms, correlations