Complete $(q+1)$-arcs in $\mathrm{PG}(2,{\mathbb{F}}_{q^6})$ from the Hermitian curve

Abstract

We prove that, if $q$ is large enough, the set of the ${\mathbb{F}}_{q^6}$-rational points of the Hermitian curve is a complete $(q+1)$-arc in $\mathrm{PG}(2,{\mathbb{F}}_{q^6})$, addressing an open case from a recent paper by Korchmáros, Szőnyi and Nagy. An algebraic approach based on the investigation of some algebraic varieties attached to the arc is used.