  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 Korchm\'aros, Sz\H{o}nyi and Nagy. An algebraic approach
based on the investigation of some algebraic varieties attached to the arc is
used.
