  In a projective plane over a finite field, complete $(k,n)$-arcs with few
characters are rare but interesting objects with several applications to finite
geometry and coding theory. Since almost all known examples are large, the
construction of small ones, with $k$ close to the order of the plane, is
considered a hard problem. A natural candidate to be a small $(k,n)$-arc with
few characters is the set $\Omega(\mathcal{C})$ of the points of a plane curve
$\mathcal{C}$ of degree $n$ (containing no linear components) such that some
line meets $\mathcal{C}$ transversally in the plane, i.e. in $n$ pairwise
distinct points. Let $\mathcal{C}$ be either the Hermitian curve of degree
$q+1$ in $\mathrm{PG}(2,q^{2r})$ with $r\ge 1$, or the rational BKS curve of
degree $q+1$ in $\mathrm{PG}(2,q^r)$ with $q$ odd and $r\ge 1$. Then
$\Omega(\mathcal{C})$ has four and seven characters, respectively. Furthermore,
$\Omega(\mathcal{C})$ is small as both curves are either maximal or minimal.
The completeness problem is investigated by an algebraic approach based on
Galois theory and on the Hasse-Weil lower bound. Our main result for the
Hermitian case is that $\Omega(\mathcal{C})$ is complete for $r\ge 4$. For the
rational BKS curve, $\Omega(\mathcal{C})$ is complete if and only if $r$ is
even. If $r$ is odd then the uncovered points by the $(q+1)$-secants to
$\Omega(\mathcal{C})$ are exactly the points in $\mathrm{PG}(2,q)$ not lying in
$\Omega(\mathcal{C})$. Adding those points to $\Omega(\mathcal{C})$ produces a
complete $(k,q+1)$-arc in $\mathrm{PG}(2,q^r)$, with $k=q^r+q$. The above
results do not hold true for $r=2$ and there remain open the case $r=3$ for the
Hermitian curve, and the cases $r=3,4$ for the rational BKS curve. As a by
product we also obtain two results of interest in the study of the Galois
inverse problem for $\mathrm{PGL}(2,q)$.
