  Let $m$ be a positive integer and $q$ be a prime power. For large finite base
fields $\mathbb F_q$, we show that any curve can be used to produce a complete
$m$-arc as long as some generic explicit geometric conditions on the curve are
verified. To show the effectiveness of our theory, we derive complete $m$-arcs
from hyperelliptic curves and from Artin-Schreier curves.
