On complete $m$-arcs

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.