On planar arcs of size (q+3)/2

The subject of this paper is the study of small complete arcs in PG(2, q), for q odd, with at least (q + 1)/2 points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal work. This gives an alternative to Pellegrino's long proof which was obtained in a series of papers in the 1980s. As a corollary of our analysis, we obtain a counterexample to a misconception in the literature.