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.