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

TL;DR

This paper provides a concise proof regarding the completeness of small arcs in projective planes, specifically those with many points on a conic, offering an alternative to previous lengthy proofs and correcting a misconception.

Contribution

It presents a short, comprehensive proof of the completeness of certain arcs in PG(2,q), addressing an open problem and correcting prior misconceptions.

Findings

Confirmed completeness of arcs with at least (q+1)/2 points on a conic

Provided an alternative, shorter proof to Segre's open problem

Counterexample to a misconception in existing literature

Abstract

The subject of this paper is the study of small complete arcs in $\mathrm{PG}(2,q)$, 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 [20]. 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 [6].