The subject of this paper is the study of small complete arcs in $\text{PG}\left(2,q\right)$, for $q$ odd, with at least $\left(q+1\right)∕2$ 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.

中文翻译：

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在大小为 ( q + 3 ) ∕ 2 的平面弧上

本文的主题是研究小完整弧 $\text{PG}\left(2,q\right)$， 为了 $q$ 奇怪的，至少 $\left(q+1\right)∕2$点在一个圆锥上。我们对 Segre 在他的开创性工作中留下的完整性问题给出了一个简短的综合证明。这为 Pellegrino 在 1980 年代的一系列论文中获得的长期证明提供了替代方案。作为我们分析的推论，我们获得了一个反例，以解决文献中的误解。