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On planar arcs of size ( q + 3 ) ∕ 2
Author(s) -
Günay Gülizar,
Lavrauw Michel
Publication year - 2021
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21793
Subject(s) - counterexample , mathematics , conic section , corollary , completeness (order theory) , combinatorics , subject (documents) , conjecture , series (stratigraphy) , discrete mathematics , calculus (dental) , geometry , mathematical analysis , computer science , medicine , paleontology , dentistry , library science , biology
Abstract 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.