Construction and Classification of Complete $(k,3)$-arcs from a Ceva 6-Figure in $PG(2,4)$

Year 2025, Volume: 7 Issue: 2, 46 - 51, 29.12.2025

Elif Altıntaş Kahriman , Ayşe Bayar

https://izlik.org/JA69PZ64XH

Abstract

This study investigates complete $(k,3)$-arcs generated from a given Ceva 6-figure in the projective plane $PG(2,4)$. The analysis reveals a unique complete $(7,3)$-arc obtained by adding the center point of the Ceva 6-figure, forming a Fano subplane, and eight distinct complete $(9,3)$-arcs constructed by adjoining three points on distinct 2-secant lines. No complete $(8,3)$-arc constructed from the given Ceva 6-figure exists. These results emphasize the combinatorial significance of Ceva-based configurations in finite projective planes and contribute to the systematic understanding of arc structures in finite geometry.

Keywords

projective plane , (k,3)-arc , complete arc , Ceva 6-figure , finite geometry , PG(2,4)

$PG(2,4)$ Projektif Düzleminde Ceva 6-Figüründen Elde Edilen Tam $(k,3)$-arkların İnşası ve Sınıflandırılması

https://izlik.org/JA69PZ64XH

Abstract

Bu çalışma, $PG(2,4)$ projektif düzleminde verilen bir Ceva 6-figüründen üretilen tam $(k,3)$-arkları araştırmaktadır. Yapılan analiz, Ceva 6-figürünün merkez noktasının eklenmesiyle bir Fano alt-düzlemi oluşturan tek bir tam $(7,3)$-arkın elde edildiğini ve farklı 2-kesen doğrular üzerinde yer alan üç noktanın eklenmesiyle sekiz farklı tam $(9,3)$-arkın oluşturulabildiğini göstermektedir. Verilen Ceva 6-figüründen türetilmiş herhangi bir tam $(8,3)$-arkın var olmadığı da belirlenmiştir. Bu sonuçlar, sonlu projektif düzlemlerde Ceva temelli konfigürasyonların kombinatoryal önemini vurgulamakta ve sonlu geometride ark yapılarının sistematik olarak anlaşılmasına katkı sağlamaktadır.

Keywords

projektif düzlem , (k,3)-ark , tam ark , Ceva 6-figür , sonlu geometri , PG(2,4)

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