Inherited conics in Hall planes

Aart Blokhuis; István Kovács; Gábor P. Nagy; Tamás Szőnyi

doi:10.1016/j.disc.2018.12.009

Inherited conics in Hall planes

Aart Blokhuis, István Kovács, Gábor P. Nagy (Corresponding author), Tamás Szőnyi

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Abstract

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.

Original language	English
Pages (from-to)	1098-1107
Number of pages	10
Journal	Discrete Mathematics
Volume	342
Issue number	4
DOIs	https://doi.org/10.1016/j.disc.2018.12.009
Publication status	Published - 1 Apr 2019

Keywords

Arcs
Finite projective planes
Hall planes

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abstract = "The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchm{\'a}ros on Desargues configurations with perspective triangles inscribed in a conic.",

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AU - Nagy, Gábor P.

AU - Szőnyi, Tamás

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AB - The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.

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