Inherited conics in Hall planes

TL;DR

This paper investigates when conics in projective planes over finite fields retain their properties as arcs in Hall planes derived from them, exploring combinatorial aspects and using classical geometric lemmas.

Contribution

It characterizes conditions under which conics in PG(2,q) remain arcs in Hall planes and analyzes their combinatorial properties, extending understanding of inherited conics in non-Desarguesian planes.

Findings

Conditions for conics to be arcs in Hall planes

Combinatorial properties of inherited conics

Application of Segre-Korchmárós lemma in this context

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 ${\rm PG}(2,q)$ 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\'aros on Desargues configurations with perspective triangles inscribed in a conic.