Generalizing Korchm\'aros--Mazzocca arcs

TL;DR

This paper extends the concept of Korchmárós--Mazzocca arcs in finite projective planes by generalizing intersection properties and exploring their existence, structure, and variants in different algebraic settings.

Contribution

It introduces a broader class of arcs with generalized intersection properties and characterizes their structure, including conditions for the existence of nuclei and variants mod p.

Findings

Classified all examples when m or t is not divisible by p.

Described all generalized arcs in PG(2,p^n) for various parameters.

Proved existence of nuclei under certain conditions.

Abstract

In this paper, we generalize the so called Korchm\'aros--Mazzocca arcs, that is, point sets of size $q+t$ intersecting each line in $0, 2$ or $t$ points in a finite projective plane of order $q$. For $t\neq 2$, this means that each point of the point set is incident with exactly one line meeting the point set in $t$ points. In $\mathrm{PG}(2,p^n)$, we change $2$ in the definition above to any integer $m$ and describe all examples when $m$ or $t$ is not divisible by $p$. We also study mod $p$ variants of these objects, give examples and under some conditions we prove the existence of a nucleus.