Generalized Hyperfocused Arcs in $PG(2,p)$

TL;DR

This paper investigates the properties of generalized hyperfocused arcs in projective planes over prime fields, establishing size restrictions and exploring connections to the cylinder conjecture in Galois geometry.

Contribution

It proves that such arcs in $PG(2,p)$ can only have sizes 1, 2, or 4, revealing new constraints on their structure.

Findings

Generalized hyperfocused arcs in $PG(2,p)$ are limited to sizes 1, 2, or 4.

The study links these arcs to the strong cylinder conjecture in Galois geometries.

Provides insights into the structure and limitations of arcs in finite projective planes.

Abstract

A {\em generalized hyperfocused arc} $\mathcal H $ in $PG(2,q)$ is an arc of size $k$ with the property that the $k(k-1)/2$ secants can be blocked by a set of $k-1$ points not belonging to the arc. We show that if $q$ is a prime and $\mathcal H$ is a generalized hyperfocused arc of size $k$, then $k=1,2$ or 4. Interestingly, this problem is also related to the (strong) cylinder conjecture [Ball S.: The polynomial method in Galois geometries, in Current research topics in Galois geometry, Chapter 5, Nova Sci. Publ., New York, (2012) 105-130], as we point out in the last section.