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

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.