Generalizing blocking semiovals in finite projective planes

TL;DR

This paper introduces a new concept called blocking set with the $r_infty$-property in finite projective planes, generalizing blocking semiovals, and explores their sizes through new theoretical insights into finite geometries.

Contribution

It defines the $r_infty$-property for blocking sets, generalizes the notion of blocking semiovals, and develops new theory linking blocking sets in projective and affine planes.

Findings

Characterization of the existence of blocking sets with the $r_infty$-property

New theoretical connections between projective and affine plane blocking sets

Determination of possible sizes of such blocking sets

Abstract

Blocking semiovals and the determination of their (minimum) sizes constitute one of the central research topics in finite projective geometry. In this article we introduce the concept of blocking set with the $r_\infty$-property in a finite projective plane $\text{PG}(2,q)$, with $r_\infty$ a line of $\text{PG}(2,q)$ and $q$ a prime power. This notion greatly generalizes that of blocking semioval. We address the question of determining those integers $k$ for which there exists a blocking set of size $k$ with the $r_\infty$-property. To solve this problem, we build new theory which deeply analyzes the interplay between blocking sets in finite projective and affine planes.