Plane curves giving rise to blocking sets over finite fields

TL;DR

This paper explores the relationship between plane algebraic curves over finite fields and blocking sets, identifying which curves can produce blocking sets and constructing examples using number theory.

Contribution

It characterizes when plane curves over finite fields can generate blocking sets and provides explicit constructions for certain degrees.

Findings

Irreducible low-degree curves do not produce blocking sets.

Refined results for cubic and quartic curves.

Constructed smooth plane curves of degree up to 4p^{3/4}+1 with blocking sets.

Abstract

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by R\'edei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over $\mathbb{F}_p$ of degree at most $4p^{3/4}+1$ whose points form blocking sets.