Proportion of blocking curves in a pencil

TL;DR

This paper studies the proportion of curves in a pencil over finite fields whose rational points form blocking sets, translating geometric problems into combinatorial ones and analyzing fixed and growing degree cases.

Contribution

It introduces a new combinatorial approach to analyze blocking sets in pencils of curves over finite fields, considering both fixed and growing degrees.

Findings

For pencils with growing degree, the problem reduces to combinatorial disjoint blocking sets.

When the degree is fixed, specific properties of blocking sets are characterized.

The paper establishes bounds on the number of such curves in the pencil.

Abstract

Let $\mathcal{L}$ be a pencil of plane curves defined over $\mathbb{F}_q$ with no $\mathbb{F}_q$-points in its base locus. We investigate the number of curves in $\mathcal{L}$ whose $\mathbb{F}_q$-points form a blocking set. When the degree of the pencil is allowed to grow with respect to $q$, we show that the geometric problem can be translated into a purely combinatorial problem about disjoint blocking sets. We also study the same problem when the degree of the pencil is fixed.