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Blocking sets from a union of plane curves

Shamil Asgarli, Dragos Ghioca, Chi Hoi Yip

TL;DR

The paper studies blocking sets in ${\mathbb{P}}^2({\mathbb{F}}_q)$ that arise from unions of plane curves. Using Lang-Weil bounds and dual-curve analysis, it proves a logarithmic lower bound on the number of geometrically irreducible conics needed to block all lines in odd characteristic, and a Chebotarev-density framework showing that any fixed bound on the number of curves fails for large $q$; it also provides two complementary constructions achieving blocking with roughly $c_d\log q$ curves: a randomized method yielding about $4\log q$ curves and an explicit pencil-based approach giving an upper bound proportional to $\log q$. The work blends arithmetic geometry with combinatorial covering arguments to advance understanding of Erdős-type blocking sets in finite geometries and offers both probabilistic and explicit avenues for assembling small blocking families.

Abstract

Motivated by a question of Erdős on blocking sets in a projective plane that intersect every line only a few times, several authors have used unions of algebraic curves to construct such sets in $\mathbb{P}^2(\mathbb{F}_q)$. In this paper, we provide new constructions of blocking sets in $\mathbb{P}^2(\mathbb{F}_q)$ from a union of geometrically irreducible curves of a fixed degree $d$. We also establish lower bounds on the number of such curves required to form a blocking set. Our proofs combine tools from arithmetic geometry and combinatorics.

Blocking sets from a union of plane curves

TL;DR

The paper studies blocking sets in that arise from unions of plane curves. Using Lang-Weil bounds and dual-curve analysis, it proves a logarithmic lower bound on the number of geometrically irreducible conics needed to block all lines in odd characteristic, and a Chebotarev-density framework showing that any fixed bound on the number of curves fails for large ; it also provides two complementary constructions achieving blocking with roughly curves: a randomized method yielding about curves and an explicit pencil-based approach giving an upper bound proportional to . The work blends arithmetic geometry with combinatorial covering arguments to advance understanding of Erdős-type blocking sets in finite geometries and offers both probabilistic and explicit avenues for assembling small blocking families.

Abstract

Motivated by a question of Erdős on blocking sets in a projective plane that intersect every line only a few times, several authors have used unions of algebraic curves to construct such sets in . In this paper, we provide new constructions of blocking sets in from a union of geometrically irreducible curves of a fixed degree . We also establish lower bounds on the number of such curves required to form a blocking set. Our proofs combine tools from arithmetic geometry and combinatorics.
Paper Structure (5 sections, 11 theorems, 46 equations)

This paper contains 5 sections, 11 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Constructions from conics
  3. Blocking families and Chebotarev Density Theorem
  4. First Proof of Theorem \ref{['thm:main']}: randomized construction
  5. Second Proof of Theorem \ref{['thm:main']}: explicit construction

Key Result

Theorem 1.2

Let $q$ be an odd prime power. There is a constant $c_0>0$ such that no blocking set in $\mathbb{P}^2(\mathbb{F}_q)$ can be constructed from a union of fewer than $c_0\log q$ conics.

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: Slavov
  • proof : Proof of Theorem \ref{['thm:conics']}
  • Claim 2.2
  • proof : Proof of claim
  • Remark 2.3
  • Theorem 3.1
  • ...and 15 more