On the algebraic properties of the Böröczky configuration

2025/10/19 by Jake Kettinger, Kettinger, Jake, Shahriyar Roshan-Zamir +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2510.17029

openalex publication_date 2025/10/19 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

Abstract

The Böröczky configuration of lines and (multiple) points exhibits extremal behavior in commutative algebra and combinatorics. Examples of this appear in the context of the containment problem for ordinary and symbolic powers and the proof of the Dirac-Motzkin conjecture by Green and Tao. This paper studies the algebraic properties of Böröczky configurations for arbitrary values of n. Our results compute the Waldschmit constant of the defining ideal of these configurations. Moreover, we use the weighted projective plane ℙ(1,2,3) to give an upper bound for the degree of the minimal generators of their ideal. Finally, this construction is applied to an elliptic curve in ℙ² to give a new counterexample to the containment I⁽³⁾⊆ I².

Citations

• Singular loci of reflection arrangements and the containment problem
• On the parameter space of Böröczky configurations
• Configurations of Linear Spaces of Codimension Two and the Containment Problem
• An introduction to varieties in weighted projective space
• Containment Counterexamples for ideals of various configurations of points in \bf P^N
• On sets defining few ordinary lines
• On Sets Defining Few Ordinary Lines
• Are symbolic powers highly evolved?
• Comparing powers and symbolic powers of ideals

Related

• On the parameter space of Böröczky configurations
• On the containment I⁽³⁾ ⊆ I² and configurations of triple points in Böröczky line arrangements
• Containment problem and combinatorics
• Demailly's Conjecture and the Containment Problem
• Chudnovsky's Conjecture and the stable Harbourne-Huneke containment