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Line arrangements with many triple points

Lukas Kühne, Tomasz Szemberg, Halszka Tutaj-Gasińska

TL;DR

The paper constructs an infinite family of line arrangements in characteristic $2$ whose intersections are exclusively triple points, challenging finiteness conjectures. It combines matroid realization spaces and Steiner triple systems to classify realizability, showing that many STS-derived combinatorics fail to realize as line arrangements over any field, while providing explicit infinite constructions in characteristic $2$. The authors compute maximal triple-point counts for up to $19$ lines, proving several nonexistence results (e.g., no $12$-line arrangement with $20$ triple points) and giving precise attainability (e.g., $19$ lines with $57$ triple points over $\mathbb{F}_{11}$). The work leverages an algorithmic realization approach and symmetry analyses to map the landscape of triple-point line arrangements, contributing both new constructions and detailed extremal data relevant to incidence geometry and combinatorial design theory.

Abstract

In this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number of such arrangements, regardless of the characteristic. Leveraging the theory of matroids and employing computer algebra software, we rigorously examine the existence and non-existence across various characteristics of line arrangements with up to 19 lines maximizing the number of triple intersection points.

Line arrangements with many triple points

TL;DR

The paper constructs an infinite family of line arrangements in characteristic whose intersections are exclusively triple points, challenging finiteness conjectures. It combines matroid realization spaces and Steiner triple systems to classify realizability, showing that many STS-derived combinatorics fail to realize as line arrangements over any field, while providing explicit infinite constructions in characteristic . The authors compute maximal triple-point counts for up to lines, proving several nonexistence results (e.g., no -line arrangement with triple points) and giving precise attainability (e.g., lines with triple points over ). The work leverages an algorithmic realization approach and symmetry analyses to map the landscape of triple-point line arrangements, contributing both new constructions and detailed extremal data relevant to incidence geometry and combinatorial design theory.

Abstract

In this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number of such arrangements, regardless of the characteristic. Leveraging the theory of matroids and employing computer algebra software, we rigorously examine the existence and non-existence across various characteristics of line arrangements with up to 19 lines maximizing the number of triple intersection points.
Paper Structure (7 sections, 7 theorems, 35 equations)

This paper contains 7 sections, 7 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids and their realizations
  4. Steiner triple systems
  5. Arrangements with only triple points in characteristic $2$
  6. Line arrangements with the maximal number of triple points
  7. Sporadic example

Key Result

Proposition 1.2

For $s\geq 1$ let where $\varepsilon(s)=1$ if $s\equiv 5\mod 6$ and $\varepsilon(s)=0$ otherwise. Then

Theorems & Definitions (16)

  • Proposition 1.2: Schönheim
  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • ...and 6 more