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On connected subgraph arrangements

Lorenzo Giordani, Tilman Möller, Paul Mücksch, Gerhard Roehrle

TL;DR

The paper analyzes connected subgraph hyperplane arrangements ${\mathscr A}_G$, establishing projective uniqueness and a combinatorial basis for freeness within this class. It provides a comprehensive classification of freeness, factoredness, and the stronger MAT-freeness/inductive factorizations, tying these to graph families such as paths, cycles, almost-paths, and path-with-triangle graphs. It advances understanding of accuracy, asphericity, and combinatorial formality, proving that many ${\mathscr A}_G$ are formal and that MAT-freeness implies accuracy. The work also develops the theory of ideal subarrangements in this setting, proving MAT-freeness and inductive factorization for important families and analyzing the rank-generating functions of the regions poset, with conjectural extensions to broader graph classes. Overall, the results illuminate deep connections between graph structure and the algebraic/topological properties of the associated hyperplane arrangements, with implications for Terao’s conjecture within this specialized but rich class.

Abstract

Recently, Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph $G$, so called connected subgraph arrangements $A_G$. In this note we strengthen some of the result from their work and prove new ones for members of this class. For instance, we show that aspherical members withing this class stem from a rather restricted set of graphs. Specifically, if $A_G$ is an aspherical connected subgraph arrangement, then $A_G$ is free with the unique possible exception when the underlying graph $G$ is the complete graph on $4$ nodes.

On connected subgraph arrangements

TL;DR

The paper analyzes connected subgraph hyperplane arrangements , establishing projective uniqueness and a combinatorial basis for freeness within this class. It provides a comprehensive classification of freeness, factoredness, and the stronger MAT-freeness/inductive factorizations, tying these to graph families such as paths, cycles, almost-paths, and path-with-triangle graphs. It advances understanding of accuracy, asphericity, and combinatorial formality, proving that many are formal and that MAT-freeness implies accuracy. The work also develops the theory of ideal subarrangements in this setting, proving MAT-freeness and inductive factorization for important families and analyzing the rank-generating functions of the regions poset, with conjectural extensions to broader graph classes. Overall, the results illuminate deep connections between graph structure and the algebraic/topological properties of the associated hyperplane arrangements, with implications for Terao’s conjecture within this specialized but rich class.

Abstract

Recently, Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph , so called connected subgraph arrangements . In this note we strengthen some of the result from their work and prove new ones for members of this class. For instance, we show that aspherical members withing this class stem from a rather restricted set of graphs. Specifically, if is an aspherical connected subgraph arrangement, then is free with the unique possible exception when the underlying graph is the complete graph on nodes.

Paper Structure

This paper contains 20 sections, 27 theorems, 37 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. Hyperplane arrangements
  4. Free arrangements
  5. MAT-Free arrangements
  6. Accurate arrangements
  7. Nice arrangements
  8. Inductively factored arrangements
  9. $K(\pi,1)$-arrangements
  10. Local Properties
  11. Projective Uniquenes: Proof of Theorem \ref{['thm:AGiscomb']}
  12. Accuracy: Proof of Theorem \ref{['thm:accurateAG']}
  13. Inductive Freeness: Evidence for Conjecture \ref{['conj:freeAG2']}
  14. Inductive Factoredness: Proof of Theorem \ref{['thm:factoredAG2']}
  15. Aspherical ${\mathscr A}_G$: Proof of Theorem \ref{['thm:kpi1']}
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $G$ be a connected graph. Then ${\mathscr A}_G$ is projectively unique.

Figures (4)

  • Figure 1: The graphs $G_1$ up to $G_8$
  • Figure 2: Projective pictures of the arrangements ${\mathscr B}={\mathscr B}(0)$ and ${\mathscr B}(4)$ for $z=1$ with the "simple triangle" shaded in.
  • Figure 3: Minimal connected graphs $G$ for which ${\mathscr A}_G$ fails to be $K(\pi,1)$.
  • Figure :

Theorems & Definitions (74)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1: cuntzkuehne:subgraphs
  • ...and 64 more