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Projective dimension of weakly chordal graphic arrangements

Takuro Abe, Lukas Kühne, Paul Mücksch, Leonie Mühlherr

TL;DR

This work characterizes when the projective dimension of the derivation module of a graphic arrangement is at most one, tying it precisely to weakly chordal graphs. By combining Terao's $B$-polynomial framework and Abe's $B$-sequence surjectivity results with a deletion-contraction analysis, the authors prove $\mathrm{pd}(D(\mathcal{A}(G)))\le 1$ if and only if $G$ is weakly chordal, with equality to 1 when $G$ is not chordal. They further establish that antihole graphs $C_\ell^C$ (for $\ell \ge 6$) yield $\mathrm{pd}(D(\mathcal{A}(C_\ell^C)))=2$ and provide an explicit minimal free resolution. Collectively, these results extend the classical freeness criterion for graphic arrangements (chordality) to a sharp homological bound and open new questions about Betti numbers and higher projective dimensions for broader graph classes.

Abstract

A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.

Projective dimension of weakly chordal graphic arrangements

TL;DR

This work characterizes when the projective dimension of the derivation module of a graphic arrangement is at most one, tying it precisely to weakly chordal graphs. By combining Terao's -polynomial framework and Abe's -sequence surjectivity results with a deletion-contraction analysis, the authors prove if and only if is weakly chordal, with equality to 1 when is not chordal. They further establish that antihole graphs (for ) yield and provide an explicit minimal free resolution. Collectively, these results extend the classical freeness criterion for graphic arrangements (chordality) to a sharp homological bound and open new questions about Betti numbers and higher projective dimensions for broader graph classes.

Abstract

A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.
Paper Structure (8 sections, 25 theorems, 77 equations, 2 figures)

This paper contains 8 sections, 25 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries -- Graph Theory
  3. Preliminaries -- Hyperplane Arrangements
  4. Projective dimension
  5. Terao's polynomial B
  6. Weakly chordal graphic arrangements
  7. Graphic arrangements of antiholes
  8. Open problems

Key Result

Theorem 1.2

The module $D(\mathcal{A}(G))$ is free if and only if the graph $G$ is chordal, i.e., $G$ does not contain a chordless cycle with four or more vertices.

Figures (2)

  • Figure 1: The triangular prism of KungSchenck on the left is the same as ${C_6}^C$ on the right.
  • Figure 2: Two graphs with 7 vertices, whose graphic arrangements have projective dimension 3.

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem 1.2: Edelman
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 45 more