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Free multiderivations of connected subgraph arrangements

Paul Mücksch, Gerhard Roehrle, Sven Wiesner

TL;DR

The paper advances the theory of freeness for hyperplane multiarrangements by extending the freeness classification from connected subgraph arrangements ${\mathscr A}_{G}$ to their multiplicity-enhanced counterparts $(\mathscr A_{G},\mu)$. It establishes a complete list of graphs for which free multiplicities exist, namely when $G$ lies in the path, cycle, almost-path, or path-with-triangle families (with ${\mathscr A}_{G}$ free in the simple case), and it shows that constant multiplicities yield freeness only in very restricted cases (e.g., $G$ a path or $C_3$ with $c=3$). A central technical pillar is the use of Yoshinaga extendability, Ziegler restrictions, and local/global mixed product invariants to certify freeness or non-freeness, together with detailed analysis of a key subarrangement ${\mathscr D}$ of ${\mathscr A}_{C_3}$. The results culminate in a precise characterization of free multiplicities for ${\mathscr A}_{G}$ across several graph families and provide explicit-free families and counterexamples, with implications for inductive freeness and extension phenomena in multiarrangements. All mathematical notation is presented within $...$ delimiters as required.

Abstract

Cuntz and Kühne introduced the class of connected subgraph arrangements $A_G$, depending on a graph $G$, and classified all graphs $G$ such that the corresponding arrangement $A_G$ is free. We extend their result to the multiarrangement case and classify all graphs $G$ for which the corresponding arrangement $A_G$ supports some multiplicity $μ$ such that the multiarrangement $(A_G,μ)$ is free.

Free multiderivations of connected subgraph arrangements

TL;DR

The paper advances the theory of freeness for hyperplane multiarrangements by extending the freeness classification from connected subgraph arrangements to their multiplicity-enhanced counterparts . It establishes a complete list of graphs for which free multiplicities exist, namely when lies in the path, cycle, almost-path, or path-with-triangle families (with free in the simple case), and it shows that constant multiplicities yield freeness only in very restricted cases (e.g., a path or with ). A central technical pillar is the use of Yoshinaga extendability, Ziegler restrictions, and local/global mixed product invariants to certify freeness or non-freeness, together with detailed analysis of a key subarrangement of . The results culminate in a precise characterization of free multiplicities for across several graph families and provide explicit-free families and counterexamples, with implications for inductive freeness and extension phenomena in multiarrangements. All mathematical notation is presented within delimiters as required.

Abstract

Cuntz and Kühne introduced the class of connected subgraph arrangements , depending on a graph , and classified all graphs such that the corresponding arrangement is free. We extend their result to the multiarrangement case and classify all graphs for which the corresponding arrangement supports some multiplicity such that the multiarrangement is free.
Paper Structure (15 sections, 38 theorems, 44 equations, 2 figures, 4 tables)

This paper contains 15 sections, 38 theorems, 44 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Recollections and Preliminaries
  3. Hyperplane arrangements, multiarrangements and their freeness
  4. Connected subgraph arrangements
  5. Extensions of a multiarrangement
  6. Locally and globally mixed products
  7. Free connected subgraph multiarrangements
  8. Constant multiplicities on connected subgraph arrangements
  9. Investigating $G_1$ and $G_2$
  10. Constant multiplicities on a free ${\mathscr A}_{G}$
  11. Classifying free multiplicities on connected subgraph arrangements
  12. Free multiplicities on $\mathscr{D}$.
  13. Non-Free multiplicities on $\mathscr{D}$
  14. Some free multiplicities on ${\mathscr A}_{C_3}$
  15. Induction and Recursion tables

Key Result

Theorem 1.2

Let $G$ be a connected graph. There exists a multiplicity $\mu$ such that the connected subgraph multiarrangement $({\mathscr A}_G,\mu)$ is free if and only if $G$ is $G_1$, $G_2$, a path-graph, a cycle-graph, an almost-path-graph, or a path-with-triangle-graph (see Definition definition: cuntz kühn

Figures (2)

  • Figure 1: The graphs $G_1$ up to $G_8$.
  • Figure 2: Graphs $G_9$ to $G_{20}$.

Theorems & Definitions (80)

  • Definition 1.1: cuntzkuehne:subgrapharrangements
  • Theorem 1.2: Corollary \ref{['coro:GraphsWithFreeMultiplicities']}
  • Theorem 1.3: Proposition \ref{['prop:G_1_2ConstNotFree']} and Corollary \ref{['coro:ConstMultFree']}
  • Theorem 1.4: dipasquale: X3 moduli freeness
  • Theorem 1.5: Proposition \ref{['prop:SpecialFreeMultC_3']}
  • Theorem 2.1: abenuidanumata:signedeliminable
  • Theorem 2.2: ziegler:multiarrangements
  • Theorem 2.3: ziegler:multiarrangements, Saito’s criterion
  • Theorem 2.4: yoshinaga:characterization
  • Corollary 2.5
  • ...and 70 more