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On homological properties of some Cynk-Szemberg octic hyperplane arrangements

Marek Janasz, Piotr Pokora

Abstract

In this paper we study Cynk-Szemberg octic hyperplane arrangements from the perspective of homological properties of their derivation modules. In particular, we define the notion of the type of hyperplane arrangements that will be used in our characterization of rigid Cynk-Szemberg octic hyperplane arrangements. Moreover, we deliver a combinatorial non-freeness criterion for essential hyperplane arrangements in $\mathbb{C}^{4}$.

On homological properties of some Cynk-Szemberg octic hyperplane arrangements

Abstract

In this paper we study Cynk-Szemberg octic hyperplane arrangements from the perspective of homological properties of their derivation modules. In particular, we define the notion of the type of hyperplane arrangements that will be used in our characterization of rigid Cynk-Szemberg octic hyperplane arrangements. Moreover, we deliver a combinatorial non-freeness criterion for essential hyperplane arrangements in .

Paper Structure

This paper contains 6 sections, 3 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cynk--Szemberg octic hyperplane arrangements
  4. A combinatorial non-freeness criterion for essential hyperplane arrangements
  5. Rigid Cynk--Szemberg octic hyperplane arrangements
  6. A few remarks about homological properties of one-parameter families of Cynk--Szemberg octics

Key Result

Theorem 2.5

Assume that $\mathcal{A}$ is free with exponents ${\rm exp}(\mathcal{A}) = (d_{1}, \ldots, d_{n})$. Then which also means that for free arrangements their exponents are determined by the intersection lattice.

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: Terao's factorization
  • Definition 2.6: Type of a hyperplane arrangement
  • Example 2.7: Free hyperplane arrangements
  • Example 2.8: Nearly free hyperplane arrangements, DS
  • Definition 3.1
  • Definition 3.2
  • ...and 10 more