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Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices

Shota Maehara

Abstract

In the theory of hyperplane arrangements, M. Wakefield and S. Yuzvinsky utilized a square matrix in their research on the exponents of $2$-dimensional multiarrangements. Using such a matrix, they showed that the exponents of $2$-dimensional multiarrangements are as close as possible in general position for any fixed balanced multiplicity. In this article, we introduce a matrix similar to that of Wakefield and Yuzvinsky and explore further applications to the exponents. In fact, the exponents of $2$-dimensional multiarrangements are determined by whether the corresponding matrices have full rank. As one of our main results, we introduce a new class of $2$-dimensional arrangements for which the exponents are as close as possible for any balanced multiplicities, except for the constant one multiplicity. We also proceed with the classification of $B_2$-exponents, and we provide an alternative proof for some known results on the exponents.

Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices

Abstract

In the theory of hyperplane arrangements, M. Wakefield and S. Yuzvinsky utilized a square matrix in their research on the exponents of -dimensional multiarrangements. Using such a matrix, they showed that the exponents of -dimensional multiarrangements are as close as possible in general position for any fixed balanced multiplicity. In this article, we introduce a matrix similar to that of Wakefield and Yuzvinsky and explore further applications to the exponents. In fact, the exponents of -dimensional multiarrangements are determined by whether the corresponding matrices have full rank. As one of our main results, we introduce a new class of -dimensional arrangements for which the exponents are as close as possible for any balanced multiplicities, except for the constant one multiplicity. We also proceed with the classification of -exponents, and we provide an alternative proof for some known results on the exponents.

Paper Structure

This paper contains 7 sections, 17 theorems, 24 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Introduction to WY-matrix
  4. Definition of the WY-matrix
  5. Tuple
  6. Reformulation of WY-matrix
  7. Proof of the main results

Key Result

Theorem 1.2

Let $n\geq2$ be an arbitrary natural number. Then, there exists a $2$-multiarrangement $(\mathcal{A},m)$ such that $|\mathcal{A}|=n$, $Q(\mathcal{A},m)=x^{m_1}y^{m_2}\prod_{i=3}^{n}(x-s_iy)^{m_i}$ where $s_i\in\mathbb{Z}$ and $\Delta(\mathcal{A}, m+k\cdot(\delta_{\ker x}+\delta_{\ker y}))=0$ for any

Theorems & Definitions (55)

  • Conjecture 1.1: Terao's Conjecture
  • Conjecture 1.2: Coxeter $B_2$
  • Theorem 1.2
  • Example 1.3
  • Remark 1.4
  • Theorem 1.5: Coxeter $B_2$ -(i)
  • Example 1.6
  • Theorem 1.7: Coxeter $B_2$ -(ii)
  • Theorem 1.8
  • Definition 2.1
  • ...and 45 more