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New $μ$-Zariski pairs of surface singularities

Christophe Eyral, Masaharu Ishikawa, Mutsuo Oka, Öznur Turhan

Abstract

To the best of the authors' knowledge, all previously known examples of $μ$-, $μ^*$-, link-, or ordinary Zariski pairs of surface singularities in $\mathbb{C}^3$ consist of (possibly weighted) Lê-Yomdin singularities. In this paper, we present an example of a $μ$-Zariski pair involving surface singularities that are not of Lê-Yomdin type.

New $μ$-Zariski pairs of surface singularities

Abstract

To the best of the authors' knowledge, all previously known examples of -, -, link-, or ordinary Zariski pairs of surface singularities in consist of (possibly weighted) Lê-Yomdin singularities. In this paper, we present an example of a -Zariski pair involving surface singularities that are not of Lê-Yomdin type.

Paper Structure

This paper contains 8 sections, 5 theorems, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of the theorem
  3. Proof of Proposition \ref{['pos']}
  4. Proof of Theorem \ref{['mt']}
  5. Condition (1)
  6. Condition (2)
  7. Condition (3)
  8. Effect of dropping the assumption \ref{['pseudoLYA']}

Key Result

Proposition 2.3

For any element $g\in\mathcal{W}(\Gamma)$, the following two assertions hold true. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Newton boundary $\Gamma$
  • Figure 2: Newton boundary $\Gamma(g^H)$
  • Figure 3: Subdivision $\Sigma^*$

Theorems & Definitions (15)

  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Conjecture 2.8
  • Lemma 4.1
  • proof
  • ...and 5 more