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Newton weighted-Lê-Yomdin polynomials and $μ$-Zariski pairs of surface singularities

Christophe Eyral, Masaharu Ishikawa, Mutsuo Oka

Abstract

We investigate surface singularities defined by weighted-Lê-Yomdin polynomials, with a particular focus on a specific subclass that we refer to as Newton weighted-Lê-Yomdin polynomials. In particular, using polynomials in this subclass, we develop a method to construct new $μ$-Zariski pairs of surface singularities.

Newton weighted-Lê-Yomdin polynomials and $μ$-Zariski pairs of surface singularities

Abstract

We investigate surface singularities defined by weighted-Lê-Yomdin polynomials, with a particular focus on a specific subclass that we refer to as Newton weighted-Lê-Yomdin polynomials. In particular, using polynomials in this subclass, we develop a method to construct new -Zariski pairs of surface singularities.

Paper Structure

This paper contains 24 sections, 10 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and preliminary observations
  3. Weighted projective plane and affine coordinate charts
  4. Weighted homogenization
  5. Milnor number and generic plane section
  6. The sets $\mathcal{W}_{P,d}(\Delta)$ and $\mathcal{W}'_{P,d}(\Delta)$
  7. Main result of this section
  8. Newton weighted-Lê-Yomdin polynomials
  9. The set $\mathcal{W}'_{P,d}(\Delta,\Xi)$
  10. The sets $\mathcal{W}"_{P,d,m}(\Delta)$ and $\mathcal{W}"_{P,d,m}(\Delta,\Xi)$ of weak-Newton weighted-Lê-Yomdin polynomials
  11. The set $\mathcal{W}"_{P,d,m}(\Delta,\Xi)_{\textnormal{ND}}$ of Newton weighted-Lê-Yomdin polynomials
  12. Constructibility of the sets $\mathcal{W}'_{P,d}(\Delta,\Xi)$ and $\mathcal{W}"_{P,d,m}(\Delta,\Xi)$
  13. $\mu$-Zariski pairs of surface singularities
  14. Zariski pairs of curves in weighted projective planes
  15. From Zariski pairs of curves in $\mathbb{P}^2$ to those in $\mathbb{P}^2(P)$
  16. ...and 9 more sections

Key Result

Theorem 3.2

Let $f\in\mathcal{W}'_{P,d}(\Delta)$, and let $H=\{(x,y,z)\in\mathbb{C}^3\, ;\, z=ax+by\}$ be a generic plane for $V(f)$ through the origin.

Figures (3)

  • Figure 1: The Newton boundary $\Gamma(f(x,y,0))$
  • Figure 2: The edge $AB$
  • Figure 3: The Newton boundary $\Gamma(f^H)$

Theorems & Definitions (33)

  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 4.1
  • Example 4.2
  • Example 4.3
  • Definition 4.4
  • Definition 4.5
  • Definition 5.1
  • Proposition 5.2
  • ...and 23 more