Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Moduli Parameters of Complex Singularities with Non-Degenerate Newton Boundary

Janko Boehm, Magdaleen S. Marais, Gerhard Pfister

Abstract

Our recent extension of Arnold's classification includes all singularities of corank up to two equivalent to a germ with a non-degenerate Newton boundary, thus broadening the classification's scope significantly by a class which is unbounded with respect to modality and Milnor number. This method is based on proving that all right-equivalence classes within a mu-constant stratum can be represented by a single normal form derived from a regular basis of a suitably selected special fiber. While both Arnold's and our preceding work on normal forms addresses the determination of a normal form family containing the given germ, this paper takes the next natural step: We present an algorithm for computing for a given germ the values of the moduli parameters in its normal form family, that is, a normal form equation in its stable equivalence class. This algorithm will be crucial for understanding the moduli stacks of such singularities. The implementation of this algorithm, along with the foundational classification techniques, is implemented in the library arnold.lib for the computer algebra system Singular.

Moduli Parameters of Complex Singularities with Non-Degenerate Newton Boundary

Abstract

Our recent extension of Arnold's classification includes all singularities of corank up to two equivalent to a germ with a non-degenerate Newton boundary, thus broadening the classification's scope significantly by a class which is unbounded with respect to modality and Milnor number. This method is based on proving that all right-equivalence classes within a mu-constant stratum can be represented by a single normal form derived from a regular basis of a suitably selected special fiber. While both Arnold's and our preceding work on normal forms addresses the determination of a normal form family containing the given germ, this paper takes the next natural step: We present an algorithm for computing for a given germ the values of the moduli parameters in its normal form family, that is, a normal form equation in its stable equivalence class. This algorithm will be crucial for understanding the moduli stacks of such singularities. The implementation of this algorithm, along with the foundational classification techniques, is implemented in the library arnold.lib for the computer algebra system Singular.
Paper Structure (6 sections, 11 theorems, 99 equations, 1 algorithm)

This paper contains 6 sections, 11 theorems, 99 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definitions and Preliminary Results
  3. Classification of Singularities with Non-Degenerate Newton Boundary
  4. Determining the Values of the Moduli Parameters in the Normal From of a germ with a Non-Degenerate Newton Boundary
  5. Proof of Algorithm \ref{['alg:normalformeq']}
  6. Next steps

Key Result

Theorem 3.1

Let $f\in\mathbb C\{x,y\}$ be a germ with a non-degenerate Newton boundary at the origin. A miniversal, equisingular unfolding is given by where $m$ is the modality of $f$, and $g_1,\ldots,g_m$ represent a regular basis for $Q_f$ on and above $\Gamma(f)$.

Theorems & Definitions (40)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • Remark 2.11
  • ...and 30 more