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Counting $D_4$ singularities in the image of a wave front

C. Muñoz-Cabello, J. J. Nuño-Ballesteros, R. Oset Sinha

TL;DR

The paper addresses counting isolated $D_4$ singularities arising in stable frontal perturbations of a corank-2 wave front $f:(\\mathbb{C}^3,0) \to (\\mathbb{C}^4,0)$. It adapts Mond's stable perturbation approach by viewing the image of a generic perturbation $f_t$ as the discriminant of a smooth map germ $H_t:(\\mathbb{C}^5,0) \to (\\mathbb{C}^4,0)$, and shows that the number of $D_4$ points equals the $\\mathbb{C}$-dimension of a certain algebra. For a corank-2 wave front written as $f(u,v,w)=(u,p(u,v,w),q(u,v,w),r(u,v,w))$, this algebra is $\\mathscr{O}_3/\\langle p_v,p_w,q_v,q_w \rangle$, whose complex dimension counts the $D_4$ singularities in a generic frontal perturbation. The work builds on discriminant-counting frameworks from the $MMR$ schemes, connects to Nash lift and integral corank formalisms for wave fronts, and provides a concrete, computable invariant for corank-2 singularities, with potential applications to related frontal invariants such as the frontal Milnor number.

Abstract

We give a formula to count the number of $D_4$ singularities in a stable frontal perturbation of a corank $2$ wave front singularity $f\colon (\mathbb{C}^3,0) \to (\mathbb{C}^4,0)$ using Mond's method of stable perturbations of map germs. For a generic germ of corank $2$ wave front $f\colon (\mathbb{C}^3,S) \to (\mathbb{C}^4,0)$, the image of a stable deformation $f_t$ of $f$ exhibits $A_k$ singularities with $k \leq 4$, their transverse intersections and the aforementioned $D_4$ singularities for $0 < |t| \ll 1$. By interpreting the image of $f_t$ as the discriminant (the image of the critical point set) of a smooth map germ $H_t\colon (\mathbb{C}^5,0) \to (\mathbb{C}^4,0)$, we define an algebra whose dimension over $\mathbb{C}$ is equal to the number of $D_4$ points in the image of $f_t$.

Counting $D_4$ singularities in the image of a wave front

TL;DR

The paper addresses counting isolated singularities arising in stable frontal perturbations of a corank-2 wave front . It adapts Mond's stable perturbation approach by viewing the image of a generic perturbation as the discriminant of a smooth map germ , and shows that the number of points equals the -dimension of a certain algebra. For a corank-2 wave front written as , this algebra is , whose complex dimension counts the singularities in a generic frontal perturbation. The work builds on discriminant-counting frameworks from the schemes, connects to Nash lift and integral corank formalisms for wave fronts, and provides a concrete, computable invariant for corank-2 singularities, with potential applications to related frontal invariants such as the frontal Milnor number.

Abstract

We give a formula to count the number of singularities in a stable frontal perturbation of a corank wave front singularity using Mond's method of stable perturbations of map germs. For a generic germ of corank wave front , the image of a stable deformation of exhibits singularities with , their transverse intersections and the aforementioned singularities for . By interpreting the image of as the discriminant (the image of the critical point set) of a smooth map germ , we define an algebra whose dimension over is equal to the number of points in the image of .
Paper Structure (2 sections, 1 theorem, 7 equations, 1 figure)

This paper contains 2 sections, 1 theorem, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Wave fronts

Key Result

Proposition 2.4

Let $f\colon (\mathbb{C}^n,S) \to (\mathbb{C}^{n+1},0)$ be a smooth map germ with ramification ideal $\mathcal{R}(f)$. Then $f$ is a frontal map germ if and only if $\mathcal{R}(f)$ is a principal ideal.

Figures (1)

  • Figure 1: Projections of the $D_4^+$ (top) and $D_4^-$ (bottom) singularities onto the hyperplane $X=0$. Source: Arnold_I, § 22.1.

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Ishikawa_Survey
  • Remark 2.5
  • Example 2.6