Counting $D_4$ singularities in the image of a wave front
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912371214123008 |
|---|---|
| author | Muñoz-Cabello, C. Nuño-Ballesteros, J. J. Sinha, R. Oset |
| author_facet | Muñoz-Cabello, C. Nuño-Ballesteros, J. J. Sinha, R. Oset |
| contents | 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 $f_t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07644 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting $D_4$ singularities in the image of a wave front Muñoz-Cabello, C. Nuño-Ballesteros, J. J. Sinha, R. Oset Algebraic Geometry Primary 32S30, Secondary 32S25, 58K60 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 $f_t$. |
| title | Counting $D_4$ singularities in the image of a wave front |
| topic | Algebraic Geometry Primary 32S30, Secondary 32S25, 58K60 |
| url | https://arxiv.org/abs/2505.07644 |