Lipschitz geometry of the image of finite mappings
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908472648400896 |
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| author | Ballesteros, Juan José Nuño Prado, Vinícius de Oliveira Sanchis, Guillermo Peñafort Sampaio, José Edson |
| author_facet | Ballesteros, Juan José Nuño Prado, Vinícius de Oliveira Sanchis, Guillermo Peñafort Sampaio, José Edson |
| contents | This paper is devoted to the study of the LNE property in complex analytic hypersurface parametrized germs, that is, the sets that are images of finite analytic map germs from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$. We prove that if the multiplicity of $f$ is equal to his generic degree, then the image of $f$ is LNE at 0 if and only if it is a smooth germ. We also show that every finite corank 1 map is sattisfies the previous hypothesis. Moreover, we show that for an injective map germ $f$ from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$, the image of $f$ is LNE at 0 if and only if $f$ is an embedding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21405 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lipschitz geometry of the image of finite mappings Ballesteros, Juan José Nuño Prado, Vinícius de Oliveira Sanchis, Guillermo Peñafort Sampaio, José Edson Algebraic Geometry Complex Variables 14B05 (Primary) 32S05 (Secondary) This paper is devoted to the study of the LNE property in complex analytic hypersurface parametrized germs, that is, the sets that are images of finite analytic map germs from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$. We prove that if the multiplicity of $f$ is equal to his generic degree, then the image of $f$ is LNE at 0 if and only if it is a smooth germ. We also show that every finite corank 1 map is sattisfies the previous hypothesis. Moreover, we show that for an injective map germ $f$ from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$, the image of $f$ is LNE at 0 if and only if $f$ is an embedding. |
| title | Lipschitz geometry of the image of finite mappings |
| topic | Algebraic Geometry Complex Variables 14B05 (Primary) 32S05 (Secondary) |
| url | https://arxiv.org/abs/2507.21405 |