Singularities of mappings on ICIS and applications to Whitney equisingularity
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arXiv
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| Format: | Preprint |
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2021
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| author | Conejero, R. Giménez Nuño-Ballesteros, J. J. |
| author_facet | Conejero, R. Giménez Nuño-Ballesteros, J. J. |
| contents | We study germs of analytic maps $f:(X,S)\rightarrow(\mathbb{C}^p,0)$, when $X$ is an ICIS of dimension $n<p$. We define an image Milnor number, generalizing Mond's definition, $μ_I(X,f)$ and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs $f_t\colon(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instabilities in terms of the constancy of the $μ_I^*$-sequences of $f_t$ and the projections $π\colon D^2(f_t)\to\mathbb{C}^n$, where $D^2(f_t)$ is the ICIS given by double point space of $f_t$ in $\mathbb{C}^n\times\mathbb{C}^n$. The $μ_I^*$-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2108_00743 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Singularities of mappings on ICIS and applications to Whitney equisingularity Conejero, R. Giménez Nuño-Ballesteros, J. J. Algebraic Geometry 58K15 (Primary) 32S30, 58K40 (Secondary) We study germs of analytic maps $f:(X,S)\rightarrow(\mathbb{C}^p,0)$, when $X$ is an ICIS of dimension $n<p$. We define an image Milnor number, generalizing Mond's definition, $μ_I(X,f)$ and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs $f_t\colon(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instabilities in terms of the constancy of the $μ_I^*$-sequences of $f_t$ and the projections $π\colon D^2(f_t)\to\mathbb{C}^n$, where $D^2(f_t)$ is the ICIS given by double point space of $f_t$ in $\mathbb{C}^n\times\mathbb{C}^n$. The $μ_I^*$-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices. |
| title | Singularities of mappings on ICIS and applications to Whitney equisingularity |
| topic | Algebraic Geometry 58K15 (Primary) 32S30, 58K40 (Secondary) |
| url | https://arxiv.org/abs/2108.00743 |