Newton non-degenerate $μ$-constant deformations admit simultaneous embedded resolutions
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| Format: | Preprint |
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2020
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| author | Leyton-Álvarez, Maximiliano Mourtada, Hussein Spivakovsky, Mark |
| author_facet | Leyton-Álvarez, Maximiliano Mourtada, Hussein Spivakovsky, Mark |
| contents | Let $\mathbb{C}^{n+1}_o$ denote the germ of $\mathbb{C}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $\mathbb{C}^{n+1}_o$ and $W$ a deformation of $V$ over $\mathbb{C}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we will prove that $W$ is a $μ$-constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$, for example, the topological triviality of the family $W$ and the fact that the natural morphism $(W(\mathbb{C}_o)_m)_{red} \rightarrow \mathbb{C}_{o}$ is flat, where $W(\mathbb{C}_o)_m$ is the relative space of $m$-jets. On the way tothe proof of our main result, we give a complete answer to a question ofArnold on the monotonicity of Newton numbers in the case of convenientNewton polyhedra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_10316 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Newton non-degenerate $μ$-constant deformations admit simultaneous embedded resolutions Leyton-Álvarez, Maximiliano Mourtada, Hussein Spivakovsky, Mark Algebraic Geometry 14B05, 14B07, 14B25, 14E15, 14E25, 32S05, 32S10, 32S15 Let $\mathbb{C}^{n+1}_o$ denote the germ of $\mathbb{C}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $\mathbb{C}^{n+1}_o$ and $W$ a deformation of $V$ over $\mathbb{C}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we will prove that $W$ is a $μ$-constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$, for example, the topological triviality of the family $W$ and the fact that the natural morphism $(W(\mathbb{C}_o)_m)_{red} \rightarrow \mathbb{C}_{o}$ is flat, where $W(\mathbb{C}_o)_m$ is the relative space of $m$-jets. On the way tothe proof of our main result, we give a complete answer to a question ofArnold on the monotonicity of Newton numbers in the case of convenientNewton polyhedra. |
| title | Newton non-degenerate $μ$-constant deformations admit simultaneous embedded resolutions |
| topic | Algebraic Geometry 14B05, 14B07, 14B25, 14E15, 14E25, 32S05, 32S10, 32S15 |
| url | https://arxiv.org/abs/2001.10316 |