On Milnor-Orlik's theorem and admissible simultaneous good resolutions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916514543697920 |
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| author | Eyral, Christophe Oka, Mutsuo |
| author_facet | Eyral, Christophe Oka, Mutsuo |
| contents | Let $f$ be a (possibly Newton degenerate) weighted homogeneous polynomial defining an isolated surface singularity at the origin of $\mathbb{C}^3$, and let $\{f_s\}$ be a generic deformation of its coefficients such that $f_s$ is Newton non-degenerate for $s\not=0$. We show that there exists an ''admissible'' simultaneous good resolution of the family of functions $f_s$ for all small $s$, including $s=0$ which corresponds to the (possibly Newton degenerate) function $f$. As an application, we give a new geometrical proof of a weak version of the Milnor-Orlik theorem that asserts that the monodromy zeta-function of $f$ (and hence its Milnor number) is completely determined by its weight, its weighted degree and its Newton boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06384 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Milnor-Orlik's theorem and admissible simultaneous good resolutions Eyral, Christophe Oka, Mutsuo Algebraic Geometry 14B05, 32S05, 32S25, 32S45 Let $f$ be a (possibly Newton degenerate) weighted homogeneous polynomial defining an isolated surface singularity at the origin of $\mathbb{C}^3$, and let $\{f_s\}$ be a generic deformation of its coefficients such that $f_s$ is Newton non-degenerate for $s\not=0$. We show that there exists an ''admissible'' simultaneous good resolution of the family of functions $f_s$ for all small $s$, including $s=0$ which corresponds to the (possibly Newton degenerate) function $f$. As an application, we give a new geometrical proof of a weak version of the Milnor-Orlik theorem that asserts that the monodromy zeta-function of $f$ (and hence its Milnor number) is completely determined by its weight, its weighted degree and its Newton boundary. |
| title | On Milnor-Orlik's theorem and admissible simultaneous good resolutions |
| topic | Algebraic Geometry 14B05, 32S05, 32S25, 32S45 |
| url | https://arxiv.org/abs/2412.06384 |