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arXiv · 2412.06384

On Milnor-Orlik's theorem and admissible simultaneous good resolutions

Christophe Eyral·Mutsuo Oka
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Abstract

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.

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Christophe Eyral, Mutsuo Oka. 2024-12-09. On Milnor-Orlik's theorem and admissible simultaneous good resolutions. https://arxiv.org/abs/2412.06384

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