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

TL;DR

This paper proves the existence of an admissible simultaneous resolution for a family of weighted homogeneous polynomials with isolated singularities, and offers a new geometric proof of a Milnor-Orlik theorem variant relating monodromy to Newton data.

Contribution

It establishes the existence of an admissible simultaneous good resolution for deformations of weighted homogeneous polynomials, extending Milnor-Orlik's theorem through a geometric approach.

Findings

Existence of simultaneous good resolution for small deformations.

New geometric proof of a weak Milnor-Orlik theorem.

Monodromy zeta-function determined by Newton data.

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.