On Milnor–Orlik’s theorem and admissible simultaneous good resolutions
Volume 134 / 2025
Annales Polonici Mathematici 134 (2025), 107-118
MSC: Primary 14B05; Secondary 32S05, 32S25, 32S45
DOI: 10.4064/ap250326-15-7
Published online: 12 August 2025
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.
