On Milnor–Orlik’s theorem and admissible simultaneous good resolutions

Let f be a (possibly Newton degenerate) weighted homogeneous polynomial defining an isolated surface singularity at the origin of C3, and let {fs} be a generic deformation of its coefficients such that fs is Newton non-degenerate for s/=0. We show that there exists an “admissible” simultaneous good resolution of the family of functions fs 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.