The Milnor fiber of the singularity f(x,y) + zg(x,y) = 0

TL;DR

This paper describes the topology of the Milnor fiber and monodromy for a class of plane curve singularities, providing formulas for the monodromy zeta function and Euler characteristic based on topological data.

Contribution

It offers a topological description of the Milnor fiber and monodromy for singularities of the form f+zg=0, depending only on the topological type of the defining plane curves.

Findings

Explicit description of the Milnor fiber boundary.

Simple formula for the monodromy zeta function.

Euler characteristic of the Milnor fiber derived.

Abstract

We give a description of the Milnor fiber and the monodromy of a singularity of the form f+zg = 0 where f and g define plane curves and have no common components. The description depends only on the topological type of the two plane curve germs defined by f and g. In particular, this gives a description of the boundary of the Milnor fiber. As a corollary, we give a simple formula for the monodromy zeta function and the Euler characteristic of the fiber.