Milnor fibre boundary of a non-isolated surface singularity

TL;DR

This paper develops a method to explicitly compute the boundary of the Milnor fibre of certain surface singularities as a plumbed 3-manifold, including the monodromy characteristic polynomial, with applications to isolated complete intersection singularities.

Contribution

It introduces an explicit procedure to determine the boundary of the Milnor fibre and monodromy for non-isolated surface singularities, extending to ICIS cases with open book decompositions.

Findings

Boundary of Milnor fibre as a plumbed 3-manifold

Characteristic polynomial of algebraic monodromy computed

Open book decompositions for ICIS cases analyzed

Abstract

Let f be a hypersurface surface local singularity whose zero set has 1-dimensional singular locus. We develop an explicit procedure that provides the boundary of the Milnor fibre of f as an oriented plumbed 3-manifold. The method provides the characteristic polynomial of the algebraic monodromy as well. Moreover, for any analytic germ g such that the pair (f,g) is an isolated complete intersection singularity, the (multiplicity system of the) open book decomposition of the boundary with binding determined by g and pages determined by the argument of g is also computed. In order to do this, we have to establish key results regarding the horizontal and vertical monodromies of the transversal type singularities associated with the singular locus of f and of the ICIS (f,g). The theory is supported by many examples. E.g. the case of homogeneous singularities (including the case of arrangements) is detailed completely. A list of especially peculiar examples, and also a list of related open problems is given.