Topology of smoothings of non-isolated singularities of complex surfaces

Abstract

We prove that the boundaries of the Milnor fibers of smoothings of non-isolated reduced complex surface singularities are graph manifolds. Moreover, we give a method, inspired by previous work of Némethi and Szilard, to compute associated plumbing graphs.

Boundary of the Milnor fiber of a Newton non degenerate surface singularity

“…In [FdBMN14], Fernández de Bobadilla & Menegon Neto prove the same result for a larger context, and in [NS12], Némethi & Szilárd give a constructive proof for the case of reduced holomorphic functions f : (C 3 , 0) → (C, 0). It is this last proof that we extend here and in [Cur19a] to study the boundary of the Milnor fiber of a Newton non degenerate function f : (X σ , 0) → (C, 0), where X σ is the 3-dimensional normal toric variety associated to a cone σ, and V (f ) ⊃ Sing(X). In this setting, we adapt the general procedure of [Cur19a] to produce an algorithm for the description of the boundary of the Milnor fiber of f , extending the work of Oka in [Oka87] for non degenerate isolated singularities in C 3 .…”

“…We recall here some of the main material of [Cur19a], concerning the structure of the boundary of the Milnor fiber of the smoothing of an arbitrary reduced complex surface singularity. One can consult this article and the references therein for more details.…”

“…In the language developped in [Cur19a], point 1 implies that the data of the so-called switches is not required, and point 2 implies that a covering data of…”

“…• In Section 2, we recall the main result of [Cur19a], on which we base the algorithm described in this work,…”