The Boundary of the Milnor Fiber of Hirzebruch Surface Singularities

Abstract

We give the first (as far as we know) complete description of the boundary of the Milnor fiber for some non-isolated singular germs of surfaces in C 3 . We study irreducible (i.e. gcd (m, k, l) = 1) non-isolated (i.e. 1 ≤ k < l) Hirzebruch hypersurface singularities in C 3 given by the equation z m − x k y l = 0. We show that the boundary L of the Milnor fiber is always a Seifert manifold and we give an explicit description of the Seifert structure. From it, we deduce that: 1) L is never diffeomorphic to the boundary of the normalization.2) L is a lens space iff m = 2 and k = 1.3) When L is not a lens space, it is never orientation preserving diffeomorphic to the boundary of a normal surface singularity.

The boundary of the Milnor fibre of complex and real analytic non-isolated singularities

“…E3 (6,4) E5 (4,6) In the holomorphic case f g, the part of the Milnor fibre F η inside each box of type V ij is a disjoint union of gcd(a i + b i , a j + b j ) cylinders, and the part of the Milnor fibre inside each box of type V i is an (a i + b i )-covering of a sphere minus r i disks, with Euler characteristic (a i + b i )(2 − r i ). Then: F η ∩ V 13 (and F η ∩ V 15 ) is two cylinders; F η ∩ V 23 (and F η ∩ V 45 ) is five cylinders; F η ∩ V 1 is two cylinders; F η ∩V 2 (and F η ∩V 4 ) is five disks; F η ∩V 3 (and F η ∩V 5 ) is a compact surface of genus 2 with boundary eight circles.…”

“…The theorem was announced first by F. Michel and A. Pichon [5], but its proof contained a gap. In [8] F. Michel, A. Pichon and C. Weber provided a proof valid for some classes of non-isolated singularities. The first complete proof of the general (reduced holomorphic) case of the theorem was provided by A. Nemethi and A. Szilard in [13], where even an algorithm to compute the graph describing the Waldhausen manifold is given.…”

Topology of smoothings of non-isolated singularities of complex surfaces

“…In 2016, they published the details of their proof in [MP16]. In collaboration with Weber, they provided explicit plumbing graphs for several classes of singularities: Hirzebruch surface singularities in [MPW07], and the so-called suspensions (f = g(x, y)+ z n ) in [MPW09]. Fernández de Bobadilla and Menegon Neto, in [FdBMN14], proved it in the context of smoothings of non-isolated and not necessarily reduced singularities whose total space has an isolated singularity, for a function of the form f · g, with f and g holomorphic.…”

The boundary of the Milnor fibre of certain non-isolated singularities

“…However, the boundary of the Milnor fibre is still a plumbed 3manifold, and one expects that its plumbing graph codifies considerable information about the germ. ∂F can be obtained by surgery of two pieces: one of them is the boundary of the resolution of the normalization, the other one is related with the transversal singularities associated with the singular curves of the hypersurface singularity [25,22,11]. In particular, the boundary of the Milnor fibre plays the same crucial role as in the isolated singularity case (in fact, it is the unique object in this case, which might fulfill this role): it is the first step in the description of the Milnor fibre, and it is the bridge in the direction of the resolution and the transversal types of the components of the singular locus.…”

“…For several examples in the literature see e.g. [22] (homogeneous singularities, cylinders of plane curves, f = zf ′ (x, y), f = f ′ (x a y b , z)), [26] (f = g(x, y) + zh(x, y)); or for other classes consult also [12] and [1].…”