The boundary of the Milnor fiber of the singularity $$f(x,y) + zg(x,y) = 0$$ f ( x , y ) + z g ( x , y ) = 0
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The boundary of the Milnor fiber of the singularity f (x, y) + zg(x, y) = 0 Baldur Sigurðsson1 Received: 15 May 2019 / Revised: 21 October 2019 / Accepted: 6 December 2019 © Springer Nature Switzerland AG 2020
Abstract Let f , g ∈ C{x, y} be germs of functions defining plane curve singularities without common components in (C2, 0) and let (x, y, z) = f (x, y) + zg(x, y). We give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber of in terms of a common resolution for f and g. We give an example of a choice for f and g yielding a boundary of a Milnor fiber having more than one irreducible component. Keywords Milnor fiber boundary · Plumbed manifold · Constructive algorithm Mathematics Subject Classification 32S25 · 14J17 · 57M50
1 Introduction It is known that the boundary of the Milnor fiber of any hypersurface singularity in (C3, 0) is a plumbed manifold. This was stated by Michel and Pichon in [6] and proved by separate methods by Némethi and Szilárd [12] and Michel and Pichon [7]. A stronger statement for certain real analytic map germs was proved by de Bobadilla and Neto [5]. As these theorems rely on resolution of singularities, they do not easily provide an explicit description of a plumbing graph describing the boundary. Calculations have been carried out, however for some particular singularities and families: for Hirzebruch singularities [8], suspensions of isolated plane curves [9], and in the many examples of [12].
The author was partially supported by the ERCEA 615655 NMST Consolidator Grant, by the Basque Government through the BERC 2014–2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323 during the writing of this article.
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Baldur Sigurðsson [email protected] Instituto de Matemáticas, Universidad Nacional Autónoma de México, Col. Lomas de Chamilpa, 62210 Cuernavaca, Morelos, Mexico
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B. Sigurðsson
In the case of a hypersurface singularity given by the equation (x, y, z) = f (x, y) + zg(x, y) = 0, where f , g are singular germs with no common factors (but not necessarily reduced), we give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber in terms of the graph associated with an embedded resolution of the plane curve singularities defined by f and g. For the explicit statement, see Theorem 6.3 and the construction in Sect. 6. Furthermore, the algorithm provides a multiplicity system associated with the function z described in Sect. 7. This is obtained from an explicit description of the Milnor fiber by the author [14]. Singularities of the form f (x, y) + zg(x, y) are closely related to the deformation theory of sandwitched singularities, see [3]. The article is organized as follows. In Sect. 2 we recall the results of [14] and fix notation related to the resolution graph of f and g. In Sect. 3 we define plumbed manifolds and prove some useful lemmas related to them. In Sect. 4 we introduce families of multiplicities and dual multip
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