Octave Curmi
email: octave.curmi@univ-amu.fr

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

We give in this work an explicit combinatorial algorithm for the description of the Milnor fiber of a Newton non degenerate surface singularity as a graph manifold. This is based on a previous work by the author describing a general method for the computation of the boundary of the Milnor fiber of any reduced non isolated singularity of complex surface.

Introduction

The study of Milnor fibers of complex-analytic functions, which began in the second half of the 20th century, gave rise to a rich interaction between algebra and topology. One of its interesting aspects is that it can be used to provide equations for sophisticated topological objects, such as exotic spheres, see [Bri00, p 46-48] and [START_REF] Hirzebruch | Singularities and exotic spheres[END_REF]. See also [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF]Chapter 9], and [Sea19, Section 1].

This potential fabric of exotic spheres led Milnor to study further the topology of hypersurface singularities, and eventually to write his famous 1968 book [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF], aimed at the study of isolated singularities of hypersurfaces of C n , V (f ) = {f = 0}. In this work he introduced two equivalent fibrations, using respectively the levels of f /|f | on spheres centered at the critical points of f and the levels of f in C n .

The second of these two fibrations has been extended to more general contexts by Lê, see [Lê77], and is known as the Milnor-Lê fibration. However, it may produce singular generic fibers, due to the singularities of the ambient space. Hamm,in [Ham71], provided a setting in which the Milnor-Lê fibration is actually a smoothing of V (f ), that is, a way to put V (f ) in a flat family of analytic spaces, where the generic space is smooth. Namely, if (X, 0) is a germ of equidimensional complex analytic space, and f is any holomorphic function on (X, 0) such that V (f ) ⊃ ΣX, then the function f provides a smoothing of the singularity (V (f ), 0), where ΣX denotes the singular locus of X.

The study of the whole Milnor fiber is very hard, even for isolated singularities, and there is only a few types of singularities where a description of the full fiber is known. It is the case for the Kleinean sigularities A, D, E, where the Milnor fiber is unique and diffeomorphic to the minimal resolution (see Brieskorn,[START_REF] Brieskorn | Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen[END_REF]), as well as for the singularities of normal toric surfaces, with a description by surgery (see Lisca,[Lis08] and Némethi & Popescu-Pampu, [START_REF] Némethi | On the Milnor fibres of cyclic quotient singularities[END_REF]), and for sandwich singularities (De Jong & Van Straten, [START_REF] Jong | Deformation theory of sandwiched singularities[END_REF]). As for nonisolated singularities, the only known case is that of hypersurface singularities of the form {f (x, y) + z • g(x, y) = 0}, see Sigurðsson,[Sig16]. One can also refer to the survey [Gre19] for more about the topology of smoothings and deformations of singularities.

On the other hand, the study of the boundary ∂F of the Milnor fiber of a smoothing has been a very active area of research in the last decades. For an isolated singularity ∂F is unique, and equal to the link of the singularity. In today's words, Mumford proved in [Mum61] that the link of any isolated singularity of complex surface is a graph manifold, that is, a manifold describable using a decorated graph whose vertices represent fibrations in S 1 over compact surfaces. It is Waldhausen, in [START_REF] Waldhausen | Eine Klasse von 3-dimensionalen Mannigfaltigkeiten[END_REF], that later introduced this vocabulary and began studying this class of manifolds in itself. Furthermore, since the work of Grauert ([Gra62]), one knows exactly which graph manifolds appear as links of isolated singularities of complex surfaces. However, this strong point is tempered by the fact that one still does not know, for example, which of these manifolds appear as links of singularities of hypersurfaces of C 3 .

Still, one would like to get an analogous result for boundaries of Milnor fibers associated to non isolated singularities. The first steps towards the comprehension of the topology of these manifolds were made by Randell [START_REF] Randell | On the topology of non-isolated singularities[END_REF], then Siersma in [Sie91], [Sie01], who computed the homology of the boundary ∂F of the Milnor fiber in certain cases, and characterized the cases in which ∂F is a rational homology sphere.

Concerning the general topology of this manifold, a series of results were aimed at proving that the boundary of the Milnor fibers associated to a non isolated singularity is, again, a graph manifold, see [MP03], [MP04] and [MP16] as well as [MPW07] for the special case of Hirzebruch surface singularities, and [MPW09] for the so-called suspensions (f = g(x, y)+z n ). In [START_REF] Fernández De Bobadilla | The boundary of the Milnor fibre of complex and real analytic non-isolated singularities[END_REF], the authors prove the same result for a larger context, and in [NS12], the authors give a constructive proof for the case of reduced holomorphic functions f : (C 3 , 0) → (C, 0). This is this last proof that we extend here and in [Cur19] 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 [Cur19] to produce an explicit combinatorial algorithm for the description of the boundary of the Milnor fiber of f , extending the work of Oka in [START_REF] Oka | On the resolution of the hypersurface singularities[END_REF] for non degenerate isolated singularities in C 3 .

We prove the following Theorem. Let (X, 0) be the germ at its vertex of a 3-dimensional toric variety defined by a 3-dimensional cone in a weight lattice. Let f : (X, 0) → (C, 0) be a Newton-nondegenerate function whose zero locus contains Sing(X). Then the boundary ∂F of its Milnor fiber is a graph manifold determined by the local Newton polyhedron of f .

The purely combinatorial nature of the description we make of the manifold ∂F opens the way for the computation of a great number of examples through a future implementation in a computer program. But it also calls for more theoretic work, such as for example extending what is done in [START_REF] Braun | Invariants of Newton non-degenerate surface singularities[END_REF], where the authors do the opposite work, retrieving a possible Newton polyhedron of a function f : C 3 → C having a given graph manifold as boundary of Milnor fiber, under the hypothesis that this manifold is a rational homology sphere. Another exploitation of this method would be to answer the ambitious question of which manifolds can appear as boundaries of Milnor fibers of non degenerate surface singularities. In this direction, our algorithm already provides the following obstruction, see Proposition 5.4:

Proposition. The normal form of the plumbing graph of the Milnor fiber of a Newton non degenerate singularity of complex surface is planar.

The article is organized in the following way:

• In Section 2, we recall the main material of [Cur19],

• in Section 3, we give the essential definitions and results of toric geometry which are involved in our construction,

• Section 4 is dedicated to the construction of the graph Γ (C tot ), which is one of the main ingredients for the description of the manifold ∂F in the general case,

• in Section 5, we show why the data of Γ (C tot ) is sufficient to proceed to the rest of the computation.

I would like to thank Patrick-Popescu Pampu for his support and for sharing his vision of toric geometry.

I thank also the AGT team in Aix-Marseille Université and the ANR LISA for allowing me to attend many enriching events.

Reminder of the general result

We recall here some of the main material of [Cur19], one can consult this article and the references therein for more details.

Milnor fibration on a singular variety

In the sequel, a variety will mean a reduced and equidimensional complex analytic space.

Let (X, 0) be a germ of complex variety of dimension 3, and let f : (X, 0) → (C, 0) be a germ of holomorphic function on (X, 0) assumed not to divide zero, that is, not to vanish on any irreducible component of (X, 0). The function f defines a germ (V (f ), 0) of hypersurface on (X, 0), where V (f ) := {x ∈ X, f (x) = 0}. Denote by ΣV (f ) the singular locus of V (f ), and by ΣX the one of X.

In the sequel, if (X, 0) is a germ, X will denote a representative of this germ.

Definition 2.1. We say that a real analytic function ρ :

X → R + defines 0 in X if 0 is isolated in ρ -1 (0), i.e. if there is another representative X ⊂ X of (X, 0) such that ρ -1 |X (0) = {0}. Theorem 2.2. (H. Hamm, [Ham71, Satz 1.6], Lê [Lê77, Theorem 1.1])
Given a real analytic function ρ defining 0 in X, and ε > 0, denote X ε := X ∩ ρ -1 ([0, ε)), and S ε := X ∩ {ρ = ε}. Let f : (X, 0) → (C, 0) be a germ of holomorphic function, such that X \ V (f ) is smooth. Then there exists ε 0 > 0, such that ∀ 0 < ε ε 0 , ∃ δ ε > 0 such that ∀ 0 < δ δ ε , the following two maps are diffeomorphic smooth fibrations:

• f |f | : S ε \ V (f ) → S 1 • f : ∂ ({|f | = δ} ∩ X ε ) → ∂ (D δ )
, where D δ denotes the closed disc of radius δ around 0 in C.

Definition 2.3. The first of the two fibrations above is referred to as the Milnor fibration of f , and the second one is called the Milnor-Lê fibration. The closure of the fiber of the Milnor-Lê fibration is called the Milnor fiber of the germ of function f .

Remark 2.4. The Milnor-Lê fibration is also sometimes referred to as the Milnor fibration.

Using transversality arguments, one may show that the diffeomorphism type of the Milnor fiber does not depend on the chosen representative, so we speak about the Milnor fiber of the germ of function f ∈ O X,0

Graph manifolds

We explain here the definition of graph manifold (also called plumbed manifolds) that we refer to. For details, one can consult the foundational articles [Mum61] and [START_REF] Waldhausen | Eine Klasse von 3-dimensionalen Mannigfaltigkeiten[END_REF], as well as the article [START_REF] Neumann | A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves[END_REF] for a description of the so-called plumbing calculus and other topological considerations on graph manifolds.

Recall that the Euler number of a fibration in S 1 over a surface is an integer e that characterizes entirely the total space of the fibration up to homeomorphism. See [GS99, Example 4.6.5] for more details.

Definition 2.5. An orientable plumbing graph is a graph Γ with decorations of the following type:

• Each edge is decorated by a ⊕ or symbol.

• Each vertex v is decorated by an Euler number e v ∈ Z and a genus [g v ], g v ∈ N.

Remark 2.6. When representing plumbing graphs, we may omit the decorations ⊕ and [0].

Definition 2.7. (Graph manifold)

If Γ is an orientable plumbing graph, define the oriented graph manifold M Γ associated to Γ in the following way: for each vertex v of Γ decorated by ([g v ], e v ), let M v be the S 1 -fibration of Euler number e v over the closed smooth surface B v of genus g v . Pick an orientation of the base and the fibers so that, taken together, they give the orientation of M v . Now, let λ v be the number of times the vertex v appears as endpoint of an edge. Remove from

B v disjoint open disks (D i ) 1 i λv ,
= ∂D × S 1 in M b v , T = ∂D × S 1 in M b v
, and glue T and T according to the matrix 0 1 1 0 , being the sign on the edge.

Finally, we use the convention

M Γ 1 Γ 2 = M Γ 1 #M Γ 1

Main aspects of the general result

We recall here the main steps of the general proof, and show how they relate to the different steps of the computation shown in Sections 4 and 5 in the setting of toric geometry.

Our computation of the boundary ∂F of the Milnor fiber of f involves a second function g : (X, 0) → (C, 0), called companion of f , which is generic with respect to f , and can be chosen to be the restriction to X of a linear form on (C N , 0) ⊃ (X, 0). Using this function, one defines, for k ∈ 2N, the germ If k is large enough, (S k , 0) is an isolated singularity, and its link ∂S k is diffeomorphic to the boundary ∂F of the Milnor fiber of f . From this point, the goal is therefore to describe the manifold ∂S k . This is done in analogy with the case of an isolated singularity of complex surface, by exhibiting a resolution Π : ( S , E) → (S k , 0) of S k whose exceptional locus is a transversal union of smooth orientable closed real analytic surfaces in a 4-dimensional oriented real analytic manifold S , and using Theorem A.3.

(S k , 0) := ({f = |g| k }, 0).
However, the existence of such a resolution is not obvious, in this category we expect the preimage of the origin to be 3-dimensional. We build an ad hoc resolution in several steps.

• The first step is a modification r X : ( X, r -1 X (0) → (X, 0) of the ambient germ which is adapted to the pair (f, g). In particular, we ask the following: denote D := r -1 X (V f •g ), and by D 0 the union of the irreducible components of D sent on the origin by r X . Denote by D f,ex the union of the components of D sent on curves in V (f ), and by V (f ) and V (g) the strict transforms of V (f ) and V (g) by r X respectively. Finally, denote

D f := D f,ex ∪ V (f ).
Finally, denote

C tot := (D f ∩ D 0 ) ∪ (D f ∩ V (g)).
Then we require that D is a simple normal crossings divisor at C tot .

Denote

C := (D f ∩ D 0 ) ∪ D f,ex ∩ V (g) ,
the union of the compact irreducible components of C tot .

In these conditions, define S k to be the strict tranform of S k by r X . We obtain a modification r S :

( S k , C ) → (S k , 0).
In this work, this first modification is the one that we build in Section 4, by refining the cone σ corresponding to the ambient variety X = X σ .

• Then one needs to resolve the singularities of S k , which are located along C . This is done by normalizing S k , and then comparing locally the normalization S k N of S k with the normalization of a singularity of complex surface, to resolve the last singularities. The normalization step corresponds to the demultiplication of vertices and edges in Subsection 5.1, while the resolution of the remaining isolated singularities corresponds to the replacement of some edges by bamboos, according to Appendix B.

The local equations of S k along points of C tot are determined by multiplicities of the pullbacks of f and g to X. If D i is an irreducible component of D, denote respectively m i , n i the multiplicities of f • r X and g • r X on D i . The following decorated graph will contain all the necessary data for the description of the exceptional locus E of the resolution of S k : Definition 2.9. Denote Γ(C tot ) the dual graph of the configuration of complex curves C tot .

The decorated graph Γ (C tot ) is obtained from Γ(C tot ) in the following way:

• If C is an irreducible component of D 1 ∩ D 2 , where D 1 ∈ D f and D 2 ∈ D 0 ∪ D g , decorate v
C , the vertex corresponding to C, with the triple (m 1 ; m 2 , n 2 ), and with its genus [g], in square brackets. If D 1 ∈ V (f ) and D 2 ∈ V (g), then the vertex associated to the non-compact curve C is an arrowhead.

• Decorate each edge e p of Γ(C tot ) corresponding to a double point p of C tot with ⊕ if this point is on exactly one component of D f , and with if it is on exactly two different components of D f .

Tools of toric geometry

We give in this Section the necessary material to proceed to Section 4 and the construction of Γ (C tot ). What follows is mainly based on the book [START_REF] Oka | Non-degenerate complete intersection singularity[END_REF] of Oka. To go further, one can consult the books [START_REF] Oda | Convex bodies and algebraic geometry[END_REF] of Oda, or [START_REF] Cox | Toric varieties[END_REF] of Cox, Little, Schenk. See also the foundational article [START_REF] Oka | On the resolution of the hypersurface singularities[END_REF].

Basic definitions

Let us start with some ad hoc definitions.

Definition 3.1. In this work, a 3-dimensional lattice N is a free group of rank 3, that is, isomorphic to (Z 3 , 0). Define the integral length of an element u ∈ N as

l(u) = max{n ∈ N, ∃ v ∈ N \ {0} such that u = n • v}. An element of N is called primitive if its integral length is equal to 1. A cone σ in N is a subset of N R := N ⊗ Z R of the form v 1 , . . . , v k := {r 1 • v 1 + • • • + r k • v k , r 1 , . . . , r k ∈ R + } ⊆ N R .
A cone σ is called stronlgy convex if it contains no linear subspace of N R .

A strongly convex cone is called regular if it can be generated by a family of elements of N that can be completed in a basis of the Z-module N . We denote by τ σ the fact that a cone τ is a face of σ, of any dimension. Definition 3.2. We will usually denote by M := N = Hom(N, Z) Z 3 the dual lattice of N . For σ a convex cone of dimension d in N R , let us define its dual cone

σ := {m ∈ M R , ∀ n ∈ σ, m, n 0} ⊂ M R
and its orthogonal

σ ⊥ := {m ∈ M R , ∀ n ∈ σ, m, n = 0} ⊂ σ .
Note that if σ is of dimension 3, σ is regular if and only if σ is regular.

Definition 3.3. If σ is a strongly convex cone in N , define X σ := Spec C[σ ∩ M ]
the 3-dimensional affine variety associated to it.

Remark 3.4. A family m 1 , • • • , m k ∈ M generating the semigroup σ ∩ M provides an embedding X σ → Spec (C[χ m 1 , • • • , χ m k ]) = C k x 1 ,••• ,x k .
Proposition 3.5. Closed points of X σ correspond to semigroup morphisms (S σ , +) → (C, •). Denote T N the set of those morphisms whose image is contained in C * . The set T N is (non canonically) isomorphic to (C * ) 3 and dense in X σ . This group is called the (algebraic) torus associated to N . The action of the group T N on itself extends to a continuous action on the whole variety X σ , making T N the unique n-dimensional orbit.

On closed points, the action is defined as on T N , by multiplication: if φ ∈ X σ and

ψ ∈ T N , then φ • ψ ∈ X σ is defined by ∀ m ∈ M, (φ • ψ)(m) = φ(m) • ψ(m). Proposition 3.6. If τ ⊂ σ, then the inclusion C[σ ] ⊂ C[τ ]
gives rise to a canonical birational morphism of algebraic varieties X τ → X σ . This morphism is an injection if and only if τ is a face (of any dimension) of σ. Definition 3.7. A fan F in N R is a finite set of strongly convex cones such that:

1. If σ ∈ F , any face of σ is in F . 2. The intersection of two cones of F is a face of each.
The support |F | of the fan F is the union σ∈F σ of the cones composing it. Definition 3.8. A fan F in N R defines a toric variety X F in the following way: take the disjoint union of the X σ 's, for all σ in F , and, if σ and σ are cones of F , glue X σ and X σ along X σ∩σ .

X F := σ∈F X σ X σ ∼ X σ∩σ X σ
Proposition 3.9. The actions of the torus T N on each X σ glue together, in agreement with the gluing of the X σ 's, giving rise to a global action of the torus on X F , under which the torus T N is the unique n-dimensional orbit. Furthermore T N is open and dense in X F .

Proposition 3.10. (Decomposition into orbits.) Let σ be a strongly convex polyhedral cone of dimension d belonging to a fan F in N R , where N is n-dimensional. Let τ 1 , . . . , τ k be the faces of σ of codimension 1.

Then

O σ := X σ \ X τ i is an orbit of X F under the action of T N . It is made of all morphisms (σ ∩ M, +) → (C, +) that are different from zero exactly on σ ⊥ ∩ M . In other words, a point of O σ is a maximal ideal of C[σ ∩ M ] containing the monomial χ m ∈ C[σ ∩ M ] if and only if m / ∈ σ ⊥ ∩ M .
Furthermore, the variety O σ can be seen as

O σ = Spec(C[σ ⊥ ∩ M ])
providing, non-canonically, the isomorphism

O σ (C * ) n-d . Remark 3.11. In particular, O {0} = T N . Furthermore, if F = σ, then O σ is the unique minimal-dimensional orbit in X σ . If dim(σ) = 3, it is called the origin of X σ , and denoted 0 Xσ .
With this notation, if σ is of maximal dimension, we get the germ of toric variety (X σ , 0 Xσ ), or (X σ , 0), whose local ring of germs of holomorphic functions is

O Xσ,0 = C{σ ∩ M }.
Considering this for any cone of F will give us a decomposition of X F into orbits under the action of the torus, a cone of F corresponding to an orbit:

X F = σ∈F O σ .
Proposition 3.12. (Closure of an orbit.)

The closure in X F of an orbit corresponding to a cone τ ∈ F is made of the union of the orbits corresponding to cones of F having τ as a face:

O τ = τ σ O σ .
The following proposition is essential for the sequel and shows how one can read on a fan the singularities of the associated variety.

Proposition 3.13. The variety X F is smooth at a point p ∈ O σ if and only if σ is regular.

One can encode certain modifications of a given variety X F through a refinement of F :

Definition 3.14. A refinement of a fan F in N R is another fan F in N R such that: |F | = |F | and ∀ σ ∈ F , ∃ σ ∈ F such that σ ⊂ σ.
A refinement of a cone is a refinement of the fan formed by its faces.

The following definition is a matter of vocabulary and is introduced for pure reasons of convenience in the exposition made in Section 4. Definition 3.15. Let F be a refinement of a fan F . If τ is a cone of F , we call minimal containing cone of τ in F the minimal-dimensional cone of F containing τ .

If τ, σ are two cones such that τ ⊂ σ, the minimal containing face of τ in σ is the minimal-dimensional face of σ containing τ . A cone τ is said to be in the interior of σ if its minimal containing face in σ is σ itself. Definition 3.16. Let F be a fan in N , and F a refinement of F .

The toric morphism Π F ,F : X F → X F associated to this refinement is obtained by gluing the morphisms given by the inclusions of cones of F in the cones of F , defined in Proposition 3.6.

Proposition 3.17. The morphism Π F ,F is a modification of X F . It has the following combinatorial property: if σ ∈ F , let σ be the minimal containing cone of σ in F . Then

Π F ,F (O σ ) ⊂ O σ . The critical locus E F ,F of Π F ,F is exactly the union τ ∈F ,τ / ∈F O τ ∈ X F
of orbits of X F corresponding to new cones, and the discriminant locus

∆(Π F ,F ) is τ ∈F ,τ / ∈F O τ ∈ X F ,
the union of orbits of X F corresponding to cones that have been subdivided.

modification associated to a germ of function

Let (X σ , 0) be the germ of affine normal variety associated to a 3-dimensional strongly convex cone.

Definition 3.18.

• Let f = m i ∈σ ∩M a i χ m i ∈ C{σ ∩ M }. The support of f is defined as Supp(f ) := a i =0 {m i } ⊂ σ ∩ M .
• The Local Newton polyhedron of f at the origin of X σ is defined as

LNP(f ) := Conv Supp(f ) + σ
where "+ denotes the Minkowski sum in M .

This definition of Newton polyhedron for general germs may be found in [Ste83, Definition 5], or [PPS13, Definition 8.7].

The following lemma is a direct consequence of the definitions, and allows one to easily read some parts of the zero locus of a function defined on a germ of toric variety.

τ ∆ := {v ∈ σ, ∆ v = ∆} = {v ∈ σ/∀ m ∈ ∆, m, v = h v (f )} is a cone of codimension d contained in σ. The set F f := {τ ∆ , ∆ ⊂ LNP(f )} is the fan associated to f . The morphism Π F f : X F f → X σ coming from the refinement of σ is called the modifi- cation of X σ associated to f .
Let us describe the behaviour of the strict tranform V (f ) of V (f ) under this modification, and more generally, under any toric modification factorizing through Π F f .

Let F be a refinement of F f , and τ ⊂ σ a cone of F . Denote

Π F : (X F , E F ) → (X σ , 0)
the modification associated to F , and

f = f • Π F the pullback of f by this modification. Let ∆ τ := {m ∈ LNP(f ), ∀ v ∈ τ, m, v = h v (f )} .
Note that we may have τ = τ and still ∆ τ = ∆ τ . However, the assumption that F is a refinement of F f ensures that this definition makes sense. In fact, if γ is the minimal containing cone of τ in F f , then ∆ τ = ∆ γ .

Note also that dim(∆ τ ) codim(τ ), and that τ σ ⇒ ∆ σ ⊂ ∆ τ .

We have a first lemma, showing how one can read from LNP(f ) the multiplicities of f along some divisors in X F :

Lemma 3.22. If τ = v R + for some primitive vector v in N , then the multiplicity of f along O τ is equal to h v (f ).
Definition 3.23. For a face ∆ of LNP(f ), of any dimension, we define f ∆ , f truncated relatively to the face ∆, as the function obtained by keeping only the terms of f corresponding to points of this face:

f ∆ := m i ∈∆ a i χ m i .
In some sense, which is made more precise for example in [START_REF] Oka | Non-degenerate complete intersection singularity[END_REF], f ∆τ is the equation of the intersection of V (f ) with the orbit O τ of X F .

We therefore have the following:

Lemma 3.24. If F is a refinement of F f , and τ ∈ F , then

V (f ) ∩ O τ = ∅ iff dim ∆ τ 1.
The following definitions describe a condition in which the strict tranform V (f ) of V (f ) behaves in a nice way with respect to the modification associated to F .

Definition 3.25. A function f ∈ C{σ ∩ M } is called suitable if V (f ) does not contain any 2-dimensional orbit of X σ , or equivalently, if Supp(f ) has points in each 2-dimensional face of σ . Definition 3.26. • A suitable germ of function f ∈ C{σ ∩ M } is said to be nonde- generate relatively to a compact face ∆ ⊂ LNP(f ) if and only if V (f ∆ ) is smooth in (C * ) n .
• A suitable germ of function is said to be Newton-nondegenerate, or NND if it is nondegenerate relatively to every compact face of its local Newton polyhedron.

Proposition 3.27. Let f ∈ C{σ ∩ M } be a NND germ of analytic function on (X σ , 0), and F a refinement of F f , such that X F is smooth along O τ and ∆ τ is compact. Then the intersection V (f ) ∩ O τ is smooth, and at any point p of this intersection, V (f ), p is a germ of smooth hypersurface of X F intersecting O τ transversally.

Furthermore, the following will be heavily used in Section 4:

Proposition 3.28. Let f be a Newton-nondegenerate function on X σ , and F be a refinement of F f . Let τ be a cone of F , such that ∆ τ is compact. Then,

• If codim(τ ) = 1, denote l(∆ τ ) the integral length of this face in M . Then Card V (f ) ∩ O τ = l(∆ τ ).
We also assume that f is a Newton non degenerate function, in particular it is suitable, see Definitions 3.25 and 3.26.

In this section, we explain how to compute the decorated graph Γ (C tot ) of Definition 2.9 from the cone σ and the Newton polyhedron LN P (f ) of f . We can already make the following remark:

Remark 4.1. Let τ be a face of σ of dimension 2. Then V (f ) ⊃ O τ iff there is a 1dimensional cone of F f whose minimal containg face in σ is τ .

In particular, if τ is not regular, then the condition V (f ) ⊃ Sing(X) implies that there is a cone of dimension 1 in F f whose minimal containing cone is τ .

Universal companion polyhedron

Although there is no chance to describe a function g that will be a companion of every Newton-nondegenerate function f on X, there exists an object which will be compatible with every function f . Definition 4.2. For any sequence G = (m 1 , • • • , m k ) generating the semigroup σ ∩ M , denote

P (G) := Conv   1 i k m i + σ   .
In the sequel, we fix such a family G. By Remark 3.4, the sequence G leads to an embedding (X, 0) → C k x 1 ,••• ,x k . From this viewpoint, P (G) is in fact the local Newton polyhedron of the restriction to X of a generic linear form of C k x 1 ,••• ,x k . Now, the fact that the restriction of a generic linear form is a companion of f implies: Lemma 4.3. For any germ of function f on (X, 0), there is a function g that is a companion of f and such that LN P (g) = P (G).

Remark 4.4. One can note that the Newton-nondegeneracy of f is not required for this lemma. However, for what follows, the hypothesis of Newton-nondegeneracy is central.

From now on, fix g : (X, 0) → (C, 0) a Newton non degenerate companion of f such that LN P (g) = P (G).

Remark 4.1 has the following consequence for g: Remark 4.5. There is no 1-dimensional cone of F g whose minimal containing face in σ is 2-dimensional.

In the next subsections, we construct a fan F refining σ and such that the associated modification r := Π F : X F → X is adapted to the pair (f, g). We will follow this construction on the following example:

Example 4.6. Denote M := Z 3 , and let

u 1 =   0 1 2   , u 2 =   0 1 0   , u 3 =   1 1 -1   , u 4 =   1 0 0   be vectors in R 3 , and σ = u 1 , u 2 , u 3 , u 4 R + .
Let X := X σ be the 3-dimensional toric variety corresponding to σ. The cone σ is not simplicial, hence X is singular at the origin. Furthermore, the face τ 1,2

:= u 1 , u 2 R + of σ is singular, hence X is singular along 0 τ 1,2 .
The cone σ is generated by the vectors

U =   1 0 1   , V =   0 1 1   , W =   1 0 0   , X =   0 2 -1   .
The face of σ corresponding to the face τ 1,2 of σ is the ray generated by W . Furthermore, the semigroup

S σ = σ ∩ M is generated by U, V, W, X and Y =   0 1 0   .
The relations between these vectors provide the description

X σ = Spec C[U, V, W, X, Y ] (Y 3 -XV, U X -W Y 2 , V W -U Y )
Figure 1 shows the local Newton polyhedron of the restriction g to X of a generic linear form in C 5 U,V,W,X,Y . Full lines represent the cone σ, while dashed lines represent the axes of coordinates, left for clarity.

Consider the function

f ∈ C[σ ∩ M ] given by f = χ 1 1 2 + χ 2 0 1 + χ 0 2 0 + χ 0 4 -2 = U V + U W + Y 2 + X 2 .
Note that V (f ) ⊃ Sing(X), since f admits no multiple of W in its support. Figure 2 shows the local Newton polyhedron of f , where we kept again the coordinate axes and the cone σ . Full round marks represent the elements of Supp(f ), while empty square marks represent the other points of M in the compact faces of LN P (f ).

The modification associated to the polyhedra

Denote F := F f •g the fan associated to the germ f • g. This fan is the minimal refinement of both F f and F g . Denote X = X F and r := Π F f •g : X → X the modification associated to this refinement.

Note that we have the following commutative diagram:

X X F f X Fg X Π F f Π Fg Π F ,F f Π F ,Fg r Denote D := r -1 (V f •g )
, and in the same way, V (f ), V (g) the strict transforms of Definition 4.8. We call cutting cone of a fan F with support σ any 2-dimensional cone γ of F with 1-dimensional faces τ 1 , τ 2 so that 1. the minimal containing face of τ 1 in σ is σ,

V (f ), V (

and the minimal containing cone of τ

2 is 2-dimensional. If γ is a cutting cone of F , then O γ is an irreducible curve of D f,ex ∩ D 0 .
The denomination cutting cone is chosen in reference to the so called "cutting edges" introduced in [NS12, Definition 7.2.2].

Example 4.9. The fact that the fan F f •g is the minimal refinement of both F f and F g implies that it can be computed by "superposing" the two fans F f and F g , as in figure 3.

This figure is to be understood as the cone over the plane figure. In this figure, thick lines indicate the 2-dimensional cones which are contained in 2-dimensional cones of F f , dashed lines indicate the 2-dimensional cones which are contained in 2-dimensional cones of F g , so the thick dashed line indicates a 2-dimensional cone which is in both.

Hollow round marks correspond to 2-dimensional orbits of X

D f,ex . 0 @ 1 0 0 1 A 0 @ 0 1 2 1 A 0 @ 0 1 0 1 A 0 @ 1 1 -1 1 A 0 @ 0 3 4 1 A 0 @ 1 2 2 1 A 0 @ 1 1 1 1 A 0 @ 1 1 0 1 A 0 @ 1 2 -1 1 A 0 @ 3 1 -1 1 A Figure 3: The fan F .
At this step, we do not yet have a modification of X σ respecting all the required axioms.

We still need to refine some cones of F .

First refinement of F

We obtain a new fan F by, first, refining regularly the 2-dimensional cones which are contained in 2-dimensional cones of F f . This is done in a canonical way, see for example [START_REF] Oka | On the resolution of the hypersurface singularities[END_REF] or [START_REF] Popescu-Pampu | The geometry of continued fractions and the topology of surface singularities[END_REF]. These cones correspond to 1-dimensional orbits of X F which are intersected by V (f ). Then, one needs to add new 2-dimensional cones in a non-canonical way in order to get a collection of cones respecting the definition of a fan, see Definition 3.7. Furthermore, for convenience, we require that this last addition is made without introducing new cutting cones. This is always possible. Denote by F the fan obtained at the end of this process.

Example 4.10. In our example, the 2-dimensional cones of F in question are already regular, so the fan F is the same as F .

By analogy with the previous subsection, we introduce the notations D 0 , V (f ), V (g), D f,ex , and r : X F → X σ .

The fan F

Here again, the components of D 0 , V (f ), V (g), D f,ex can be read on F as in Discussion 4.7, and if γ is a cutting cone of F , then O γ is an irreducible curve of D f,ex ∩ D 0 . This is what motivates the final refinement of F .

Denote F a fan obtained by refining the 3-dimensional cones of F having a cutting cone as a face. These cones correspond to points of X F which are on curves of D f,ex ∩ D 0 , whence the need for X F to be smooth here.

There is no canonical way to refine regularly a 3-dimensional cone, but a first necessary step is the refining of its 2-dimensional faces. For a description of the general process, see [START_REF] Oka | On the resolution of the hypersurface singularities[END_REF].

Our construction ensures the following Lemma 4.11. The modification r X := Π F : X := X F → X is a modification of X adapted to the pair (f, g).

Example 4.12. Figure 4 shows a possible F for our example. The new cutting cones are represented by continuous thin lines. We erased one end of the corresponding segments on the drawing to reflect the fact that the corresponding curves do not intersect, as will be explained in the next subsection. The other new 2-dimensional cone is represented with a dotted line. Other codes are the same as previously.

Reading Γ (C tot )

Let us explain now how the decorated configuration C tot can be read from the fan F , together with LNP(f ) and LNP(g).

0 @ 1 0 0 1 A 0 @ 0 1 2 1 A 0 @ 0 1 0 1 A 0 @ 1 1 -1 1 A 0 @ 0 3 4 1 A 0 @ 1 2 2 1 A 0 @ 1 1 1 1 A 0 @ 1 1 0 1 A 0 @ 1 2 -1 1 A 0 @ 3 1 -1 1 A 0 @ 0 1 1 1 A 0 @ 2 1 -1 1 A 0 @ 0 2 3 1 A Figure 4: The fan F . Recall that C tot = (D f ∩ D 0 ) ∪ D f ∩ V (g) can be decomposed as (D f,ex ∩ D 0 ) ∪ V (f ) ∩ D 0 ∪ V (f ) ∩ V (g) ∪ D f,ex ∩ V (g) .
Since F g contains no cutting cone, the last intersection is empty. There are therefore three types of curves to be read.

We introduce the following notations: Definition 4.13. For τ ∈ F , denote repectively ∆ τ (f ) and ∆ τ (g) the corresponding faces of LN P (f ) and LN P (g).

To make short, if the dimension of those faces is 1 or less, denote

l τ (f ) := l(∆ τ (f )) and l τ (g) := l(∆ τ (g))
with the convention that the length of a 0-dimensional face is 0.

If dim(∆ τ (f )) 2, denote i τ the number of interior points of ∆ τ (f ), with the convention that the number of interior points of a face of dimension 0 or 1 is 0.

Finally, a cone τ in the interior of σ is called

pertinent if dim(∆ τ (f )) 1, that is, if V (f ) ∩ O τ = ∅.
Discussion 4.14. (Two types of compact curves.)

1. The first type is made of curves in D f,ex ∩ D 0 , called exceptional curves Such curves are of the form O γ , where γ is a cutting cone of F .

Two such curves intersect if and only if the corresponding cones are faces of a same 3-dimensional cone of F . The nature of their intersection point can be read on the fan: it is ⊕ if and only if they are in the same component of D f,ex .

The multiplicity decorations are obtained in the following way: let γ be a cutting cone, and τ 1 ⊂

• σ, τ 2 be its 1-dimensional faces. Then O γ has multiplicity decoration (h τ 1 (f ); h τ 2 (f ), h τ 2 (g)). Each exceptional curve is rational.

2. The second type is made of curves of V (f )∩D 0 , called strict tranform curves.

They correspond to 1-dimensional pertinent cones τ of F .

Furthermore, in these conditions, the intersection

O τ ∩ V (f ) is • an irreducible curve C τ of genus i τ if dim(∆ τ (f )) = 2. • a disjoint union C τ = C τ 1 • • • C τ lτ (f ) of l τ (f ) irreducible rational curves if dim(∆ τ (f )) = 1. Each connected component of C τ has multiplicity decoration (1; h τ (f ), h τ (g)).
Furthermore, let τ 1 , τ 2 be two 1-dimensional such cones. The possibly disconnected curves C τ 1 and C τ 2 intersect if and only if τ 1 and τ 2 are faces of the same 2-dimensional pertinent cone γ. In these conditions,

Card(C τ 1 ∩ C τ 2 ) = l γ (f )
and these intersection points are all of type ⊕.

In this situation, if dim(τ 1 ) = 1, then l τ 1 (f ) = l γ (f ), and each connected component of C τ 1 intersects C τ 2 in exactly one point.

Lemma 4.15. (Intersection of curves of the first and of the second type.)

Let C 1 = O γ be an exceptional curve, and C 2 = C τ be a strict transform curve. Then

C 1 ∩ C 2 = ∅ ⇔ τ ≺ γ and γ is pertinent.
In these conditions,

Card(C 1 ∩ C 2 ) = l γ (f )
and each intersection point is of type . Now, one needs to add to the picture the non compact curves of the intersection V (f ) ∩ V (g).

Pick generic coefficients of g so that for any 2-dimensional orbit O τ of X intersected by V (f ), the truncations f ∆τ (f ) and g ∆τ (g) verify the hypothesis of Lemma 3.31. is entirely encoded in Γ (C tot ). Indeed, one has the following direct consequences of our construction:

Lemma 5.1.

1. Every non-rational curve of C tot has a multiplicity decoration of the form (1; m 2 , n 2 ).

2. Each cycle in the graph Γ (C tot ) contains a vertex having a multiplicity decoration of the form (1; m 2 , n 2 ).

In the language developped in [Cur19], point 1 implies that the data of the so-called where ν i indicates that an edge is repeated ν i times. Then replace v C by the union of n C := gcd(m 1 , • • • , m s+t ) vertices, each one of them decorated by the same genus g C verifying

switches
n C • (2 -2 • g C ) = 2 -2g - s+t i=3 ν i • gcd(m 1 , m 2 ) + s+t i=3 gcd(m 1 , m 2 , m i ) • ν i ,
and a mutliplicity decoration given by µ C := m 2 • n 1 gcd(m 1 , m 2 ) .

2.

• An edge of type ⊕, of the form Example 5.2. Figure 6 shows the graph Γ µ (E tot ) of our example. Here, the different bamboos are simple and do not bring any new vertices, but the upper vertex gave rise to two vertices, and the edge ending at this vertex has also produced two edges. The multiplicity decorations on Γ µ (E tot ) correspond to the multiplicities of the pullback of the function g on each irreducible surface of (g • Π) -1 (0).

(m 2 ; m 1 , n 1 ) (m 2 ; m 3 , n 3 ) [χ] [χ ] v C v C ⊕ is replaced by d := gcd(m 1 , m 2 , m 3 ) identical strings of type Str ⊕ m 2 d ; m 1 d , m 3 d |0; n 1 , n 3 . • An edge of type , of the form (m 2 ; m 1 , n 1 ) (m 3 ; m 1 , n 1 ) [χ] [χ ] v C v C is

The plumbing graph Γ(∂F )

A plumbing graph Γ(∂F ) is obtained from Γ µ (E tot ) by replacing the multiplicity decorations (µ i ) by the self-intersections k i of the components of the preimage of the origin by Π in S , erasing the arrowhead vertices, and keeping the genus and edge decorations.

Let v be a vertex of Γ µ (E tot ), with multiplicity µ. Let v 1 , • • • , v n be the adjacent vertices (including arrowheads), with respective multiplicities µ 1 , • • • , µ n , and denote 1 , • • • , n the decorations of the corresponding edges. Then, following Lemma A.8, the self-intersection decoration k of v in Γ(∂F ) is obtained through the equality

k • µ = n i=1 i • µ i .
Example 5.3. Figure 7 shows the plumbing graph for ∂F obtained after applying the step described previously.

Figure 8 shows the normal form of this graph. Note that, unlike in the case of isolated singularities, this graph does not have a negative definite incidence matrix.

(1)

[1] ⊖ ⊖ ⊖ f 4× ⊖ ⊖ ⊖ (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1)
(1) (1)

(1) The plumbing calculus does not preserve planarity. However, the process of normalization of a given plumbing graph preserves this property. Therefore, a first consequence of our method of computation is the following: Proposition 5.4. The normal form of the plumbing graph of the Milnor fiber of a Newton non degenerate singularity of complex surface is planar.

⊖ [1] Figure 6: The graph Γ µ (E tot ). [1] ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ [1] 1 1 1 1 2 -5 -1 0 4 -2 -2 -2

Appendices

A Tubular neighbourhoods and graph manifolds

Graph manifolds can appear naturally as boundaries of neigbourhoods of complex curves in smooth complex surfaces, see [Mum61]. In our context, they will appear as boundaries of neighbourhoods of real surfaces inside a smooth 4-manifold.

Definition A.1. (Simple configuration of surfaces, and its plumbing dual graphs.)

Let S be a 4-dimensional oriented real analytic manifold. A simple configuration of compact real analytic surfaces in S is a subset E ⊂ S such that:

1. E = f inite E i , such that each E i is an oriented closed smooth real analytic surface.

For all

i = j = k = i, the intersection E i ∩ E j ∩ E k is empty.
3. For all i = j, the intersection E i ∩ E j is either empty or transverse. In particular, it is a finite union of points.

In this setting, one defines a plumbing dual graph Γ S (E) of E in S by decorating its dual graph in the following way:

1. Decorate each vertex v E i by the self-intersection e i of E i in S and by the genus [g i ] of the surface E i .

2. Decorate each edge of Γ(E) corresponding to the intersection point p of E i ∩ E j , by ⊕ if the orientation of E i followed by the orientation of E j is equal to the orientation of S at p, and by otherwise.

Definition A.2. (See [Dur83].)
Let E be a simple configuration of compact orientable real analytic surfaces in an oriented 4-dimensional real analytic manifold S . In this context, we call rug function any real analytic proper function ρ : S → R + such that ρ -1 (0) = E.

Theorem A.3. Let E be a simple configuration of compact orientable real analytic surfaces in an oriented 4-dimensional real analytic manifold S , that admits a rug function ρ as in Definition A.2. Then, for ε > 0 small enough, the boundary of the oriented 4-manifold {ρ ε} is orientation-preserving homeomorphic to the graph manifold associated to the graph Γ S (E).

Definition A.4. In this setting, the manifold {ρ ε} is called a tubular neighbourhood of E.

About the proof of Theorem A.3. This theorem can be seen as an extension of what is done in [Mum61], in the case of a configuration of complex analytic curves in a smooth complex surface.

In our case, observe first that we can extend the definition of rug functions to semianalytic functions ρ, and still have a unique homeomorphism type for the boundary of the neighbourhood {ρ ε} of E for ε > 0 small enough, following the proof of [Dur83, Proposition 3.5]. Now, one can build by hand a semi-analytic neighbourhood whose boundary is homeomorphic to the manifold Γ S (E). This is done by building a rug function for each irreducible component E i of E, providing a tubular neighbourhood T i of each E i whose boundary is an S 1 -bundle of Euler class e i over E i . One then plumbs those bundles using appropriate normalizations of the rug functions, building a semi-analytic neighbourhood of E which is homeomorphic to the desired graph manifold.

Remark A.5. Note that the decorations on the edges of the graph Γ S (E) depend on the orientations of the surfaces E i . However, if the surfaces E i are only orientable, the different possible plumbing dual graphs still encode the same graph manifold, see move [R0] of the plumbing calculus in [START_REF] Neumann | A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves[END_REF].

We end this section with a tool that can be used to compute the self-intersections of the irreducible components of E: Definition A.6. Let E = f inite E i be a simple configuration of compact oriented real analytic surfaces in a 4-dimensional real analytic manifold S . A real analytic function g : S → C is called adapted to E if 1. E tot := g -1 (0) is a simple configuration of oriented, not necessarily compact, real analytic surfaces, such that E tot ⊃ E. Lemma A.8. (Computing self-intersections.) Let S , E, g, E tot be as in Definition A.6. Let E (1) be an irreducible component of E. Then the self-intersection e (1) of the surface E (1) in S verifies the following condition: let p 1 , • • • , p n be the intersection points of E (1) with other components of E tot , p j ∈ E j ∩ E (1) , where the same component may appear several times. Then n (1) • e (1) = n i=1 i • n i where i ∈ {-1, +1} refers to the sign associated to the intersection p i in the following sense: if p ∈ E i ∩ E j , associate +1 to p if and only if the combination of the orientations of E i and E j at p provides the ambient orientation of S .

(a

Proof. The proof follows the standard argument in the holomorphic category. The difference of the two members of the equation is the intersection number of E (1) with the cycle defined by g = 0. This cycle is homologous with that defined by a nearby level of g, which does not meet E (1) any more. The intersection number being invariant by homology, one gets the desired result.

In order to make this argument rigorous, one has to work in convenient tubular neighborhoods of E and to look at the cycles defined by the levels of g in the homology of the tube relative to the boundary.

  consequently removing as many open solid tori from M v . Each ∂D i is oriented as boundary component of B v \ D i . Denote by M b v the resulting circle bundle with boundary. Denote ∂M b v = T i a disjoint union of tori. For every edge between the vertices v and v , glue the manifolds M b v and M b v in the following way: pick boundary components T

Lemma 3. 19 .

 19 The ideal I(τ ) ⊂ C{σ ∩ M } of functions cancelling on the orbit O τ of X σ is made of the functions f ∈ C{σ ∩ M } such that Supp(f ) ∩ τ ⊥ = ∅. Definition 3.20. Let v ∈ σ, and f ∈ C{σ ∩ M }. Define the height of LNP(f ) in the direction v to be h v (f ) := min m∈LNP(f ) m, v ∈ R + . Definition 3.21. Let σ be a cone in N , and f ∈ C{σ ∩ M }. Let ∆ be a face of LNP(f ) of dimension d. Then the set

Figure 1 :

 1 Figure 1: LN P (g).

  g) by r. Denote also D f,ex the union of components of D whose images by r are curves of V (f ), and D 0 := r

Figure 2 :

 2 Figure 2: LN P (f ).

F

  whose closures are irreducible components of D 0 intersected by V (f ).Hollow square marks correspond to 2-dimensional orbits of X F whose closures are irreducible components of D f,ex . Remarks 4.1 and 4.5 imply in fact that any 1-dimensional cone whose minimal containing face in σ is 2-dimensional correpsonds to a component of

  replaced by d := gcd(m 1 , m 2 , m 3 ) identical strings of type Str m 1 d ; m 2 d , m 3 d |n 1 ; 0, 0 . The notations Str are explained in Appendix B. In both cases, the number d of new edges is a multiple of the number of new vertices, and the edges are distributed uniformly on the vertices.

Figure 7 :

 7 Figure 7: The graph Γ(∂F ).

Figure 8 :

 8 Figure 8: The normal form of Γ(∂F ).

  is unnecessary, and point 2 implies that a covering data of Γ (C tot ) determines a unique graph, see[START_REF] Némethi | Resolution graphs of some surface singularities[END_REF] Theorem 1.20].As in [NS12, 10.2] (see also [Cur19] for more details), a plumbing graph for the boundary ∂F of the Milnor fiber of f is obtained from Γ (C tot ) by the following two steps:5.1 The multiplicity graph Γ µ (E tot )Denote Γ µ (E tot ) the graph obtained from Γ (C tot ) in the following way:1. Let v C be a vertex of Γ (C tot ) corresponding to a curve C of genus g, such that the union of v C and the adjacent vertices is of the form

	(m 1 ; m 3 , n 3 )			(m s+1 ; m 2 , n 2 )
	(m 1 ; m s , n s ) [g 3 ]	. . .	ν s ⊕ ⊕ ν 3	(m 1 ; m 2 , n 2 ) [g]	ν s+t ν s+1	(m s+t ; m 2 , n 2 ) [g s+1 ] . . .
	[g s ]					[g s+t ]

  ) For any component E i of E tot , ∀ p ∈ E i \ j =i E j , there is a neighbourhood U p of p in S and complex coordinates (x p , y p ) on U p such that U p ∩ E i = {x p = 0} and n i ∈ N * such that g = x n i p • ϕ where ϕ : U p → C is a unit at p. (b) For any components E i of E, E j of E tot , ∀ P ∈ E i ∩ E j ,there is a neighbourhood U p of p in S and complex coordinates (x p , y p ) on U p such that U p ∩E i = {x p = 0}, U p ∩ E k = {y p = 0}, and n i , n k ∈ N * such that g = x n i p y Definition A.7. In this setting, the integer n i of point 2a of Definition A.6, independent of the point p ∈ E i \

	n j

p • ϕ where ϕ : U p → C is a unit at p. j =i E j , is called the multiplicity of g on E i , denoted m E i (g).

• If codim(τ ) = 2, denote i(∆ τ ) the number of points of M in the interior of ∆ τ . Then C(τ ) := V (f ) ∩ O τ is a smooth curve. Furthermore, if dim(∆ τ ) = 2, this curve is irreducible and g(C(τ )) = i(∆ τ ).

If dim(∆ τ ) = 1, C(τ ) is a disjoint union of l(∆ τ ) smooth curves of genus 0.

Remark 3.29. Let us remind that, in both cases, if dim(∆ τ ) = 0, then V (f ) ∩ O τ = ∅.

Let us conclude this section with a lemma giving a precise meaning to the genericity of Newton-nondegenerate functions.

Lemma 3.30. Let f be a holomorphic function on X σ . Then in the space of coefficients of all those functions h such that LNP(h) = LNP(f ), those which are non-degenerate are Zariski-dense.

Finally, we will need a corollary of the Bernstein-Koushnirenko-Khovanskii Theorem (see [CLO98, Theorem 7.5.4]), in dimension 2. We will need the following notations:

for some lattice M of dimension 2, denote P (f ) := Conv(Supp(f )). Denote by V (P ) the lattice volume of a polyhedron P , for which a simplex is of volume 1. Finally, if P 1 , P 2 are two polyhedra in M , denote

V ol(P 1 + P 2 ) -V ol(P 1 ) -V ol(P 2 ) 2 called the mixed volume of P 1 and P 2 .

Lemma 3.31. Let M be a 2-dimensional lattice, and let f ∈ C[M ] be a Newton-nondegenerate function. Denote P 1 := P (f ). Then for any convex polyhedron P 2 with vertices in M , a generic choice of coefficients for the elements of P 2 ∩ M will provide a function g such that P (g) = P 2 and, denoting P = P 1 + P 2 , the compactifications V (f ) and V (g) of V (f ) and V (g) in X F P intersect transversally only on O 0 , in V (P 1 , P 2 ) points.

Construction of Γ (C tot )

Let (X, 0) = (X σ , 0 Xσ ) be the germ of normal 3-dimensional toric variety associated to a rational strongly conve cone σ ⊂ M R , where M is a 3-dimensional lattice. Let σ sing denote the union of non regular faces of σ, i.e. the union of faces τ of σ corresponding to orbits O τ along which X is singular. Let f : (X, 0) → (C, 0) be a holomorphic function such that Supp(f ) ∩ σ sing = ∅. This guarantees that V (f ) := {f = 0} ⊃ Sing(X), and therefore that f is a smoothing of V (f ). Definition 4.16. Denote V (τ ) the mixed 2-dimensional volume of ∆ τ (f ) and ∆ τ (g).

With this choice of g, we can access the rest of the configuration C tot . Indeed, Lemma 4.17. (Adding non-compact curves)

Let C τ be a strict transform curve of C , possibly disconnected. Then

each of these intersection points being an intersection point of C τ with a curve in

Each of these points is of type ⊕, and, in Γ (C tot ), the new curves are represented by arrowheads decorated with (1; 0, 1).

Example 4.18. Figure 5 shows the graph Γ (C tot ) of our example. The representation of the graph reflects the disposition of the fan F of figure 4. Following the classical convention, we do not indicate the decorations ⊕.

(1; 4; 1) (4; 4; 1) (1; 2; 1) (1; 2; 1) (1; 2; 1) (2; 2; 1) (1; 2; 1) (1; 0; 1) (1; 0; 1)

Final step of the computation and sufficiency of C tot

In order to get a plumbing graph for the boundary of the Milnor fiber of f , one needs to modify the graph Γ (C tot ). Unlike in the general case, the necessary data for this operation

B Hirzebruch-Jung chains

We introduce here the Hirzebruch-Jung chains (or strings) that occur in the final step of our resolution of ( S k , 0). For more details, one can consult [BPVdV84, III.5], [NS12, 4.3] or [START_REF] Popescu-Pampu | The geometry of continued fractions and the topology of surface singularities[END_REF].

For a, b, c ∈ N, (a, b, c) denotes gcd(a, b, c). Suppose that d := (a, b, c) = 1. In these conditions, the normal surface (V, p) := {x a = y b z c }, 0

N is an isolated singularity of complex surface. We describe here a graph of resolution of this singularity, decorated with the self-intersections of the irreducible components of the exceptional divisor and the multiplicities of the pullback of the function g(x, y, z) = x n 1 y n 2 z n 3 .

be defined by the relation

Then the graph of Figure 9 is a graph of resolution of (V, 0). Left and right-hand arrows represent, respectively, the pullbacks of the z and the y-axes. The numbers between parentheses are the multplicities of the pullback of the function g(x, y, z) = x n 1 y n 2 z n 3 on the irreducible components of its total transform. We denote by Str ⊕ (a; b, c|n 1 ; n 2 , n 3 ), resp. Str (a; b, c|n 1 ; n 2 , n 3 ) the chain of Figure 9 with each edge decorated by ⊕, resp. , and the self-intersection decorations removed.