THE LINK OF {f(x, y) + z^{n} = 0} AND ZARISKI’S CONJECTURE

ROBERT MENDRIS AND ANDRÁS NÉMETHI

Abstract. We consider suspension hypersurface singularities of type g = f(x, y) + z^{n}, where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski’s conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated hypersurface singularity is a rational homology 3-sphere then it determines the equisingularity type, the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.

1. Introduction.

In the last decades an intense research effort has been concentrated on the following problem: what kind of analytic invariants or smoothing invariants (if they exist) can be determined from the topology of a normal surface singularity (X, x).

Some of the results became already classical. E.g., Mumford’s result which states that (X, x) is smooth if and only if the fundamental group of the link L_{X} is trivial [15]. Or its generalization by Neumann [25] which claims that the oriented homeomorphism type of the link contains the same information as the resolution graph of (X, x). Or Artin’s computations of the multiplicity and the embedded dimension of rational singularities [2, 3]; and their generalizations by Laufer [11] for minimally elliptic singularities, and by S. S.-T. Yau [40] for some elliptic singularities.

In general, these questions are very difficult, even if we restrict ourselves to some special families, e.g. to complete intersections or hypersurface singularities; and even if we permit ourselves to use, instead of the topology of (X, x), a richer topological information, e.g. in the case of hypersurfaces the embedded topological type. For example, Zariski conjectured three decades ago that the embedded topological type of an isolated hypersurface singularity determines its multiplicity [42]. This was verified till now only for quasi-homogeneous singularities [9, 33, 39, 41] (and some other sporadic cases).

In a different direction, the conjecture of Neumann and Wahl [28] about a possible connection between the Casson invariant of the link (provided that it is an integral homology sphere) and the signature of the Milnor fiber opened new windows for the theory.

Recently, the subject revives even with a larger intensity, see e.g. [18, 29, 30, 22, 23, 24], their introductions and listed references. Basically, these articles claim that if the link of a ℚ-Gorenstein singularity is a rational homology sphere, then it codifies an extremely rich analytic information about the singularity.

2000 Mathematics Subject Classification. 32S.

Key words and phrases. suspension singularities, plane curve singularities, Newton pairs, resolution graphs, cyclic covers, Zariski’s conjecture, multiplicity, Milnor number, links of singularities.

The second author is partially supported by NSF grant DMS-0071820.

The present article is in the spirit of the above efforts. We will consider the family of suspension hypersurface singularities of type f(x, y) + z^{n}, where f is an irreducible plane curve singularity.

For this family, we not only answer positively both main conjectures, (namely the Zariski’s conjecture, and also the possibility to recover the main analytic and smoothing invariants from the link), but we succeed to obtain much sharper statements.

The main result of the paper is the following (cf. 5.4):

Theorem 1. Let f : (ℂ^{2}, 0) → (ℂ, 0) be an irreducible plane curve singularity with Newton pairs {(p_{i}, q_{i})}_{i=1}^{s} and let n ≥ 2 be an integer. Let L_{X} be the link of the hypersurface singularity (X, 0) = ({f(x, y) + z^{n} = 0}, 0). Then, except for two pathological cases S1 and S2 (which are described completely in 5.1 and 5.3, and can be characterized perfectly in terms of L_{X}), from the link L_{X} one can recover completely the Newton pairs of f and the integer n (provided that we disregard the “z-axis ambiguity”, cf. 5.2.) In both exceptional cases the links have non-trivial first Betti numbers. In particular, the above statement holds without any exception provided that the link is a rational homology sphere.

On the other hand, in the cases S1 and S2, the link together with the Milnor number of the hypersurface singularity f + z^{n} determine completely the Newton pairs of f and the integer n (cf. the two addendums in 5.1 and 5.3).

Here some remarks are in order.

(1) L_{X} determines the number s of Newton pairs of f in all the cases.

The exceptional case S1 appears when s = 1, and the corresponding singularities have the equisingular type of some special Brieskorn singularities. This case can be easily classified.

The exceptional case S2 appears for s = 2 with some other strong additional restrictions. In this case any link L_{X} can be realized by at most two possible pairs (f, n). This case again is completely clarified.

In all other cases, e.g. when s ≥ 3, the theorem assures uniqueness. This is slightly surprising. At the beginning of our study, we expected here more and more complicated special families providing interesting coincidences for their links. But, it turns out that this is not the case: if the plumbing graph of the link (or equivalently, the resolution graph of (X, x)) has more and more complicated structure, then it becomes more “over-determined”, and it leaves no room for any ambiguity for f and n.

In fact, in order to reach our goal, it was sufficient to consider a rather limited information about this graph: the determinants of its maximal strings and the determinants of some subgraphs with only one rupture vertex. Except for the two pathological cases, in all other cases already these determinants determine all the Newton pairs and n.

(2) In general, it is very difficult to characterize those resolution graphs (or links) which can be realized by, say, hypersurface singularities, or complete intersections, or by any family of germs defined by some analytic property.

Our proof gives a complete characterization of those graphs which can be realized as resolution graphs of some {f +z^{n} = 0} for some irreducible f. Indeed, the proof is a precise recipe how one can recover the Newton pairs of f and n. If one runs this algorithm (the steps of the proof of 5.4) for an arbitrary minimal resolution graph, and at some point it fails, then the graph definitely is not of this type. If the algorithm goes through and provides some candidates for the Newton pairs of f and for n, then one has to compute the minimal resolution graph of f + z^{n} = 0 (using e.g. the algorithm of [21], appendix 1, see also below) and compare it with the initial graph. If they are the same, then the answer is yes. But it could happen that these two graphs are not the same (since our algorithm is based on a very limited number of determinants: these determinants could be the same even if the graphs are not). In this case again the answer is no.

(3) For the case when f is arbitrary (i.e. reduced), but n = 2, and the link is rational homology sphere, Laufer established the uniqueness in [12].

In [34, 35], A. Pichon proved by a different method that for any fixed n, any link can be realized in a finitely many ways as the link of f + z^{n} = 0 (in her case f is reduced too). Some of our partial results (after some – sometimes nontrivial – identifications) can be compared with some of her results. E.g. our last formula from 3.5 can be compared with Proposition 3 [34], and our theorem 4.1 with theorem (6.4) of [35] (formulated in terms of Waldhausen decomposition). Nevertheless, we decide to present our arguments as well, in order to provide a self-contained proof and uniform picture.

(4) We expect that a possible generalization of the above result for reduced f inevitably will contain a much larger list of pathological cases, with a lot of numerical, case by case verifications in the proof. For some more concrete “warnings” in this case, see 5.7.

Now we return back to our main theorem and its corollaries.

Corollary 1. Assume that g(x, y, z) = f(x, y) + z^{n} is a suspension hypersurface singularity with f irreducible, and not of type described in the pathological cases S1 and S2 (cf. 5.1 and 5.3). Then the link L_{X} of (X, 0) = ({g = 0}, 0) determines completely the following data:

(1) (X, 0) up to equisingular deformation (within the class of suspension of irreducible plane curve singularities).

(2) The embedded topological type of (X, 0) (i.e. the embedding L_{X} ⊂ S^{5}), in particular, the Milnor fibration and all the homological package derived from it.

(3) All the equivariant Hodge numbers associated with the vanishing cohomology of g, in particular, the geometric genus of (X, 0).

(4) The multiplicity of g.

In particular, if L_{X} is a rational homology sphere, then L_{X} determines all the four data (1-4). If g is a pathological case listed in S1 or S2, then L_{X} together with the Milnor number of g determines completely (1-4).

Indeed, the Newton pairs of f determine the equisingularity type of f. (2-4) follows from the description of the corresponding invariants for plane curve singularities and different “Sebastiani-Thom type” formulae; see e.g. [1, 37, 16, 19]. In fact, recently in [24], the geometric genus (together with the Milnor number and the signature of the Milnor fiber) was computed in terms of the Seiberg-Witten invariant of the link, provided that the link is a rational homology sphere.

It is well-known that the Milnor number of g can be determined from the embedded topological type of g (a fact firstly noticed by Teissier, see also [41]). Therefore, we get:

Corollary 2. [Zariski’s conjecture for this family] The multiplicity of g = f(x, y) + z^{n} (f irreducible) is determined by the embedded topological type of g.

Corollary 1 (see also Theorem 2 below) motivates the following conjecture.

Conjecture. Let g : (ℂ^{3}, 0) → (ℂ, 0) be an isolated hypersurface singularity whose link L_{X} is a rational homology sphere. Then the fundamental group of the link characterizes completely the equisingularity type, the embedded topological type, the equivariant Hodge numbers and the multiplicity of g. More generally, if the link of a ℚ-Gorenstein singularity (X, 0) is a rational homology sphere, then the multiplicity of (X, 0) is determined by the (oriented) homeomorphism type of the link.

Notice that the last general property is true for rational [2, 3] and elliptic [11, 40, 18] singularities. The first part can be verified in the following cases. If g = f + z^{n} with f irreducible, then Conjecture is true by Theorem 1 above (see also [25] for the relation between L_{X} and its fundamental group). If f is arbitrary but n = 2, then it is true by [12].

The next theorem provides a positive answer for weighted homogeneous singularities. Since its proof is short and involves a different terminology than the body of the paper, we decided to separate the whole statement at the end of this introduction.

Theorem 2. The above conjecture is true for any weighted homogeneous hypersurface singularity.

Proof. The Poincaré polynomial p_{A}(t) of the singularity can be determined from the link by [6] or [36] (see also [26]). Then, by a recent result of Ebeling [7] (Theorem 1) follows that the characteristic polynomial Δ(t) of the algebraic monodromy can be recovered from the link L_{X}. Indeed, notice that ψ_{A}(t) used by Ebeling is link-invariant as well. By [7, Theorem 1], Δ(t) = [ψ_{A}(t)p_{A}(t)]^{∗}, where ∗ stays for K. Saito’s duality. In general, ∗ is defined in terms of some integer h (cf. [7, page 3]), but here one can verify (using e.g. [14]) that if ψ_{A}(t)p_{A}(t) = ∏_{m}(1 − t^{m})^{χ_{m}}, then one can take h :=lcm{m : χ_{m} ≠ 0}; hence Δ(t) = ∏_{k|h}(1 − t^{k})^{−χ_{h/k}} for this h.

Then, by [39] (see also [32]), we obtain from L_{X} the weights, multiplicity and the embedded topological type. The statement about the Hodge data follows from [38]. □

2. Preliminaries about resolution graphs

2.1. Definitions/Notations. Assume that (X, x) is a normal surface singularity, f : (X, x) → (ℂ, 0) is a germ of an analytic function, and ϕ is an embedded resolution of the pair (f^{−1}(0), x) ⊂ (X, x). We denote by E the exceptional divisor of ϕ and by S the strict transform of f^{−1}(0). Let ∪_{w∈W}E_{w} (resp. ∪_{a∈A}S_{a}) be the irreducible decomposition of E (resp. S). We will assume that W ≠ ∅, any two irreducible components of E have at most one intersection point, and no irreducible exceptional divisor has a self–intersection points. The (good) dual embedded resolution graph associated with ϕ will be denoted by Γ(X, f). Its vertices V = W ∐ A consist of the non-arrowhead vertices W and arrowhead vertices A. Any w ∈ W is decorated by the self–intersection e_{w} and genus g_{w} of E_{w}; and any v ∈ V by the multiplicity m_{v} of f. In all our graph–diagrams, we put the multiplicities in parentheses (e.g.: (3)) and the genera in brackets (e.g.: [3]), with the convention that we omit [0].

If we delete the arrows and multiplicities of the graph Γ(X, f), we get a possible (good) resolution graph of (X, x).

If Γ is a decorated graph (with or without arrowheads), then for w ∈ W we denote by δ_{w} the number of vertices v ∈ V adjacent to w. A vertex w ∈ W is called rupture (resp. leaf) vertex if either g_{w} > 0 or δ_{w} ≥ 3 (resp. if δ_{w} = 1).

There are many (embedded) resolutions, but they are all connected by quadratic modifications. By the above convention we can blow down a −1-curve E_{w} with g_{w} = 0 if and only if δ_{w} ≤ 2. If the (embedded) resolution graph has no such curve then we say that it is minimal. There is a unique minimal (embedded) resolution graph denoted by Γ^{min}(X) (resp. Γ^{min}(X, f)).

Let I := (E_{w} · E_{v})_{(w,v)∈W×W} be the (negative definite) intersection matrix, and write det(Γ) := det(−I) > 0. By convention, the determinant of the empty graph is 1.

We will use the following relations connecting the ingerers {m_{v}}_{v∈V} and {e_{w}}_{w∈W}. Fix a total ordering of the set W. Let m_{W} be the column vector with |W| entries {m_{w}}_{w∈W}. Similarly, define the column vector m_{A} with |W| entries whose w^{th} entry is ∑_{a∈A∩V_{w}} m_{a}. Then

I · m_{W} + m_{A} = 0.  (1)

Moreover, if Γ = Γ(X) is a tree, then the inverse matrix I^{−1} can be computed in terms of determinants of some subgraphs as follows. Consider two vertices w_{1}, w_{2} ∈ W and the shortest path which connects them. Let Γ_{w_{1}w_{2}} be the maximal (in general non-connected) subgraph of Γ which has no vertices on this path. Then the (w_{1}, w_{2})-entry of I^{−1} is given by

I_{w_{1}w_{2}}^{−1} = − det(Γ_{w_{1}w_{2}})/ det(Γ).

Let L_{X} be the (oriented) link of (X, x). It is known that L_{X} and Γ(X) codify the same amount of information. For example, L_{X} is a rational homology sphere if and only if g_{w} = 0 for any w and Γ(X) is a tree.

For more details about the notions of this paragraph, see [20] or [21].

2.2. Example. Irreducible plane curve singularities. (See e.g. [5] or [8].) In this case (X, x) ≈ (ℂ^{2}, 0), Γ(ℂ^{2}, f) is a tree, and g_{w} = 0 for any w ∈ W.

In this article we are mainly interested in irreducible plane curve singularities (i.e. when |A| = 1). Their equisingular type and link is completely characterized by the set of Newton pairs {(p_{k}, q_{k})}_{k=1}^{s} (see e.g. [8], page 49). Here (p_{k}, q_{k}) = 1, p_{k} ≥ 2, q_{k} ≥ 1 and q_{1} > p_{1}.

The minimal embedded resolution graph can be reconstructed from the Newton pairs as follows (see e.g. in [5, 8] or [17]). First determine u_{i}^{l} and v_{i}^{l} (u_{i}^{0}, v_{i}^{0} ≥ 1, and u_{i}^{l}, v_{i}^{l} ≥ 2 for l > 0) from the continued fractions:

\frac{p_{i}}{q_{i}} = u_{i}^{0} − \frac{1}{u_{i}^{1} − \frac{1}{... − \frac{1}{u_{i}^{t_{i}}}}} ; \frac{q_{i}}{p_{i}} = v_{i}^{0} − \frac{1}{v_{i}^{1} − \frac{1}{... − \frac{1}{v_{i}^{r_{i}}}}} .

Then Γ^{min}(ℂ^{2}, f) is given by

[figure: …; …; …; ⋮; ⋮; ⋮; −u_{1}^{1}; −u_{1}^{t_{1}}; −u_{2}^{0} − 1; −u_{2}^{1}; −u_{s−1}^{t_{s−1}}; −u_{s}^{0} − 1; −u_{s}^{1}; −u_{s}^{t_{s}}; −1; −v_{1}^{r_{1}}; −v_{1}^{2}; −v_{1}^{1}; −v_{s−1}^{r_{s−1}}; −v_{s−1}^{2}; −v_{s−1}^{1}; −v_{s}^{r_{s}}; −v_{s}^{2}; −v_{s}^{1}]

This has the following schematic form:

[figure: v̄_{0}; v_{1}; v_{2}; v_{s−1}; v_{s}; v̄_{1}; v̄_{2}; v̄_{s−1}; v̄_{s}; ⋯]

Here we emphasized only those vertices {v̄_{k}}_{k=0}^{s} and {v_{k}}_{k=1}^{s} which have degree δ ≠ 2. We denote the set of these vertices by W^{∗}. The dash-line between two such vertices replaces a string [figure: ⋯] . In our discussions below, the corresponding self-intersections will be less important, but the multiplicities of the vertices v ∈ W^{∗} will be crucial. They can be easily described in terms of the integers {a_{k}}_{k=1}^{s}:

a_{1} = q_{1} and a_{k+1} = q_{k+1} + p_{k+1}p_{k}a_{k} if s − 1 ≥ k ≥ 1.  (1)

Then again, (p_{k}, a_{k}) = 1 for any k. Clearly, the two sets of pairs {(p_{k}, q_{k})}_{k=1}^{s} and {(p_{k}, a_{k})}_{k=1}^{s} determine each other completely. In fact, the set of pairs {(p_{k}, a_{k})}_{k=1}^{s} constitutes the set of decoration of the so called splice, or Eisenbud-Neumann diagram of f, cf. [8], page 51. Then by [8], section 10, one has:

\begin{matrix}m_{v_{k}} = a_{k}p_{k}p_{k+1} ⋯ p_{s} & for 1 ≤ k ≤ s; \\ mv̄_{0} = p_{1}p_{2} ⋯ p_{s}; \\ mv̄_{k} = a_{k}p_{k+1} ⋯ p_{s} & for 1 ≤ k ≤ s − 1; \\ mv̄_{s} = a_{s}. \end{matrix}  (2)

2.3. Cyclic coverings. For any isolated plane curve singularity f we write X_{f,n} := {z^{n} = f(x, y)} ⊂ (ℂ^{3}, 0). Then (x, y, z) ↦ (x, y) induces a ℤ_{n}−Galois covering of (X_{f,n} , 0) over (ℂ^{2}, 0), and the z-projection induces a germ (X_{f,n} , 0) → (ℂ, 0), still denoted by z.

From any fixed (good) embedded resolution graph Γ(ℂ^{2}, f) of f and the integer n, one can construct a possible dual resolution graph Γ(X_{f,n} , z), resp. Γ(X_{f,n}). A precise algorithm is given in [19] (or [21]; for a more general situation, see [20]).

Assume that in this algorithm, we start with the minimal (good) embedded resolution graph Γ^{min}(ℂ^{2}, f) of f. Then the output graph provided by the algorithm (without any modification by any blow up or down) will be called the canonical embedded resolution graph of (X_{f,n} , z), and will be denoted by Γ^{can}(X_{f,n} , z). (The name is motivated by [12], where Laufer proved that the above algorithm for a plane curve singularity f provides exactly the canonical resolution of X_{f,n} in the sense of Zariski, provided that n = 2.)

3. The embedded resolution graph Γ^{can}(X_{f,n} , z).

In this section we make the above mentioned algorithm for Γ^{can}(X_{f,n} , z) explicit. We assume that n ≥ 2 and f is irreducible.

First, we fix some notations. Recall that {(p_{k}, q_{k})}_{k=1}^{s} denotes the set of Newton pairs of f, and the integers {a_{k}}_{k=1}^{s} are defined in 2.2(1). Additionally, we define

• d_{k} := (n, p_{k+1}p_{k+2} ⋯ p_{s}) for 0 ≤ k ≤ s − 1, and d_{s} := 1;

• h_{k} := d_{k−1}/d_{k} = (p_{k}, n/d_{k}) and p′_{k} := p_{k}/h_{k} for 1 ≤ k ≤ s;

• h̃_{k} := (a_{k}, n/d_{k}) and a′_{k} := a_{k}/h̃_{k} for 1 ≤ k ≤ s.

In order to run the algorithm, besides 2.2(2), one also needs some additional data about Γ(ℂ^{2}, f). For the convenience of the reader we collect them in the next lemma. For the proof, see e.g. [27] or the proof of (3.2) in [16]. Set M_{w} := gcd{m_{v} | v = w or v is adjacent to w}.

3.1. Lemma. In Γ^{min}(ℂ^{2}, f) one has: Mv̄_{k} = mv̄_{k} (0 ≤ k ≤ s); M_{v_{k}} = p_{k+1} ⋯ p_{s} (1 ≤ k ≤ s − 1); and M_{v_{s}} = 1.

Moreover, for any 1 ≤ k ≤ s, fix two integers i_{k} and j_{k} with a_{k}i_{k} + p_{k}j_{k} = 1. Then the multiplicities of the three vertices adjacent to v_{k}, modulo m_{v_{k}} = a_{k}p_{k}p_{k+1} ⋯ p_{s}, are: (−i_{k}a_{k}p_{k+1} ⋯ p_{s}; −j_{k}p_{k}p_{k+1} ⋯ p_{s}; p_{k+1} ⋯ p_{s}) if k ≤ s − 1, and (−i_{s}a_{s}; −j_{s}p_{s}; 1) if k = s.

Now we start to discuss Γ^{can}(X_{f,n} , z). By the algorithm, this graph can be considered as a “covering” q : Γ^{can}(X_{f,n} , z) → Γ^{min}(X, f). Above w ∈ W(Γ^{min}(ℂ^{2}, f)) there are (M_{w}, n) vertices v′ ∈ q^{−1}(w) of Γ^{can}(X_{f,n} , z), each with some multiplicity m_{v′} and genus g_{v′}. For a fixed w ∈ W^{∗}(Γ^{min}(ℂ^{2}, f)), the Galois action guarantees that the integers m_{v′} and g_{v′} do not depend on the choice of v′ ∈ q^{−1}(w), but only on w. Sometimes, we denote them by m_{w} and g_{w}. The reader can easily verify that 2.2(2), 3.1 and the algorithm gives:

3.2. Corollary. (a) For any irreducible f and n, Γ^{can}(X_{f,n} , z) is a tree with

#q^{−1}(v_{s}) = 1  #q^{−1}(v_{k}) = h_{k+1} ⋯ h_{s}  (1 ≤ k ≤ s − 1)
#q^{−1}(v̄_{s}) = h̃_{s}  #q^{−1}(v̄_{k}) = h̃_{k}h_{k+1} ⋯ h_{s} (1 ≤ k ≤ s − 1)

#q^{−1}(v̄_{0}) = h_{1} ⋯ h_{s}.

(b) mv̄_{0} = p′_{1}p′_{2} ⋯ p′_{s}

mv̄_{k} = a′_{k}p′_{k+1} ⋯ p′_{s} (1 ≤ k ≤ s − 1)
mv̄_{s} = a′_{s}
m_{v_{k}} = a′_{k}p′_{k}p′_{k+1} ⋯ p′_{s} (1 ≤ k ≤ s).

(c)

\begin{matrix}gv̄_{k} = 0 & (0 ≤ k ≤ s) \\ g_{v_{k}} = (h_{k} − 1)(h̃_{k} − 1)/2 & (1 ≤ k ≤ s). \end{matrix}

In particular, the link of X_{f,n} is a rational homology sphere if and only if (h_{k}−1)(h̃_{k}−1) = 0 for any 1 ≤ k ≤ s.

The graph Γ^{can}(X_{f,n} , z) has the following schematic form (where the dash-lines replace strings as above, and we omit the genera and the self-intersections):

[figure: {z = 0}; ⋮; h_{1}; ⋮; h_{1}; ⋮; h_{1}; ⋯; h̃_{1}; ⋯; h̃_{1}; ⋯; h̃_{1}; ⋮; ⋮; h_{s−1}; ⋮; ⋮; h_{s}; ⋯; ⋯; h̃_{s−1}; ⋯; h̃_{s}; ⋯; ⋯; ⋯; ⋯]

3.3. Example. Assume that s = 1 and write p = p_{1} and a = a_{1}. Take n such that h = (p, n) and h̃ = (a, n) = 1. Then X_{f,n} can be identified with the Brieskorn hypersurface singularity {(x, y, z) ∈ ℂ^{3} : x^{a} + y^{p} + z^{n} = 0}. Then the link is a Seifert 3-manifold with Seifert invariants: a, a, ⋯, a, p/h, n/h (a appearing h times, hence all together there are h + 2 special fibers corresponding to the h + 2 arms, cf. the above graph-diagram). These numbers also give (up to a sign) the determinants of the corresponding arms of the graph Γ(X_{f,n}). For more details about Seifert manifolds and their plumbings, see e.g. [10, 31].

3.4. The maximal strings of Γ^{can}(X_{f,n} , z). The next goal is to compute the determinants of the maximal strings of Γ^{can}(X_{f,n} , z). For this, fix a vertex w ∈ W^{∗}(Γ^{min}(ℂ^{2}, f)) and v′ ∈ q^{−1}(w). Consider the shortest path in Γ^{can}(X_{f,n} , z) which connects v′ and the arrowhead.

If w ≠ v_{s}, then on this path there is at least one rupture vertex of Γ^{can}(X_{f,n} , z). Let v′′ (v′′ ≠ v′) be the closest one to v′. If w = v_{k} (1 ≤ k ≤ s − 1), then let Γ(v′) be the string which contains all the vertices between v′ and v′′ (excluding v′ and v′′), and all the edges connecting them. If w = v̄_{k} (1 ≤ k ≤ s), then Γ(v′) is the string constructed similarly, but at this time we include v′ and its connecting edge as well. If w = v_{s}, then the above path is already a string. Let Γ(v′) be the string which contains all the vertices between v′ and the arrowhead (excluding v′), and all the edges connecting them.

In this way we have a codification of all the maximal strings of Γ^{can}(X_{f,n} , z). Notice also that the isomorphism type of the string Γ(v′) does not depend on the choice of v′ ∈ q^{−1}(w), but only on w. Therefore, sometimes it is preferable to denote this type by Γ(w). Denote by D(v′) (or by D(w)) the determinant det(Γ(v′)). If Γ(w) = ∅, then by definition D(w) = 1.

3.5. Proposition. The possible values of D(w) are the following:

D(v̄_{0}) = a′_{1}
D(v̄_{k}) = p′_{k}  (1 ≤ k ≤ s)
D(v_{s}) = n/(h_{s}h̃_{s})
D(v_{k}) = n · q_{k+1}/(d_{k−1}h̃_{k}h̃_{k+1}) (1 ≤ k ≤ s − 1)

Proof. We start with the (“difficult”) case D(v_{k}) (1 ≤ k ≤ s − 1). Using the notations of 2.2, the maximal string in Γ^{min}(ℂ^{2}, f) between v_{k} and v_{k+1} has the following form:

[figure: (m_{v_{k}}); (m_{v_{k+1}}); −u_{k+1}^{1}; −u_{k+1}^{2}; −u_{k+1}^{t_{k+1}}; ⋯]

where m_{v_{k}} = a_{k}p_{k} ⋯ p_{s} and m_{v_{k+1}} = a_{k+1}p_{k+1} ⋯ p_{s}, cf. 2.2(2). Moreover, p_{k+1}/q_{k+1} = u_{k+1}^{0} − λ/q_{k+1}, and the quotient q_{k+1}/λ gives the continued fraction [u_{k+1}^{1}, ⋯, u_{k+1}^{t_{k+1}}]. This graph can be identified with the embedded resolution graph of a germ defined on a Hirzebruch-Jung singularity. Indeed, using e.g. [20], page 102, one can verify that one can take the germ (z^{p_{k}a_{k}}y)^{p_{k+1}···p_{s}} defined on the normalization of {z^{q_{k+1}} = xy^{p_{k+1}}}. In particular, the collection of graphs {Γ(v′)}_{v′∈q^{−1}(v_{k})} is exactly the (non-connected) graph of the normalization of

X = {(x, y, z, w) : z^{q_{k+1}} = xy^{p_{k+1}} ; w^{n} = (z^{p_{k}a_{k}}y)^{p_{k+1}···p_{s}}}.

X has d_{k} = (n, p_{k+1} ⋯ p_{s}) (isomorphic) irreducible components, number which agrees exactly with #q^{−1}(v_{k}). Hence, D(v′) is the graph of the normalization of

X_{1} = {z^{q_{k+1}} = xy^{p_{k+1}} ; w^{n/d_{k}} = (z^{p_{k}a_{k}}y)^{(p_{k+1}···p_{s})/d_{k}}}.

Let us write q = q_{k+1}, p = p_{k+1}, N = n/d_{k}, r = p_{k}a_{k}, P = p_{k+1} ⋯ p_{s}/d_{k} and a = a_{k+1}. Then (q, p) = 1, (N, P) = 1, a = rp + q, and X_{1} = {z^{q} = xy^{p}; w^{N} = (z^{r}y)^{P} } in (ℂ^{4}, 0). We will verify that the determinant of the graph of X_{1} is n = \frac{Nq}{(N,r)(N,a)}.

In order to prove this last statement, first we show that we can assume P = 1. Indeed, consider the space Y_{1} := {z^{q} = xy^{p}, w = t^{P} , t^{N} = z^{r}y}. Then X_{1} and Y_{1} are birationally equivalent (eliminate t from the equation of Y_{1}), hence their normalizations are the same. On the other hand Y_{1} is isomorphic with X′_{1} := {z^{q} = xy^{p}, t^{N} = z^{r}y} (eliminate w).

Then notice that X_{1} is irreducible and its normalization is a Hirzebruch-Jung singularity. Indeed, the discriminant of the finite map ρ : (X_{1}, 0) → (ℂ^{2}, 0) induced by the (x, y)- projection is the union of the coordinate lines. Let π_{1} = ℤ^{2} be the fundamental group of {xy ≠ 0} generated by e_{1} and e_{2} representing two elementary loops around the axes x and y. Let ρ_{∗} : π_{1} → G be the monodromy representation of the restriction of ρ above {xy ≠ 0}, and ρ_{∗}|ℤ(e_{i}) be its restriction to ℤ(e_{i}), the subgroup generated by e_{i} (i = 1, 2). Then one can verify (using e.g. [4], III.5) that ℤ_{n} ≈ ker(ρ_{∗})/ ker(ρ_{∗}|ℤ(e_{1})) × ker(ρ_{∗}|ℤ(e_{2})).

In our case the Galois group G of the induced regular covering can be identified with G = {(ξ, η) ∈ ℂ^{∗} × ℂ^{∗} : ξ^{q} = 1, η^{N} = ξ^{r}}. Since ρ_{∗} is onto, ker(ρ_{∗}) has index Nq in ℤ^{2}. Moreover, ρ(e_{1}) = (exp(2πi/q), exp(2πir/(Nq))), hence ker(ρ_{∗}|ℤ(e_{1})) = k_{1}ℤ for k_{1} = Nq/(N, r). Similarly, ρ(e_{2}) = (exp(2πip/q), exp(2πia/(Nq))), hence ker(ρ_{∗}|ℤ(e_{1})) = k_{2}ℤ for k_{2} = Nq/(N, a). Hence the claim follows.

The other identities can be computed by a similar argument. But also notice that in all other cases the corresponding maximal string contains a leaf vertex of Γ^{can}(X_{f,n}). Therefore, D(v′) can be identified with the corresponding Seifert invariant, similarly as in 3.3. Hence these identities also follow from 3.3. □

3.6. Remark. In Γ^{can}(X_{f,n} , z) the following hold:

(1) If w = v̄_{k} (0 ≤ k ≤ s) then Γ(w) ≠ ∅. Indeed, Γ(w) contains at least as many vertices as the corresponding arm in Γ^{min}(ℂ^{2}, f), which is clearly not empty.

(2) The same argument is valid for any Γ(v_{k}) (1 ≤ k ≤ s − 1) provided that q_{k+1} > 1. In fact, for such w = v_{k}, Γ(v_{k}) = ∅ if and only if q_{k+1} = 1 and n = d_{k−1}h̃_{k}h̃_{k+1}.

(3) Γ(v_{s}) = ∅ if and only if n = h_{s}h̃_{s}.

Here appears a natural question: is it possible to distinguish the arms Γ(v′) (v′ ∈ q^{−1}(v̄_{0})) from the arms of type Γ(v′) (v′ ∈ q^{−1}(v̄_{1})) ? The next corollary says that if g_{v_{1}} = 0 then already their determinants are different:

3.7. Corollary. (a) If D(v̄_{0}) = D(v̄_{1}) then D(v̄_{0}) = D(v̄_{1}) = 1 and g_{v_{1}} ≠ 0.

(b) If D(v̄_{s}) = D(v_{s}) then D(v̄_{s}) = D(v_{s}) = 1.

Proof. (a) If D(v̄_{0}) = D(v̄_{1}), then a_{1}/h̃_{1} = p_{1}/h_{1} by 3.5. Since (a_{1}, p_{1}) = 1, one gets a_{1}/h̃_{1} = p_{1}/h_{1} = 1. But then h_{1} ≥ 2 and h̃_{1} ≥ 2 since a_{1} = q_{1} > p_{1} ≥ 2. (b) Similarly by 3.5 one has p_{s}/h_{s} = n/h_{s}h̃_{s}. But this two numbers are also relatively prime. □

3.8. The subgraphs Γ_{±}(v_{k}). Above we discussed the case of maximal strings of Γ^{can}(X_{f,n} , z). Obviously, one can consider the determinants of much bigger subgraphs delimited by different rupture vertices. In this way one obtains a large number of rather subtle invariants of this graph. Nevertheless, in order to recover the Newton pairs of f and the integer n from this graph, it is enough to consider only a restrictive sub-family of them.

Let us fix an integer k (1 ≤ k ≤ s). Consider the maximal subgraph of Γ^{can}(X_{f,n} , z) which does not contain any vertex from the set q^{−1}(v_{k}). It has many connected components.

The component which supports the arrowhead of Γ^{can}(X_{f,n} , z) is denoted by Γ_{+}(v_{k}). There are h̃_{k}h_{k+1} ⋯ h_{s} more components (isomorphic to each other), which contain vertices above v̄_{k}. They are strings of type Γ(v̄_{k}) (cf. 3.2(a) and 3.4). Finally, there are h_{k} ⋯ h_{s} isomorphic components containing vertices above v̄_{0}. We denote such a component by Γ_{−}(v_{k}). D_{±}(v_{k}) denotes det(Γ_{±}(v_{k})).

Obviously, Γ_{−}(v_{1}) = Γ(v̄_{0}), and Γ_{+}(v_{s}) = Γ(v_{s}) whose determinants are computed in 3.5.

3.9. Proposition. Assume that s ≥ 2.

(a) D_{−}(v_{2}) = (a′_{1})^{h_{1}−1} · (p′_{1})^{h̃_{1}−1} · a′_{2}.

(b) D_{+}(v_{s−1}) = n · D(v_{s−1})^{h_{s}−1} · D(v̄_{s})^{h̃_{s}−1}/(h_{s}h_{s−1}h̃_{s−1}).

Proof. (a) Fix a vertex v′ ∈ q^{−1}(v_{2}) and one of the graphs Γ_{−}(v′). Let w_{1} be its unique rupture point, and w_{2} denotes that vertex which was connected by an edge with v′ in Γ^{can}(X_{f,n} , z). (If Γ(v_{1}) = ∅ then w_{1} = w_{2}, but the proof is valid in this case as well.) We put back on the vertices of Γ_{−}(v′) the multiplicities of Γ^{can}(X_{f,n} , z). They will form a compatible set (i.e. will satisfy 2.1(1)) provided that we put on w_{2} an arrow with multiplicity m_{v′} = m_{v_{2}}. This graph with arrowhead and multiplicities has the following schematic form:

[figure: (m_{v_{2}}); (m_{v_{1}}); ⋮; h_{1}; ⋯; h̃_{1}]

Notice that Γ_{−}(v′) \ {w_{1}} has h_{1} + h̃_{1} + 1 connected components, h_{1} of type Γ(v̄_{0}), h̃_{1} of type Γ(v̄_{1}), and one of type Γ(v_{1}). Therefore, by 2.1 one gets:

\frac{m_{v_{1}}}{m_{v_{2}}} = −I_{w_{1}w_{2}}^{−1} = \frac{D(v̄_{0})^{h_{1}} · D(v̄_{1})h̃_{1}}{D_{−}(v_{2})} .

Now, use 3.2(b) and 3.5. For part (b) we proceed similarly, but now with the graph Γ_{+}(v_{s−1}). Its schematic form, together with the multiplicities of Γ^{can}(X_{f,n} , z), is

[figure: ⋮; h_{s}; (m_{v_{s−1}}); (m_{v_{s−1}}); (1); (m_{v_{s}}); ⋯; h̃_{s}]

If from this graph we delete its rupture point (and the arrows and multiplicities) then we get the following connected components: h_{s} of type Γ(v_{s−1}), one of type Γ(v_{s}), and h̃_{s} of type Γ(v̄_{s}). Therefore, from 2.1, similarly as above, one gets:

m_{v_{s}} = \frac{D(v_{s−1})^{h_{s}} · D(v̄_{s})h̃_{s}}{D_{+}(v_{s−1})} + h_{s} · \frac{D(v_{s−1})^{h_{s}−1} · D(v̄_{s})h̃_{s} · D(v_{s})}{D_{+}(v_{s−1})} · m_{v_{s−1}}.

Then use again 3.2(b) and 3.5 (and a_{s} = q_{s} + p_{s}p_{s−1}a_{s−1}). □

3.10. Remark. By a similar argument one can prove the next identity for k ≥ 2 (which is not needed later):

\frac{D_{−}(v_{k})}{a′_{k}} = [\frac{D_{−}(v_{k−1})}{a′_{k−1}}]^{h_{k−1}} · (a′_{k−1})^{h_{k−1}−1} · (p′_{k−1})^{h̃_{k−1}−1}.

4. From Γ^{can}(X_{f,n} , z) to Γ^{min}(X_{f,n})

Let Γ^{min}(X_{f,n} , z) be the minimal embedded resolution graph of (X_{f,n} , z). This can be obtained from Γ^{can}(X_{f,n} , z) by a sequence of blow downs (and without any blow up).

4.1. Proposition. All the rupture vertices of Γ^{can}(X_{f,n} , z) survive in Γ^{min}(X_{f,n} , z) as rupture vertices (i.e. they are not blown down in the minimalization procedure, and in Γ^{min}(X_{f,n} , z) they still live as rupture vertices).

Proof. From the theory of Hirzebruch-Jung singularities follows that a string of type Γ(v̄_{k}) (0 ≤ k ≤ s) is completely collapsed in the minimalization procedure if and only if its determinant D(v̄_{k}) equals 1. First we verify that all the rupture vertices above v_{1} will survive (as rupture vertices). Let v′ be one of them considered in Γ^{can}. It supports h_{1} strings of type Γ(v̄_{0}), h̃_{1} strings of type Γ(v̄_{1}) and another edge, denoted by e. Recall that D(v̄_{0}) = a′_{1} and D(v̄_{k}) = p′_{k}, cf. 3.5. By 3.7, if both D(v̄_{0}) and D(v̄_{1}) equal one, then g_{v′} ≠ 0. Hence v′ will be a rupture vertex in Γ^{min}(X_{f,n} , z).

If D(v̄_{0}) ≠ 1 but D(v̄_{1}) = p_{1}/h_{1} = 1 then the strings of type Γ(v̄_{0}) will survive. Their number is h_{1} = p_{1} ≥ 2. Symmetrically, if D(v̄_{1}) ≠ 1 but D(v̄_{0}) = a_{1}/h̃_{1} = 1 then h̃_{1} = a_{1} ≥ 2 strings of type Γ(v̄_{1}) will survive. If both determinants are greater than one, then all the strings will survive with total number h_{1}+h̃_{1} ≥ 2. Since the arrowhead survives, and Γ^{min}(X_{f,n} , z) is connected, the edge e will survive as well. Hence v′ has degree at least three in Γ^{min}(X_{f,n} , z).

By induction, we assume that for a fixed k, all the rupture vertices above any v_{i} ∈ W^{∗} survive for any i ≤ k − 1. We show that this is the case for v_{k} as well. For this, fix an arbitrary v′ ∈ q^{−1}(v_{k}). First notice that by the inductive step, the h_{k} subgraphs Γ_{−}(v′) will survive (they cannot be completely contracted since they contain rupture points that survive). Similarly as above, since the arrowhead survives, the edge connecting v′ with Γ_{+}(v_{k}) will also survive. If D(v̄_{k}) = 1, then h_{k} = p_{k} ≥ 2. If D(v̄_{k}) ≠ 1, then all the graphs Γ(v̄_{k}) will survive. Hence, in any case δ_{v′} ≥ 3 in Γ^{min}(X_{f,n} , z). □

Now, recall that Γ^{min}(X_{f,n}) denotes the minimal (good) resolution graph of (X_{f,n} , 0). It can be obtained from Γ^{min}(X_{f,n} , z) by deleting its arrowhead (and all the multiplicities) and blowing down successively all the (−1)-curves with genus zero and new degree ≤ 2. In fact, there is exactly one case when after deleting the arrowhead of Γ^{min}(X_{f,n} , z) we do not obtain a minimal graph, and this is described completely in the next proposition. In the sequel we refer to this “pathological” situation as the “P-case”.

4.2. Proposition. Assume that by deleting the arrowhead of Γ^{min}(X_{f,n} , z) we obtain a non-minimal graph. Then Γ^{min}(X_{f,n} , z) has the following schematic form with the two left branches isomorphic and with e ≤ −3 (we omit the multiplicities). The rational (−1)-curve is the unique vertex v′ = q^{−1}(v_{s}) (which survives in Γ^{min}(X_{f,n} , z), cf. 4.1).

[figure: −1; v′; e; e]

This situation can happen if and only if n = p_{s} = 2.

In this case, Γ^{min}(X_{f,n}) is obtained from Γ^{min}(X_{f,n} , z) by deleting its arrowhead and blowing down v′. No other blow downs are necessary.

Moreover, in this case, all the vertices of Γ^{min}(X_{f,n} , z) have genus zero.

Proof. If the graph obtained from Γ^{min}(X_{f,n} , z) by deleting its arrowhead is not minimal, then the vertex v′ in Γ^{min}(X_{f,n} , z) which supports the arrowhead should be a (−1) rational curve of degree 3 in Γ^{min}(X_{f,n} , z). This can happen only if this vertex v′ is exactly the unique vertex q^{−1}(v_{s}) (and g_{v_{s}} = 0). This also shows that Γ(v_{s}) was collapsed in Γ^{min}(X_{f,n} , z), hence D(v_{s}) = n/h_{s}h̃_{s} = 1. Hence:

h_{s}h̃_{s} = n ≥ 2 and (h_{s} − 1)(h̃_{s} − 1) = g_{v_{s}} = 0.

Assume first that h_{s} = 1 and h̃_{s} > 1. Since D(v̄_{s}) = p_{s} > 1, the h̃_{s} strings Γ(v̄_{s}) are present in Γ^{min}(X_{f,n} , z). This can happen if and only if h̃_{s} = 2 and Γ_{−}(v_{s}) is collapsed completely in Γ^{min}(X_{f,n} , z). Since for s ≥ 2 the rupture points q^{−1}(v_{1}) survive in Γ^{min}(X_{f,n} , z), this can happen only if s = 1 and D(v̄_{0}) = a_{1}/h̃_{1} = 1. This shows that a_{1} = h̃_{1} = 2, which contradicts the inequality a_{1} = q_{1} > p_{1} ≥ 2.

Therefore h̃_{s} = 1 and h_{s} > 1. Since the degree of v′ (in Γ^{min}(X_{f,n} , z)) is at least 1 + h_{s} (hence 1 + h_{s} ≤ 3), one gets h_{s} = 2, and also the fact that the graphs of type Γ(v̄_{s}) are collapsed in Γ^{min}(X_{f,n} , z), hence p_{s}/h_{s} = 1. Therefore, n = h_{s} = p_{s} = 2.

Then h̃_{k} = 1, hence g_{v_{k}} = 0 for any k.

Finally, notice that e ≤ −3 (cf. the diagram) since after we blow down v′ we get a subgraph of type [figure: e + 1; e + 1] which must be negative definite.

On the other hand, one can verify easily that if n = p_{s} = 2 then the above situation always occurs. □

4.3. Remark. Assume that g_{v_{k}} = 0. Then if a family of strings supported by any fixed v′ ∈ q^{−1}(v_{k}) is collapsed completely during the minimalization procedure, then the cardinality of this family (in spite of the fact that it is missing in Γ^{min}(X_{f,n})) can be determined, and it is one. More precisely, if for k ≥ 1, the h̃_{k} graphs of type Γ(v̄_{k}) are completely collapsed, then h_{k} = p_{k} ≥ 2. Then by the genus-formula 3.2(c) one gets h̃_{k} = 1. Similarly, if k = 1 and the h_{1} graphs of type Γ(v̄_{0}) are collapsed, then h_{1} = 1.

In order to recover the Newton pairs of f and the integer n from the graph Γ^{min}(X_{f,n}), we need some information about some subgraphs G of Γ^{min}(X_{f,n}) of the following type. Each G is a connected component of Γ^{min}(X_{f,n}) \ {v} for some rupture vertex v of Γ^{min}(X_{f,n}), and it contains exactly one rupture vertex of Γ^{min}(X_{f,n}). In a general setting their precise definition is the following.

4.4. Definitions. Let Γ be a decorated tree (with self-intersections and genera {g_{v}}_{v}, without arrowheads and multiplicities). Assume that it has at least two rupture vertices.

(1) Let Γ(R) be the minimal connected subgraph of Γ which contains all the rupture vertices R of Γ. Let L(Γ(R)) be the set of leaf vertices of Γ(R). For any v ∈ L(Γ(R)) let G(v) be the maximal connected subgraph of Γ which contains v but contains no other rupture vertex of Γ. The determinant det(G(v)) is denoted by D(v).

(2) For any v ∈ L(Γ(R)), let v_{root} be the unique rupture vertex of Γ with the property that on the shortest path in Γ connecting v and v_{root} there are no other rupture vertices of Γ. Then clearly v_{root} is adjacent with a certain vertex of G(v) (in fact G(v) is one of the connected components of Γ \ {v_{root}}).

(3) For each rupture vertex v ∈ R, denote by St(v) the set of maximal strings of Γ which are supported by v (on one end) and contain a leaf vertex of Γ (on the other end). More precisely, these strings are those connected components of Γ \ Γ(R) which have an adjacent vertex with v (in Γ). We write St(v) as a disjoint union of its subsets {St_{i}(v)}_{i∈I(v)} which are the level sets of det : St(v) → ℤ. We set D_{i} := det(St) for St ∈ St_{i}(v) and #_{i} := #St_{i}(v). Then we define

D_{St}(v) := \{ \begin{matrix}∏_{i∈I(v)} D_{i}^{#_{i}} & if St(v) ≠ ∅ \\ 1 & if St(v) = ∅, \end{matrix} D_{St}^{red}(v) := \{ \begin{matrix}∏_{i∈I(v)} D_{i}^{#_{i}−1} & if St(v) ≠ ∅ \\ 1 & if St(v) = ∅, \end{matrix}

and α(v) ∈ ℚ ∪ {∞} by

α(v) = \{ \begin{matrix}∏_{i∈I(v)} #_{i} & if St(v) ≠ ∅ \\ 1 & if St(v) = ∅ and g_{v} = 0 \\ \frac{2g_{v}}{δ_{v}−2} + 1 & if St(v) = ∅ and g_{v} ≠ 0. \end{matrix}

[α(v) = ∞ if and only if the degree δ_{v} of v in Γ is 2, St(v) = ∅ and g_{v} ≠ 0.] (4) For each v ∈ L(Γ(R)) we define the β-invariant by

β(v) := \frac{D(v)}{D_{St}(v)} · \frac{α(v_{root})}{α(v)}.

4.5. In the next paragraphs we apply these definitions for Γ = Γ^{min}(X_{f,n}). Here, we prefer to regard Γ^{min}(X_{f,n}) together with Γ^{can}(X_{f,n} , z), as a minimalization of Γ^{can}(X_{f,n} , z). In particular, we will define subsets, subgraphs, etc. in Γ^{min}(X_{f,n}) as the images of welldefined subsets, subgraphs, etc. of Γ^{can}(X_{f,n} , z) by the minimalization procedure. (Of course, in the next section will be a crucial task to recover some of these sets only from the abstract graph Γ^{min}(X_{f,n}). The key result for this is the next 4.7.)

In order to avoid any confusion, for any subset of vertices of Γ^{can}(X_{f,n} , z), we will denote by π(A) the image of A by the minimalization procedure. Hence, π(A) denotes those vertices of Γ^{min}(X_{f,n}) which have ancestors in A, and survive in Γ^{min}(X_{f,n}); in some cases this set can be empty.

The following facts follow easily from the structure-results proved in section 4 and the above propositions 4.1 and 4.2.

4.6. Facts. Assume that Γ = Γ^{min}(X_{f,n}) with s ≥ 2. Then the following hold:

(a) The set L(Γ(R)) is the disjoint union of two sets R_{1} and LR_{s}, where (i) R_{1} := π(q^{−1}(v_{1})); (ii) LR_{s} := ∅ if h_{s} > 1. Otherwise LR_{s} := π(q^{−1}(v_{s})), the image by the minimalization

procedure of the unique rupture vertex of Γ^{can}(X_{f,n} , z) sitting above v_{s}. (In both cases, by 4.1 and 4.2, these sets are subsets of the rupture vertices of Γ^{min}(X_{f,n}).)

(b) The subgraphs G(v) for v ∈ L(Γ(R)) = R_{1} ∪ LR_{s} (cf. part (a)) can be identified as follows:

(i) Assume that we are not in the “P-case” with s = 2. For each v ∈ q^{−1}(v_{1}) consider the unique subgraph of type Γ_{−}(v_{root}) in Γ^{can}(X_{f,n} , z), for some v_{root} ∈ q^{−1}(v_{2}), which contains v. Then its image in Γ^{min}(X_{f,n}) by the minimalization procedure is G(v).

(ii) Assume that h_{s} = 1. For v = q^{−1}(v_{s}) consider Γ_{+}(v_{root}) in Γ^{can}(X_{f,n} , z) with v_{root} := q^{−1}(v_{s−1}). Then its image in Γ^{min}(X_{f,n}) by the minimalization procedure is G(v).

(c) α and β are constant on R_{1}.

(The motivation for the notation LR_{s} is the following: later we will use the symbol R_{s} for π(q^{−1}(v_{s})); hence LR_{s} = R_{s} if π(q^{−1}(v_{s})) is a “leaf rupture vertex”, otherwise it is empty.)

The main point is that in (a), the cases (i) and (ii) can be distinguished by the genus and β-invariant.

4.7. Proposition. Assume that Γ = Γ^{min}(X_{f,n}) with s ≥ 2.

(a) If there exists at least one v ∈ L(Γ(R)) with g_{v} ≠ 0, then R_{1} = {v ∈ L(Γ(R)) : g_{v} ≠ 0} and LR_{s} = {v ∈ L(Γ(R)) : g_{v} = 0} (LR_{s} can be empty).

(b) If g_{v} = 0 for any v ∈ L(Γ(R)) and LR_{s} ≠ ∅, then β(v) ∈ (0, ∞) and

\begin{matrix}β(v) > 2 & if v ∈ R_{1}, \\ β(v) ≤ 1/2 & if v ∈ LR_{s}. \end{matrix}

Proof. (a) g_{v} is constant on R_{1} and g_{v} = 0 for (the unique) v ∈ LR_{s} provided that LR_{s} ≠ ∅, since in this case h_{s} = 1 (cf. 3.2).

Now we prove (b). Since h_{s} = 1 we can exclude the “P-case”.

First assume that s ≥ 3.

If v ∈ R_{1} then by 3.7 and 4.3 one has α(v) = h_{1}h̃_{1}. For v_{root}, analyzing the three different cases from the definition of α(v_{root}), and using 4.3 and the genus formula, we get α(v_{root}) = h̃_{2}. On the other hand, D(v) = (a′_{1})^{h_{1}−1}(p′_{1})^{h̃_{1}−1}a′_{2} (cf. 3.9(a)) and D_{St}(v) = (a′_{1})^{h_{1}}(p′_{1})h̃_{1} (use 3.5 and notice that if a string is collapsed completely then its determinant is one). Therefore, β(v) = a_{2}/(a_{1}p_{1}) > p_{2} ≥ 2, cf. 2.2(1).

If v = π(q^{−1}(v_{s})) then α(v) = h̃_{s} (use g_{s} = 0, 4.3 and 3.7). By similar argument as above, α(v_{root}) = h̃_{s−1}. By 3.9(b) and h_{s} = 1 one has D(v) = D_{+}(v_{s−1}) = n(p′_{s})^{h̃_{s}−1}/(h_{s−1}h̃_{s−1}). By 3.5, D_{St}(v) = (p′_{s})h̃_{s} · n/(h_{s}h̃_{s}). Therefore, using again h_{s} = 1 one gets β(v) = 1/(h_{s−1}p_{s}) ≤ 1/2.

Assume that s = 2 and let v′_{i} = q^{−1}(v_{i}) (i = 1, 2). Then α(v′_{i}) = h_{i}h̃_{i} (i = 1, 2). Hence the computation of β(v′_{1}) is the same as above, and it gives a_{2}/(a_{1}p_{1}) > 2. For v′_{2} we have an additional h_{1} and we get β(v′_{2}) = 1/p_{2} ≤ 1/2. □

5. From Γ^{min}(X_{f,n}) back to f and n

Our final goal is to recover the Newton pairs of f and the integer n from the graph Γ^{min}(X_{f,n}). In general, this is not possible. Nevertheless, by our main theorem, there are only two cases when such an ambiguity appears. They are presented in the next subsections.

5.1. Example. The S1-coincidence. Assume that (X, 0) = (x^{q_{1}} + y^{p_{1}} + z^{n} = 0, 0) is a Brieskorn singularity with (q_{1}, p_{1}) = 1. Let us first analyze how one can recover the set of integers {q_{1}, p_{1}, n} from the minimal resolution graph Γ of (X, 0). In this case, the computation of the graph Γ from the integers {q_{1}, p_{1}, n} is a classical, well-known fact (cf. also with our algorithm). The graph is either a string (with all genera zero) or a star-shaped graph (where only the central vertex might have a non-zero genus). If Γ is a string, then (X, 0) is a Hirzebruch-Jung hypersurface singularity. But there is only one family of such singularities, namely the A_{q_{1}−1}-singularities provided by the integers {q_{1}, 2, 2}. In this case, q_{1} is just the determinant of the string.

There is a rich literature of star-shaped graphs and Seifert 3-manifolds, and also of their subclass given by Brieskorn hypersurface singularities. The reader is invited to consult [32], section 3, case (I) (cf. also with [10, 31]).

If one wants to recover the integers {q_{1}, p_{1}, n}, then one considers the set of strings St(v) of the central vertex v. Recall 4.4(3) for the notations. Then #I(v) ≤ 3. If #I(v) = 3 then {q_{1}, p_{1}, n} = {D_{1}#_{2}#_{3}, D_{2}#_{1}#_{3}, D_{3}#_{1}#_{2}}. If one St_{i_{0}} is missing (empty) then D_{i_{0}} = 1 and #_{i_{0}} can determined from the genus of the central vertex (see e.g. our genus formula 3.2(c) or [32] (3.5)). Hence the previous procedure still works.

Similarly, in our special situation (q_{1}, p_{1}) = 1, one can show that if two subsets St_{i} are empty, then one can still recover {q_{1}, p_{1}, n} excepting only one case, namely when St(v) consists of only one string. In our terminology, this can happen only when the string which supports the arrowhead survives and all the others are contracted (i.e. p′_{1} = a′_{1} = 1). Similar ambiguity appears when St(v) = ∅.

But all these ambiguity cases can be classified very precisely. Consider an identity of type (h_{1} − 1)(h̃_{1} − 1) = 2g > 0 and an arbitrary positive integer l. Then the triplet {q_{1}, p_{1}, n} = {h̃_{1}, h_{1}, h̃_{1}h_{1}l} provides the following graph Γ (with l − 1 (−2)-vertices):

[figure: ⋯; [g]; −1; −2; −2; −2]

Now fix l > 0 and g > 0. Then, different triplets {h̃_{1}, h_{1}, h̃_{1}h_{1}l} with (h_{1} − 1)(h̃_{1} − 1) = 2g, h_{1} > 1 and h̃_{1} > 1 provide the same graph. [For example, (3, 7, 21) and (4, 5, 20) provide the same graph consisting of a vertex with g = 6 and self-intersection −1. ]

This is the only coincidence in the case of Brieskorn singularities with (q_{1}, p_{1}) = 1. Obviously, this cannot happen if g = 0.

Addendum. Relation with the Milnor number. Notice that in those cases when Γ fails to determine the integers {q_{1}, p_{1}, n}, Γ together with the Milnor number μ of the Brieskorn singularity do determine {q_{1}, p_{1}, n}. Indeed, in the “ambiguity cases” one has (q_{1}, p_{1}, n) = (h̃_{1}, h_{1}, h̃_{1}h_{1}l), where 2g = (h_{1} − 1)(h̃_{1} − 1) > 0 and l are readable from the graph. But μ = (h_{1} − 1)(h̃_{1} − 1)(h̃_{1}h_{1}l − 1) = 2g(h̃_{1}h_{1}l − 1). This determines h̃_{1}h_{1}, and finally h_{1} and h̃_{1} (using the genus formula).

5.2. Remark. The “z-axis ambiguity”. Recall that by our general aim, we have to recover the Newton pairs of f and the integer n. In the Brieskorn case, after we recover the set (q_{1}, p_{1}, n) we have to make a choice for the z-axis. Recall that (p_{1}, q_{1}) = 1. If k integers among of (p_{1}, q_{1}), (p_{1}, n) and (q_{1}, n) equal 1, then there are k possibilities for the choice of the z-axis.

5.3. Example. The S2-coincidence. The next coincidence appears when

s = 2, and a′_{1} = p′_{1} = h̃_{2} = 1 (or equivalently, s = 2, (n, a_{2}) = 1 and (n, p_{2})a_{1}p_{1}|n ). (∗)

In this case clearly q_{1} = h̃_{1} and p_{1} = h_{1}, but the h̃_{1} strings of type Γ(v̄_{0}) and the h_{1} strings of type Γ(v̄_{1}) are not visible on the minimal graph since their determinants are one, hence they are contracted. The graph Γ^{min}(X_{f,n}) has the following schematic form, where g_{v_{1}} > 0 and we omit the self-intersections:

[figure: [g_{v_{1}}]; [g_{v_{1}}]; ⋮; h_{2}]

The strings that appear on the right correspond to Γ(v_{2}) and Γ(v̄_{2}), but in general, we cannot decide which one is which. From the graph we can read h_{2} and the genus g_{v_{1}} = (h_{1} − 1)(h̃_{1} − 1)/2, and of course, a lot of determinants.

Using h_{2}, h̃_{2}, and D(v_{1}), D_{−}(v_{2}), D_{+}(v_{1}) and the set {D(v_{2}), D(v̄_{2})}, we can also recover a_{2}, q_{2}, n/(h_{1}h̃_{1}), h_{1}h̃_{1}p_{2} and the set {p_{2}, n}, where we cannot distinguish p_{2} from n.

Notice that once we know h_{1}h̃_{1}, then using the genus formula and h̃_{1} = q_{1} > p_{1} = h_{1}, we obtain h_{1} and h̃_{1} without any ambiguity, hence (by the above equations) all the data. But for the three “variables” h_{1}h̃_{1}, p_{2}, n we know only the values n/(h_{1}h̃_{1}), h_{1}h̃_{1}p_{2} and the set {p_{2}, n}. This, in general, has two possible solutions (which correspond by a permutation of p_{2} and n). If this is the case, then it might happen that there are two different realizations of the same graph for two different pairs (f, n). But for this, both solutions should provide positive integers as candidates for the Newton pairs and n. If this is not happening then the graph is uniquely realized (see Example 3 below).

The complete discussion of all the cases when the above equations which involve D(v_{1}), D_{−}(v_{2}), D_{+}(v_{1}) and the set {D(v_{2}), D(v̄_{2})} associated with the graph provide exactly two “good” solutions for (f, n) is long and tedious, so we decided not to give it here (nevertheless we think that Example 3 illuminates completely the problem). What is important is the fact that any graph (in this family) can be realized by at most two possible pairs (f, n), and this coincidence in some cases really occurs. (Moreover, given a pair (f, n), or the graph of X_{f,n}, one can write down easily the possible candidate for the numerical data of (f′, n′), the possible pair of (f, n), with the same graph.)

In the next examples we will write {(p_{1}, q_{1}), (p_{2}, q_{2}); n} for the Newton pairs of f and the integer n. Recall that (∗) implies p_{1} = h_{1} and q_{1} = h̃_{1}.

Example 1. The two different solutions (3, 7) and (4, 5) for the genus formula (h_{1} −1)(h̃_{1} − 1) = 2 · 6 can be completed to the following two sets of invariants: {(3, 7), (20, 1); 21} and {(4, 5), (21, 1); 20}. For them the corresponding two graphs are the same:

[figure: [6]; −421; −1; −2; −2; −2; −21; ⋯]

where the number of (−2)-curves is 19. Here h_{2} = 1.

Example 2. If one wants examples with arbitrary h_{2}, then one of the possibilities is the following: one multiplies in the above data (of Example 1) p_{2} and n by the wanted h_{2}. E.g. the data {(3, 7), (40, 1); 42} and {(4, 5), (42, 1); 40} provide the same h_{2} = 2 and the same graph:

[figure: [6]; −841; [6]; −841; −1; −2; −2; −2; −21; ⋯]

where again, the number of (−2)-curves is 19.

Example 3. Assume the data {(p_{1}, q_{1}), (p_{2}, q_{2}); n} of (f, n) satisfies (∗), hence p_{1} = h_{1} and q_{1} = h̃_{1}. If (f, n) has “a pair” (f_{2}, n_{2}) (with the same graph) then the data of (f_{2}, n_{2}) has the form (cf. the above discussion) {(x, y), (n, q_{2}), p_{2}}, where x and y can be determined by the equations: xy/p_{2} = p_{1}q_{1}/n and (x − 1)(y − 1) = (p_{1} − 1)(q_{1} − 1). It is easy to write down cases when this has no integral solutions.

E.g., the data {(2, 3), (5, 1); 6} satisfies (∗), but it has no “pair”. Its minimal resolution graph can be realized in a unique way in the form f + z^{n} (f irreducible) (cf. 5.4).

Addendum. Relation with the Milnor number. Even if the same graph is realized by two different pairs (f_{1}, n_{1}) and (f_{2}, n_{2}), the corresponding Milnor numbers μ_{i} associated with the hypersurface singularities f_{i} + z^{n_{i}} (i = 1, 2) distinguish the two cases. This follows from the formula μ = [2g_{v_{1}}p_{2} + (p_{2} − 1)(a_{2} − 1)](n − 1). Since g_{v_{1}} > 0, a_{2}, np_{2} and n + p_{2} are readable from the graph, this relation determines p_{2}, hence all the numerical data.

Now we are ready to formulate and prove the main result of the article.

5.4. Theorem. Let f : (ℂ^{2}, 0) → (ℂ, 0) be an irreducible plane curve singularity with Newton pairs {(p_{i}, q_{i})}_{i=1}^{s} and let n be an integer ≥ 2. Let Γ^{min}(X_{f,n}) be the minimal (good) resolution graph of the hypersurface singularity (X_{f,n} , 0) := ({f(x, y) + z^{n} = 0}, 0). Then the following facts hold:

(a) The integer s is uniquely determined by Γ^{min}(X_{f,n}).

(b) s = 1 if and only if Γ^{min}(X_{f,n}) is either a string (with all the genera zero), or a star-shaped graph (where only the central vertex might have genus g non-zero). Moreover, f(x, y) + z^{n} has the same equisingularity type as the Brieskorn singularity x^{q_{1}} + y^{p_{1}} + z^{n}.

Γ^{min}(X_{f,n}) is a string if and only if {q_{1}, p_{1}, n} = {q_{1}, 2, 2}. If Γ^{min}(X_{f,n}) is a star-shaped graph with g = 0, then the set of integers {q_{1}, p_{1}, n} is uniquely determined. Moreover, the only ambiguity which can appear in the case g > 0 is described in 5.1.

(c) If s = 2 then it can happen that two pairs (f_{1}, n_{1}) and (f_{2}, n_{2}) (but not more) provide identical graphs Γ^{min}(X_{f,n}). If this is the case then both of them should satisfy the numerical restrictions:

(n, a_{2}) = 1, and (n, p_{2})a_{1}p_{1}|n.  (∗)

(which can be recognized from the graph as well), and (automatically) at least one of the vertices has genus g > 0. This case is described in 5.3.

(d) In all other cases (i.e. for any s ≥ 3 or for s = 2 excluding the exceptional case (∗)), Γ^{min}(X_{f,n}) determines uniquely the Newton pairs of f and the integer n (by a precise algorithm which basically constitutes the next proof).

(e) In particular, except for the two cases S1 and S2 (cf. 5.1 and 5.3), from the link one can recover completely the Newton pairs of f and the integer n (provided that we disregard the z-axis ambiguity, cf. 5.2.) In particular, this is true without any exception provided that the link is a rational homology sphere.

On the other hand, in the cases S1 and S2, the link together with the Milnor number of the hypersurface singularity f + z^{n} determines completely the Newton pairs of f and the integer n (cf. the two addendums in 5.1 and 5.3).

Proof. We denote Γ^{min}(X_{f,n}) by Γ and its rupture vertices by R. It is convenient to separate those cases when #R is small.

(Case A) Assume that R = ∅. By 4.1 the set of rupture vertices of Γ^{min}(X_{f,n} , z) is never empty. Hence, by 4.2, R = ∅ if and only if in the “P-case” we contract v′, and v′ is the unique rupture vertex of Γ^{min}(X_{f,n} , z). But Γ^{min}(X_{f,n} , z) has a unique rupture point if and only if s = 1. Therefore (cf. 4.2) this situation occurs if and only if s = 1, p_{1} = n = 2 and (q_{1}, 2) = 1; i.e. f(x, y) + z^{n} has the equisingularity type of x^{q_{1}} + y^{2} + z^{2}. Clearly, q_{1} can be recovered from the graph: it is its determinant. Cf. also with 5.1.

(Case B) Assume that #R = 1. From 4.2 is clear that in the “P-case” the number of rupture vertices of Γ is even. Hence, this case is excluded, and the number of rupture vertices of Γ^{min}(X_{f,n} , z) is also 1. By 4.1, this can happen only of s = 1. In particular, f + z^{n} is of Brieskorn type: x^{q_{1}} + y^{p_{1}} + z^{n}, with q_{1} > p_{1} ≥ 2, (p_{1}, q_{1}) = 1 and n ≥ 2 (where the case p_{1} = n = 2 is excluded, see above). This case is completely covered by 5.1.

(Case C) Assume that #R > 1. By 4.1 and 4.2 #R > 1 if and only if s ≥ 2. The proof (algorithm) consists of several steps, each step recovers some data.

(1) The set R_{1} can be determined from 4.6 and 4.7. Indeed, we start with the set L(Γ(R)) (where Γ = Γ^{min}(X_{f,n})). Then, if there exists at least one v ∈ L(Γ(R)) with g_{v} ≠ 0, then R_{1} = {v ∈ L(Γ(R)) : g_{v} ≠ 0} (cf. 4.7(a)). If g_{v} = 0 for all v, then we consider their β-invariants β(v). If they are all equal, then by 4.7(b) one gets R_{1} = L(Γ(R)). If they are not all equal, then only one can be ≤ 1/2 (corresponding to {v} = R_{s}), and all the others are > 2 (and equal to each other) corresponding to R_{1} (cf. 4.7(b)).

(2) The sets π(q^{−1}(v_{k})) (1 ≤ k ≤ s). Define a distance on the set R. If w_{1}, w_{2} ∈ R, and on the shortest path in Γ connecting them there are exactly l rupture vertices of Γ (including w_{1} and w_{2}), then we say that d(w_{1}, w_{2}) := l − 1 ≥ 0. Moreover, for any subset R′ ⊂ R and w ∈ R we define d(R′, w) as usual by min{ d(w′, w) : w′ ∈ R′}.

Then, for any k ≥ 1, we write R_{k} := {v ∈ R : d(R_{1}, v) = k − 1}. Let s′ := max{k : R_{k} ≠ ∅}. We distinguish two cases:

(a) #R_{s′} = 1. Then s = s′ and π(q^{−1}(v_{k})) = R_{k} for 1 ≤ k ≤ s.

(b) #R_{s′} > 1. This can happen exactly in the “P-case” (cf. 4.2). In this situation, s = s′ + 1, π(q^{−1}(v_{k})) = R_{k} for 1 ≤ k ≤ s − 1, and the (unique) vertex v′ = q^{−1}(v_{s}) of Γ^{min}(X_{f,n} , z) is “missing” in Γ, i.e. π({v′}) = ∅ (cf. 4.2).

For a moment we postpone the “P-case”, and we assume (a). We will come back to the “P-case” in (11).

(3) The integers {h_{k}}_{k=2}^{s} are determined by the identities h_{k} = #R_{k−1}/#R_{k}. (h_{1} will be determined later.)

(4) The sets {π(q^{−1}(v̄_{k}))}_{k=2}^{s−1} and the integers {h̃_{k}}_{k=2}^{s−1}, {p_{k}}_{k=2}^{s−1} (for s ≥ 3). Fix 2 ≤ k ≤ s − 1 and some v ∈ R_{k}. Consider the set of strings St(v) supported by v (cf. 4.4(3)).

If St(v) ≠ ∅, then #St(v) = h̃_{k} and det : St(v) → ℤ is constant with value p′_{k}. Then p_{k} = p′_{k} · h_{k}. Then this is happening for any choice of v, and π(q^{−1}(v̄_{k})) is the set of leaf vertices of Γ situated on the strings of type ∪_{v}St(v), v ∈ R_{k}.

If St(v) = ∅ then all the strings of type Γ(v̄_{k}) are collapsed in Γ, in particular π(q^{−1}(v̄_{k})) = ∅. Hence their determinants p′_{k} = p_{k}/h_{k} = 1. In particular, p_{k} = h_{k} ≥ 2. Then h̃_{k} is given by the genus formula h̃_{k} − 1 = 2g_{v}/(h_{k} − 1).

For the “ends” k = 1 and k = s we need more special computations (since we have to separate the two different types of strings which may be, or may not be “missing” from Γ). In (5) we recover h̃_{s}, in (6) and (7) n and p_{s} and the arrowhead of Γ^{min}(X_{f,n} , z) (excepting the case (∗)). In (8) we treat the invariants with index k = 1.

(5) The integer h̃_{s}. If h_{s} > 1 then the genus formula for g_{v_{s}} gives h̃_{s}. If h_{s} = 1, the strings of type Γ(v̄_{s}) cannot be collapsed, hence St(v) ≠ ∅ for {v} = R_{s}. Then h̃_{s} = α(v), cf. 4.4(3) and 3.7(b).

(6) p_{s} and n and the arrowhead of Γ^{min}(X_{f,n} , z) in the following cases:

(i) either s ≥ 3, or

(ii) s = 2 but {a′_{1}, p′_{1}} ≠ {1}

First we show that in both cases we can compute the product h_{s−1}h̃_{s−1}. Indeed, in the case (i) this follows from (3) and (4). If s = 2 then we proceed as follows. Since a′_{1} and p′_{1} are not both 1, St(v) ≠ ∅ for v ∈ R_{1}. If the determinant has two values on this set, then h_{1}h̃_{1} = α(v), cf. 4.4(3) and 3.7(a). If all the determinants are the same, then either Γ(v̄_{0}) or Γ(v̄_{1}) is collapsed. If Γ(v̄_{0}) is collapsed then a′_{1} = 1 hence h̃_{1} ≥ 2. In the second case h_{1} ≥ 2. Hence in both cases the following procedure works: take c_{1} := #St(v) (which automatically is ≥ 2), compute c_{2} by the genus formula 2g_{v_{1}} = (c_{1} − 1)(c_{2} − 1) and set h_{1}h̃_{1} = c_{1}c_{2}.

Now, we go back to p_{s} and n and the position of the arrowhead.

Notice that by 4.1 and part (2), the determinants of type D(v_{s−1}) and D_{+}(v_{s−1}) are well-defined in Γ, and their values do not change by the minimalization procedure. E.g. D_{+}(v_{s−1}) can be computed from 3.9(b). Notice also that D_{St}(v_{s}) = n(p′_{s})h̃_{s}/h_{s}h̃_{s} (cf. 3.5). Therefore

\frac{D(v_{s−1})^{h_{s}−1}D_{St}(v_{s})}{D_{+}(v_{s−1})} = \frac{h_{s−1}h̃_{s−1}p_{s}}{h_{s}h̃_{s}} .

Hence this value can be determined from the graph, a fact which is true for h_{s−1}h̃_{s−1} (see above) and h_{s} (cf. (3)) and h̃_{s} (cf. (5)) as well. Hence we get p_{s}. In particular, we can compute the string determinants D(v̄_{s}) = p′_{s} and (using D_{St}(v_{s})) D(v_{s}) = n/(h_{s}h̃_{s}) as well. This gives n too. If D(v_{s}) ≠ 1 then we put the arrow on the string with this determinant (cf. 3.7); if D(v_{s}) = 1 then we put the arrowhead on π(q^{−1}(v_{s})). In this way we recover the arrow of Γ^{min}(X_{f,n} , z).

(7) p_{s} and n and the arrowhead of Γ^{min}(X_{f,n} , z) if

(iii) h̃_{s} ≠ 1.

Consider St(v) for {v} = R_{s}, cf. 4.4(3). Then by a verification D_{St}^{red}(v) = (p′_{s})^{h̃_{s}−1}. Since h̃_{s} is determined in (5), and it is ≠ 1, one gets p′_{s}. Then we repeat the arguments of (6).

(8) The integers a_{1}, p_{1}, h_{1} and h̃_{1} in the cases when one of the conditions (i) or (ii) or (iii) is valid. Fix a vertex v ∈ R_{1} and consider St(v) as in 4.4(3).

If St(v) ≠ ∅, then #I(v) ≤ 2. If #I(v) = 2, then compute the two numbers D_{1} · #_{2} and D_{2} · #_{1}. They are the candidates for a_{1} = q_{1} and p_{1}, cf. 3.5. Since q_{1} > p_{1}, these two numbers cannot be the same. If, say, D_{1} · #_{2} > D_{2} · #_{1}, then St_{1} is the index set of π(q^{−1}(v̄_{0})) and St_{2} of π(q^{−1}(v̄_{1})). Hence h_{1} = #_{1}, h̃_{1} = #_{2}, q_{1} = D_{1} · #_{2} and p_{1} = D_{2} · #_{1}.

If there is only one level set with data D_{1} and #_{1}, then in the above argument we write D_{2} = 1 and we determine #_{2} using the genus formula 2g_{v_{1}} = (#_{1} − 1)(#_{2} − 1) (which is possible since D_{2}#_{1} = #_{1} ≥ 2). And we repeat the above argument.

Now we assume that St(v) = ∅. This can happen only if a′_{1} = p′_{1} = 1, hence q_{1} = h̃_{1} and p_{1} = h_{1}. First we determine H := h_{1}h̃_{1}.

D(v_{1}) gives an equation of type q_{2} = H · A, where A is a positive number which can be determined from the graph by the previous steps. Moreover, D_{−}(v_{2}) = a′_{2}, hence a_{2} = D_{−}(v_{2})h̃_{2} is known from the graph. Finally, a_{1}p_{1}p_{2} = Hp_{2}, where p_{2} too is known from the graph. Then the identity a_{2} = q_{2} + a_{1}p_{1}p_{2} gives a non-trivial linear equation for H.

Then h_{1}h̃_{1} = H and (h_{1}−1)(h̃_{1}−1) = 2g_{v_{1}} provides h_{1} and h̃_{1} modulo their permutation. But h̃_{1} = q_{1} > a_{1} = h_{1}, hence we get h_{1} and h̃_{1}.

(9) The integers {a_{k}}_{k=2}^{s} when one of the conditions (i) or (ii) or (iii) is valid. Once we have the position of the arrow, we have all the multiplicities {m_{v_{k}}}_{k}, hence 3.2(b) gives all the integers a′_{k}. An alternative way is to use inductively 3.10.

(10) Assume that the conditions (i), (ii) and (iii) are not valid. This means that s = 2 and a′_{1} = p′_{1} = h̃_{2} = 1. This is exactly the case of S2-coincidence treated in 5.3.

(11) The “P-case”. Now we go back to step (2), case (b). In this case #R_{s′} = 2, so write R_{s′} = {w_{1}, w_{2}}. Take the shortest path in Γ connecting w_{1} and w_{2}. Take the edge “at the middle of the path”, blow it up, and put an arrow on it. This new graph is exactly Γ^{min}(X_{f,n} , z). Set R_{s} := {v′}, where v′ is the new vertex. Then we can repeat all the above arguments.

Notice that step (6) works, since if s = 2 and a′_{1} = p′_{1} = 1, then h_{1} = p_{1} ≥ 2 and h̃_{1} = a_{1} ≥ 2 hence g_{v_{1}} > 0. But in the “P-case” all the genera are zero. (Hence step (7) is not needed.) [In fact, since in this case we already have the position of the arrow, we can compute some of the invariants much faster using the multiplicities and 3.2(b).] □

5.5. The symmetry of the graph. In the above proof we were rather meticulous in separating the possible sets π(q^{−1}(v)). The fruit of this is the following corollary (whose proof is left to the reader, and basically it is incorporated in the previous proof of the main theorem).

First recall that the cyclic covering X_{f,n} → X has a ℤ_{n} Galois action. This lifts to the level of the resolution, hence Γ^{min}(X_{f,n}) inherits a natural ℤ_{n}-action as well. The question is: has the graph Γ^{min}(X_{f,n}) any extra symmetry?

Take for example the Brieskorn case 3.3. Then the Galois action permutes cyclically (via its image ℤ_{h}) the h arms with Seifert invariants a. On the other hand, the total symmetry group of the graph is the total permutation group of these arms. So, in this sense, the symmetry group of the graph is definitely larger than the (image) of the Galois action. On the other hand, their orbits are the same. This fact is valid in general.

5.6. Corollary. Assume that σ is a (decorated graph-) automorphism of Γ^{min}(X_{f,n}) which identifies two vertices, say, v_{1} and v_{2}. Then v_{1} and v_{2} are in the same orbit with respect to the Galois action.

5.7. Final remarks about the reducible case. 1) We emphasize again that it is rather surprising (at least for the authors) that from the link of X_{f,n} one can recover the equisingularity type of f and the integer n excepting only two families, without any additional assumption about the link. Still, we expect that even for reducible f, the link of X_{f,n} contains all the information about the equisingularity type of f and the integer n “generically”, i.e. excepting a set of coincidences which is “small” inside of suspension singularities. At this moment it is extremely difficult to guess how large is the set of these exceptions. Coincidences with star-shaped graphs are known, and can be classified. But definitely there are other type of coincidences as well.

For example, the minimal resolution graphs of the suspension singularities {(x^{19}+y^{38})(x^{32}+ y^{16}) + z^{3} = 0} and {(x^{5} + y^{6})(x^{10} + y^{3}) + z^{45} = 0} are the same, namely:

[figure: [18]; [15]; −2; −2]

Moreover, the above S2-coincidence warns us that we can expect rather strange coincidences as well. We also notice that if f is reducible, the graph of X_{f,n} in general is not a tree, a fact which enlarges the possible graphs considerably.

On the other hand, we conjecture that even for reducible f, if the link of X_{f,n} is a rational homology sphere, then it determines the equsingularity type of f and the integer n in a unique way.

2) Analyzing the above results, we realize that some of them can be generalized in a natural way to the reducible case; but in some other cases we cannot even formulate the possible generalizations (e.g. in the crucial proposition 4.7).

3) 5.6 cannot be extended to the general case of non-irreducible germs. E.g., Γ^{min}(ℂ^{2}, 0) can have a symmetry (take e.g. f = (x^{2} + y^{3})(x^{3} + y^{2})) which lifts to an automorphism of Γ^{min}(X_{f,n}), which does not come from the Galois covering.

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22 Robert Mendris and András Némethi

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Department of Mathematics, Ohio State University, Columbus, OH 43210

   E-mail address: mendris@math.ohio-state.edu
   E-mail address: nemethi@math.ohio-state.edu