Dedekind sums and the signature of f(x, y) + z^{N}, II

by András Némethi The Ohio State University, Dept. of Mathematics 231 West 18th Avenue, Columbus, OH 43210, USA.

§1. Introduction

The present paper has two goals.

First, we compute the signature of the Milnor fiber associated with a singularity of type f(x, y) + z^{N}, and we prove some conjectures of A. Durfee.

A. Durfee in [4] listed some conjectures about the signature σ, the Milnor number μ and the geometric genus p_{g} of an isolated complete intersection singularity g : (C^{k+2}, 0) → (C^{k}, 0). Namely, he conjectured that p_{g} ≤ μ/6 (Conjecture 5.3 in [4], in the sequel (∗)), which is equivalent to −3σ ≥ μ+3μ_{0} (where μ_{0} is the rank of the kernel of the Milnor lattice). In particular (∗) implies a weaker conjecture (5.2 in [4]): the negativity of the signature. On the other hand, J. Wahl in [17] constructed a non-hypersurface smoothing with positive signature, showing the subtility of the problem. But, even in the hypersurface case, the conjecture still resist to the persistent proof attemps (see [22]).

In a series of papers, the author studied suspensions of hypersurface singularities (i.e. germs of type f(x_{1}, …, x_{n}) + z_{n+1}^{N}). In particular, in [7], the signature of f + z^{N} is computed in terms of eta–invariants η(f; N) associated with (the variation structure of) f and the number N: σ(f + z^{N}) = η(f; N) − N · η(f; 1). In [10], the eta–invariant η(f; N) of a plane curve singularity f, is expressed in terms of generalized Dedekind sums associated with the embedded resolution graph G_{f} of f (we recall this here in 2.3). This is a powerful relation, for example, in the particular case of Brieskorn polynomials it is equivalent with the computation of the lattice points in a tetrahedron in terms of Dedekind sums, solved in particular cases by Mordell [6], and in general by Pommersheim [13]. This formula generalizes a result (Proposition (2.5) [12]) of W. Neumann and J. Wahl, where the signature is computed when the link of g = f + z^{N} is an integer homology sphere. Using this relation (2.3), in [10], we prove that for an irreducible germ f, its eta–invariant, hence the signature of f + z^{N} too, is additive with respect to the splice decomposition of the graph G_{f}. In particular, we prove that the inequality (∗) is valid for these type of germs. Unfortunately, additivity results as in [10] (or as in [12]) are not true if f is not irreducible.

In this paper, by a different method, we attack the same problem for arbitrary f: we prove (∗) for f +z^{N} (with some minor restrictions). Actually, we provide a large list of inequalities: we compare the signature with the Milnor number μ_{f} of f (“topological” inequality), with the multiplicity ν_{f} of f (“algebraic” inequality) and the size of G_{f} (“combinatorial” inequality) (cf. section 5). (Using any of these inequalities one can prove the negativity of σ.) The method is the following. We start again with the relation which describes the signature in terms of Dedekind sums (cf. 2.3). Using some properties of these Dedekind sums, in section 3 we transform this relation in some inequalities expressed in terms of the combinatorics of the embedded resolution graph G_{f} of f. These inequalities are verified in section 4.

Now, we arrived to our second point. The second goal of the paper is less precise but more important (for the author). We are hunting for an answer of the following question: what is special in the hypersurface case? Why the Milnor fiber of a hypersurface has negative signature and a nonhypersurface maybe not? The present paper teaches us that the answer is in the particular form of the embedded resolutions of hypersurface singularities. In the case of two dimensional germs (i.e. f : (X, x) → (C, 0), with (X, x) normal surface singularity) we can be more precise: all the combinatorial relations (which imply our inequalities) proved in section 4 for plane curve singularities (i.e. with X smooth) distinquish these hypersurface germs from the general class of two-dimensional singular germs (cf. 4.25). The author expects that similar relations are valid in higher dimension, and they are responsible for the particular behaviour of hypersurface singularities.

A few more words about the relation σ(f + z^{N}) = η(f; N) − Nη(f; 1). Since η(f; N) is periodic in N, lim_{n→∞}(σ(f + z^{N})/N) = η(f; 1), and any estimate for η(f; 1) will provide inequalities for σ when N is big. In particular, any inequality of type σ(f + z^{N}) ≤ c · N (c constant) with c > −η(f; 1) is “weak” for N big, but can be very difficult for small N, because of the very irregular behaviour of η(f; N) (i.e. of the Dedekind sums). Moreover, any inequality of type σ(f + z^{N}) ≤ c · I_{f}, (where I_{f} = μ_{f}, or ν_{f} or any invariant of f), can be very weak for f with I_{f} big, but very sharp for germs at the beginning of the classification list. For germs with I_{f} small, in order to verify an inequality as above, the periodic term η(f; N) must be really computed. This explains the form of our inequalities in section 4 (and the difficulties in their proofs).

Finally, we mention the papers of Y. Xu and S. S.-T. Yau [19, 20], where they verified the inequality (∗) (actually, even a stronger version) in the case of quasihomogeneous germs g : (C^{3}, 0) → (C, 0).

§2. σ_{N} in terms of the embedded resolution graph of f

In this section we introduce some notations and we recall one of the main results of [10].

Let f : (C^{2}, 0) → (C, 0) be a germ of an analytic function which defines an isolated singularity at the origin. We consider an embedded resolution ϕ : (Y, D) → (C^{2}, f^{−1}(0)) of (f^{−1}(0), 0) ⊂ (C^{2}, 0) (here D = ϕ^{−1}(f^{−1}(0))). Let E = ϕ^{−1}(0) be the exceptional divisor and E = ∪_{w∈W}E_{w} be its decomposition in irreducible divisors. If f = ∏_{a∈A} f_{a} is the irreducible decomposition of f, then D = E ∪ ∪_{a∈A}S_{a}, where S_{a} is the strict transform of f_{a}^{−1}(0). Let G_{f} be the resolution graph of f, i.e. its vertices V = W ∐ A consist of the nonarrowhead vertices W (corresponding to the irreducible exceptional divisors), and arrowhead vertices A (correponding to the strict transform divisors of D). We will assume that no irreducible exceptional divisor has an autointersection and W ≠ 0. If two irreducible divisors corresponding to v_{1}, v_{2} ∈ V have an intersection point then (v_{1}, v_{2}) (= (v_{2}, v_{1})) is an edge of G_{f}. The set of edges is denoted by E. Since G_{f} is a tree, one has:

(2.1) #W + #A = #E + 1.

For any w ∈ W, we denote by V_{w} the set of vertices v ∈ V adjacent to w. The graph G_{f} is decorated by the autointersection (or Euler-) numbers

e_{w} := E_{w} · E_{w} for any w ∈ W.

For any v ∈ V let m_{v} be the multiplicity of f ∘ ϕ along the irreducible divisor corresponding to v. In particular, for any a ∈ A one has m_{a} = 1. The multiplicities satisfy the following relations. For any w ∈ W one has:

(2.2) e_{w}m_{w} + ∑_{v∈V_{w}} m_{v} = 0.

These relations determine the multiplicities {m_{w}}_{w∈W} in terms of the autointersection numbers {e_{w}}_{w}.

It is convenient to use the following notations:

(a) for any w ∈ W, we define M_{w} := gcd(m_{w}, m_{v_{1}}, …, m_{v_{t}}), where V_{w} = {v_{1}, …, v_{t}}; and

(b) for any e = (v_{1}, v_{2}) ∈ E, we define m_{e} := gcd(m_{v_{1}}, m_{v_{2}}).

For any a ∈ A there exists exactly one w_{a} ∈ W such that (a, w_{a}) ∈ E.

With these notations one has (cf. also [12], Proposition 2.5):

2.3. Theorem. [10] Let f : (C^{2}, 0) → (C, 0) be an isolated plane curve singularity as above. Then the signature σ_{N} of the Milnor fiber of the suspension f(x, y) + z^{N} is:

σ_{N} = η(f; N) − N · η(f; 1), where

η(f; K) = #A − 1 + ∑_{e∈E}((K, m_{e}) − 1) − ∑_{w∈W}((K, M_{w}) − 1)+

+4 · ∑_{w∈W} ∑_{v∈V_{w}} ∑_{k=1}^{m_{w}} ((\frac{km_{v}}{m_{w}})) · ((\frac{kK}{m_{w}})).

Notice also that K ↦ η(f; K) is a periodic function.

§3. σ_{N} via the reciprocity law of Dedekind sums

In this section we will use the generalization of the reciprocity law of Dedekind given by Rademacher [14, 21] in order to rewrite theorem (2.3). In the new expression we will have Dedekind sums “with denominator N” only. We separate the needed notations and facts about the Dedekind sums in the Appendix.

We start with the following easy remark. If g : V ×V → R is an arbitrary function, then:

(3.1)  ∑_{w∈W} ∑_{v∈V_{w}} g(u, v) + ∑_{a∈A} g(a, w_{a}) = ∑_{(u,v)∈E}(g(u, v) + g(v, u)).

Since s(m_{w_{a}}, N; m_{a}) = 0 for any a ∈ A (because m_{a} = 1), (3.1) gives:

(3.2) 4 ∑_{w∈W} ∑_{v∈V_{w}} s(m_{v}, N; m_{w}) = 4 ∑_{(u,v)∈E}[s(m_{v}, N; m_{u}) + s(m_{u}, N; m_{v})].

Using the reciprocity law (A.2), this expression is equal to:

∑_{e=(u,v)∈E} [−4s(m_{u}, m_{v}; N) − (N, m_{e}) + \frac{N^{2}m_{e}^{2} + m_{u}^{2}(N, m_{v})^{2} + m_{v}^{2}(N, m_{u})^{2}}{3Nm_{u}m_{v}}].

Using this equality, the formula of the eta–invariant η(f; N) given in (2.3) can be transformed in:

(3.3) η(f; N) = #A − 1 − #E − ∑_{v∈V}((N, M_{w}) − 1) − 4 ∑_{(u,v)∈E} s(m_{u}, m_{v}; N)

+ ∑_{(u,v)∈E} \frac{N}{3} \frac{m_{e}^{2}}{m_{u}m_{v}} + \frac{1}{3N} · [ ∑_{(u,v)∈E} \frac{m_{u}(N, m_{v})^{2}}{m_{v}} + \frac{m_{v}(N, m_{u})^{2}}{m_{u}} ].

For the last sum we apply again (3.1), (2.2) (and m_{a} = 1), hence it can be replaced by:

∑_{w∈W} ∑_{v∈V_{w}} \frac{m_{v}(N, m_{w})^{2}}{m_{w}} + ∑_{a∈A} \frac{m_{w_{a}}(N, m_{a})^{2}}{m_{a}} = ∑_{w∈W}(−e_{w})(N, m_{w})^{2} + ∑_{a∈A} m_{w_{a}}.

Using (2.1) and the above identity, (3.3) reads as:

(3.4) η(f; N) = −#W − ∑_{w∈W}((N, M_{w}) − 1) − 4 ∑_{(u,v)∈E} s(m_{u}, m_{v}; N)

+ ∑_{(u,v)∈E} \frac{N}{3} \frac{m_{e}^{2}}{m_{u}m_{v}} + \frac{1}{3N} [ ∑_{w∈W}(−e_{w})(N, m_{w})^{2} + ∑_{a∈A} m_{w_{a}}].

The periodicity of η(f; N) and (3.4) applied for N = 1 provide:

3.5. Theorem.

lim_{N→∞} \frac{−σ_{N}}{N} = η(f; 1) = −#W + \frac{1}{3} ∑_{(u,v)∈E} \frac{m_{e}^{2}}{m_{u}m_{v}} + \frac{1}{3}[ ∑_{w∈W}(−e_{w}) + ∑_{a∈A} m_{w_{a}}].

The expression σ_{N} = η(f; N) − N · η(f; 1) of theorem (2.3) transforms in the following formula:

3.6. Theorem.

σ_{N} = (N − 1) · #W − ∑_{w∈W}((N, M_{w}) − 1) − 4 ∑_{(u,v)∈E} s(m_{u}, m_{v}; N)

+\frac{1}{3N} · ∑_{w∈W}(−e_{w})[(N, m_{w})^{2} − N^{2}] + \frac{1 − N^{2}}{3N} ∑_{a∈A} m_{w_{a}}.

It is interesting to consider some particular cases: 3.7. Corollary.

a) Assume that (N, m_{w}) = 1 for any w ∈ W. Then:

σ_{N} = (N − 1) · #W − 4 ∑_{(u,v)∈E} s(m_{u}, m_{v}; N) + \frac{1 − N^{2}}{3N} [ ∑_{w∈W}(−e_{w}) + ∑_{a∈A} m_{w_{a}}].

b) Assume that N = 2. Then:

σ_{2} = #\{w ∈ W : 2 |̸M_{w}\} − \frac{1}{2}[ ∑_{{w∈W \atop 2 |̸mw}} (−e_{w}) + ∑_{a∈A} m_{w_{a}}].

Now, for (3.7.a) we can apply the inequality (A.3) and (2.1), and we obtain: 3.8. Corollary. Assume that (N, m_{w}) = 1 for any w ∈ W. Then:

σ_{N} ≤ \frac{1 − N}{N} (#W +#E)+ \frac{1 − N^{2}}{3N} [−4#W −#A+1+ ∑_{w∈W}(−e_{w})+ ∑_{a∈A} m_{w_{a}}].

We would like to have similar inequality for arbitrary N. We start with the following lemma: 3.9. Lemma. There exists a one-to-one function l : W → E such that l(w) = (w, v) for some v ∈ V. The proof is easy and is left to the reader.

Notice that (A.4) provides for l(w) = (w, v):

(3.10) − 4 · s(m_{w}, m_{v}; N) + \frac{1}{3N} ((N, m_{w})^{2} − N^{2}) ≤ 0.

Therefore, by (3.10) and (A.3) one has:

−4 ∑_{(u,v)∈E} s(m_{u}, m_{v}; N) + \frac{1}{3N} ((N, m_{v})^{2} − N^{2})

≤ −4 · ∑_{{(u,v)∈E \atop (u,v)∉l(W)}} s(m_{u}, m_{v}; N) ≤ \frac{(N − 1)(N − 2)}{3N}(#A − 1).

This inequality and (3.6) give: 3.11. Corollary. For any N the following inequality holds:

σ_{N} ≤ − ∑_{w∈W}((N, M_{w}) − 1) + \frac{1}{3N} · ∑_{w∈W}(−e_{w} − 1)[(N, m_{w})^{2} − N^{2}]

+\frac{1 − N}{N} #E + \frac{1 − N^{2}}{3N} [−3#W − #A + 1 + ∑_{a∈A} m_{w_{a}}].

This inequality simplified reads as: 3.12. Corollary.

a) For any N ≥ 1:

σ_{N} ≤ \frac{1 − N}{N} #E + \frac{1 − N^{2}}{3N} [−3#W − #A + 1 + ∑_{a∈A} m_{w_{a}}].

b) If N > B(f) := [∑_{w∈W}(−e_{w} − 1)(m_{w}^{2} − 1)]/(3#E), then:

σ_{N} ≤ \frac{1 − N^{2}}{3N} [−4#W − #A + 1 + ∑_{w∈W}(−e_{w}) + ∑_{a∈A} m_{w_{a}}].

Proof. For a) use −e_{w}−1 ≥ 0, and for b) (N, m_{w})^{2}−N^{2} ≤ (m_{w}^{2}−1)+(1−N^{2}). □

All our inequalities ((3.8),(3.12), cf. also (3.5) and (3.7.b)) give estimates of type (−σ_{N})/(N − 1) ≥ {a combinatorial expression in terms of G_{f}}. In the next section we will study these expressions.

§4. Inequalities satisfied by the minimal resolution graph of f

I. Preliminaries.

Let f : (C^{2}, 0) → (C, 0) be an isolated plane curve singularity. Let G_{f} be its minimal embedded resolution, with the convention that #W ≠ ∅, (which in the sequel will be called shortly: resolution). We will keep the notations of section 2 for the numerical invariants of G_{f}. Recall that for any arrow a ∈ A, w_{a} ∈ W is the unique vertex adjacent with a, and m_{w_{a}} is its multiplicity.

In this section we will present some properties of :

M_{f}^{r} := ∑_{a∈A} m_{w_{a}} − r · #W(G_{f}) (r = 2, 3, 4),

which distinguish the plane curve singularities among the singular germs f : (X, x) → (C, 0) defined on a normal surface singularity (X, x) (cf. 4.32). We will compare M_{f}^{r} with the Milnor number μ_{f} of f (“topological inequality”), with the multiplicity ν_{f} of f (“algebraic inequality”), and with the size of G_{f} (“combinatorial inequality”).

It is convenient to use the following notation: if two germs f and g have the same topological type, in particular, the same minimal embedded resolution graph G_{f} = G_{g}, then we write f ∼ g. (Notice that f can have big modularity, but G_{f} depends only on the topological embedding (f^{−1}(0), 0) ⊂ (C^{2}, 0).)

We recall the structure of G_{f} when f is irreducible. Assume that f has Newton pairs (p_{i}, q_{i})_{i=1}^{s} (p_{i} ≥ 2, q_{i} ≥ 1, q_{1} > p_{1}). The minimal embedded resolution graph of y(x^{p_{i}} + y^{q_{i}}) (q_{i} ≥ 1, p_{i} ≥ 2) has the following form:

[figure: …; ⋮; (y = 0); −u_{i}^{1}; −u_{i}^{t_{i}}; −1; (x^{p_{i}} + y^{q_{i}} = 0); −v_{i}^{r_{i}}; −v_{i}^{2}; −v_{i}^{1}]

where 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) are given by the continuous 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}^{s_{i}}}}} .

The graph G_{f} can be reconstructed (by splicing) from the graphs of y(x^{p_{i}} + y^{q_{i}}) and the numbers u_{i}^{0} as follows ([5] Appendix of chap. 1, and section 22):

[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}]

In particular:

(4.1) #W(G_{f}) = ∑_{i=1}^{s} #W(G_{y(x^{p_{i}}+y^{q_{i}})}).

If q_{i} = 1 then

(4.2) #W(G_{y(x^{p_{i}}+y^{1})}) = p_{i}.

For q_{i} ≥ 2 we have the following estimate: 4.3. Lemma. Assume that f ∼ x^{p} + y^{q} or f ∼ y(x^{p} + y^{q}), with (p, q) = 1, p ≥ 2, q ≥ 2. Then:

#W(G_{f}) ≤ (p + q + 1)/2.

Proof. First notice that G_{x^{p}+y^{q}} and G_{y(x^{p}+y^{q})} differ only by an arrow, so the number of their vertices agrees. We will use the induction over the pair (p, q). Assume that q > p. If q = p + 1, then 2#W(G_{f}) = p + q + 1. Assume q ≥ p + 2. Consider the blowing up (u, v) →^{ϕ} (x, y) = (uv, u). Then f̃ = f ∘ϕ = u^{p}(v^{p} +u^{q−p}). Now, by the inductive step and the above remark: #W(G_{f}) ≤ 1 + (p + q − p + 1)/2 ≤ (p + q + 1)/2. □ The unique arrow of G_{f} corresponds to A = {a}, and

(4.4) m_{w_{a}} = a_{s}p_{s},

where a_{s} can be computed inductively as follows:

(4.5) a_{1} = q_{1}, and for i ≥ 1 : a_{i+1} = q_{i+1} + p_{i}p_{i+1}a_{i}.

Now, assume that f is not irreducible: f = ∏_{a∈A} f_{a}, where f_{a}’s are irreducible and #A_{f} > 1 (A_{f} := A(G_{f})). It is convenient to fix an ordering of the set A.

In the next lemmas, we would like to compare the graphs G_{f} and G_{f_{a}}’s. Recall that for any germ g, G_{g} denotes its minimal embedded resolution graph. If we want to emphasize that a certain invariant is considered in a graph G, then we put G in a parenthesis near the corresponding invariant.

If E is one of the irreducible exceptional divisors of the embedded resolution ϕ : (Y, D) → (C^{2}, f^{−1}(0)) of f^{−1}(0), and g : (C^{2}, 0) → (C, 0) is any germ, then m_{E}(g, G_{f}) denotes the multiplicity of g∘ϕ along E. In particular, if E_{a} corresponds to w_{a} ∈ W in the graph G_{f}, then m_{E_{a}}(f, G_{f}) = m_{w_{a}}(G_{f}) = m_{w_{a}}. If a ≠ a′ (a, a′ ∈ A_{f}), then (cf. e.g. [3]): m_{E_{a}}(f_{a′}) = ν(f_{a}, f_{a′}), where ν(,) denotes the intersection multiplicity at the origin. With these notations one has:

(4.6)  ∑_{a∈A} m_{w_{a}}(G_{f}) = ∑_{a∈A} m_{E_{a}}(f_{a}, G_{f}) + 2 ∑_{a<a′} ν(f_{a}, f_{a′}).

Let G_{f_{a}} denote the minimal resolution graph of f_{a}. By our convention m_{w_{a}}(G_{f_{a}}) is the multiplicity of f_{a} along the irreducible divisor (in G_{f_{a}}) which intersects the strict transform of f_{a} = 0.

We say that two germs are “tangent” if their tangent cones at the origin have a common line. 4.7. Lemma. Assume that f = g · f_{0} and f′ = g · f′_{0} such that f_{0} ∼ f′_{0} are topologically equivalent irreducible germs, and f′_{0} and g are not “tangent”. Then:

(a) ∑_{a∈A_{f}} m_{E_{a}}(f_{a}, G_{f}) − 2#W(G_{f}) ≥

∑_{a∈A_{f′}} m_{E_{a}}(f′_{a}, G_{f′}) − 2#W(G_{g}) − 2#W(G_{f_{0}}) + 2 + 2T + T′,

(b) ∑_{a∈A_{f}} m_{E_{a}}(f_{a}, G_{f}) + 2ν(g, f_{0}) − r#W(G_{f}) ≥

∑_{a∈A_{f′}} m_{E_{a}}(f′_{a}, G_{f′})+2ν(g)ν(f_{0})−r#W(G_{g})−r#W(G_{f_{0}})+r +(r +2)T +2T′,

where r = 3 or 4, and T = T′ = 0, unless f_{0} is a non-smooth (resp. smooth) component tangent with a non-smooth component of g, in which case T = 1 (resp. T′ = 1). Proof. Let ϕ_{1} : (Y, E) → (C^{2}, g^{−1}(0)) be the minimal resolution of g^{−1}(0) with exceptional divisor E (and W ≠ ∅). Let X_{0} be the strict transform of f_{0}^{−1}(0) via ϕ_{1}. Let ϕ_{2} be the minimal resolution of (Y, E ∪ X_{0}) at the point E ∩ X_{0}. Notice that ϕ_{2} can be considered as a “part” of the minimal resolution tower of (C^{2}, f_{0}^{−1}(0)). Set ϕ = ϕ_{1} ∘ϕ_{2}, and let Z_{0} (resp. Z_{a}) be the strict transform of {f_{0} = 0} (resp. of the irreducible component {g_{a} = 0} of g) via ϕ.

Assume that Z_{0} ∩ Z_{a} = ∅ for any a ∈ A_{g}. First notice that ϕ provides G_{f}, ϕ_{1} provides G_{g}, and the number of irreducible exceptional divisors of ϕ_{2} is ≤ #W(G_{f_{0}}) − 1.

Now, we claim that the following inequalities hold (with k = 0):

(i)  ∑_{a∈A_{f}} m_{E_{a}}(f_{a}, G_{f}) ≥ ∑_{a∈A_{f′}} m_{E_{a}}(f′_{a}, G_{f′}) + T′ + 2k

(ii) #W(G_{f}) ≤ #W(G_{g}) + #W(G_{f_{0}}) − 1 − T + k
(iii) ν(g, f_{0}) ≥ ν(g)ν(f_{0}) + T + T′ + k.

Indeed, in general ∑_{a∈A_{f}} m_{E_{a}}(f_{a}, G_{f}) ≥ ∑_{a∈A_{f′}} m_{E_{a}}(f′_{a}, G_{f′}), but if f_{0} is smooth tangent with g, then m_{E_{f′_{0}}}(f′_{0}, G_{f′}) = 1, but m_{E_{f_{0}}}(f_{0}, G_{f}) ≥ 2, this gives (i). For (ii) notice that #W(G_{f}) ≤ #W(G_{g}) + #W(G_{f_{0}}) − 1 (see above), but if two non-smooth components are tangent then they have at least two common infinitely near points, so the minimal resolution graph satisfies (ii); (iii) is obvious.

Now, assume that Z_{0} ∩Z_{a} = P for some a ∈ A_{g}. Let D be the irreducible exceptional divisor of ϕ with P ∈ D. If the smooth germs Z_{0} and Z_{a} have contact k (k ≥ 1), then we need k more blowing ups. The last newly created exceptional divisor is denoted by D′. Then we claim thet the inequalities (i-ii-iii) are valid. Indeed: m_{D′}(g_{a}) = m_{D}(g_{a}) + k and m_{D′}(f_{0}) = m_{D}(f_{0}) + k, which proves (i), (ii) follows by the same argument as above, and finally, (iii) follows, for example, from Max Noether’s theorem (cf. e.g. [3]) which describes the intersection multiplicity in terms of the multiplicity sequence.

Finally, notice that (i-ii-iii) implies the lemma. □ 4.8. Lemma. Let f = ∏_{a∈A_{f}} f_{a} be the irreducible decomposition of f. Then:

(a) ∑_{a} m_{E_{a}}(f_{a}, G_{f})−2#W(G_{f}) ≥ ∑_{a}[m_{w_{a}}(G_{f_{a}})−2#W(G_{f_{a}})]+2(#A_{f}−1)+2T +T′,

(b)  ∑_{a} m_{w_{a}}(G_{f}) − r#W(G_{f}) ≥

∑_{a}[m_{w_{a}}(G_{f_{a}})−r#W(G_{f_{a}})]+r(#A_{f} −1)+2 ∑_{a<a′} ν(f_{a})·ν(f_{a′})+(r+2)T +2T′,

where r = 3 or 4, and T = T′ = 0 unless f has a non-smooth irreducible component tangent to some other non-smooth (resp. smooth) component, in which case T = 1 (resp. T′ = 1.) Proof. The proof is over induction, we replace step by step all the irreducible components f_{a} of f (starting with the smooth ones) by f′_{a} such that f_{a} ∼ f′_{a} and the tangent cone of f′_{a} is in a generic position with respect to the tangent cone of the other components. The inductive step is provided by (4.7). With the notation of (4.7), notice that

(4.9) #W(G_{f′}) = #W(G_{g}) + #W(G_{f′_{0}}) − 1.

Therefore, (4.7.a) reads as:

(4.10)  ∑_{a} m_{E_{a}}(f_{a}, G_{f})−2#W(G_{f}) ≥ ∑_{a} m_{E_{a}}(f′_{a}, G_{f′})−2#W(G_{f′})+2T +T′.

Moreover, using (4.6) and (4.9), (4.7.b) reads as:

(4.11)  ∑_{a} m_{w_{a}}(G_{f})−r#W(G_{f}) ≥ ∑_{a} m_{w_{a}}(G_{f′})−r#W(G_{f′})+(r+2)T +2T′.

So, the induction can be applied. Finally, notice that if the components of f have all distinct tangent cones, then ν(f_{a}, f_{a′}) = ν(f_{a})·ν(f_{a′}), m_{E_{a}}(f_{a}, G_{f}) = m_{w_{a}}(G_{f_{a}}), and #W(G_{f}) = ∑_{a} #W(G_{f_{a}}) − (#A_{f} − 1). Hence the lemma follows (again by 4.6). □

II. Examples.

We are interested in the following expressions: E_{f}^{top} := ∑_{a} m_{w_{a}} − 2#W(G_{f}) − #A + 1 − (μ_{f} − 2), E_{f}^{alg} := ∑_{a} m_{w_{a}} − 3#W(G_{f}) − (ν_{f}^{2} − 2ν_{f} − 3), E_{f}^{com} := ∑_{a} m_{w_{a}} − 4#W(G_{f}) − #A + 1.

The following table gives these invariants of the following plane curve singularities: f ∼ x (i.e. f smooth), f ∼ x^{2} + y^{2k+1}, (A_{2k}, k ≥ 1), f ∼ x^{3} + y^{3k+1}, (E_{6k}, k ≥ 1), f ∼ x^{3} + y^{3k+2}, (E_{6k+2}, k ≥ 1), f ∼ x^{2} + y^{2k}, (A_{2k−1}, k ≥ 1), f ∼ (x^{2} + y^{2k+1})(y^{2} + x^{2l+1}), (A_{2k,2l}, k ≥ 1, l ≥ 1), f ∼ y(x^{2} + y^{2k+1}), (D_{2k+3}, k ≥ 1).

f ∼  #A ∑_{a} m_{w_{a}}  #W E_{f}^{top} E_{f}^{alg}  E_{f}^{com} ∑(−e_{w})
smooth 1  1  1  1  2  -3  1
A_{2k}  1  2(2k+1) k+2  0  k-1  -6  2k+4
E_{6k}  1  3(3k+1) k+3  k-1 6k-6 5k-9 2k+7
E_{6k+2} 1  3(3k+2) k+3  k  6k-3 5k-6 2k+7
A_{2k−1} 2  4k  k  2  k+3  -1  2k-1
A_{2k,2l} 2  4l+4k+12 k+l+3 0  k+l+2 -1  2k+2l+7
D_{2k+3} 2  4k+7  k+2  1  k+1  -2  2k+4

The Milnor number in the case of A_{2k,2l} is 2k + 2l + 7, in other cases is exactly the corresponding index.

III. The “topological” inequality.

4.12. Theorem. If G_{f} is the minimal embedded resolution graph of f, and μ is the Milnor number of f, then:

(a) E_{f}^{top} := ∑_{a∈A} m_{w_{a}} − 2#W(G_{f}) − #A + 1 − (μ − 2) ≥ 0.

If f ≁ A_{n}(n ≥ 1), A_{2k,2l}(k ≥ 1, l ≥ 1), D_{2k+3}(k ≥ 1), E_{6}, then:

(b)  ∑_{a∈A} m_{w_{a}} − 4#W(G_{f}) − #A + 1 + ∑_{w∈W}(−e_{w}) ≥ μ.

Proof. Assume that f is irreducible with s = 1 and Newton pair (p, q). By (4.3) and (4.4): m_{w_{a}} − 2#W ≥ pq − (p + q + 1) = μ − 2. Assume that s ≥ 2 (cf. I). Let f_{(l)} be a germ with Newton pairs (p_{i}, q_{i})_{i=1}^{l}, (1 ≤ l ≤ s). Notice that:

(4.13) μ(f_{(l)}) = (a_{l} − 1)(p_{l} − 1) + p_{l} · μ(f_{(l−1)}).

First consider the expression P_{(l)} := p_{l}a_{l} − μ(f_{(l)}). Then P_{(1)} ≥ 4 and for l ≥ 2: P_{(l)} − p_{l}P_{(l−1)} = p_{l} + q_{l} − 1 ≥ 2. Therefore P_{(l)} ≥ P_{(l−1)} + 2, hence by induction P_{(l)} ≥ 2 + 2l for any 1 ≤ l ≤ s.

Now, by (4.1): E_{f_{(l)}}^{top} − E_{f_{(l−1)}}^{top} = p_{l}a_{l} − p_{l−1}a_{l−1} − 2#W(G_{y(x^{p_{l}}+y^{q_{l}})}) − μ(f_{(l)}) + μ(f_{(l−1)}). If q_{l} ≥ 2 then by (4.3), (4.5) and (4.13): E_{f_{(l)}}^{top} − E_{f_{(l−1)}}^{top} ≥ (p_{l} − 1)P_{(l−1)} − 2 ≥ P_{(l−1)} − 2. The same inequality is valid in the case q_{l} = 1 (use (4.2) instead of (4.3)). Therefore E_{f_{(l)}}^{top} −E_{f_{(l−1)}}^{top} ≥ 2l−2 for any 2 ≤ l ≤ s, which gives:

(4.14) for irreducible f one has: E_{f}^{top} ≥ s(s − 1).

Now assume that f = ∏_{a} f_{a}. First recall that

(4.15) μ(f) = ∑_{a} μ(f_{a}) + 2 ∑_{a<a′} ν(f_{a}, f_{a′}) − #A + 1.

Now, (4.6), (4.8.a) and (4.15) gives:

(4.16) E_{f}^{top} ≥ ∑_{a} E_{f_{a}}^{top} + 2T + T′ (cf. (4.8) for notations).

Therefore (a) follows from (4.14) and (4.15). In order to prove (b) we need to verify:

(4.17) E_{f}^{top} + ∑_{w∈W}(−e_{w}) − 2#W ≥ 2,

excepting the four cases given in the hypothesis. Consider the invariant I(G) = ∑_{w}(−e_{w}) − 2#W associated with any graph G. After a blowing up #W increases 1, and ∑(−e_{w}) with 2 or 3. Therefore, for any G, I(G) ≥ −1, but if we blew up at least one node of the exceptional divisor, then I(G) ≥ 0. Therefore, if f has at least one factor with ν(f_{a}) ≥ 2, then:

(4.18) I(G_{f}) ≥ 0; and I(G_{f}) = 0 ⟺ f ∼ A_{2k}.

Assume that (4.17) is not true for f. If f is irreducible then ν(f) ≥ 2, hence by (4.14) and (4.18) s = 1. For ν(f) ≥ 4 one has I(G_{f}) ≥ 2, hence by (II): f ∼ A_{2k} or E_{6}. (In fact, if s = 1, then I(G_{f}) = #W − [q/p] − 2.)

Assume that #A > 1, and let S be the number of smooth components. Since E_{smooth}^{top} = 1 one has S ≤ 2. If S = 2 then by (4.18) we can have no other component, hence f ∼ A_{2k−1}. If S ≤ 1, then by (4.18) I(G_{f}) ≥ 0, hence all the tangent lines are distinct (by 4.16). In this case:

(4.19) I(G_{f}) = ∑_{a} I(G_{f_{a}}) + #A − 1.

So, if S = 1, then (4.18), (4.19) and E_{smooth}^{top} = 1 implies f ∼ D_{2k+3}. If S = 0, then similarly: f ∼ A_{2k,2l}. □

IV. The “algebraic” inequality.

4.20. Theorem. If G_{f} is the minimal embedded resolution graph of f and ν is the multiplicity of f, then:

(a) E_{f}^{alg} := ∑_{a∈A} m_{w_{a}} − 3#W(G_{f}) − (ν^{2} − 2ν − 3) ≥ 0,

(b)  ∑_{a∈A} m_{w_{a}} − 3#W(G_{f}) − #A + 1 ≥ 0.

Proof. The proof is similar as the proof of (4.12). First recall that if f is irreducible then ν = p_{1} ⋯ p_{s} (recall p_{1} < q_{1}). If s + 1 then E^{alg} ≥ 0 by (II) if p_{1} = 2 or 3, and by (4.3) if p_{1} ≥ 4. If s ≥ 2, then E_{f_{(l)}}^{alg} ≥ E_{f_{(l−1)}}^{alg} by similar argument as in (4.12). If f = ∏ f_{a}, then by (4.8.b) E_{f}^{alg} ≥ ∑_{a} E_{f_{a}}^{alg}. This last inequality together with table (II) shows (b) in the case ν ≤ 3. If ν > 3 then ν^{2} − 2ν − 3 ≥ #A − 1. (The details are left to the reader.) □

V. The “combinatorial” inequality.

4.21. Theorem. Let G_{f} be the minimal embedded resolution graph of f.

(a) Assume that f ≁ A_{n}(n ≥ 1), E_{6}, E_{8}, A_{2k,2l}(k ≥ 1, l ≥ 1), D_{2k+3}(k ≥ 1). Then:

E_{f}^{com} := ∑_{a∈A} m_{w_{a}} − 4#W(G_{f}) − #A + 1 ≥ 0.

(b) E_{f}^{com} ≥ −6 for any f, and E_{f}^{com} ≥ #A(#A − 1) − 4 if #A ≥ 2.

Proof. For irreducible f with s = 1: E_{f}^{com} ≥ −6, and for p_{1} ≥ 4 : E_{f}^{com} ≥ 0 (use the table (II) and (4.3).) By similar inductive step as in (4.12): E_{f_{(l)}}^{com} ≥ E_{f_{(l−1)}}^{com} + 6. Therefore:

(4.22) E_{f}^{com} ≥ 6(s − 2) for irreducible f.

If #A ≥ 2, write f = g · f_{0} with f_{0} irreducible. With the notation of (4.7.b), if f′_{a} ≠ f′_{0}, then m_{E_{a}}(f′_{a}, G_{f′}) = m_{E_{a}}(f′_{a}, G_{g}), hence by (4.6) and (4.7.b):

(4.23) E_{f}^{com} ≥ E_{g}^{com} + E_{f_{0}}^{com} + 2ν(g) · ν(f_{0}) + 3 + 6T + 2T′.

Notice that E_{smooth}^{com} = −3. Let S be the number of smooth components of f.

Assume #A = 2. If S = 2 then f ∼ A_{2k−1}. (4.22) and (4.23) implies that if S = 1 then either E_{f}^{com} ≥ 0 or f ∼ D_{2k+3}, and if S = 0 then either E_{f}^{com} ≥ 0 or f ∼ A_{2k,2l}. In all cases E_{f}^{com} ≥ −2.

Assume that #A > 2. Notice that E_{f_{0}}^{com} + 2ν(g)ν(f_{0}) ≥ 2#A_{f} − 5, hence (4.23) gives: E_{f}^{com} ≥ E_{g}^{com} + 2(#A_{f} − 1), which provides (b). □

VI. Remarks.

4.24. In the previous inequalities the minimality of the resolution graph G_{f} is necessary. With some additional blowing ups, we (can) keep the invariants μ, ν, m_{w_{a}} constant, but increase #W arbitrarily high. 4.25. For general germs f : (X, x) → (C, 0) (where (X, x) is a normal surface singularity) it is not true anymore that ∑_{a} m_{w_{a}} is “big”. In particular, all the inequalities presented in this section are specific for plane curve singularities, and the negativity of σ(f + z^{N}) is provided exactly by the existence of this type of relations in G_{f}. We expect that similar relations exist for the embedded resolution of higher dimensional hypersurface singularities.

To see that in general ∑ m_{w_{a}} can be small, consider the following diagram with ∑ m_{w_{a}} = 4, but with #W arbitrary high.

[figure: …; -2; -2; -2; -2; -2; -2; (1); (1); (2); (2); (2); (2); (1); (1)]

§5. Inequalities satisfied by σ_{N}

For the convenience of the reader we recall the notations: #W (resp. #E) denotes the number of vertices (resp. edges) of the minimal embedded resolution graph G_{f} of f, A_{f} is the number of irreducible components of f, ν_{f} is its multiplicity and μ_{f} is its Milnor number.

I. “Topological” inequalities

Let ϵ_{f} be 1 if f ∼ A_{n}(n ≥ 1), A_{2k,2l}(k ≥ 1, l ≥ 1), D_{2k+3}(k ≥ 1), E_{6}, otherwise ϵ_{f} = 0. 5.1. Theorem. (a)

σ(f + z^{2}) ≤ −\frac{1}{2}(μ_{f} + #A − 1).

(b) For any f and N:

σ(f + z^{N}) ≤ \frac{1 − N}{N} #E + \frac{1 − N^{2}}{3N} (μ_{f} − #W − 2).

(c) If N > B(f) (cf. 3.12.b), then:

σ(f + z^{N}) ≤ \frac{1 − N^{2}}{3N} (μ_{f} − 3ϵ_{f}).

(d) If (N, m_{w}) = 1 for any w ∈ W(G_{f}), then:

σ(f + z^{N}) ≤ \frac{1 − N}{N} (#W + #E) + \frac{1 − N^{2}}{3N} (μ_{f} − 3ϵ_{f}).

and with the same assumption (below p_{g} is the geometric genus):

σ(f + z^{N}) ≤ −μ(f + z^{N})/3 or, equivalently: p_{g}(f + z^{N}) ≤ μ(f + z^{N})/6.

Notice that in the inequality (5.1.a) the coefficient 1/2 of μ is the same as in the inequality σ ≤ −μ/2 − μ_{0} of Tomari [16]. Proof. By (2.2) there are at least two vertices w such that m_{w} is odd, hence (3.7.b) and (4.12.a) give (a). (b-c) follows from (4.12) and (3.12); and the first part of (d) from (4.12) and (3.8).

Recall the standard relations (for a germ g : (C^{3}, 0) → (C, 0)): μ = μ_{0} + μ_{+} + μ_{−}, σ = μ_{+} − μ_{−} and 2p_{g} = μ_{0} + μ_{+}. The hypothesis (N, m_{w}) = 1 for any w implies that μ_{0}(f + z^{N}) = 0 (use, e.g. A’Campo’s theorem about the characteristic polynomial of f in terms of m_{w}’s). Therefore σ ≤ −μ/3 is equivalent to p_{g} ≤ μ/6. We already know from the first part of (d) that these inequalities are valid for f ≁ A_{n}, A_{2k,2l}, D_{2k+3}, E_{6}. In the sequel we verify these cases.

Let g : (C^{3}, 0) → (C, 0) be an isolated singularity with Milnor lattice L_{g}. Let g_{t} be a deformation of g such that g_{0} = g and g_{t}, for t ≠ 0 small, has k singular points with Milnor lattices L_{1}, …, L_{k}. Then there is a natural embedding ⊕_{i}L_{i} ↪ L_{g}. If c is the codimension of this embedding then: σ(g) ≤ c+∑_{i} σ(L_{i}). In the suspension case, any deformation f_{t} of f provides an embedding in L(f + z^{N}).

Now we verify the exceptional cases. If f ∼ A_{n} then σ ≤ −μ/3 follows from (5.1.a). Actually, (5.1.a) implies: (∗) σ(x^{2} + y^{2k+1} + z^{N}) ≤ −k(N − 1). If f ∼ D_{2k+3}, then if we put the components in generic position, we obtain an embedding A_{2k} ⊕ A_{1} ⊕ A_{1} ↪ D_{2k+3}. Therefore σ(D_{2k+3} + z^{N}) ≤ σ(A_{2k} + z^{N}) + σ(A_{1} + z^{N}) + (N − 1) ≤ −k(N − 1) − 2(N − 1) + (N − 1) < μ/3 (here we used (∗)). If f ∼ A_{2k,2l}, one has the embedding A_{2k} ⊕ A_{2l} ⊕ 4A_{1} ↪ A_{2k,2l}. Recall that μ(A_{2k,2l}) = 2k+2l+7. Now (∗) gives easily the result if k+l ≥ 4, otherwise we use σ(x^{2} + y^{3} + z^{N}) ≤ −4(N − 1)/3 ( provided that (N, 2) = 1) which can be easily verified by Brieskorn’s formula [2]. The case f ∼ E_{6} is again Brieskorn type and its verification is left to the reader. □

II. “Algebraic” and “combinatorial” inequalities, and the limit lim_{N→∞} −σ_{N}/N

Similarly as in the “topological case” we can a list of inequalities corresponding to the different cases. In the next theorem we present only a few, the interested reader can formulate all the others:

5.2. Theorem. For any f and N:

σ(f + z^{N}) ≤ \frac{1 − N}{N} #E + \frac{1 − N^{2}}{3N} (ν_{f}^{2} − 2ν_{f} − 3 − #A + 1).

σ(f + z^{N}) ≤ \frac{1 − N}{N} #E + \frac{1 − N^{2}}{3N} (#W + #A(#A − 1) − 6).

σ(f + z^{N}) ≤ \frac{1 − N}{N} #E ≤ 0.

Proof. Use (3.12), (4.20) and (4.21). Our first inequality in (5.2) is similar to the main result of [1]. 5.3. Theorem. The limit lim_{N→∞}(−σ_{N})/N = η(f; 1) satisfies:

η(f; 1) ≥ (μ_{f} + #W + #A − 3 + B)/3,

where B := ∑_{w∈W}(−e_{w}) − 2#W ≥ −1; and also:

η(f; 1) ≥ (3#W + #A − 8)/3.

Proof. Use (3.5) and (4.12).

III. Conjectures

For any f : (C^{2}, 0) → (C, 0) and N conjecturaly the following inequalities hold:

1. σ(f + z^{N}) ≤ −\frac{N + 1}{3N}μ(f + z^{N}).

2. η(f; 1) + η(f; N) − η(f; N + 1) ≥ 0

(which would imply the monotonity: σ_{N+1} ≤ σ_{N}.)

3. η(f; 1) + η(f; N) − η(f; N + 1) ≥ μ_{f}/3,

(which would imply, for example, the inequality σ ≤ −μ/3 for f + z^{N}).

Appendix

In this appedix we separate some properties of the generalized Dedekind sums [14, 21]. They are defined for arbitrary non–zero integers a, b, c by:

s(b, c; a) = ∑_{k=1}^{a}((\frac{kb}{a}))((\frac{kc}{a})),

where ((x)) is the usual function defined via the fractional part {x} as:

((x)) = \{ \begin{matrix}\{x\} − 1/2 & if x ∉ Z \\ 0 & otherwise. \end{matrix}

Below (a_{1}, ⋯, a_{n}) denotes the greatest common divisor of these numbers. By similar argument as in [10] (A.1), one can prove that:

(A.1) s(b, c; a) = (a, b, c) · s(\frac{b(a, b, c)}{(a, b)(b, c)}, \frac{c(a, b, c)}{(a, c)(b, c)}; \frac{a(a, b, c)}{(a, b)(a, c)}).

The famous generalization of the reciprocity law of Dedekind given by Rademacher [14] (see also [21]) asserts that: if a, b, c are strict positive, mutually coprime integers, then: s(b, c; a) + s(c, a; b) + s(b, a; c) = −1/4 + (a^{2} + b^{2} + c^{2})/12abc. Using (A.1), this relation for arbitrary, strict positive integers reads as:

(A.2) s(b, c; a)+s(c, a; b)+s(b, a; c) = −\frac{(a, b, c)}{4}+\frac{a^{2}(b, c)^{2} + b^{2}(a, c)^{2} + c^{2}(b, a)^{2}}{12abc} .

The inequality (A.3) of [10] together with (A.1) of this paper provide: Corollary: For any integers a, b, c (a > 0) one has:

(A.3) | s(b, c; a) |≤ \frac{(a − 1)(a − 2)}{12a}.

(A.4) | s(b, c; a) |≤ \frac{a^{2} − (a, b)^{2}}{12a} .

References

Math., 2, 1-14 (1966).

Math. Ann., 232, 85-98 (1978).

[1] Ashikaga, T.: The Signature of the Milnor fiber of Complex Surface Singu-

larities on Cyclic Coverings, preprint, (1995).

[2] Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten, Invent.

[3] Brieskorn E. and Knörrer H.: Plane algebraic curves, Birkhäuser, 1986.

[4] Durfee, A.: The Signature of Smoothings of Complex Surface Singularities,

[5] Eisenbud, D. and Neumann, W.: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Studies 110, Princeton University Press, 1985.

[6] Mordell, L.J.: Lattice points in a tetrahedron and generalized Dedekind sums, J. Indian Math., 15, 41-46 (1951).

[7] Némethi, A.: The equivariant signature of hypersurface singularities and eta– invariant, Topology, 34, 243-259 (1995).

[8] Némethi, A.: The eta–invariant of variation structures I, Topology and its Applications, 67, 95-111 (1995).

[9] Némethi, A.: The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compositio Math., 98, 23-41 (1995).

[10] Némethi. A.: Dedekind sums and the signature of f(x, y)+z^{N}, preprint 1996.

[11] Neumann, W.: Splicing Algebraic Links, Advanced Studies in Pure Math., 8, 349-361 (1986). (Proc U.S.-Japan Seminar on Singularities 1984)

[12] Neumann, W. and Wahl, J.: Casson invariant of links of singularities, Comment. Math. Helv. 65, 58-78 (1991).

[13] Pommersheim, J. E.: Toric varieties, lattice points and Dedekind sums, Math. Ann, 295, 1-24 (1993).

[14] Rademacher, H.: Generalization of the Reciprocity formula for Dedekind sums, Duke Math. Journal, 21, 391-397 (1954).

[15] Rademacher, H. and Grosswald, E.: Dedekind sums, The Carus Math. Monographs, 16, 1972.

[16] Tomari, M.: The inequality 8p_{g} ≤ μ for hypersurface two–dimensional isolated double points, Math. Nachr., 164, 37-48 (1993).

[17] Wahl, J.: Smoothings of normal surface singularities, Topology, 20, 219-246 (1981).

[18] Xu, Y. and Yau, S.S.T.: The inequality μ ≥ 12p_{g} − 4 for hypersurface weakly elliptic singularities, Contemporary Math., 90, 375-344 (1989).

[19] Xu, Y. and Yau, S.S.-T.: Durfee’s conjecture and coordinate free characterization of homogeneous singularities, J. Diff. Geometry 37, 375-396 (1993).

[20] Xu, Y. and Yau, S.S.-T.: A sharp estimate of the number of integral points in a tetrahedron, J. reine angew. Math., 423, 199-219 (1992).

[21] Zagier, D.: Higher dimensional Dedekind sums, Math. Ann., 202, 149-172 (1973).

[22] Weber, C. (Ed.): Noeuds, tresses et singularités, Monographie No 31 de L’Enseignement Mathématique, 1983. (Problem C. of W. Neumann)