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

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

§1. Introduction

Almost twenty years ago, A. Durfee conjectured [4] that the signature σ(g) of the Milnor fiber associated with an isolated complete intersection singularity g : (C^{k+2}, 0) → (C^{k}, 0) is negative. On the other hand, J. Wahl found a smoothing of a (non–complete intersection) singularity with positive signature [25]. But matematicians working in singularity theory firmly believe in Durfee’s conjecture (at least) in the case of hypersurfaces (see, for example, the list of open problems in [30]).

Actually, in the same paper, [4] Durfee conjectured a much-much stronger inequality, namely that the geometric genus p_{g} and the Milnor number μ satisfy:

(∗) p_{g} ≤ μ/6.

Using some examples, e.g. the germ x^{a} + y^{a} + z^{a}, it is easy to verify that the coefficient 1/6 in this inequality is the optimal one (cf. (4.5) and the end of (4.10)).

For quasi-homogeneous hypersurface singularities, Y. Xu and S. S.-T. Yau verified the inequality (∗) [27, 28] (even a stronger version, see belove). In this paper we will prove the inequality (∗) for singularities of type f(x, y) + z^{N}, where f is an irreducible plane curve singularity (Theorem 5.1), and also a sharper inequality for Brieskorn singularities. Moreover, in section 5, we verify that for any reducible curve singularity f = ∏_{i=1}^{r} f_{i} (f_{i} irreducible) one has:

−3σ(f + z^{N}) ≥ (N − 1)(∑_{i} μ(f_{i}) + 3r − 3) ≥ (N − 1)(m_{0}(f) + 2r − 3) ≥ 0,

where m_{0}(f) denotes the multiplicity of f. In particular, σ(f + z^{N}) < 0 for any f.

As a first step, we find a formula for the signature of f + z^{N} (f arbitrary) in terms of generalized Dedekind sums associated with the multiplicities of the irreducible exceptional divisors of the embedding resolution of f (Theorem 2.3). The proof involves results about plane curve and suspension singularities: σ(f + z^{N}) is computed via the eta–invariant of f ([9, 10]), the eta–invariant is computed from the spectral pairs (or equivariant Hodge numbers) of f and the number N (as in [9]), and finally, the set of spectral pairs of f can be computed from the embedded resolution graph of f (see [23] and [14]).

This formula (2.3) generalizes Proposition (2.5) of W. Neumann and J. Wahl [17], where the signature of g = f(x, y) + z^{N} is computed in the case when the link of g is an integer homology sphere (equivalently: f is irreducible, and N is relatively prime with the Newton pairs of f, cf. [loc.cit.]). (Unfortunately, the method of [17] cannot be used in our general situation, from this reason we developed the new approach.)

Theorem 2.3 applied for the particular case of Brieskorn singularities x^{a}+ y^{b} +z^{c} (via Brieskorn formula of the signature [3]) gives (see 4.6) the number of lattice points in the open tetrahedron (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c) in terms of Dedekind sums. This provides a new proof of this famous lattice point problem, which was solved by Mordell [8] in the case when (a, b, c) are pairwise relative prime numbers, and recently by Pommersheim [18] in the general case. (For the importance of this problem in the theory of toric varieties, see [18].)

The general result of section 2, applied for irreducible germs f, gives in section 3 nice inductive formulas (see 3.2). In particular, it is proven that the signature of f + z^{N} is a sum of signatures of Brieskorn type singularities (see 3.3). (If N is generic, then we recover the result (2.5) of [17], mentioned above.) This provides the inductive step in the proof of Durfee’s conjecture. On the other hand, we emphasize, that the inequality (∗) for a Brieskorn singularity does not imply the general case: we need stronger inequalities for the Brieskorn case.

For quasi-homogeneous singularities g : (C^{3}, 0) → (C, 0) (in particular, for Briekorn singularities too), Y. Xu amd S. S.-T. Yau [27, 28] proved the inequality 6p_{g} ≤ μ − m_{0}(g) + 1 (which is stronger that (∗), but not sufficient strong for our inductive step). Our sharp inequality is presented in Theorem 4.1. Its proof is technical, it uses identities and inequalities about Dedekind sums (e.g. the reciprocity law). These facts are separated in the Appendix.

We mention (as already [27, 28] suggest) that the difficult part in the proof of Durfee’s conjecture is not the computation of the signature in terms of Dedekins sums, but to find good estimates for the Dedekind sums.

In the last years, different inequalities of type p_{g} ≤ c·μ are proven (where c is some constant) for particular classes of singularities (see, e.g. [24, 26]).

Notice that our class is rather general with respect to these classes.

Recently appeared a preprint of Ashikaga [1] in which he proves that −3σ(f + z^{N}) has at least the order of m_{0}(f)^{2} · N. Our approach in this article is more conceptual, shorter and gives stronger results (actually the optimal one) in the case when f is irreducible. The case f reducible is discussed in [12]. We mention, that the negativity of the signature of f + z^{N} (f arbitrary) is proved in [13] by a short, rather elementary argument.

We will use the following notations: μ_{0} is the dimension of the kernel of the Milnor lattice L of a singularity, μ_{+} (resp. μ_{−}) is the dimension of a maximal subspace where L is positive (negative) definite. The generalized Dedekind sum (cf. [19], [29]) s(b, c; a) is defined by:

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

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

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

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

2.1. 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.

For any w ∈ W, we denote by V_{w} the set of vertices v ∈ V adjacent to w. Set δ_{w} = #V_{w} for any w ∈ W. If δ_{w} > 2, then w ∈ W is called “rupture point”. The set of rupture points is denoted by R.

The graph G_{f} is decorated by the self–intersection (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.

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}}). With these notations one has:

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

σ_{N}(f) = 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∈R} ∑_{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. The last term is a sum of generalized Dedekind sums.

Proof: Let F be the Milnor fiber of f and (H; b, h, V) the variation structure associated with f (see [9, 10, 11]). This means that H = H_{1}(F, C); b : H → H^{∗} corresponds to the intersection form ⟨, ⟩ : H⊗H → C via b(x)(y) = ⟨x, y⟩; h : H → H is the monodromy operator and V : H^{∗} → H is the variation map (here H_{1}(F, ∂F) is identified with the dual space H^{∗} via the perfect pair H_{1}(F) ⊗ H_{1}(F, ∂F) → C).

For any natural number K we define V (K) := (I+h+⋯+h^{K−1})∘V. Then the system (H; b, h^{K}, V (K)) has a spectral decomposition ⊕_{χ}(H_{χ}; b_{χ}, (h^{K})_{χ}, V (K)_{χ}) with respect to the automorphism h^{K}; i.e. H_{χ} is the χ–generalized eigenspace of h^{K}, and the spectral decomposition is compatible with the extra-structure (b, V (K)) (see [9]).

For any χ we define:

η(f; K)_{χ} = \{ \begin{matrix}(1 − 2c) · signature(ib_{χ}) & if χ = e^{2πic}, 0 < c < 1; \\ −signature[(1 + (h^{K})_{χ}^{−1})V (K)_{χ}] & if χ = 1. \end{matrix}

Now, for any K > 0, the eta–invariant of f is defined by [9, 10]:

η(f; K) = ∑_{χ} η(f; K)_{χ}.

In [9, 10] it is proved that: σ_{N}(f) = η(f; N) − N · η(f; 1).

The eta–invariant η(f; K) can be computed from the spectral pairs of f. First we recall that the set of spectral pairs Spp(f) ∈ Z[Q × Z] of f codifies the equivariant Hodge numbers {h_{λ}^{pq}} of the mixed Hodge structure of the vanishing cohomology of f via the relation [22]:

(2.4) Spp(f) = ∑_{(α,ω)} h_{exp(−2πiα)}^{1+[−α],ω+s_{α}−1−[−α]} · (α, ω),

where s_{α} = 0 if α ∉ Z, and = 1 otherwise (cf. 5.20 [9]).

The set of spectral (or characteristic) numbers is defined ([22]) by

Sp(f) = ∑ α ∈ Z[Q] (the sum over the spectral pairs (α, ω)).

Then we set (cf. 5.20. [9]):

Σp_{λ,±}(f) = #{c| c is a spectral number with e^{−2πic} = λ, and (−1)^{[c]} = ±1}.

Then the general result (5.21) [9] in the plane singularity case reads as:

(2.5) η(f; K) = ∑_{{λ^{K} =1 \atop λ≠1}} h_{λ}^{11}− ∑_{{λ^{K} ≠1 or λ=1 \atop λ=e^{−2πic};0≤c<1}} (1−2\{Kc\})(Σp_{λ,−}(f)−Σp_{λ,+}(f)).

In particular, η(f; K) can be computed explicitely from the spectral pairs. On the other hand, the set of spectral pairs can be computed in terms of the embedded resolution graph G_{f} (see [23] and [14]).

The formula (6.5) in [14] (via the transformation (2.11) [14]) reads as:

Spp(f) = (#A − 1)(0, 1) + ∑_{e∈E} ∑_{0<k<m_{e}}((\frac{k}{m_{e}} , 0) + (−\frac{k}{m_{e}} , 2))

− ∑_{w∈W} ∑_{{0<k<mw \atop R^{k}_{w}=0}} ((\frac{k}{m_{w}} , 0) + (−\frac{k}{m_{w}} , 2))

+ ∑_{w∈W} ∑_{{0<k<mw \atop R^{k}_{w}≠0}} (R_{w}^{k} − 1)((1 − \frac{k}{m_{w}} , 1) + (−1 + \frac{k}{m_{w}} , 1)),

where R_{w}^{k} := ∑_{v∈V_{w}}{k · m_{v}/m_{w}}. Notice that Σp_{1,−} = 0, Σp_{1,+} = h_{1}^{11} = #A − 1. In order to compute ∑_{λ^{K}=1,λ≠1} h_{λ}^{11} in (2.4), we need spectral pairs with [−α] = 0, s_{α} = 0, ω = 2 and αK ∈ Z. Hence, by the above formula:

(2.6)  ∑_{λ^{K}=1,λ≠1} h_{λ}^{11} = ∑_{e} ∑_{{0<k<me \atop me|Kk}} 1 − ∑_{w} ∑_{{0<k<mw \atop R^{k}_{w}=0,mw|Kk}} 1

which is exactly ∑_{e}((K, m_{e}) − 1) − ∑_{w}((K, M_{w}) − 1). In the sum Σ := Σ_{λ^{K}≠1}(1−2{Kc})(Σp_{λ,−} −Σp_{λ,+}), a set of numbers of type ∑_{k}(k/m, ω) have no contribution, therefore:

Σ = ∑_{w} ∑_{k} R_{w}^{k}((1 − 2\{K(1 − \frac{k}{m_{w}})\}) − (1 − 2\{K \frac{k}{m_{w}} \})) = 4 ∑_{w} ∑_{k} R_{w}^{k}((\frac{kK}{m_{w}})).

In this sum the non-rupture points have trivial contribution. Indeed, by (2.2), if δ_{w} ≤ 2, then ∑_{v∈V_{w}}((km_{v}/m_{w})) = 0 for any k.

Notice that ∑_{k}((\frac{Kk}{m_{w}})) = 0, so the result follows. □

2.7. Remark. If f is irreducible (#A = 1), or if the monodromy of f has finite order, then

∑_{e∈E}((N, m_{e}) − 1) − ∑_{w∈W}((N, M_{w}) − 1) = 0.

Indeed, the above expression in the proof of (2.3) is identified with ∑_{λ^{N} =1,λ≠1} h_{λ}^{11} (cf. 2.6). But for λ ≠ 1, the Hodge number h_{λ}^{11} is exactly the number of Jordan blocks of the monodromy operator of f with eigenvalue λ and size two. If f is irreducible then this number is zero by a result of Lê [6]. This follows also from [15, 16] (cf. also [5]).

§3. σ_{N}(f) for irreducible germs f

It is well-known that some invariants of plane curve singularities behave additively with respect to the splicing of their embedded resolution graph (or with respect to their rupture points). We ask the following natural question: is the signature σ_{N}(f) of f(x, y) + z^{N} additive in this sense? Reformulating this: is σ_{N}(f) a sum of signatures of some Brieskorn type singularities? Actually, the isometric structure or the Seifert form of f is “additive” only if f is irreducible (cf. [5] (15.3)), therefore we can expect a nice answer of this type only in this case.

Consider an irreducible germ f with Newton pairs (p_{i}, q_{i})_{i=1}^{s} (cf. [5] page 49). Define also the integers {a_{i}}_{i=1}^{s} by:

(3.1) a_{1} = q_{1}, and a_{i+1} = q_{i+1} + p_{i+1}p_{i}a_{i} if i ≥ 1.

We will denote by σ(a, b, c) (respective by μ(a, b, c)) the signature (respective the Milnor number (a − 1)(b − 1)(c − 1)) of the Brieskorn singularity g(x, y, z) = x^{a} + y^{b} + z^{c}.

3.2. Theorem. Assume that f is irreducible with Newton pairs (p_{i}, q_{i})_{i=1}^{s}. If d_{i} = (N, p_{i+1} ⋯ p_{s}) for 1 ≤ i < s and d_{s} = 1, then:

σ_{N}(f) = ∑_{i=1}^{s} d_{i} · σ(a_{i}, p_{i}, N/d_{i}).

Proof. By (2.3) and (2.7):

η(f; N) = 4 · ∑_{w∈R} ∑_{v∈V_{w}} s(m_{v}, N; m_{w}).

The multiplicities {m_{w}}_{w∈R} and {m_{v}}_{v∈V_{w}} can be computed easily using the Eisenbud–Neumann splicing graph of f [5] (page 51). The “rupture points” {w_{1}, …, w_{s}} of this grapf correspond to the rupture points of G_{f} (for the correspondence, see [loc. cit. chap. V]); and the multiplicity m_{w_{i}} is the multipilicity of a generic fiber of the i^{th} Seifert component. By [loc. cit. chap. III]: m_{w_{i}} = a_{i}p_{i}p_{i+1} ⋯ p_{s}. Since f is irreducible, each rupture point has exactly three adjacent vertices in G_{f}. Their multiplicities {m_{v}} can be computed as follows (see [16] or [21] p. 127). Since (a_{i}, p_{i}) = 1, we can consider two integers u_{i} and v_{i} such that a_{i}u_{i} + p_{i}v_{i} = 1. Then the multiplicities {m_{v}}_{v∈V_{w_{i}}} (modulo m_{w_{i}}) are:

(−u_{i}a_{i}p_{i+1} ⋯ p_{s}, −v_{i}p_{i}p_{i+1} ⋯ p_{s}, p_{i+1} ⋯ p_{s}).

On the other hand, the multiplicity of the unique rupture point of G(x^{a_{i}} + y^{p_{i}}) is a_{i}p_{i}, and (by the same argument) the multipilicities of the adjacent vertices are (−u_{i}a_{i}, −v_{i}p_{i}, 1) (cf. the proof of (4.1) too). Therefore, by (A.1) η(f; N) = ∑_{i=1}^{s} d_{i} · η(x^{a_{i}} + y^{p_{i}}; N/d_{i}). Now (2.3) applied for f and x^{a_{i}} + y^{p_{i}} gives the result. □ 3.3. Corollary. Fix an irreducible singularity f with Newton pairs (p_{i}, q_{i})_{i=1}^{s} as above. For 1 ≤ l ≤ s let f_{(l)} : (C^{2}, 0) → (C, 0) be an irreducible singularity with Newton pairs (p_{i}, q_{i})_{i=1}^{l}. Then for any 1 < l ≤ s one has:

σ_{N}(f_{(l)}) = σ(a_{l}, p_{l}, N) + (N, p_{l}) · σ_{N′}(f_{(l−1)}),

where N′ = N/(N, p_{l}). Proof. Use (3.2) and the identity (N, ab) = (N, a) · (N/(N, a), b). □ 3.4. Remark. (3.3) can be compared with the following relation. For simplicity, we denote μ_{0}(f + z^{N}) by μ_{0,N}(f). Then one has:

μ_{0,N}(f_{(l)}) = μ_{0}(a_{l}, p_{l}, N) + (N, p_{l}) · μ_{0,N′}(f_{(l−1)}),

where N′ = N/(N, p_{l}) as above.

In order to see this, notice that μ_{0,N}(g) is #{λ eigenvalue of the monodromy of g; λ^{N} = 1, λ ≠ 1} (counted with multiplicities). Now, the above relation follows from the following formula of the characteristic polynomials of the monodromy operators:

P_{f_{(l)}}(t) = P_{x^{a_{l}}+y^{p_{l}}}(t) · P_{f_{(l−1)}}(t^{p_{l}}).

3.5. Remark. Compare (3.3) and (3.4) with the corresponding inductive formula of the Milnor numbers. (Below μ_{N}(f_{(l)}) denotes the Milnor number of f_{(l)} + z^{N}):

μ_{N}(f_{(l)}) = μ(a_{l}, p_{l}, N) + p_{l} · μ_{N}(f_{(l−1)}).

Notice that μ_{N} preserves the corresponding information from f_{(l−1)} with “higher weight” (p_{l} versus (N, p_{l})).

§4. Inequalities for Brieskorn singularitites

It is well-known that the signature σ(a, b, c) of Brieskorn singularities is negative. Therefore, by (3.2), for any irreducible singularity f, the signature of f + z^{N} is negative. In the next section we will prove a stronger (and in some sense, optimal) inequality: −3σ(f + z^{N}) ≥ μ(f + z^{N}) + 3μ_{0}(f + z^{N}). A short analysis of the inductive formulas (3.3), (3.4) and (3.5) shows that the same inequality for Brieskorn singularities does not imply the general one. The needed, stronger inequality, which is valid for Brieskorn singularities is given in the next theorem.

4.1. Theorem. Assume that a ≤ b ≤ c. Then:

−3σ(a, b, c) ≥ μ(a, b, c) · \frac{a + 1}{a} + (c − 1) · \frac{a − 1}{a}.

In particular, for any 2 ≤ a ≤ c and 2 ≤ b one has:

−3σ(a, b, c) ≥ μ(a, b, c) + (a − 1)(c − 1)b/a.

Proof. We will apply (2.3) for the germ f(x, y) = x^{a} + y^{b} and N = c. We will use the notations (a, b) = d, a = dp, b = dq. Obviously q ≥ p ≥ 1 and N ≥ dq.

The germ f has exactly d irreducible components. There is only one rupture point r with multiplicity m_{r} = dpq.

If p > 1 then in the minimal resolution (or splicing graph) δ_{r} = d + 2. If q > 1 and p = 1 (respectively p = q = 1) then in the minimal graph δ_{r} = d + 1 (respectively δ_{r} = d). But also in these two special cases we can assume δ_{r} = d + 2, if we work with non-minimal graphs (i.e. in the splicing graph we introduce edges weighted with 1). (Actually, if p = q = 1, then σ(a, a, c) is given in (4.5), and the inequality follows easily from the properties of the Dedekind sums listed in the Appendix.)

Since (2.7) is valid, by (2.3):

η(f, N) = d − 1 + 4 ∑_{v∈V_{r}} s(m_{v}, N; dpq).

The multiplicities {m_{v}} (v ∈ V_{r}) can be computed as follows ([16] or [21] p.127, cf. also the proof of (3.2)). Set α_{q} ·p+α_{p} ·q = 1. Then the d+2 multiplicities, modulo m_{r}, are: (−α_{p}dq, −α_{q}dp, 1, …, 1). Therefore [η(f; N)−(d− 1)]/4 = s(−α_{p}dq, N; dpq)+s(−α_{q}dp, N; dpq)+d·s(1, N; dpq). Using the properties of the Dedekind sums: s(α_{p}dq, N; dpq) = (dq, N)·s(α_{p}, N/(dq, N); p) = (dq, N) · s(1, α_{p}^{−1}N/(dq, N); p) = (dq, N) · s(1, qN/(dq, N); p), where α_{p}^{−1} is the inverse of α_{p} modulo p (cf. A.1). By similar computation of the second term, (and again by (A.1)) one has:

(4.2) \frac{1}{4}η(f; N) = \frac{d − 1}{4} −(dq, N)s(\frac{qN}{(dq, N)}, 1; p)−(dp, N)s(\frac{pN}{(dp, N)}, 1; q)+

+d · s(\frac{N}{(dpq, N)}, 1; \frac{dpq}{(dpq, N)}).

In the case N = 1 we can apply the reciprocity law (A.2), once for the first two terms and then for the third one (notice that s(dpq, 1; 1) = 0). We obtain:

(4.3) η(f; 1) = \frac{d^{2}p^{2}q^{2} − p^{2} − q^{2} + 1}{3pq}.

Using the relation σ_{N}(f) = η(f; N) − Nη(f; 1), we have to verify that:

3η(f; N) ≤ 3Nη(f; 1) − (dp − 1)(dq − 1)(N − 1)\frac{dp + 1}{dp} − (N − 1)\frac{dp − 1}{dp}.

The right hand side R(N, d) of this inequality (via (4.3)) is :

R(N, d) = N[dp − \frac{p}{q} + \frac{1}{pq} − 1] + d^{2}pq − dp − \frac{q}{p} + 1.

First notice that we can assume that (N, d) = 1. Indeed, with the notation (N, d) = u consider the plane curve singularity f′ = x^{a/u} + y^{b/u} and N = uN′, d = ud′. In the next lines, we show that the inequality applied for f′ and N′ implies the inequality for f and N (i.e. 3η(f′; N′) ≤ R(N′, d′) implies 3η(f, N) ≤ R(N, d)). For this notice that 3η(f; N) = −3 + u(3η(f′; N′) + 3), hence we have to show that 3η(f′; N′) ≤ R(N′, d′) implies

R(uN′, ud′) + 3 − u · (3η(f′; N′) + 3) ≥ 0.

The left hand side of this last inequality (via the definition of R) has the form Q(u) = Au^{2} +Bu+C with A ≥ 0, 2A ≥ C. Moreover, 3η(f′; N′) ≤ R(N′, d′) is equivalent with Q(1) ≥ 0. Now, Q(1) = A + B + C ≥ 0 and 2A ≥ C imply Q(u) ≥ Q(1) for any u ≥ 2. In this discussion, if a = u, then f′ defines a smooth germ, but in this case Q(1) = 0 hence Q(u) ≥ 0, so the wanted inequality is proven and we can stop. If a > u, then f′ is still a singular germ and Q(u) ≥ Q(1) shows that it is enough to verify (4.1) for f′ and N′ only.

So, in the sequel we will assume (d, N) = 1 (and q ≥ p ≥ 1, d ≥ 1, N ≥ dq and dp ≥ 2).

Using (A.3) one has 3η(f; N) ≤ S, where:

S = −1 + (q, N)(p − 3 + \frac{2}{p}) + (p, N)(q − 3 + \frac{2}{q}) + \frac{d^{2}pq}{(p, N)(q, N)}.

Consider the function S(x, y) = −1 + xA + yB + C/(xy), for 1 ≤ x ≤ q, 1 ≤ y ≤ p (with C > 0). Since S is convex its maximum value is max{S(1, 1), S(1, p), S(q, 1), S(q, p)}. If C ≥ pB and C ≥ qA, then S(1, 1) ≥ S(1, p) and S(1, 1) ≥ S(q, 1). In our case A = p − 3 + 2/p, B = q − 3 + 2/q and C = d^{2}pq, hence S ≤ max{S(1, 1), S(q, p)}. Notice that R(N, d) is an increasing function in N, so it is enough to verify max{S(1, 1), S(q, p)} ≤ R(dq, d).

The inequality S(q, p) ≤ R(dq, d) is always true. Indeed, write the inequality in the form Q(d) = Ad^{2} + Bd + C ≥ 0, and verify that 2A + B ≥ 0. Hence Q(d) ≥ Q(1). Now, Q(1) ≥ 0 if p ≥ 2. If p = 1, then d ≥ 2 (because a = dp ≥ 2), and Q(d) ≥ Q(2) (because 2A + B ≥ 0), and Q(2) ≥ 0 is easy.

The inequality S(1, 1) ≤ R(dq, d) is true unless d = 1 and p = 2. The proof is similar as above; Q(2) ≥ 0 always, and Q(1) ≥ 0 if p ≥ 3.(It is helpful to notice that the expressions Q(1) and Q(2) are increasing functions in q, so it is enough to consider the case q = p.)

If d = 1 and p = 2, then S(1, 1) ≤ R(2q, 1), hence in this case also S ≤ R(2q, d) is valid.

Therefore, the only case which is not covered by the above discussion is: d = 1, p = 2 and q ≤ N ≤ 2q. In this case the needed inequality 3η(f; N) ≤ R(N, d) reads as (cf. (4.2) and (A.1)):

(4.4) − 12(2, N) · s(\frac{2N}{(2, N)}, 1; q) + 12 · s(N, 1; 2q) ≤ R(N, 1).

For the first Dedekind sum we will use again (A.3). Since 2q ≥ N, for the second Dedekind sum, we obtain a better estimate if we transform it (by the reciprocity law (A.2)) in a Dedekind sum with denominator N, and we use (A.3) for this latter one:

12s(N, 1; 2q) = −12s(2q, 1; N) − 3 + \frac{4q^{2} + N^{2} + (N, 2q)^{2}}{2qN}

≤ −12s(\frac{2q}{(2, N)}, 1; \frac{N}{(2, N)}) − 3 + \frac{4q^{2} + N^{2} + 2qN}{2qN}

≤ \frac{N}{(2, N)} + \frac{2(2, N)}{N} + \frac{2q}{N} + \frac{N}{2q} − 5.

Now, we can consider the two cases (2, N) = 1 and (2, N) = 2, and (4.4) follows in both cases by an elementary computation. □ 4.5. Example.

(a) If a = b, then by the proof of (4.1) one has:

σ(a, a, c) = a − 1 + 4a · s(c, 1; a) − c(a^{2} − 1)/3.

In particular, if a = b = c, then −3σ(a, a, a) = (a − 1)(a^{2} + a − 3). Then (4.1) reads as (a − 1)(a^{2} + a − 3) ≥ a(a − 1)^{2}.

(b) If a = 2 and 2 ≤ b ≤ c, then −6σ(a, b, c) ≥ (c − 1)(3b − 2), in particular, σ(a, b, c) ≤ −μ(a, b, c)/2.

Actually, for any germ g = f(x, y)+z^{2}, Tomari [24] proved the inequality σ ≤ −μ/2. Notice that our coefficient (a + 1)/3a of μ in (4.1) generalizes Tomari’s coefficient 1/2 (case a = 2). 4.6. Remark. The invariant μ_{0}(x^{a} +y^{b} +z^{c}) can be computed as follows (see, for example, [7]):

(4.7) μ_{0} = \frac{(a, b)(a, c)(b, c)}{(a, b, c)} − (a, b) − (a, c) − (b, c) + 2.

Using the identities μ = μ_{0} + μ_{+} + μ_{−} and σ = μ_{+} − μ_{−} and Brieskorn’s formula about the signature [3], the computation of σ(a, b, c) is equivalent to the computation of the lattice points in the open tetrahedron with vertices (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c) (which is exactly μ_{+}/2). Via these relations, our theorem (2.3) applied for Brieskorn singularities (more precisely (4.2) and (4.3)) gives the number of these lattice points in terms of Dedekind sums. In the case of pairwise relative prime numbers, this was computed by Mordell [8], and in general case recently by Pommersheim [18]. 4.8. By Durfee’s formula [4], the geometric genus p_{g} of an isolated singularity f : (C^{3}, 0) → (C, 0) satisfies 2 · p_{g} = μ_{0} + μ_{+}. 4.9. Corollary. Assume a ≤ b ≤ c. Then the geometric genus of x^{a} +y^{b} +z^{c} satisfies:

p_{g} ≤ \frac{1}{6}(1 − \frac{1}{2a})μ + \frac{1}{4}μ_{0} − \frac{(c − 1)(a − 1)}{12a}.

Proof. Use (4.1) and the relation: 4p_{g} = μ + σ + μ_{0}. □ The next result proves Durfee’s conjecture in the case of Brieskorn singularities. 4.10. Theorem. For a singularity x^{a} + y^{b} + z^{c} the following holds:

p_{g} ≤ \frac{1}{6}μ.

Proof. The verification of the following fact is elementary but tedious. We leave it to the reader (use (4.7)). Fact. If a ≤ b ≤ c and (a, b, c) ∉ {(a, a, a), (a, a, 2a), (a, 2a, 2a)}, then

μ_{0} ≤ \frac{(c − 1)(a − 1)b}{3a}.

This inequality together with (4.9) implies the result excepting the three cases which appear in the fact. In these exceptional cases σ and μ_{0} can be computed by (4.5) and (4.7). The values of p_{g} are:

a(a − 1)(a − 2)/6, a(a − 1)(a − 2)/3, a(a − 1)(4a − 5)/6,

hence the inequality follows. □

§5. Inequalities for singularities of type f(x, y) + z^{N}

We start with the following theorem. 5.1. Theorem. Assume that f : (C^{2}, 0) → (C, 0) is irreducible. Then the invariants p_{g}, σ, μ and μ_{0} of f + z^{N} satisfy:

p_{g} ≤ \frac{1}{6}μ or equivalently − 3σ ≥ μ + 3μ_{0}.

Proof. Consider the numerical invariants (p_{i}, q_{i})_{i=1}^{s}, (a_{i})_{i=1}^{s} as in §3, and construct the germs f_{(l)} as in (3.3). We will use induction over l. The case l = 1 follows from (4.10). For the inductive step, we need two facts. 5.2. Lemma. a) For any l ≥ 2 one has: a_{l} > 1 + 3p_{l} + p_{l} · μ(f_{(l−1)}).

b) μ_{0}(a_{l}, p_{l}, N) ≤ N − 1. Proof. a) Use induction and (3.1). The inductive step is: a_{l+1} ≥ 1 + p_{l+1}[(p_{l} −1)(a_{l} −1)+p_{l} +a_{l} −1] ≥ 1+p_{l+1}[(p_{l} −1)(a_{l} −1)+4p_{l} +p_{l}μ(f_{(l−1)})] = 1 + p_{l+1}μ(f_{(l)}) + 4p_{l}p_{l+1}. Here we used μ(f_{(l)}) = p_{l}μ(f_{l−1}) + (p_{l} − 1)(a_{l} − 1). For b) use (a_{l}, p_{l}) = 1 in (4.7). □ Using these, we have the following inequalities:

−3σ(a_{l}, p_{l}, N) ≥^{(4.1)} μ(a_{l}, p_{l}, N) + (p_{l} − 1)N(a_{l} − 1)/p_{l}

≥^{(5.2)} μ(a_{l}, p_{l}, N) + (p_{l} − 1)Nμ(f_{(l−1)}) + 3μ_{0}(a_{l}, p_{l}, N).

Above, we used the second inequality from (4.1) (a = p_{l}, b = N and c = a_{l}). Now, the verification of the inductive step (in the proof of 5.1) is easy (use (3.3),(3.4) and (3.5)). □

5.3. When f is reducible, we will need the following inequality. Let g : (C^{3}, 0) → (C, 0) be an isolated singularity with Milnor lattice L_{g}. If g_{t} is 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 an embedding ⊕_{i=1}^{k}L_{i} ↪ L_{g}. If c is the codimension of this embedding, then: σ(g) ≤ c + ∑_{i} σ(L_{i}). 5.4. Now, assume that f : (C^{2}, 0) → (C, 0) defines a reducible isolated singularity. Let f = ∏_{i=1}^{r} f_{i} be its decomposition in irreducible factors. Set f_{t} = ∏_{i} f_{i} ∘ T_{i}(t), where T_{i}(t) = Id + tA_{i} is a generic affine transformation with T_{i}(0) = Id. The degeneration f_{t} + z^{N} of f + z^{N} gives an embedding:

⊕_{i=1}^{r}L(f_{i} + z^{N}) ⊕ (∑_{i<j} m_{0}(f_{i}, f_{j})) · L(x^{2} + y^{2} + z^{N}) ↪ L(f + z^{N}).

(Above m_{0}(,) denotes the intersection multiplicity at the origin.) Since

μ(f) = ∑_{i} μ(f_{i}) + 2 ∑_{i<j} m_{0}(f_{i}, f_{j}) − r + 1,

the codimension c of the embedding is (N − 1)(∑_{i<j} m_{0}(f_{i}, f_{j}) − r + 1). Therefore, (5.3) gives: 5.5. Theorem. Assume that f = ∏_{i=1}^{r} f_{i}. Then:

σ(f + z^{N}) ≤ ∑_{i=1}^{r} σ(f_{i} + z^{N}) + (1 − r)(N − 1).

The inequality −3σ ≥ μ for f_{i} + z^{N} (cf. 5.1) implies: 5.6. Corollary. For arbitrary isolated plane curve singularity f one has:

−3σ(f + z^{N}) ≥ (N − 1)(∑_{i=1}^{r} μ(f_{i}) + 3r − 3).

Notice that μ(f_{i}) ≥ (m_{0}(f_{i}) − 1)^{2} ≥ m_{0}(f_{i}) − 1, hence the inequality from the introduction follows.

§6. Appendix

For definitions, properties about (generalized) Dedekind sums, see [20, 19, 29]. Using the identity

∑_{t=0}^{d−1}((\frac{x + t}{d} · n)) = (n, d) · ((\frac{n}{(n, d)} · x)),

it is not difficult to prove that

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

(Write (a, b) = d, a′ = a/d, x = r/a′ and k = r + a′t with 0 ≤ r ≤ a′ − 1 and 0 ≤ t ≤ d − 1.)

Even more generally, if k divides a and b, a′ = a/k, b′ = b/k, c′ = c/(c, k), then

s(b, c; a) = (c, k) · s(b′, c′; a′).

If (a, b) = 1, then the sum s(b, 1; a) is the classical Dedekind sum s(b, a). The famous reciprocity law of Dedekind asserts that for any two numbers a and b (if they are not relative prime numbers, use (A.1)):

(A.2) s(b, 1; a) + s(a, 1; b) = −\frac{1}{4} + \frac{a^{2} + b^{2} + (a, b)^{2}}{12ab} .

Notice that s(b, c; 1) = 0, hence by (A2) s(1, 1; a) = (a − 1)(a − 2)/(12a). Moreover: Lemma. (see, e.g. [2], page 208) Assume that a > 0 and (a, b) = 1. Then:

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

Proof. By Cauchy–Schwartz inequality: |s(b, 1; a)| = | ∑((bk/a))((k/a))| ≤ (∑((bk/a))^{2} ∑((k/a))^{2})^{1/2} = s(1, 1; a). □

Actually, using (A.1), the above inequality is valid even if (a, b) ≠ 1.

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e-mail address: nemethi@math.ohio-state.edu