Surface Singularities on Cyclic Coverings and an Inequality for the Signature

Tadashi Ashikaga

Introduction.

Let (B, P_{0}) be a germ of a plane curve singularity g(x, y) = 0 at the origin of C^{2}. By a germ (V, P) of an n-fold cyclic cover (n ≥ 2) branched along (B, P_{0}), we mean a germ of 2-dimensional isolated hypersurface singularity defined by the equation f(x, y, z) := z^{n} + g(x, y) at the origin of C^{3}, which we simply call a singularity of cyclic type. The aim of this paper is to describe analytic and topological invariants of (V, P) via a certain algebro-geometric method. Especially, for the signature of the Milnor fiber σ(V, P), we prove a certain inequality, which naturally induce an affirmative answer to the negativity conjecture for the signature posed by Durfee [D, p.96] in the case of a singularity of cyclic type.

In §1, we investigate the invariants of a complete surface S which is a cyclic covering over a nonsingular surface, where we admit non-normal singularities on S. Especially we define the “Milnor number” of such a non-normal surface in some sense, and prove a Noether type formula in Proposition 1.5.

In §2, we first realize a singularity of cyclic type on a complete surface, and resolve it by the method of Hirzebruch [Hi1]. Next we express the Milnor number and the geometric genus of (V, P) in terms of the non-reduced divisor which naturally appears at the final step of the resolution process by using Proposition 1.5 and the Esnault-Viehweg formula [E, V].

We note that, by the reason that these invariants are essentially described in A’Campo [Ac2] and in Esnault[E], our contribution in §§1, 2 is only to propose another formalism. Our motivation of this formulation is to generalize to the arbitorary covering degree of Horikawa’s method [Ho, §2] for double covering in some sence.

In §3, we define the improved singularity (V^{∨}, P^{∨}) of (V, P) which has the following property: The number of times needed of blow-ups of the branch curve of (V^{∨}, P^{∨}) such that the reduced scheme of its total transform has normal crossing is one less than that of (V, P) in general. We also calculate the difference of the invariants between (V, P) and (V^{∨}, P^{∨})

The method of the construction of the improved singularity is a combinatorial interpretation of the “local perturbation method” on the way of the embedded resolution process of a plane curve singularity appeared in A’Campo [Ac1], and has some natural connection to A’Campo and Gusein-Zade theory [Ac1, G]. We will publish the topological meaning of this process as well as another application in our forthcoming paper.

In §4, by using Hirzebruch-Durfee-Laufer’s formula [Hi2, D, L] and our previous result, we prove a certain formula for σ(V, P) in Proposition 4.7.

§§5 and 6 are devoted to estimate the signature. Our method is, roughtly speaking, to show that the signature of the improved singularity is not less than that of the original singularity. We note that, after producing our process inductively, we finally reach the singularity (W, P) whose branch curve is an ordinary singularity. Therefore we can compare σ(V, P) with σ(W, P). Our main result is the following:

Theorem. Let (B, P_{0}) be a germ of a plane curve singularity. Let (C, P_{0}) be a germ of an ordinary plane curve singularity whose multiplicity coincides with the multiplicity of (B, P_{0}). Let (V, P) and (W, P) be germs of n-fold cyclic cover branched along (B, P_{0}) and (C, P_{0}), respectively. Let σ(V, P) (resp. σ(W, P)) be the signature of the Milnor fiber of (V, P) (resp. (W, P)). Then we have

σ(V, P) ≤ σ(W, P).

Furtherover, the equality σ(V, P) = σ(W, P) holds if and only if (B, P_{0}) itself is an ordinary singularity.

We remark that σ(W, P) is explicitely calculated and its negativity is well-known. Therefore the above theorem induce an answer to the problem of Durfee [D,p.96] in the case of a singularity of cyclic type as follows:

Corollary. Let (V, P) be a germ of analytic function f(x, y, z) = z^{n} + g(x, y) such that (V, P) defines at most an isolated singularity. Then we have

σ(V, P) ≤ 0.

Furthermore, the equality σ(V, P) = 0 holds if and only if (V, P) is a germ of a nonsingular point.

As a topological approach to the signature of a singularity of cyclic type, Neumann and Wahl [NW] show that, under the assumption that the link L of the singularity is a Zhomology sphere, σ coincides with the (1/8)-time the Casson invariant of L. As they proved in [NW, Proposition 2.1], the assumption for L to be a homology sphere is somewhat strong, and so it seems an interesting problem to avoid it by extending the notion of the Casson invariant. We note that, in the double point case z^{2} + g(x, y), it is classically known that σ coincides with the signature of the symmetrized Seifert form of the classical link of the branch curve g(x, y), which is the compound torus link (cf.[Sh]). In this case, since the signature of the Seifert form changes sign when the link changes into its mirror image, the negativity of σ means that the mirror image of a classical algebraic link cannot be algebraic. Therefore one can imagine that our result corresponds to some feature of the “one-sided chirality” of the non-classical link arising from such a surface singularity.

From another point of view, the Durfee problem is directly connected with the upper bound problem of the geometric genus of the singularity as is mentioned in [D, p.97]. For instance, we refer to [XY1, XY2, To1, To2, As] for some types of singularities, whereas Wahl [W] found a non-complete intersection singularity whose signature of the Milnor fiber is positive.

On the other hand, general and powerful approachs for the signature of singularity have been done via mixed Hodge theory and Seifert geometry ( [Ne, St1, St2, SSS] etc.).

For the signature of weighted homogeneous singularities, many work have been done ([Br, HZ, HM, St2, E, XY2, FMS, NW] etc.).

For recent global study of cyclic coverings of surfaces, we refer Sakai[Sa]. For singularities on Galois coverings, we refer Tsuchihashi [Ts].

Acknowledgement. I specially thank to Professor Masataka Tomari and Professor Kazuhiro Konno for available discussions and advices. The proof of Lemma 3.7 is due to Tomari. I also thank to Professor Noboru Nakayama, Professor Shouetsu Ogata, Professor Tadashi Tomaru, Professor Tohsuke Urabe and Professor Mikio Furushima for useful advices for certain parts of the paper. I thank to Professor Tadao Oda and Professor Makoto Namba for encouragement.

I thank to Max-Planck-Institute für Mathematik in Bonn for its hospitality, where the paper was revised.

§ 1. Cyclic coverings.

1.1 Let W be a complete nonsingular analytic surface and L ⟶ W a line bundle on W. For an integer n greater than 1, let B be a divisor on W which is linearly equivalent to nL. We construct an n-fold cyclic cover S of W branched along B in the total space V (L) of L in a usual manner as follows:

Suppose B is defined by the equation b_{i} = 0 on a chart U_{i} of W. Let ξ_{i} be the fiber coordinate of V (L) ⟶ W on a local trivialization U_{i} × C. Then the equations ξ_{i}^{n} + b_{i} = 0 on U_{i} × C for all i patched and define the surface S on V (L).

We remark that if B is reduced, then S is normal.

Let \overline{π} : \overline{V (L)} = P(O_{W} (L) ⨁ O_{W}) ⟶ W be the associated P^{1}-bundle. It is convienient to consider the above S as a hypersurface on \overline{V (L)}. The dualizing sheaf ω_{S} is isomorphic to O\overline{V (L)}(K\overline{V (L)} + S) ⨂ O_{S} and the self-intersection number ω_{S}^{2} is defined as (K\overline{V (L)} + S)^{2}S on \overline{V (L)}.

Lemma 1.2. Let S be an n-fold cyclic cover of W on V (L). Then we have

(i) χ(O_{S}) = \frac{1}{4}n(n − 1)K_{W} L + \frac{1}{12}n(n − 1)(2n − 1)L^{2} + nχ(O_{W}), (ii) ω_{S}^{2} = n{(n − 1)L + K_{W} }^{2}.

Proof We first prove the assertion (ii). We set X = \overline{V (L)} and let T be the tautological line bundle on X. Since K_{X} is linearly equivalent to −2T + \overline{π}^{∗}(K_{W} + L), we have

ω_{S}^{2} = \{(n − 2)T + \overline{π}^{∗}(K_{W} + L)\}^{2} · nT (on X)
= (n − 2)^{2}nL^{2} + 2n(n − 2)K_{W} L + n(K_{W} + L)^{2} (on W)
= n{(n − 1)L + K_{W} }^{2}

by the formula T^{3} = T (\overline{π}^{∗}L)^{2}.

Next we prove the assertion (i). Since we have

\begin{aligned}R^{1}\overline{π}_{∗}O_{X}(K_{X} + S) & = & 0, \\ \overline{π}_{∗}O_{X}(K_{X} + S) & ≃ & Symm^{n−2}(O ⨁ O(L)) ⨂ O(K_{W} + L) ≃ ⨁_{j=1}^{n−1} O(K_{W} + jL), \end{aligned}

it follows from Leray’s spectral sequence and the Riemann-Roch formula that

χ(X, O(K_{X} + S)) = ∑_{j=1}^{n−1} χ(W, O(K_{W} + jL)) = \frac{1}{2} ∑_{j=1}^{n−1}(jK_{W} L + j^{2}L^{2}) + (n − 1)χ(O_{W}) = \frac{1}{4}n(n − 1)K_{W} L + \frac{1}{12}n(n − 1)(2n − 1)L^{2} + (n − 1)χ(O_{W}). (1.2.1)

On the other hand, it follows from the exact sequence 0 ⟶ O(K_{X}) ⟶ O(K_{X} + S) ⟶ O(ω_{S}) ⟶ 0 that

\begin{aligned}χ(O_{S}) & = & χ(ω_{S}) = χ(O(K_{X} + S)) − χ(O(K_{X})) \\ & = & χ(O(K_{X} + S)) + χ(O_{W}). (1.2.2) \end{aligned}

Therefore the assertion (i) follows. Q.E.D.

Definition 1.3. (i) For a divisor B on W, we define the number μ(B) by

μ(B) = −2χ(O_{B}) + e(B),

where e(B) is the topological Euler number of the reduced scheme B_{red} of B. (ii) For an n-fold cyclic cover S of W branched along B, we define the number μ(S) by

μ(S) = (n − 1)μ(B)

Lemma 1.4. (S.L.Tan) If B is a reduced divisor on W, then the number μ(B) coincides with the total Milnor number of B, i.e. the sum of the Milnor numbers of all the isolated singularities on B.

Moreover in this case, for an n-fold cyclic cover S of W branched along B (which is automatically normal), the number μ(S) coincides with the total Milnor number of S.

Proof The first assertion is proved in [Ta, Lemma 1.1] by Milnor’s formula [M, p.85]. The second assertion is clear from the first assertion because the equation of an isolated singularity \overline{P} on S is written as f(x, y, z) = z^{n} + g(x, y) and we have

μ(\overline{P}) = dim C\{x, y, z\}/(f_{x}, f_{y}, f_{z})
= (n − 1)dim C{x, y}/(f_{x}, f_{y}) = (n − 1)μ(P),

where P is the singularity defined by g(x, y) = 0. Q.E.D.

Proposition 1.5. (The Noether type formula)

Let S be an n-fold cyclic cover of W with branch locus B on V (L). Then we have

χ(O_{S}) = \frac{1}{12}(ω_{S}^{2} + e(S) + μ(S)).

Proof Since S is topologically obtained by patching n copies of W along B for n − 1 times, it follows from the Mayer-Vietoris exact sequence that

e(S) = ne(W) − (n − 1)e(B).  (1.5.1)

On the other hand, by the Riemann-Roch formula and the duality theorem, it follows from the exact sequence 0 ⟶ O(K_{W}) ⟶ O(K_{W} + B) ⟶ O(ω_{B}) ⟶ 0 that

χ(O_{B}) = χ(O(ω_{B})) = −χ(O(K_{W} + B)) + χ(O(K_{W})) = −\frac{1}{2}(K_{W} + B)B.

Therefore it follows from Lemma 1.2 and the usual Noether formula for W that

χ(O_{S}) − \frac{1}{12}(ω_{S}^{2} + e(S)) = \frac{1}{12}n^{2}(n − 1)L^{2} + \frac{1}{12}n(n − 1)K_{W} L + \frac{1}{12}(n − 1)e(W)
= \frac{1}{12}(n − 1)(K_{W} B + B^{2} + e(B))
= \frac{1}{12}(n − 1)\{−2χ(O_{B}) + e(B)\} = \frac{1}{12}μ(S).

Q.E.D.

Remark 1.6. In the situation of Proposition 1.5, we add the assumption that S is normal. Then the assertion directly follows from Laufer’s formula [L] (see Brenton [B]).

§ 2. The invariant formula via the Hirzebruch-Jung resolution.

Let (V, P) be a germ of 2-dimensional isolated hypersurface singularity defined at the origin of C^{2} by the equation

f(x, y, z) = z^{n} + g(x, y),

where g(x, y) is an analytic function with respect to the variables x, y, and n is an integer greater than 1. Such a singularity (V, P) is said to be of cyclic type in this paper. The aim of this section is to describe the geometric genus p_{g}(V.P) and the Milnor number μ(V, P) in terms of the data on a branch curve which naturally appears in the process of the Hirzebruch-Jung resolution of (V, P). We note that this resolution process is a special case of the original one introduced by Hirzebruch [Hi1] for the solution of the resolution problem of any 2-dimensional analytic space.

Lemma 2.1. Let (V, P) be as above. Then there exist a nonsingular surface W_{0}, a line bundle L_{0} on W_{0} and an n-fold cyclic cover π_{0} : S_{0} ⟶ W_{0} on V (L_{0}) such that

(i) there exists a point \overline{P_{0}} ∈ S_{0} such that the germ of the singularity at \overline{P_{0}} of S_{0} coincides with (V, P), (ii) the reduced scheme of the branch locus B_{0} of π_{0} has normal crossing except in a neighborhood of P_{0} = π_{0}(\overline{P_{0}}).

Proof By Artin’s theorem [Ar], we may assume that g(x, y) is algebraic. Let D be the closure in P^{2} of the affine curve g(x, y) = 0. The curve D is reduced since (V, P) is an isolated singularity. Let Q_{0} ∈ P^{2} be the point corresponding to the origin of C^{2} in the open immersion C^{2} ↪ P^{2}.

We choose general hyperplanes H_{1}, ⋯, H_{t} of P^{2} which do not pass through Q_{0} in such a way that the degree of the reduced divisor

D^{♯} := D + H_{1} + ⋯ + H_{t}

is a multiple of n. We set deg D^{♯} = nα and L^{♯} = O_{P^{2}}(α) for α ∈ Z. Let π^{♯} : S^{♯} ⟶ P^{2} be the n-fold cyclic cover branched along D^{♯} on V (L^{♯}). Let {Q_{0}, Q_{1}, ⋯, Q_{s}} (s ≥ 0) be the set of all isolated singularities on D^{♯}. The fiber (π^{♯})^{−1}(Q_{i}) consists of one point for a fixed i (0 ≤ i ≤ s), which we denote by \overline{Q_{i}}. Then S^{♯} is a normal surface whose singularities are on \overline{Q_{0}}, ⋯, \overline{Q_{s}}. Moreover, the germ of the singularity of S^{♯} at \overline{Q_{0}} coincides with (V, P).

Now let τ^{♯} : W_{0} ⟶ P^{2} be the succession of blow-ups whose centers are infinitely near points of Q_{1}, ⋯, Q_{s} such that the reduced scheme of the divisor

B_{0} := (τ^{♯})^{∗}D^{♯}

has normal crossing except in a neighborhood of P_{0} := (τ^{♯})^{−1}(Q_{0}). Set L_{0} = (τ^{♯})^{∗}O_{P^{2}}(α). Let π_{0} : S_{0} ⟶ W_{0} be the n-fold cyclic cover branched along B_{0} on V (L_{0}), and we put \overline{P_{0}} = π_{0}^{−1}(P_{0}). Then the desired properties are satified. Q.E.D.

2.2. Let V (L_{0}) ⊃ S_{0} ⟶ W_{0} be the n-fold cyclic cover branched along B_{0} as in Lemma 2.1. Let W_{0} ⟵^{τ_{1}} W_{1} ⟵^{τ_{2}} ⋯ ⟵^{τ_{r}} W_{r} be the succession of blow-ups at infinitely near points of P_{0} such that the reduced scheme (B_{r})_{red} of the total transform B_{r} of B_{0} by τ_{1} ∘⋯∘τ_{r} has normal crossing. We take the smallest such number r which enjoys the above property and fix it from now on.

We note that, if r coincides with 0, then B_{0} defines an ordinary double point at P_{0}. In this case, (V, P) is a rational double point of type A_{n−1}, whose properties concerning our problem are well-known. From now on, we assume r ≥ 1 in this section.

For 1 ≤ i ≤ r, let S_{i} be the fiber product of S_{i−1} and W_{i} over W_{i−1}, and let τ̂_{i} : S_{i} ⟶ S_{i−1} and π_{i} : S_{i} ⟶ W_{i} be the natural maps. We set L_{i} = τ_{i}^{∗}L_{i−1}. Then S_{i} is realized on V (L_{i}) and the map τ̂_{i} is also induced by the restriction of the natural map V (L_{i}) ⟶ V (L_{i−1}). Let ρ′ : S̃ ⟶ S_{r} be the normalization. We set ρ = π_{r} ∘ ρ′. Then the surface S̃ has at most isolated cyclic quotient singulatities. Let ρ′′ : S^{∗} ⟶ S be the resolution of such singularities. We note that this process ρ′′ is performed by using the socalled Hirzebruch-Jung string ([Hi1], [BPV, Chap.III]). Therefore we obtain the following diagram:

V (L^{♯})  V (L_{0})  V (L_{1})  V (L_{r})
∪  ∪  ∪  ∪
S^{♯}  \underleftarrow{_{τ^{♯}}} S_{0}  \underleftarrow{_{τ_{1}}} S_{1} ⟵ ⋯ \underleftarrow{_{τ_{r}}} S_{r}  ⟵^{ρ′} S̃ ⟵^{ρ′′} S^{∗}
↓  ↓  ↓  ↓
P^{2}  ⟵^{τ^{♯}} W_{0}  ⟵^{τ_{1}} W_{1} ⟵ ⋯ ⟵^{τ_{r}} W_{r}

The nonsingular surface S^{∗} itself is not uniquely determined by the germ (V, P) since it depends on the compactification S^{♯} in Lemma 2.1. Although locally, i.e. over an open neighborhood which is the inverse image of the affine neighborhood of the original singularity, the structure of S^{∗} is uniquely determined. Moreover, the above process is also uniquely determined up to order in the selection of the centers of blow-ups τ_{1}, ⋯, τ_{r}. We call this process the Hirzebruch-Jung resolution of (V, P).

For later use, we introduce some definitions and notation. For 0 ≤ j < i ≤ r, we put τ_{j,i} = τ_{j+1} ∘ ⋯ ∘ τ_{i} : W_{i} ⟶ W_{j}. For 1 ≤ i ≤ r, let B_{i} (resp. B̃_{i}) be the total transform (resp. the proper transform) of B_{0} by τ_{0,i}. Then B_{i} coincides with the branch locus of the n-fold cyclic cover π_{i} : S_{i} ⟶ W_{i}. Let P_{i−1} ∈ W_{i−1} be the center of the blow-up τ_{i}, and let m_{i} = mult_{P_{i−1}}B_{i−1} be the multiplicity of B_{i−1} at P_{i−1}. Let E_{i,i} = τ_{i}^{−1}(P_{i−1}) be the exceptional curve of τ_{i}, and let E_{i,j} be the proper transform of E_{i,i} by τ_{i,j} for i < j ≤ r. Let E_{i} be the (scheme-theoretic) exceptional divisor for τ_{0,i}. Then we have the irreducible decomposition

E_{i} = ∑_{j=1}^{i} m_{j}E_{j,i}.

We put E_{i} = (E_{i})_{red}. Moreover we decompose B_{0} into B′_{0} + B′′_{0}, where B′_{0} is the divisor consisting of the components of B_{0} which pass through P_{0} and B′′_{0} = B_{0} − B′_{0}. Note that B′_{0} is a reduced divisor since P_{0} is an isolated singularity. We formally write the irreducible decompositions as B′_{0} = ∑_{j=−r′}^{0} E_{j,0} and B′′_{0} = ∑_{j=−r′′}^{−r′−1} m_{j}E_{j,0}. For 1 ≤ i ≤ r and −r′′ ≤ j ≤ 0, let E_{j,i} be the proper transform of E_{j,0} by τ_{0,i}. We also formally put m_{j} = 1 for −r′ ≤ j ≤ 0. Then we have the irreducible decomposition

B_{i} = ∑_{j=−r′′}^{i} m_{j}E_{j,i}, B̃_{i} = ∑_{j=−r′′}^{0} m_{j}E_{j,i}.

Now we look at the branch curve B_{r} = ∑_{j=−r′′}^{r} m_{j}E_{j,r} on W_{r}. The following lemma essentially comes from the Esnault-Viehweg formula:

Lemma 2.3. In the above situation, we have

(i) χ(O_{S_{r}}) = χ(O_{S_{0}}) = χ(O_{S^{♯}}), ω_{S_{r}}^{2} = ω_{S_{0}}^{2} − nr,

(ii) χ(O_{S}) − χ(O_{S_{r}}) = \frac{1}{2} ∑_{i=1}^{n−1} ∑_{j=−r′′}^{r} [\frac{m_{j}i}{n}] \{2 + E_{j,r}^{2} + ∑_{k=−r′′}^{r} [\frac{m_{k}i}{n}] E_{k,r}E_{j,r}\} ,

(iii) χ(O_{S^{∗}}) = χ(O_{S}),

where [m_{j}i/n] is the greatest integer not exceeding the number m_{j}i/n.

Proof Since K_{W_{i}} = τ_{i}^{∗}K_{W_{i−1}} + E_{i,i} and L_{i} = τ_{i}^{∗}L_{i−1} for 1 ≤ i ≤ r, we easily have χ(O_{S_{i}}) = χ(O_{S_{i−1}}) and ω_{S_{i}}^{2} = ω_{S_{i−1}}^{2} −n by Lemma 1.2. Therefore we have χ(O_{S_{r}}) = χ(O_{S_{0}}) and ω_{S_{r}}^{2} = ω_{S_{0}}^{2} − nr. We remark that the birational morphism S_{0} ⟶ S^{♯} is similarly constructed as for S_{r} ⟶ S_{0} with respect to the isolated singularities {Q_{i}}_{1≤i≤s} which appear in the proof of Lemma 2.1. Therefore we have χ(O_{S_{0}}) = χ(O_{S^{♯}}) similarly.

The assertion (iii) is clear because ρ′′ is the resolution of isolated rational singularities. It remains to prove (ii). For 0 ≤ i ≤ n − 1, let L^{(i)} be the line bundle on W_{r} defined by

L^{(i)} = L_{r}^{⊗i} ⨂ O_{W_{r}}(− ∑_{j=−r′′}^{r} [\frac{m_{j}i}{n}] E_{j,r}).

Since (B_{r})_{red} has normal crossing, it follows from [E, Lemma 2] or [V, §1] that

ρ_{∗}O_{S} ≃ ⨁_{i=0}^{n−1}(L^{(i)})^{∨},

where (L^{(i)})^{∨} is the dual bundle of L^{(i)}.

Since R^{q}ρ_{∗}O_{S} vanishes for q ≥ 1, it follows from the Riemann-Roch formula that

χ(O_{S}) = χ(ρ_{∗}O_{S}) = ∑_{i=0}^{n−1} χ(W_{r}, O(−iL_{r} + ∑_{j} [\frac{m_{j}i}{n}] E_{j,r}))

= ∑_{i=0}^{n−1} \{\frac{1}{2}(−iL_{r} + ∑_{j} [\frac{m_{j}i}{n}] E_{j,r})(−iL_{r} + ∑_{j} [\frac{m_{j}i}{n}] E_{j,r} − K_{W_{r}}) + χ(O_{W_{r}})\}

= \frac{1}{2} ∑_{i=0}^{n−1} iL_{r}(iL_{r} + K_{W_{r}}) + nχ(O_{W_{r}})

+\frac{1}{2} ∑_{i=1}^{n−1} ∑_{j=−r′′}^{r} [\frac{m_{j}i}{n}] \{−\frac{2i}{n}B_{r}E_{j,r} + 2 + E_{j,r}^{2} + ∑_{k=−r′′}^{r} [\frac{m_{k}i}{n}] E_{k,r}E_{j,r}\} ,

On the other hand, by (1.2.1) and (1.2.2), we have

χ(O_{S_{r}}) = \frac{1}{2} ∑_{i=0}^{n−1} iL_{r}(iL_{r} + K_{W_{r}}) + nχ(O_{W_{r}}).

Moreover if E_{j,r} is an exceptional curve for τ^{♯} ∘ τ_{1} ∘ ⋯ ∘ τ_{r}, then we have B_{r}E_{j,i} = 0. If E_{j,r} is not an exceptional curve for τ^{♯} ∘ τ_{1} ∘ ⋯ ∘ τ_{r}, then we have [m_{j}i/n] = 0 (1 ≤ i ≤ n − 1) by m_{j} = 1. Therefore we obtain the assertion (ii). Q.E.D.

Corollary 2.4.

p_{g}(V, P) = −\frac{1}{2} ∑_{i=1}^{n−1} ∑_{j=1}^{r} [\frac{m_{j}i}{n}] \{2 + E_{j,r}^{2} + ∑_{k=1}^{r} [\frac{m_{k}i}{n}] E_{k,r}E_{j,r}\} .

Proof We put τ = τ^{♯} ∘ τ_{0,r} : W_{r} ⟶ P^{2}. We first decompose the set of integers J = {−r′′, −r′′ + 1, ⋯, r} into the disjoint union J_{E} ⨿ J_{B}, where

J_{E} = {j ∈ J | m_{j} ≥ 2}, J_{B} = {j ∈ J | m_{j} = 1}.

Since D^{♯} is a reduced divisor on P^{2}, ∑_{j∈J_{B}} E_{j,r} coincides with the proper transform B̃_{r} of D^{♯} by τ, and ∑_{j∈J_{E}} m_{j}E_{j,r} coincides with the exceptional divisor of ∪_{ℓ=0}^{s}Q_{ℓ} for τ. We next decompose J_{E} into the disjoint union ⨿_{0≤ℓ≤s}J_{ℓ}, where

J_{ℓ} = {j ∈ J_{E} | τ(E_{j,r}) = Q_{ℓ}}.

It follows from Lemma 2.3 that ∑_{ℓ=0}^{s} p_{g}(\overline{Q_{ℓ}}) = χ(O_{S^{♯}}) − χ(O_{S^{∗}})

= −\frac{1}{2} ∑_{i=1}^{n−1} ∑_{j=−r”}^{r} [\frac{m_{j}i}{n}] \{2 + E_{j,r}^{2} + ∑_{k=−r”}^{r} [\frac{m_{k}i}{n}] E_{k,r}E_{j,r}\}

= −\frac{1}{2} ∑_{ℓ=0}^{s} ∑_{i=1}^{n−1} ∑_{j∈J_{ℓ}} [\frac{m_{j}i}{n}] \{2 + E_{j,r}^{2} + ∑_{k∈J_{ℓ}} [\frac{m_{k}i}{n}] E_{k,r}E_{j,r}\} .

We remark that, for any isolated singularity, its geometric genus is a local invariant and is independent of the compactification of its ambient space. Hence p_{g}(\overline{Q_{ℓ}}) is determined by the resolution data of \overline{Q_{ℓ}} themselves, i.e. we have

p_{g}(\overline{Q_{ℓ}}) = −\frac{1}{2} ∑_{i=1}^{n−1} ∑_{j∈J_{ℓ}} [\frac{m_{j}i}{n}] \{2 + E_{j,r}^{2} + ∑_{k∈J_{ℓ}} [\frac{m_{k}i}{n}] E_{k,r}E_{j,r}\} .

Since we have p_{g}(\overline{P}) = p_{g}(\overline{Q_{0}}) and J_{0} = \{1, ⋯, r\}, the assertion is proved. Q.E.D.

Lemma 2.5.

\begin{aligned}μ(V, P) & = & (n − 1) \{−r + ∑_{i=1}^{r}(m_{i} − 1)(m_{i}E_{i,r}^{2} − 2) \\ & & + ∑_{−r′≤i≤r,1≤j≤r,i<j,E_{i,r}∩E_{j,r}≠∅}(2m_{i}m_{j} − 1)\} . \end{aligned}

Proof For brevity, we write E_{i} := E_{i,r} for −r′′ ≤ i ≤ r.

Since e(W_{r}) = e(W_{0}) + r and e(B_{r}) = e(B_{0}) + r, we have e(S_{r}) = e(S_{0}) + r by (1.5.1). Therefore it follows from Proposition 1.5 and Lemma 2.3 that

μ(S_{r}) − μ(S_{0}) = 12{χ(O_{S_{r}}) − χ(O_{S_{0}})} − (ω_{S_{r}}^{2} − ω_{S_{0}}^{2}) − {e(S_{r}) − e(S_{0})} = (n − 1)r. (2.4.1)

Now for an integer i with 0 ≤ i ≤ r, we define a divisor B^{(i)} on W_{r} by

B^{(i)} = ∑_{j=−r′′}^{−r′−1} m_{j}E_{j} + ∑_{j=−r′}^{i} E_{j} + ∑_{j=i+1}^{r} m_{j}E_{j}.

Then we have B_{r} = B^{(0)} and B̃_{r} + E_{r} = B^{(r)}. Now we claim

μ(B_{r}) − μ(B̃_{r} + E_{r}) = 2 ∑_{i=1}^{r}(m_{i} − 1)(E_{i}B^{(i)} − 1) + ∑_{i=1}^{r}(m_{i} − 1)(m_{i} − 2)E_{i}^{2}.  (2.4.2)

Indeed, if m_{i} = 1, then we clearly have χ(O_{B^{(i−1)}}) = χ(O_{B^{(i)}}). Let us assume m_{i} ≥ 2. Then for any integer k (1 ≤ k ≤ m_{i} − 1), we consider the exact sequence

0 ⟶ O_{E_{i}}(−B^{(i)} − (k − 1)E_{i}) ⟶ O_{B^{(i)}+kE_{i}} ⟶ O_{B^{(i)}+(k−1)E_{i}} ⟶ 0.

We note that E_{i} is isomorphic to P^{1} for 1 ≤ i ≤ r. Therefore, by using the above sequence inductively and by the Riemann-Roch formula, we have

χ(O_{B^{(i−1)}}) = χ(O_{B^{(i)}+(m_{i}−1)E_{i}}) = χ(O_{B^{(i)}}) + ∑_{k=0}^{m_{i}−2} χ(O_{E_{i}}(−B^{(i)} − kE_{i}))
= χ(O_{B^{(i)}}) + ∑_{k=0}^{m_{i}−2}{E_{i}(−B^{(i)} − kE_{i}) + 1}
= χ(O_{B^{(i)}}) − (m_{i} − 1)(E_{i}B^{(i)} − 1) − \frac{1}{2}(m_{i} − 1)(m_{i} − 2)E_{i}^{2}.

From this, we obtain (2.4.2).

Next we compare μ(B_{0}) with μ(B̃_{r} + E_{r}). Let U_{0} be a suitable affine neighborhood of P_{0} in W_{0} and set U_{r} = τ_{0,r}^{−1}(U_{0}). Then the restricted curves B_{0}|_{W_{0}\U_{0}} and (B̃_{r} + E_{r})|_{W_{r}\U_{r}} are mutually isomorphic. On the other hand, E_{r} and B̃_{r} meet transversally at λ := E_{r}B̃_{r} points. Moreover E_{r} has just r − 1 ordinally double points as its singularities. Therefore the set of singularities on (B̃_{r} + E_{r})|_{U_{r}} consists of r + λ − 1 ordinally double points, while the set of singulariries on B_{0}|_{U_{0}} consists of only P_{0} itself. Hence it follows from Lemma 1.4 that

μ(B̃_{r} + E_{r}) − μ(B_{0}) = r + λ − 1 − μ(P_{0}).  (2.4.3)

Therefore from (2.4.2) and (2.4.3), we have

μ(B_{r}) − μ(B_{0}) = ∑_{i=1}^{r}{2(m_{i} − 1)(E_{i}^{2} − 1) + (m_{i} − 1)(m_{i} − 2)E_{i}^{2}}

+2 ∑_{i=1}^{r}(m_{i} − 1)( ∑_{j=−r′}^{i−1} E_{j} + ∑_{j=i+1}^{r} m_{j}E_{j})E_{i} + r + λ − 1 − μ(P_{0})

= ∑_{i=1}^{r}(m_{i} − 1)(m_{i}E_{i}^{2} − 2) + ∑_{−r′≤i≤0,1≤j≤r,E_{i}∩E_{j}≠∅}{(2m_{j} − 1) + 1}

+ ∑_{1≤i<j≤r,E_{i}∩E_{j}≠∅}{2(m_{i} − 1)m_{j} + 2(m_{j} − 1) + 1} − μ(P_{0})

= ∑_{i=1}^{r}(m_{i} − 1)(m_{i}E_{i}^{2} − 2) + ∑_{−r′≤i≤r,1≤j≤r,i<j,E_{i}∩E_{j}≠∅}(2m_{i}m_{j} − 1) − μ(P_{0}).

From this and (2.4.1), we obtain the assertion. Q.E.D.

Example 2.6. Let (V, P) be the singularity defined by z^{45} + x(x^{2} + y^{3}) = 0. The total transform of the embedded resolution of the singularity x(x^{2} + y^{3}) = 0 is written by ∑_{i=−1}^{3} m_{i}E_{i} where E_{1}^{2} = −3, E_{2}^{2} = −2, E_{3}^{2} = −1, E_{3}E_{2} = E_{3}E_{1} = E_{2}E_{0} = E_{3}E_{−1} = 1, m_{−1} = m_{0} = 1, m_{1} = 3, m_{2} = 5 and m_{3} = 9. Therefore we have p_{g}(V, P) = 35 and μ(V, P) = 308 by Corollary 2.4 and Lemma 2.5.

§ 3. Improved singularities.

Let (V, P) be a germ of 2-dimensional isolated singularity of cyclic type. The aim of this section is to define a new singularity (V^{∨}, P^{∨}) whose difficulty compared with (V, P) is improved in “a unit” in the following sense: If we complete the process of blow-ups of the base surfaces just r times in the Hirzebruch-Jung resolution of (V, P), then the corresponding process for (V^{∨}, P^{∨}) is completed in at most r − 1 times. Next we calculate the difference of invariants between (V, P) and (V^{∨}, P^{∨}).

3.1. We go back to the situation of 2.2. We assume r ≥ 2 in this section. For 2 ≤ i ≤ r, we consider the blow-up τ_{i} : W_{i} ⟶ W_{i−1} with center P_{i−1}. We set β_{i−1} = mult_{P_{i−1}} B̃_{i−1}. According to the position of P_{i−1} in E_{i−1}, we classify τ_{i} into the following two types:

(A) Assume that P_{i−1} is a nonsingular point of E_{i−1}. There exists an integer φ_{1}(i) with 1 ≤ φ_{1}(i) ≤ i − 1 which is uniquely determined by i such that P_{i−1} is contained in E_{φ_{1}(i),i−1}. We have

m_{i} = m_{φ_{1}(i)} + β_{i−1}.

In this case, τ_{i} is said to be of type A.

(B) Assume that P_{i−1} is a double point of E_{i−1}. There exist integers φ_{1}(i) and φ_{2}(i) with 1 ≤ φ_{1}(i) < φ_{2}(i) ≤ i − 1 determined by i such that P_{i−1} coincides with E_{φ_{1}(i),i−1} ∩ E_{φ_{2}(i),i−1}. We have

m_{i} = m_{φ_{1}(i)} + m_{φ_{2}(i)} + β_{i−1}.

In this case, τ_{i} is said to be of type B.

3.2. We consider the blow up τ_{r} : W_{r} ⟶ W_{r−1}. We define a new curve D_{r−1} on a neighborhood of E_{r−1} in W_{r−1} in the following way:

(A) Assume τ_{r} is of type A. Since τ_{r} is the last blow-up in order to obtain the normal crossing property for (B_{r})_{red} near E_{r}, the local branch of B̃_{r−1} at P_{r−1} consists of β_{r−1} nonsingular components C_{1}, ⋯, C_{β_{r−1}} such that the tangent lines at P_{r−1} of C_{1}, ⋯, C_{β_{r−1}} and E_{φ_{1}(r)} are mutually distinct. Now we take any nonsingular analytic curves C′_{1}, ⋯, C′_{β_{r−1}} locally defined in a suitable neighborhood U of E_{r−1} in W_{r−1} such that

(i) C′_{1}, ⋯, C′_{β_{r−1}} do not intersect one another, (ii) C′_{i} intersects E_{r−1} transversally at one point in E_{φ_{1}(r),r−1} \(E_{φ_{1}(r),r−1} ∩ Sing(E_{r−1})) for 1 ≤ i ≤ β_{r−1}. Then we define a new reduced curve D_{r−1} on U by

D_{r−1} = B̃_{r−1}|_{U} − ∑_{i=1}^{β_{r−1}} C_{i} + ∑_{i=1}^{β_{r−1}} C′_{i},

where B̃_{r−1}|_{U} is the restriction of B̃_{r−1} to U. See Figure 1.

[figure: C_{1}; C_{2}; C_{β_{r−1}}; P_{r−1}; E_{φ_{1}(r),r−1}; C′_{β_{r−1}}; C′_{2}; C′_{1}] Figure 1:

(B) Assume τ_{r} is of type B. The local branch of B̃_{r−1} at P_{r−1} consists of β_{r−1} nonsingular components C_{1}, ⋯, C_{β_{r−1}} such that the tangent lines at P_{r−1} of C_{1}, ⋯, C_{β_{r−1}}, E_{φ_{1}(r)} and E_{φ_{2}(r)} at P_{r−1} are mutually distinct. We take nonsingular analytic curves C′_{1}, ⋯, C′_{β_{r−1}}, C′′_{1}, ⋯, C′′_{β_{r−1}} defined on U such that

(i) C′_{1}, ⋯, C′_{β_{r−1}}, C′′_{1}, ⋯, C′′_{β_{r−1}} do not intersect one another, (ii) C′_{i} intersects E_{r−1} transversally at one point in E_{φ_{1}(r),r−1} \(E_{φ_{1}(r),r−1} ∩ Sing(E_{r−1})), and C′′_{i} intersects E_{r−1} transversally at one point in E_{φ_{2}(r),r−1} \(E_{φ_{2}(r),r−1} ∩ Sing(E_{r−1})) for 1 ≤ i ≤ β_{r−1}. Then we define a new reduced curve D_{r−1} on U by

D_{r−1} = B̃_{r−1}|_{U} − ∑_{i=1}^{β_{r−1}} C_{i} + ∑_{i=1}^{β_{r−1}} C′_{i} + ∑_{i=1}^{β_{r−1}} C′′_{i}.

as in Figure 2.

[figure: C_{1}; C_{2}; C_{β_{r−1}}; E_{φ_{2}(r),r−1}; E_{φ_{1}(r),r−1}; P_{r−1}; C′_{1}; C′_{2}; C′_{β_{r−1}}; C′′_{1}; C′′_{2}; C′′_{β_{r−1}}] Figure 2:

For 0 ≤ i ≤ r − 2, we put

D_{i} = (τ_{i,r−1})_{∗}D_{r−1}.

The curve D_{i} is defined in a neighborhood of E_{i} in W_{i} for 1 ≤ i ≤ r − 1, and is reduced by our construction. The curve D_{0} is defined in a neighborhood of P_{0} ∈ W_{0} and is also reduced.

Remark 3.3. The above construction of D_{r−1} is a combinatorial interpretation of the following: If we could perturb the germ at P_{r−1} of B̃_{r−1} to the transversal direction against E_{r−1}, we would obtain D_{r−1}.

This idea is already seen in A’Campo [Ac1]. Furthermore this is directly related to the method of T.S.Röbbecke [R]. (This was pointed out to the author by Professor Brieskorn.) For more general treatment of the argument of this type, we refer Oka [O].

Definition 3.4. As in the above situation, we put g^{∨}(x, y) = 0 to be the defining equation of D_{0} at P_{0}. We also define a germ of 2-dimensional isolated singularity (V^{∨}, P^{∨}) of cyclic type to be the germ at the origin of C^{3} defined by the equation

f^{∨}(x, y, z) := z^{n} + g^{∨}(x, y) = 0.

We call (V^{∨}, P^{∨}) an improved singularity of (V, P).

Example 3.5. If g(x, y) = x^{2} + y^{5}, then we have g^{∨}(x, y) = x^{2} + y^{4} as in Figure 3.

Example 3.6. Since the definition of g^{∨}(x, y) depends on the order of blow-ups to resolve g(x, y), if g(x, y) has many local analytic branches, then g^{∨}(x, y) is not uniquely determined. For example, if g(x, y) = (x^{2}+y^{3}){(x+y)^{3}+y^{6}}, then we easily have g^{∨}(x, y) = (x^{2} + y^{2}){(x + y)^{3} + y^{6}} or g^{∨}(x, y) = (x^{2} + y^{3}){(x + y)^{3} + y^{3}}.

3.7. The Hirzebruch-Jung resolution of (V^{∨}, P^{∨}) is produced in the following way: We may assume that g^{∨}(x, y) is algebraic as a germ of an isolated singularity by Artin [Ar]. Then by the same argument as in Lemma 2.1, there is an n-fold cyclic cover π′_{0} : S′_{0} ⟶ W′_{0} on V (L′_{0}) for some line bundle L′_{0} with the branch locus B_{0}^{∨} such that

(i) there exist a point P′_{0} ∈ W′_{0} and an open neighborhood U′ of P′_{0} in W′_{0} such that the restricted curve B_{0}^{∨}|_{U′} coincides with D_{0}|_{U} for some open neighborhood U of P_{0} in W_{0}. Especially the germ of the singularity of S′_{0} at \overline{P′_{0}} = π_{0}^{′−1}(P′_{0}) coincides with (V^{∨}, P^{∨}). (ii) (B_{0}^{∨})_{red} has normal crossing on the locus W′_{0} \ U′. Since D_{0} is the image of the local curve D_{r−1} induced by the succession of blow-downs of the ambient spaces τ_{0,r−1} : W_{r−1} ⟶ W_{0}, the embedded resolution process for the singularity P′_{0} is produced by the same process locally isomorphic to these (r − 1)-fold blow-ups. In

other words, there exists a succession of blow-ups W′_{0} ⟵^{τ′_{1}} W′_{1} ⟵^{τ′_{2}} ⋯ ⟵^{τ′_{r−1}} W′_{r−1} which satisfies the following properties: For 1 ≤ i < j ≤ r − 1, we set τ′_{i,j} = τ′_{i+1} ∘ ⋯ ∘ τ′_{j}. Then:

(i) For any i (1 ≤ i ≤ r − 1), (τ′_{0,i})^{−1}(U′) is isomorphic to τ_{0,i}^{−1}(U). (ii) Let B̃_{i}^{∨} be the proper transform of B_{0}^{∨} by τ′_{0,i}. Then B̃_{i}^{∨} |_{(τ′_{0,i})^{−1}(U′)} is isomorphic to D_{i} |_{τ_{0,i}^{−1}(U)}.

Let B_{i}^{∨} be the total transform of B_{0}^{∨} by τ′_{0,i} (0 ≤ i ≤ r − 1). Then the reduced scheme of B_{r−1}^{∨} has normal crossing. Let π′_{r−1} : S′_{r−1} ⟶ W′_{r−1} be the n-fold cyclic cover branched along B_{r−1}^{∨} on V (τ_{0,r−1}^{′∗}L′_{0}). Let S′_{r−1} ⟵^{ρ′′′} Ŝ ⟵^{ρ′′′′} S^{∗∗} be the composite of the normalization and the resolution of cyclic quotient singularities on Ŝ. This complete the process of the Hirzebruch-Jung resolution of (V^{∨}, P^{∨}).

Incidentally in the resolution process for (V^{∨}, P^{∨}), the minimal number of times of blowups so that the reduced scheme of the proper transform of the branch curve has normal crossing need not coincide with r −1 in general. For instance in Example 3.5, we have r = 4 for (V, P) while r = 2 for (V^{∨}, P^{∨}). However, it does not matter for our argument, because we do not need the minimality of the resolution.

From now on, since we only consider local properties satisfied in the neighborhoods of the exceptional sets of P_{0} and P′_{0}, we identify W′_{i} with W_{i} and τ′_{i} with τ_{i}, respectively, for 0 ≤ i ≤ r −1, and use the symbols P_{i}, E_{i} and E_{i,j} as in the previous case. That is, P_{i−1} and E_{i} are respectively the center and the exceptional set of the blow-up τ′_{i} = τ_{i}, and E_{i,j} (1 ≤ i < j ≤ r − 1) is the proper transform of E_{i} by τ′_{i,j} = τ_{i,j}. We set E_{r−1}^{∨} = B_{r−1}^{∨} − B̃_{r−1}^{∨}. The support of E_{r−1}^{∨} coincides with E_{r−1} = ∑_{k=1}^{r−1} E_{k,r−1}. We set E^{∨} := (π′_{r−1}∘ρ′′′∘ρ′′′′)^{−1}(E_{r−1}), which is the exceptional set of the singularity (V^{∨}, P^{∨}) by this resolution.

Now the proof of the following lemma is due to Masataka Tomari.

\begin{aligned}[figure] & x^{2} + y^{5} & x^{2} + y^{3} & y + x^{2} & −1 & −2 & −2 & −2 & −1 & −2 & −3 & −1 & −1 & −2 & E_{1,1} & E_{1,2} & E_{1,3} & E_{1,4} & E_{2,2} & E_{2,3} & E_{2,4} & E_{3,3} & E_{4,4} & E_{3,4} & C_{1} & C′_{1} & C′′_{1} & ↑ & ↑ & ↑ & ↑ & [figure] & [figure] & x^{2} + y^{4} & x^{2} + y^{2} & −1 & −2 & −2 & −1 & −2 & −1 & ↑ & ↑ & ↑ & [figure] & [figure] \\ & & & & Figure 3: \end{aligned}

14 Lemma 3.8. (Tomari) (i) For 0 ≤ i ≤ r − 2, we have

mult_{P_{i}} B̃_{i}^{∨} = mult_{P_{i}} B̃_{i}.

(ii) The divisor E_{r−1}^{∨} coincides with the divisor E_{r−1}.

Proof We prove (i). Let

τ_{i,r−1}^{∗}m_{P_{i}} = O( ∑_{j=i+1}^{r−1} n_{j}E_{j,r−1})

be the pull-back of the maximal ideal m_{P_{i}} at P_{i}. Since B̃_{r−1} (resp. B̃_{r−1}^{∨}) is the proper transform of B̃_{i} (resp. B̃_{i}^{∨}) by τ_{i,r−1}, it is well-known (and easily proved by induction) that

mult_{P_{i}} B̃_{i} = B̃_{r−1} · τ_{i,r−1}^{∗}m_{P_{i}}, mult_{P_{i}} B̃_{i}^{∨} = B̃_{r−1}^{∨} · τ_{i,r−1}^{∗}m_{P_{i}}.

Therefore if τ_{r} is of type A, then we have

mult_{P_{i}} B̃_{i} − mult_{P_{i}} B̃_{i}^{∨} = ∑_{j=i+1}^{r−1} n_{j}E_{j,r−1}(B̃_{i} − B̃_{i}^{∨})
= n_{φ_{1}(r)}E_{φ_{1}(r),r−1}(∑_{k=1}^{β_{r−1}} C_{k} − ∑_{k=1}^{β_{r−1}} C′_{k}) = 0.

If τ_{r} is of type B, then we also have

mult_{P_{i}} B̃_{i} − mult_{P_{i}} B̃_{i}^{∨}
= (n_{φ_{1}(r)}E_{φ_{1}(r),r−1} + n_{φ_{2}(r)}E_{φ_{2}(r),r−1})(∑_{k=1}^{β_{r−1}} C_{k} − ∑_{k=1}^{β_{r−1}} C′_{k} − ∑_{k=1}^{β_{r−1}} C′′_{k}) = 0.

Hence the assertion (i) is proved.

Next we prove (ii). Let E_{r−1}^{∨} = ∑_{i=1}^{r−1} m′_{i}E_{i,r−1} be the irreducible decomposition. It suffices to prove

m_{i} = m′_{i}  (3.8.1)

for 1 ≤ i ≤ r − 1. We prove this by induction on i. If i = 1, it follows from the assertion (i) that

m′_{1} = mult_{P_{0}} B̃_{0}^{∨} = mult_{P_{0}} B̃_{0} = m_{1}.

Assume (3.8.1) is satisfied for any j with 1 ≤ j ≤ i − 1. If τ_{i} is of type A, then it follows from the assertion (i) that

m′_{i} = m′_{φ_{1}(i)} + mult_{P_{i−1}} B̃_{i−1}^{∨} = m_{φ_{1}(i)} + mult_{P_{i−1}} B̃_{i−1} = m_{i}.

If τ_{i} is of type B, we also have

m′_{i} = m′_{φ_{1}(i)} + m′_{φ_{2}(i)} + mult_{P_{i−1}} B̃_{i−1}^{∨} = m_{φ_{1}(i)} + m_{φ_{2}(i)} + mult_{P_{i−1}} B̃_{i−1} = m_{i}.

Thus the assertion (ii) is also proved. Q.E.D.

Now we compare the invariants of the improved singularity with those of the original one. For a rational number α, we set ⟨α⟩ = α − [α], and call it the fractional part of α.

Lemma 3.9. (a) Assume τ_{r} is of type A. Then we have

p_{g}(V, P) − p_{g}(V^{∨}, P^{∨}) = \frac{β_{r−1}(n − 1)}{12n}\{β_{r−1}(2n − 1) − 3n\}

−\frac{β_{r−1}}{n} ∑_{i=1}^{n−1} i (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}(r)}i}{n}⟩) − \frac{1}{4}\{gcd(m_{r}, n) − gcd(m_{φ_{1}(r)}, n)\}

+\frac{1}{2} ∑_{i=1}^{n−1} (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}(r)}i}{n}⟩)^{2}.

(b) Assume τ_{r} is of type B. Then we have

p_{g}(V, P) − p_{g}(V^{∨}, P^{∨}) = \frac{β_{r−1}(n − 1)}{12n}\{β_{r−1}(2n − 1) − 3n\}

−\frac{β_{r−1}}{n} ∑_{i=1}^{n−1} i (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}(r)}i}{n}⟩ − ⟨\frac{m_{φ_{2}(r)}i}{n}⟩)

−\frac{1}{4} \{n + gcd(m_{r}, n) − gcd(m_{φ_{1}(r)}, n)) − gcd(m_{φ_{2}(r)}, n)\}

+\frac{1}{2} ∑_{i=1}^{n−1} (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}(r)}i}{n}⟩ − ⟨\frac{m_{φ_{2}(r)}i}{n}⟩)^{2} .

Proof For brevity, we write β = β_{r−1}, φ_{i} = φ_{i}(r) (i = 1, 2), E_{r} = E_{r,r} d_{r} = gcd(m_{r}, n), d_{φ_{1}} = gcd(m_{φ_{1}(r)}, n) and so on.

We first assume τ_{r} is of type A. It follows from Corollary 2.4 and Lemma 3.8 that

p_{g}(V, P) − p_{g}(V^{∨}, P^{∨}) = −\frac{1}{2} ∑_{i=1}^{n−1} [\frac{m_{φ_{1}}i}{n}] \{E_{φ_{1},r}^{2} − E_{φ_{1},r−1}^{2} + [\frac{m_{φ_{1}}i}{n}](E_{φ_{1},r}^{2} − E_{φ_{1},r−1}^{2})

+[\frac{m_{r}i}{n}]E_{φ_{1},r}E_{r}\} − \frac{1}{2} ∑_{i=1}^{n−1} [\frac{m_{r}i}{n}] (E_{r}^{2} + 2 + [\frac{m_{φ_{1}}i}{n}] E_{r}E_{φ_{1},r} + [\frac{m_{r}i}{n}] E_{r}^{2})

= \frac{1}{2} ∑_{i=1}^{n−1} ([\frac{m_{r}i}{n}] − [\frac{m_{φ_{1}}i}{n}]) ([\frac{m_{r}i}{n}] − [\frac{m_{φ_{1}}i}{n}] − 1)

= \frac{1}{2} ∑_{i=1}^{n−1} \{\frac{βi}{n} − (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}}i}{n}⟩)\} \{\frac{βi}{n} − 1 − (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}}i}{n}⟩)\}

= \frac{β(n − 1)}{12n} \{β(2n − 1) − 3n\} − \frac{β}{n} ∑_{i=1}^{n−1} i (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}}i}{n}⟩)

+\frac{1}{2} ∑_{i=1}^{n−1} (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}}i}{n}⟩) + \frac{1}{2} ∑_{i=1}^{n−1} (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}}i}{n}⟩)^{2} .

We set n = d_{r}ñ and m_{r} = d_{r}m̃_{r}. Since ñ and m̃_{r} are mutually prime, we have

∑_{i=1}^{n−1} ⟨\frac{m_{r}i}{n}⟩ = ∑_{i=1}^{d_{r}_{n}−1} ⟨\frac{m̃_{r}i}{n}⟩ = d_{r} ∑_{j=1}^{_{n}−1} \frac{j}{ñ} = \frac{n − d_{r}}{2} .

By the above argument, we have

∑_{i=1}^{n−1} (⟨\frac{m_{r}i}{n}⟩ − ⟨\frac{m_{φ_{1}}i}{n}⟩) = −\frac{1}{2} (d_{r} − d_{φ_{1}}) .

Hence the assertion (a) is proved.

In the case of type B, we have the assertion (b) by the similar calculation.

Q.E.D.

Lemma 3.10. (a) Assume τ_{r} is of type A. Then we have

μ(V, P) − μ(V^{∨}, P^{∨}) = (n − 1)β_{r−1}(β_{r−1} − 1).

(b) Assume τ_{r} is of type B. Then we have

μ(V, P) − μ(V^{∨}, P^{∨}) = (n − 1)β_{r−1}^{2}.

Proof We use the shortened symbol as in Lemma 3.9. We use Lemmas 2.5 and 3.8. Assume τ_{r} is of type A. Since m_{r} is equal to m_{φ_{1}} + β, we have

\begin{matrix}\frac{1}{n−1}\{μ(V, P) − μ(V^{∨}, P^{∨})\} = (m_{r} − 1)(m_{r}E_{r}^{2} − 2) + (m_{φ_{1}} − 1)m_{φ_{1}}(E_{φ_{1},r}^{2} − E_{φ_{1},r−1}^{2}) \\ +(2m_{r}β − 1) + (2m_{r}m_{φ_{1}} − 1) − (2m_{φ_{1}}β − 1) − 1 = β^{2} − β. \end{matrix}

Thus the assertion (a) is proved.

In the case of type B, we have the assertion (b) by the similar calculation. Q.E.D.

Next we calculate the difference of the first Betti number of the exceptional sets. We consider E_{r} on W_{r}. For each i (1 ≤ i ≤ r), we set

I(i) = {j| − r_{0} ≤ j ≤ r, j ≠ i, E_{j} ∩ E_{i} ≠ ∅}.

Let c_{i} be the great common divisor of n, m_{i} and all of m_{j} (j ∈ I(i)).

Lemma 3.11. (a) Assume τ_{r} is of type A. Then we have

b_{1}(Ẽ) − b_{1}(E^{∨}) = (β_{r−1} − 1){gcd(m_{r}, n) − gcd(m_{φ_{1}(r)}, n)}
+c_{φ_{1}(r)} − gcd(m_{r}, m_{φ_{1}(r)}, n).

(b) Assume τ_{r} is of type B. Then we have

b_{1}(Ẽ) − b_{1}(E^{∨}) = β_{r−1}{gcd(m_{r}, n) − ∑_{k=1}^{2} gcd(m_{φ_{k}(r)}, n) + 1} + ∑_{k=1}^{2} c_{φ_{k}(r)}
−1 − ∑_{k=1}^{2} gcd(m_{r}, m_{φ_{k}(r)}, n) + gcd(m_{φ_{1}(r)}, m_{φ_{2}(r)}, n).

Proof Adding to the simplified notation as in Lemma 3.9, we further write d_{φ_{1},r} = gcd(m_{φ_{1}}, m_{r}, n) and so on.

Assume τ_{r} is of type A. We first consider S^{∗} ⟶ W_{r}. The number of components in Ẽ which dominate E_{φ_{1}} is c_{φ_{1}}. (e.g. Tsuchihashi [Ts,§3].) We write these components

by Ẽ_{φ_{1}}^{(1)}, ⋯, Ẽ_{φ_{1}}^{(c_{φ_{1}})}. For each i (1 ≤ i ≤ c_{φ_{1}}), the natural map Ẽ_{φ_{1}}^{(i)} ⟶ E_{φ_{1}} is a cyclic covering of degree d_{φ_{1}}/c_{φ_{1}} whose ramification points are on the inverse image of E_{φ_{1}} ∩ E_{j} (j ∈ I(φ_{1})). Since the number of the ponts on the inverse image of E_{φ_{1}} ∩ E_{j} is d_{φ_{1},j}/c_{φ_{1}} and the ramification index at these points is d_{φ_{1}}/d_{φ_{1},j}, it follows from the Riemann-Hurwiz formula that

∑_{i=1}^{c_{φ_{1}}} b_{1}(Ẽ_{φ_{1}}^{(i)}) = 2c_{φ_{1}} − d_{φ_{1}} − d_{φ_{1},r} + ∑_{j∈I(φ_{1}),j≠r}(d_{φ_{1}} − d_{φ_{1},j}).

Similarly there is an unique component Ẽ_{r} of E which dominate E_{r} and we have

b_{1}(Ẽ_{r}) = 2 − d_{r} − d_{φ_{1},r} + β(d_{r} − 1).

Moreover each Ẽ_{φ_{1}}^{(i)} (1 ≤ i ≤ c_{φ_{1}}) and Ẽ_{r} is connected by d_{φ_{1},r}/c_{φ_{1}} Hirzebruch-Jung strings. Therefore they contribute

c_{φ_{1}} (\frac{d_{φ_{1},r}}{c_{φ_{1}}} − 1) = d_{φ_{1},r} − c_{φ_{1}}

to b_{1}(E) as loops of the dual graph of E.

Next we consider S^{∗∗} ⟶ W_{r−1}. There is a unique component Ẽ_{φ_{1},r−1} of E^{∨} which dominate E_{φ_{1},r−1} and we have

b_{1}(Ẽ_{φ_{1},r−1}) = (β − 2)(d_{φ_{1}} − 1) + ∑_{j∈I(φ_{1}),j≠r}(d_{φ_{1}} − d_{φ_{1},j}).

It is verified that the other components of E and E^{∨} do not contribute to the difference of the first Betti numbers. Therefore the assertion (a) easily follows.

In the case of type B, we have the assertion (b) by the similar calculation. Q.E.D.

§ 4. The signature of the Milnor fiber.

For a germ of 2-dimensional isolated hypersurface singularity (V̄ , P̄), the signature of smoothing σ(V̄ , P̄) is defined as follows: Let F and ∂F be the Milnor fiber of (V̄ , P̄) and its natural real boundary, respectively. Then σ(V̄ , P̄) is defined to be the signature of the quadratic form

H^{2}(F, ∂F; R) × H^{2}(F, ∂F; R) ⟶^{α_{1}} H^{2}(F, ∂F; R) × H^{2}(F; R) ⟶^{α_{2}} H^{4}(F, ∂F; R) ≃ R,

where the map α_{1} is induced by the natural map between the second factors and α_{2} is induced by the cup product. By using essentially the Hirzebruch theory of the signature defect [Hi2], Durfee [D] proposed some formula for σ. By combining the formulas [D, Corollary 2.1] and Laufer [L], we have:

Theorem 4.1. (Hirzebruch, Durfee, Laufer)

σ(V̄ , P̄) = −μ(V̄ , P̄) + 4p_{g}(V̄ , P̄) − b_{1}(Ē),

where Ē is the exceptional set of a good resolution of (V̄ , P̄).

Let (V, P) be a germ of 2-dimensional isolated singularity of cyclic type. The aim of this section is to propose a formula for σ(V, P) by using the method in the previous sections.

We consider the Hirzebruch-Jung resolution of (V, P), and use the same notation as in §§ 2 and 3. We first formally set (V_{[r]}, P_{[r]}) = (V, P). If r ≥ 2, then for any integer i with 1 ≤ i ≤ r − 1, we inductively define a new germ (V_{[i]}, P_{[i]}) of 2-dimensional isolated singularity of cyclic type to be the improved singularity (V_{[i]}, P_{[i]}) := (V_{[i+1]}^{∨}, P_{[i+1]}^{∨}) of (V_{[i+1]}, P_{[i+1]}). We note that the blow-up τ_{i} is of type A (resp. type B) as a step of the Hirzebruch-Jung resolution of (V, P) if and only if the i-th blow up of the Hirzebruch-Jung resolution of (V_{[i]}, P_{[i]}) is of type A (resp. type B). Therefore by Lemmas 3.9, 3.10 and 3.11 and Theorem 4.1, we easily obtain the following:

Lemma 4.2. We simply write d_{i} = gcd(m_{i}, n), d_{i,φ_{1}(i)} = gcd(m_{i}, m_{φ_{1}(i)}, n), c_{φ_{1}(i)} = gcd(m_{φ_{1}(i)}, m_{j_{1}}, ⋯, m_{j_{k}}, n) where I(φ_{1}(i)) = {j_{1}, ⋯, j_{k}} and so on. For 2 ≤ i ≤ r, we have:

(a) If τ_{i} is of type A, then

\begin{matrix}σ(V_{[i]}, P_{[i]}) − σ(V_{[i−1]}, P_{[i−1]}) = − (\frac{n}{3} − \frac{1}{3n}) β_{i−1}^{2} − β_{i−1}(d_{i} − d_{φ_{1}(i)}) + d_{i,φ_{1}(i)} \\ +2 ∑_{j=1}^{n−1} (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{1}}j}{n} ⟩)^{2} − \frac{4β_{i−1}}{n} ∑_{j=1}^{n−1} j (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{1}}j}{n} ⟩) − c_{φ_{1}(i)}. \end{matrix}

(b) If τ_{i} is of type B, then

σ(V_{[i]}, P_{[i]}) − σ(V_{[i−1]}, P_{[i−1]}) = − (\frac{n}{3} − \frac{1}{3n}) β_{i−1}^{2} − (β_{i−1} + 1)(n + d_{i} − d_{φ_{1}(i)}

−d_{φ_{2}(i)}) + d_{φ_{1}(i),i} + d_{φ_{2}(i),i} − d_{φ_{1}(i),φ_{2}(i)} + 2∑_{j=1}^{n−1} (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{1}}j}{n} ⟩ − ⟨\frac{m_{φ_{2}}j}{n} ⟩)^{2}

−\frac{4β_{i−1}}{n} ∑_{j=1}^{n−1} j (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{1}}j}{n} ⟩ − ⟨\frac{m_{φ_{2}}j}{n} ⟩) − c_{φ_{1}(i)} − c_{φ_{2}(i)} + 1.

Next we consider (V_{[1]}, P_{[1]}). We note that mult(V_{[1]}, P_{[1]}) = mult(V, P) = β_{0} by Lemma 3.7.

Lemma 4.3.

σ(V_{[1]}, P_{[1]}) = −\frac{1}{3} (β_{0}^{2} − 3β_{0} − 2) n − β_{0}d_{β_{0}} − 1 + \frac{β_{0}^{2} + d_{β_{0}}^{2}}{3n} − \frac{4β_{0}}{n} ∑_{j=1}^{n−1} j ⟨\frac{β_{0}j}{n} ⟩ .

Proof By Lemma 2.1, we construct an n-fold cyclic cover π_{1,0} : S_{1,0} ⟶ W_{1,0} such that the germ (S_{1,0}, π_{1,0}^{−1}(P_{0})) (P_{0} ∈ W_{1,0}) coincides with (V_{[1]}, P_{[1]}). Let

S_{1,0} ⟵ S_{1,1} ⟵^{ρ′_{1}} S̃_{1} ⟵^{ρ′′_{1}} S_{1}^{∗}
↓  ↓
W_{1,0} ⟵^{τ_{1,1}} W_{1,1}

be the Hirzebruch-Jung resolution of (V_{[1]}, P_{[1]}). Then the branch divisor B_{1,1} of π_{1,1} : S_{1,1} ⟶ W_{1,1} is written in a neighborhood of E_{1} = τ_{1,1}^{−1}(P_{0}) on W_{1,1} as

B_{1,1} = β_{0}E_{1} + C_{1} + ⋯ + C_{β_{0}} (locally)

such that C_{i} (1 ≤ i ≤ β_{0}) do not intersect one another and each of C_{i} (1 ≤ i ≤ β_{0}) intersects E_{1} transversally at one point. Therefore by Corollary 2.4 and Lemma 2.5, we easily have

p_{g}(V_{[1]}, P_{[1]}) = \frac{β_{0}^{2}(n − 1)(2n − 1)}{12n} − \frac{β_{0}(n − 1)}{4} − \frac{β_{0}}{n} ∑_{j=1}^{n−1} j ⟨\frac{β_{0}j}{n} ⟩

+\frac{n − d_{β_{0}}}{4} + \frac{(n − d_{β_{0}})(2n − d_{β_{0}})}{12n}.

μ(V_{[1]}, P_{[1]}) = (n − 1)(β_{0} − 1)^{2}.

We set Ẽ_{1} = (π_{1} ∘ρ′_{1} ∘ρ′′_{1})^{−1}(E_{1}), which is the exceptional set of (V_{[1]}, P_{[1]}) by this resolution. Then we have

b_{1}(Ẽ_{1}) = 2 − 2d_{β_{0}} + β_{0}(d_{β_{0}} − 1).

Therefore from the above equalities and Theorem 4.1, the assertion follows. Q.E.D.

4.4. We go back to the Hirzebruch-Jung resolution of (V, P) in 2.2. We further introduce some definitions and notation. For any integer i with 1 ≤ i ≤ r, let T (i) = {R ∈ E_{i,i} ∩ B̃_{i}} be the set of points at which E_{i,i} and B̃_{i} intersect. We decompose T (i) into a disjoint union T_{1}(i) ⨿ T_{2}(i) where

\begin{aligned}T_{1}(i) & = & \{R ∈ T (i) | R ∉ E_{j,i} (1 ≤ ∀j ≤ i − 1) and I_{R}(B̃_{i}, E_{i,i}) = 1\}, \\ T_{2}(i) & = & \{R ∈ T (i) | R ∈ E_{j,i} (1 ≤ ∃j ≤ i − 1) or I_{R}(B̃_{i}, E_{i,i}) ≥ 2\}. \end{aligned}

Here I_{R}(B̃_{i}, E_{i,i}) is the intersection number of B̃_{i} and E_{i,i} at R. We denote the cardinality of the set T_{1}(i) by

θ_{i} = ♯T_{1}(i).

Since the multiplicity β_{i−1} coincides with ∑_{R∈T (i)} I_{R}(B̃_{i}, E_{i,i}), we have

0 ≤ θ_{i} ≤ β_{i−1}.

Moreover if r ≥ 2, then we decompose the set {2, ⋯, r} into a disjoint union A ⨿ B where A = {2 ≤ i ≤ r | τ_{i} is of type A }, B = {2 ≤ i ≤ r | τ_{i} is of type B }.

Lemma 4.5. We have

(i) β_{0} ∑_{j=1}^{n−1} j ⟨\frac{m_{1}j}{n} ⟩ + ∑_{i∈A} β_{i−1} ∑_{j=1}^{n−1} j (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{1}(i)}j}{n} ⟩)

+ ∑_{i∈B} β_{i−1} ∑_{j=1}^{n−1} j (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{1}(i)}j}{n} ⟩ − ⟨\frac{m_{φ_{2}(i)}j}{n} ⟩) = ∑_{i=1}^{r} θ_{i} ∑_{j=1}^{n−1} j ⟨\frac{m_{i}j}{n} ⟩ ,

(ii) β_{0}d_{1} + ∑_{i∈A} β_{i−1}(d_{i} − d_{φ_{1}(i)}) + ∑_{i∈B} β_{i−1}(d_{i} − d_{φ_{1}(i)} − d_{φ_{2}(i)}) = ∑_{i=1}^{r} θ_{i}d_{i}.

Proof We set G(x) = ∑_{j=1}^{n−1} j ⟨\frac{xj}{n} ⟩ for an integer x. For 1 ≤ i ≤ r, we define subsets J_{A}(i) and J_{B}(i) of {1, ⋯, r} by

J_{A}(i) = {j ∈ Z, i + 1 ≤ j ≤ r | τ_{j} is of type A and φ_{1}(j) = i},

J_{B}(i) = {j ∈ Z, i + 1 ≤ j ≤ r | τ_{j} is of type B and φ_{1}(j) = i or φ_{2}(j) = i}. We put

a_{i} = β_{i−1} − ∑_{j∈J_{A}(i)∪J_{B}(i)} β_{j−1}.

Then it is clear that

\begin{matrix}β_{0}G(m_{1}) + ∑_{i∈A} β_{i−1}\{G(m_{i}) − G(m_{φ_{1}(i)})\} + ∑_{i∈B} β_{i−1}\{G(m_{i}) − G(m_{φ_{1}(i)}) \\ −G(m_{φ_{2}(i)})\} = ∑_{i=1}^{r} a_{i}G(m_{i}). \end{matrix}

Therefore it suffices to prove

a_{i} = θ_{i} (1 ≤ i ≤ r).

We note that

θ_{i} = β_{i−1} − ∑_{R∈T_{2}(i)} I_{R}(B̃_{i}, E_{i,i}) .

For any fixed R ∈ T_{2}(i), let

R_{0} = R, R_{1} = E_{i,i_{1}} ∩ B̃_{i_{1}}, ⋯, R_{s} = E_{i,i_{s}} ∩ B̃_{i_{s}} (i < i_{1} < ⋯ < i_{s} ≤ r)

be all the infinitely near points of R. Then by Max Noether’s theorem (e.g.,[BK,§8]), we have

I_{R}(B̃_{i}, E_{i,i}) = ∑_{k=0}^{s} mult_{R_{k}} B̃_{i_{k}} · mult_{R_{k}}E_{i,i_{k}} = ∑_{k=0}^{s} β_{i_{s}}.

From this, we have

∑_{R∈T_{2}(i)} I_{R}(B̃_{i}, E_{i,i}) = ∑_{j∈J_{A}(i)∪J_{B}(i)} β_{j−1}.

Hence we obtain a_{i} = θ_{i}, and the assertion (i) is proved.

To prove (ii), we just replace the definition of G(x) by G(x) = gcd(x, n).

Q.E.D.

4.6. We introduce some definitions. We first set

F (1) = −\frac{1}{3}(β_{0}^{2} − 3β_{0} − 2)n − 1 + \frac{β_{0}^{2} + d_{β_{0}}^{2}}{3n} − \frac{4θ_{1}}{n} ∑_{j=1}^{n−1} j ⟨\frac{β_{0}j}{n} ⟩ − θ_{1}d_{β_{0}}.

For i ∈ A, set

\begin{aligned}F_{A}(i) & = & −\frac{1}{3} (n − \frac{1}{n}) β_{i−1}^{2} + d_{i,φ_{1}(i)} + 2 ∑_{j=1}^{n−1} (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{1}(i)}j}{n} ⟩)^{2} \\ & & −\frac{4θ_{i}}{n} ∑_{j=1}^{n−1} j ⟨\frac{m_{i}j}{n} ⟩ − θ_{i}d_{i} − c_{φ_{1}(i)}. \end{aligned}

For i ∈ B, set

F_{B}(i) = −\frac{1}{3} (n − \frac{1}{n}) β_{i−1}^{2} − (β_{i−1} + 1)n − d_{i} + d_{φ_{1}(i)} + d_{φ_{2}(i)} + d_{i,φ_{1}(i)}

+d_{i,φ_{2}(i)} − d_{φ_{1}(i),φ_{2}(i)} + 2 ∑_{j=1}^{n−1} (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{1}(i)}j}{n} ⟩ − ⟨\frac{m_{φ_{2}(i)}j}{n} ⟩)^{2}

−\frac{4θ_{i}}{n} ∑_{j=1}^{n−1} j ⟨\frac{m_{i}j}{n} ⟩ − θ_{i}d_{i} − c_{φ_{1}(i)} − c_{φ_{2}(i)} + 1.

Proposition 4.7. For a 2-dimensional isolated singularity (V, P) of cyclic type with r ≥ 1, we have

σ(V, P) = F (1) + ∑_{i∈A} F_{A}(i) + ∑_{i∈B} F_{B}(i).

Proof Since we have

σ(V, P) = σ(V_{[1]}, P_{[1]}) + ∑_{i=2}^{r}{σ(V_{[i]}, P_{[i]}) − σ(V_{[i−1]}, P_{[i−1]})},

the assertion follows from Lemmas 4.2, 4.3 and 4.5. Q.E.D.

§ 5. The estimate concerning the fractional part symbol.

In order to estimate the formula in Proposition 4.7, we preprare some calculations concerning the fractional part symbol ⟨ ⟩.

Lemma 5.1. Let n_{1}, n_{2} be natural numbers which are not less than 2, and we set s = gcd(n_{1}, n_{2}) and N = lcm(n_{1}, n_{2}) (the least common multiple). Let s_{1}, s_{2} be any natural numbers with (n_{1}, s_{1}) = 1 and (n_{2}, s_{2}) = 1. Then we have

∑_{i=0}^{N−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ⟨\frac{s_{2}i}{n_{2}}⟩ = \frac{1}{4s}\{(n_{1} − 1)(n_{2} − 1) − (s − 1)^{2}\} + ∑_{i=0}^{s−1} ⟨\frac{s_{1}i}{s}⟩ ⟨\frac{s_{2}i}{s}⟩ .

Proof We set ñ_{1} = n_{1}/s and ñ_{2} = n_{2}/s. Since N = ñ_{1}ñ_{2}s = n_{1}ñ_{2}, we have

\begin{matrix}∑_{i=0}^{N−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ⟨\frac{s_{2}i}{n_{2}}⟩ = ∑_{i=0}^{n_{1}−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ∑_{j=0}^{_{n_{2}}−1} ⟨\frac{s_{2}(i + jn_{1})}{n_{2}}⟩ \\ = ∑_{i=0}^{n_{1}−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ∑_{j=0}^{_{n_{2}}−1} ⟨\frac{1}{ñ_{2}} (s_{2}ñ_{1}j + [\frac{s_{2}i}{s}]) + \frac{1}{ñ_{2}} ⟨\frac{s_{2}i}{s}⟩⟩ . \end{matrix}

Since s_{2}ñ_{1} and ñ_{2} are mutually prime, the numbers s_{2}ñ_{1}j + [s_{2}i/s] (0 ≤ j ≤ ñ_{2} − 1) define mutually distinct equivalence classes in Z/ñ_{2}Z, i.e. the set {s_{2}ñ_{1}j+[s_{2}i/s]|0 ≤ j ≤ ñ_{2}−1} is congruent modulo ñ_{2} to {0, 1, ⋯, ñ_{2} −1} neglecting its order. The sets {s_{1}i|0 ≤ i ≤ n_{1} −1} and {0, 1, ⋯, n_{1} − 1} are congruent modulo n_{1} in the same sence. Therefore we have

∑_{i=0}^{N−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ⟨\frac{s_{2}i}{n_{2}}⟩ = ∑_{i=0}^{n_{1}−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ∑_{k=0}^{_{n_{2}}−1} ⟨\frac{k}{ñ_{2}} + \frac{1}{ñ_{2}} ⟨\frac{s_{2}i}{s}⟩⟩

= ∑_{i=0}^{n_{1}−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ∑_{k=0}^{_{n_{2}}−1} (\frac{k}{ñ_{2}} + \frac{1}{ñ_{2}} ⟨\frac{s_{2}i}{s}⟩) = ∑_{i=0}^{n_{1}−1} ⟨\frac{s_{1}i}{n_{1}}⟩ (\frac{ñ_{2} − 1}{2} + ⟨\frac{s_{2}i}{s}⟩)

= \frac{(n_{1} − 1)(ñ_{2} − 1)}{4} + ∑_{i=0}^{_{n_{1}}s−1} ⟨\frac{s_{1}i}{ñ_{1}s}⟩ ⟨\frac{s_{2}i}{s}⟩ .

Since (ñ_{1}s, s_{1}) = (s, s_{2}) = 1, by the same argument as above, we have

∑_{i=0}^{_{n_{1}}s−1} ⟨\frac{s_{1}i}{ñ_{1}s}⟩ ⟨\frac{s_{2}i}{s}⟩ = \frac{(s − 1)(ñ_{1} − 1)}{4} + ∑_{i=0}^{s−1} ⟨\frac{s_{1}i}{s}⟩ ⟨\frac{s_{2}i}{s}⟩ .

From these, the assertion follows. Q.E.D.

Lemma 5.2. Let s be any natural number and let s_{1}, s_{2} be natural numbers with (s, s_{1}) = (s, s_{2}) = 1. Then we have

\frac{(s + 1)(s − 1)}{6s} ≤ ∑_{i=0}^{s−1} ⟨\frac{s_{1}i}{s}⟩ ⟨\frac{s_{2}i}{s}⟩ ≤ \frac{(s − 1)(2s − 1)}{6s}.

Moreover the left hand side equality holds if and only if s_{1} + s_{2} ≡ 0 (mod s), and the right hand side equality holds if and only if s_{1} ≡ s_{2} (mod s).

Proof For a permutation

τ = ( \begin{matrix}1, & 2, & ⋯, & s − 1 \\ τ(1), & τ(2), & ⋯, & τ(s − 1) \end{matrix} ) ,

we define the function

F (τ) := ∑_{i=1}^{s−1} iτ(i).

We easily obtain that, when τ moves all the permutations of (1, 2, ⋯, s − 1), the minimal value of F (τ) attains if and only if τ coincides with

( \begin{matrix}1, & 2, & ⋯, & s − 1 \\ s − 1, & s − 2, & ⋯, & 1 \end{matrix} ) ,

and the maximal value of F (τ) attains if and only if τ coincides with the identity permutation.

Now for any natural number α, we write ≪ α ≫ as the remainder divided by s. We put

σ_{s_{1},s_{2}} = ( \begin{matrix}≪ s_{1} ≫, & ≪ 2s_{1} ≫, & ⋯, & ≪ (s − 1)s_{1} ≫ \\ ≪ s_{2} ≫, & ≪ 2s_{2} ≫, & ⋯, & ≪ (s − 1)s_{2} ≫ \end{matrix} ) .

By (s, s_{1}) = (s, s_{2}) = 1, σ_{s_{1},s_{2}} is considered as a permutation of (1, 2, ⋯, s − 1). Moreover we have

∑_{i=0}^{s−1} ⟨\frac{s_{1}i}{s}⟩ ⟨\frac{s_{2}i}{s}⟩ = \frac{F (σ_{s_{1},s_{2}})}{s^{2}}.

Therefore the assertion easily follows. Q.E.D.

Lemma 5.3. Let n be an integer greater than 2 and let M_{1}, M_{2} be natural numbers with M_{1} ≢ 0, M_{2} ≢ 0 (mod n). Then we have

∑_{i=1}^{n−1} (⟨\frac{M_{1}i}{n}⟩ − ⟨\frac{M_{2}i}{n}⟩)^{2} ≤ \frac{(n − 1)(n − 2)}{3n}.

Proof For j = 1, 2, we set d_{j} = gcd(M_{j}, n), n_{j} = n/d_{j} and s_{j} = M_{j}/d_{j}. We have n_{j} ≥ 2 by M_{j} ≢ 0 (mod n). Moreover we set s = gcd(n_{1}, n_{2}), N = lcm(n_{1}, n_{2}) and ñ_{j} = n_{j}/s. We have N = ñ_{1}ñ_{2}s. We set d̃= n/N. From d_{1} = d̃ñ_{2} and d_{2} = d̃ñ_{1}, we have

∑_{i=1}^{n−1} (⟨\frac{M_{1}i}{n}⟩ − ⟨\frac{M_{2}i}{n}⟩)^{2} = d̃ ∑_{i=1}^{N−1} (⟨\frac{ñ_{2}s_{1}i}{N}⟩ − ⟨\frac{ñ_{1}s_{2}i}{N}⟩)^{2} .

Now we claim

∑_{i=1}^{N−1} (⟨\frac{ñ_{2}s_{1}i}{N}⟩ − ⟨\frac{ñ_{1}s_{2}i}{N}⟩)^{2} ≤ \frac{(N − 1)(N − 2)}{3N}.  (5.3.1)

The assertion (5.3.1) implies the assertion of the Lemma, because we have

\frac{(n − 1)(n − 2)}{3n} − \frac{d̃(N − 1)(N − 2)}{3N} = \frac{(d̃− 1)(3d̃N − d̃− 1)}{3d̃N} ≥ 0.

It suffices to prove (5.3.1). Since (n_{1}, s_{1}) = (n_{2}, s_{2}) = 1, we have

\begin{matrix}∑_{i=1}^{N−1} (⟨\frac{ñ_{2}s_{1}i}{N}⟩ − ⟨\frac{ñ_{1}s_{2}i}{N}⟩)^{2} = ñ_{2} ∑_{i=1}^{n_{1}−1} ⟨\frac{s_{1}i}{n_{1}}⟩^{2} + ñ_{1} ∑_{i=1}^{n_{2}−1} ⟨\frac{s_{2}i}{n_{2}}⟩^{2} − 2 ∑_{i=1}^{N−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ⟨\frac{s_{2}i}{n_{2}}⟩ \\ = \frac{ñ_{2}(n_{1} − 1)(2n_{1} − 1)}{6n_{1}} + \frac{ñ_{1}(n_{2} − 1)(2n_{2} − 1)}{6n_{2}} − 2 ∑_{i=1}^{N−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ⟨\frac{s_{2}i}{n_{2}}⟩ . \end{matrix}

Therefore it easily follows from Lemmas 5.1 and 5.2 that

\frac{(N − 1)(N − 2)}{3N} − ∑_{i=1}^{N−1} (⟨\frac{ñ_{2}s_{1}i}{N}⟩ − ⟨\frac{ñ_{1}s_{2}i}{N}⟩)^{2}

= (\frac{(N − 1)(2N − 1)}{6N} − \frac{ñ_{2}(n_{1} − 1)(2n_{1} − 1)}{6n_{1}}) + (\frac{(N − 1)(2N − 1)}{6N}

−\frac{ñ_{1}(n_{2} − 1)(2n_{2} − 1)}{6n_{2}}) + 2 \{ ∑_{i=1}^{N−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ⟨\frac{s_{2}i}{n_{2}}⟩ − \frac{(N − 1)(N + 1)}{6N}\}

≥ \frac{1}{6N} \{(3N − ñ_{2} − 1)(ñ_{2} − 1) + (3N − ñ_{1} − 1)(ñ_{1} − 1)\}

+2 \{\frac{(n_{1} − 1)(n_{2} − 1) − (s − 1)^{2}}{4s} + \frac{(s − 1)(s + 1)}{6s} − \frac{(N − 1)(N + 1)}{6N}\}

= \frac{ñ_{1}ñ_{2}(ñ_{1}ñ_{2} − 1)s^{2} − (ñ_{1} + ñ_{2})^{2} + 4}{6ñ_{1}ñ_{2}s} ≥ \frac{(ñ_{1}^{2} − 1)(ñ_{2}^{2} − 1) − 3ñ_{1}ñ_{2} + 3}{6ñ_{1}ñ_{2}} ≥ 0.

by ñ_{1} ≥ 2 and ñ_{2} ≥ 2. Hence the assertion is proved. Q.E.D.

We slightly generalize a part of Lemma 5.2.

Lemma 5.4 Let n, M_{1}, M_{2} be any natural numbers. Then we have

∑_{i=1}^{n−1} ⟨\frac{M_{1}i}{n}⟩ ⟨\frac{M_{2}i}{n}⟩ ≤ \frac{(n − 1)(2n − 1)}{6n} .

Proof If M_{1} ≡ 0 or M_{2} ≡ 0 (mod n), then the assertion is clear. Assume M_{1} ≢ 0 and M_{2} ≢ 0. We denote by d_{j}, s_{j}, n_{j}, ñ_{j}, ( j = 1, 2) s, N and d̃ as in the proof of Lemma 5.3. We easily have

\begin{aligned}\frac{(n − 1)(2n − 1)}{6n} & ≥ & \frac{d̃(N − 1)(2N − 1)}{6N} , \\ ∑_{i=1}^{n−1} ⟨\frac{M_{1}i}{n}⟩ ⟨\frac{M_{2}i}{n}⟩ & = & d̃ ∑_{i=1}^{N−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ⟨\frac{s_{2}i}{n_{2}}⟩ . \end{aligned}

Moreover it follows from Lemmas 5.1 and 5.2 that

\frac{(N − 1)(2N − 1)}{6N} − ∑_{i=1}^{N−1} ⟨\frac{s_{1}i}{n_{1}}⟩ ⟨\frac{s_{2}i}{n_{2}}⟩

≥ \frac{(sñ_{1}ñ_{2} − 1)(2sñ_{1}ñ_{2} − 1)}{6sñ_{1}ñ_{2}} − \frac{(sñ_{1} − 1)(sñ_{2} − 1)}{4s} − \frac{(s − 1)(2s − 1)}{6s}

= \frac{ñ_{1}ñ_{2} − 1}{12}s + \frac{ñ_{1} + ñ_{2} − 2}{4} + (\frac{1}{6ñ_{1}ñ_{2}} − \frac{1}{6}) \frac{1}{s} ≥ 0.

Hence the assertion follows. Q.E.D.

Lemma 5.5. Let n, M be natural numbers with n ≥ 2. Then we have

\frac{4}{n} ∑_{i=1}^{n−1} i ⟨\frac{Mi}{n}⟩ + gcd(M, n) ≥ \frac{2n}{3} + 1 − \frac{2}{3n} .

Proof The calculation is similar as above, and we omit it.

Remark 5.6. The statement of lemmas in this section can be translated into the language of the Dedekind sum (e.g. Hirzebruch-Zagier [HZ]).

§ 6. The estimate of the signature.

The aim of this section is to prove the following:

Theorem 6.1. Let (C, P_{0}) be a germ of an isolated plane curve singularity of multiplicity m. Let {C_{1}, ⋯, C_{t}} be the set of local analytic branches of C at P_{0}. Let {C_{i_{1}}, ⋯, C_{i_{θ}}} be the subset of {C_{1}, ⋯, C_{t}} such that; (⋆) For any j (1 ≤ j ≤ θ), C_{i_{j}} is nonsingular at P_{0} and the tangent line of C_{i_{j}}

at P_{0} is mutually distinct from the tangent line of any other one of {C_{1}, ⋯, C_{t}}. The number θ = θ(C, P_{0}) is defined to be the maximal number which enjoy the property (⋆). If such a subset {C_{i_{1}}, ⋯, C_{i_{θ}}} is empty, then we set θ = 0.

Now let (V, P) be the germ of the singularity on n-fold cyclic cover (n ≥ 2) branched along (C, P_{0}), i.e. (V, P) is defined by the equation f(x, y, z) = z^{n} + g(x, y) where g(x, y) is the defining equation of (C, P_{0}). Let σ(V, P) be the signature of the Milnor fiber of (V, P). Then we have

\begin{aligned}σ(V, P) & ≤ & −\frac{1}{3}(m^{2} − 3m − 2)n − 1 + \frac{m^{2} + gcd(m, n)^{2}}{3n} \\ & & −\frac{4θ}{n} ∑_{i=1}^{n−1} i ⟨\frac{mi}{n}⟩ − θ · gcd(m, n). (6.1.1) \end{aligned}

where ⟨mi/n⟩ = mi/n − [mi/n] is the fractional part of mi/n.

Furthermore, the equality of (6.1.1) holds if and only the singularity (V, P) satisfies the property θ = m, that is, (C, P_{0}) is an ordinary singularity.

It is clear that Theorem 6.1 implies the Theorem in the introduction.

Corollary 6.2. Let (V, P) be a germ of analytic function f(x, y, z) = z^{n} + g(x, y) such that (V, P) defines at most an isolated singularity. Then we have

σ(V, P) ≤ 0.

Furthermore, the equality σ(V, P) = 0 holds if and only if (V, P) is a germ of a nonsingular point.

Proof If the multiplicity of (V, P) is less than 4, the assertion is known ([To1][A]). Therefore we may assume n ≥ 4 and m ≥ 4. Then we have σ(V, P) < 0 by an easy calculation. Q.E.D.

6.3. We start to prove Theorem 6.1. We produce the Hirzebruch-Jung resolution of (V, P). We first assume r = 0. Then (C, P_{0}) is an ordinary double point and (V, P) is a rational double point of type A_{n−1}. By p_{g} = 0, μ = n − 1 and b_{1} = 0, Theorem 4.1 imply σ = −n + 1 in this case. On the other hand, we have

∑_{j=1}^{n−1} j ⟨\frac{2j}{n} ⟩ = \{ \begin{matrix}(n − 1)(7n − 5)/24, & if n is odd \\ (n − 2)(7n − 4)/24, & if n is even. \end{matrix}

by an easy calculation. Since we have m = θ = 2, the right hand side of (6.1.1) also coincides with −n + 1, which prove the assertion in this case. (Another proof: Although the branch curve has normal crossing, we blow up the base surface at one time and use Proposition 4.7 by settig r = 1.)

Next we assume r = 1. In the notation of § 4, we note that θ = θ_{1} and m = β_{0}. Moreover, F (1) in Proposition 4.7 coincides with the right hand side term of (6.1.1). Therefore the equality of (6.1.1) holds. We remark that the condition r = 1 occurs if and only if θ coincides with m.

From now on, we assume r ≥ 2. By Proposition 4.7, it suffices to prove

∑_{i∈A} F_{A}(i) + ∑_{i∈B} F_{B}(i) < 0.  (6.3.1)

We prepare several lemmas to prove (6.3.1).

Lemma 6.4. Assume i ∈ A. If m_{i} ≢ m_{φ_{1}(i)} (mod n), then

F_{A}(i) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 2)(β_{i−1} + 2) + Max \{−\frac{5}{12}n − 3 + \frac{8}{3n}, −\frac{7}{12}n + \frac{4}{3n}\} .

If m_{i} ≡ m_{φ_{1}(i)} (mod n), then there exists an integer k ≥ 1 with β_{i−1} = kn such that

F_{A}(i) ≤ −\frac{1}{3}k^{2}n^{2} (n − \frac{1}{n}) + d_{i} − 1.

Especially if β_{i−1} ≥ 2, then we always have F_{A}(i) < 0.

Proof We first assume m_{i} ≢ 0, m_{φ_{1}(i)} ≢ 0 and m_{i} ≢ m_{φ_{1}(i)} (mod n). Since we have d_{i,φ_{1}(i)} ≤ gcd(n, n/2, n/4) = n/4 and c_{φ_{1}(i)} ≥ 1, it follows from Lemma 5.3 that

F_{A}(i) ≤ −\frac{4}{3} (n − \frac{1}{n}) − \frac{1}{3} (n − \frac{1}{n}) (β_{i−1}^{2} − 4) + \frac{n}{4} − 1 + 2 (\frac{n}{3} − 1 + \frac{2}{3n})
= −\frac{5}{12}n − 3 + \frac{8}{3n} − \frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 2) (β_{i−1} + 2) .

Next we assume m_{i} ≢ 0 and m_{φ_{1}(i)} ≡ 0 (mod n). We have ∑_{j=1}^{n−1}⟨m_{i}j/n⟩^{2} = (n − d_{i})(2n − d_{i})/6n. Therefore we have

F_{A}(i) ≤ −\frac{2}{3}n + \frac{4 + d_{i}^{2}}{3n} − \frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 2) (β_{i−1} + 2)
≤ −\frac{7}{12}n + \frac{4}{3n} − \frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 2) (β_{i−1} + 2)

by d_{i} ≤ n/2. In the case of m_{i} ≡ 0 and m_{φ_{1}(i)} ≢ 0 (mod n), we have the same assertion by a similar argument. Therefore we obtain the former assertion of the Lemma.

Assume m_{i} ≡ m_{φ_{1}(i)} (mod n). Then we also obtain the latter assertion of the Lemma by a similar calculation. Q.E.D.

Lemma 6.5. We have

F_{B}(i) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 2) (β_{i−1} + 2) − (β_{i−1} − 2)n + \frac{d_{φ_{1}(i)}^{2}}{3n}

+Max \{−n − 4 + \frac{2}{n}, −\frac{5n}{3} − 1 + \frac{4}{3n}\}

for any i ∈ B. Especially if β_{i−1} ≥ 2, then we have F_{B}(i) < 0.

Proof We set

F^{⋆} = −\frac{1}{3} (n − \frac{1}{n}) β_{i−1}^{2} − (β_{i−1} + 1)n + d_{φ_{1}(i)} + d_{i,φ_{2}(i)} − 1

+2 ∑_{j=1}^{n−1} (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{2}(i)}j}{n} ⟩)^{2} + 2 ∑_{j=1}^{n−1} ⟨\frac{m_{φ_{1}(i)}j}{n} ⟩^{2} ,

F^{⋆⋆} = −d_{i} + d_{φ_{2}(i)} + d_{i,φ_{1}(i)} − d_{φ_{1}(i),φ_{2}(i)}

−4 ∑_{j=1}^{n−1} (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{2}(i)}j}{n} ⟩) ⟨\frac{m_{φ_{1}(i)}j}{n} ⟩ .

Then we claim

F^{⋆} ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 1)(β_{i−1} + 1) − (β_{i−1} − 1)n − \frac{2}{3}n − 1 + \frac{1 + d_{φ_{1}(i)}^{2}}{3n} , (6.5.1)
F^{⋆⋆} ≤ Max \{\frac{5}{3}n − 3 + \frac{2}{3n}, n\} .  (6.5.2)

Since we have F_{B}(i) ≤ F^{⋆} +F^{⋆⋆}, the assertions (6.5.1) and (6.5.2) easily imply the assertion of the Lemma.

We prove (6.5.1). Assume d_{i,φ_{2}(i)} ≤ n/3. Then it follows from Lemma 5.3 that

F^{⋆} ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 1)(β_{i−1} + 1) − \frac{1}{3} (n − \frac{1}{n}) − (β_{i−1} − 1)n − 2n

+d_{φ_{1}(i)} + \frac{n}{3} − 1 + +\frac{2(n − 1)(n − 2)}{3n} + \frac{(n − d_{φ_{1}(i)})(2n − d_{φ_{1}(i)})}{3n}

= −\frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 1)(β_{i−1} + 1) − (β_{i−1} − 1)n − \frac{2}{3}n − 3 + \frac{5 + d_{φ_{1}(i)}^{2}}{3n} .

Assume d_{i,φ_{2}(i)} > n/3. Then one of the following cases occur:

(i) “ m_{i} ≡ 0, m_{φ_{2}(i)} ≡ n/2 (mod n) ” or “ m_{i} ≡ n/2, m_{φ_{2}(i)} ≡ 0 (mod n) ” , (ii) m_{i} ≡ m_{φ_{2}(i)} ≡ n/2 (mod n), (iii) m_{i} ≡ m_{φ_{2}(i)} ≡ 0 (mod n). We consider the case (i). Then the integer n is even, and we have

∑_{j=1}^{n−1} (⟨\frac{m_{i}j}{n} ⟩ − ⟨\frac{m_{φ_{2}(i)}j}{n} ⟩)^{2} = ∑_{j=1}^{n−1} ⟨\frac{j}{2}⟩^{2} = \frac{n}{8}.

Therefore we obtain

F^{⋆} ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 1)(β_{i−1} + 1) − (β_{i−1} − 1)n − \frac{11}{12}n − 1 + \frac{1 + d_{φ_{1}(i)}^{2}}{3n} .

In the case of (ii), we have

F^{⋆} ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 1)(β_{i−1} + 1) − (β_{i−1} − 1)n − \frac{7}{6}n − 1 + \frac{1 + d_{φ_{1}(i)}^{2}}{3n} .

In the case of (iii), we have

F^{⋆} ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{i−1} − 1)(β_{i−1} + 1) − (β_{i−1} − 1)n − \frac{2}{3}n − 1 + \frac{1 + d_{φ_{1}(i)}^{2}}{3n} .

From these, we obtain the assertion (6.5.1).

Next we prove (6.5.2). First we have

F^{⋆⋆} ≤ d_{φ_{2}(i)} − 1 + 4 ∑_{j=1}^{n−1} ⟨\frac{m_{φ_{1}(i)}j}{n} ⟩ ⟨\frac{m_{φ_{2}(i)}j}{n} ⟩ .

We assume d_{φ_{2}(i)} ≤ n/3. By Lemma 5.4, we have

F^{⋆⋆} ≤ \frac{n}{3} − 1 + 4 · \frac{(n − 1)(2n − 1)}{6n} = \frac{5n}{3} − 3 + \frac{2}{3n}.

We assume d_{φ_{2}(i)} = n/2. We set n = 2n′. We have

F^{⋆⋆} ≤ \frac{n}{2} − 1 + 2 ∑_{k=1}^{n′} ⟨\frac{m_{φ_{1}(i)}(2k − 1)}{n}⟩ .

Since the number m_{φ_{1}(i)}(2k−1)/n+m_{φ_{1}(i)}{2(n′−k+1)−1}/n is an integer for 1 ≤ k ≤ [n/2], we have

⟨\frac{m_{φ_{1}(i)}(2k − 1)}{n}⟩ + ⟨\frac{m_{φ_{1}(i)}\{2(n′ − k + 1) − 1\}}{n}⟩ ≤ 1.

Therefore we have

F^{⋆⋆} < \frac{n}{2} − 1 + 2 ([\frac{n′}{2} ] + 1) ≤ n + 1.

If d_{φ_{2}(i)} = n, then we have

F^{⋆⋆} ≤ n − 1.

From these, we obtain the assetion (6.5.2). Q.E.D.

6.6 Assume τ_{i_{0}+1} is of type B and β_{i_{0}} = 1 for some i_{0} (2 ≤ i_{0} ≤ r − 1). By our construction, there are integers v and w with 1 ≤ v < w ≤ i_{0} such that τ_{i_{0}+1} is the blow-up at the center P_{i_{0}} = E_{v,i_{0}} ∩ E_{w,i_{0}}, and the curve B̃_{i_{0}} is nonsingular at P_{i_{0}}. We denote by C̃ the unique local branch of B̃_{i_{0}} at P_{i_{0}}. Then one of the following two conditions (I) and (II) is satisfied : (I) i) C̃ intersect both E_{v,i_{0}} and E_{w,i_{0}} transversally at P_{i_{0}}, or

ii) C̃ is tangential to E_{v,i_{0}} at P_{i_{0}} of order t ≥ 2. (II) C̃ is tangential to E_{w,i_{0}} at P_{i_{0}} of order t ≥ 2. From now on, we consider that the case i) of (I) is the special case of the case ii) of (I) by setting t = 1.

We consider the curve B̃_{w−1} on W_{w−1}. In both cases (I) and (II), there exists a local branch C of B̃_{w−1} at the center P_{w−1} such that the proper transform of C by τ_{w} is isomorphic to C̃. More precisely, if i_{0} is greater than w, then any of the centers of the blow-ups τ_{w+1}, ⋯, τ_{i_{0}} does not coincide with the infinitely near point of P_{w−1}, and so these blow-ups do not matter the analysis of the singularity at P_{i_{0}}. Therefore by changing the order of the blow-ups if necessary, we may assume that the number w coincides with i_{0}.

Now assume that the case (I) occure. Then the following two conditions are satisfied (see Figure 4) : I-a) C is nonsingular and contacts at P_{w−1} to E_{v,w−1} of order t + 1, and I-b) If other local branches of B̃_{w−1} at P_{w−1} exist, then any tangent line of them does not coincides with E_{v,w−1}.

C  x + y^{t+1} E_{v,w−1} P_{w−1}  C̃  E_{v,w} E_{w,w} x + y^{t} −1 ⟵^{τ_{w}} Figure 4:

On the other hand, if the case (II) occure, then the following conditions are satisfied (see Figure 5) : II-a) C defines at P_{w−1} a tangential (t, t+1)-cusp to E_{v,w−1}, i.e. the local analytic equation of C at P_{w−1} is given by x^{t} + y^{t+1} = 0, where E_{v,w−1} is locally given by x = 0, and II-b) If other local branches of B̃_{w−1} at P_{w−1} exist, then any tangent line of them does not coincides with E_{v,w−1}.

E_{v,w−1} C x^{t} + y^{t+1} E_{v,w} E_{w,w} y + x^{t} C̃ −1 ⟵^{τ_{w}} Figure 5:

Assume that the case (I) occure. By changing the order of the blow-ups if necessary, we may assume that the successive blow-ups τ_{w+1}, ⋯, τ_{w+t} are produced at the infinitely near points of P_{w−1}, and therefore τ_{w+j} (1 ≤ j ≤ t) satisfies the following:

(i) τ_{w+j} is the blow-up of type B with the center P_{w+j} = E_{v,w+j} ∩ E_{w+j,w+j}, (ii) B̃_{w+j} is nonsingular at P_{w+j} and is tangent to E_{v,w+j} of order t − j at P_{w+j}. After these bow-ups, the curve B̃_{w+t} + E_{w+t} has normal crossing in the pull back of a neighborhood of P_{w}.

If the case (II) occure, then we have the same argument by replacing the definition of P_{w+j} by E_{w,w+j} ∩ E_{w+j,w+j}.

Lemma 6.7 In the above situation , we further assume that τ_{w} is of type A. If C̃ satisfies the codition (I), then

\begin{matrix}F_{A}(w) + ∑_{j=1}^{t} F_{B}(w + j) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{w−1} − 1)(β_{w−1} + 1) \\ + (−\frac{2}{3}n − 1 + \frac{1 + d_{v}^{2}}{3n} ) (t − 1) − \frac{1}{3}n − d_{v} − 3 + \frac{d_{w}^{2} + 2d_{v}^{2} + 4}{3n} < 0. \end{matrix}

If C̃ satisfies the codition (II), then

\begin{matrix}F_{A}(w) + ∑_{j=1}^{t} F_{B}(w + j) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{w−1} − 1)(β_{w−1} + 1) \\ + (−\frac{2}{3}n − 1 + \frac{1 + d_{w}^{2}}{3n} ) (t − 1) − \frac{1}{3}n − d_{w} − 3 + \frac{d_{v}^{2} + 2d_{w}^{2} + 4}{3n} < 0. \end{matrix}

Proof First we assume that C̃ satisfies the condition (I). In this case, we have φ_{1}(w + j) = v (0 ≤ j ≤ t), φ_{2}(w + j) = w + j − 1 (1 ≤ j ≤ t), θ_{w+j} = 0 (1 ≤ j ≤ t − 1), θ_{w+t} = 1, β_{w+j−1} = 1 (1 ≤ j ≤ t) and c_{φ_{1}(w+j)} = c_{φ_{2}(w+j)} = 1 (1 ≤ j ≤ t). We have

\begin{aligned}F_{A}(w) & ≤ & −\frac{1}{3} (n − \frac{1}{n}) β_{w−1}^{2} + d_{v,w} − 1 + \frac{(n − d_{w})(2n − d_{w})}{3n} \\ & & +\frac{(n − d_{v})(2n − d_{v})}{3n} − 4 ∑_{k=1}^{n−1} ⟨\frac{m_{w}k}{n} ⟩ ⟨\frac{m_{v}k}{n} ⟩ . \end{aligned}

On the other hand, it follows from (6.5.1) and Lemma 5.5 that

∑_{j=1}^{t} F_{B}(w + j) = ∑_{j=1}^{t} \{−\frac{1}{3} (n − \frac{1}{n}) − 2n + d_{v} + d_{w+j−1,w+j} − 1

+2 ∑_{k=1}^{n−1} (⟨\frac{m_{w+j}k}{n} ⟩ − ⟨\frac{m_{w+j−1}k}{n} ⟩)^{2} + 2 ∑_{k=1}^{n−1} ⟨\frac{m_{v}k}{n} ⟩^{2}\}

+ ∑_{j=1}^{t} \{d_{w+j−1} − d_{w+j} − d_{w+j−1,v} + d_{w+j,v} − 4 ∑_{k=1}^{n−1} (⟨\frac{m_{w+j+1}k}{n} ⟩ − ⟨\frac{m_{w+j}k}{n} ⟩)

× ⟨\frac{m_{v}k}{n} ⟩\} − \frac{4}{n} ∑_{k=1}^{n−1} k ⟨\frac{m_{w+t}k}{n} ⟩ − d_{w+t}

≤ (−\frac{2}{3}n − 1 + \frac{1 + d_{v}^{2}}{3n} ) t + d_{w} − d_{w+t} − d_{v,w} + d_{w+t,v} + 4 ∑_{k=1}^{n−1} ⟨\frac{m_{w}k}{n} ⟩ ⟨\frac{m_{v}k}{n} ⟩

−4 ∑_{k=1}^{n−1} ⟨\frac{m_{w+t}k}{n} ⟩ ⟨\frac{m_{v}k}{n} ⟩ − (\frac{2}{3}n + 1 − \frac{2}{3n})

≤ (−\frac{2}{3}n − 1 + \frac{1 + d_{v}^{2}}{3n} ) (t − 1) + d_{w} − d_{v,w} + 4 ∑_{k=1}^{n−1} ⟨\frac{m_{w}k}{n} ⟩ ⟨\frac{m_{v}k}{n} ⟩

−\frac{4}{3}n − 2 + \frac{d_{v}^{2} + 3}{3n} . (6.7.1)

Therefore by an easy calculation, the left hand side inequality of the Lemma follows.

We prove the negativity. The claim is clear except for the case

β_{w−1} = 1 and d_{w}^{2} + 2d_{v}^{2} > n^{2}.  (∗)

Assume the codition (∗). The inequality d_{w}^{2} + 2d_{v}^{2} > n^{2} holds if and only if “d_{w} = n or d_{v} = n”. Suppose d_{w} = n. It follows from m_{v} = m_{w} − β_{w−1} = m_{w} − 1 that d_{v} = 1. Therefore we have

−\frac{1}{3}n − d_{v} − 3 + \frac{d_{w}^{2} + 2d_{v}^{2} + 4}{3n} = −4 + \frac{2}{n} < 0.

The case d_{v} = n is similar, and we omit it.

Secondly we assume that C̃ satisfies the condition (II). In this case, we have φ_{1}(w +j) = w (1 ≤ j ≤ t), φ_{2}(w + 1) = v, φ_{2}(w + j) = w + j − 1 (2 ≤ j ≤ t), θ_{w+j} = 0 (1 ≤ j ≤ t − 1) and θ_{w+t} = 1. By the similar calculation as above, we obtain the desired result. (We omit it.) Q.E.D.

Corollary 6.8. Assume that τ_{i} is of type A and β_{i−1} = 1 for some i (2 ≤ i ≤ r). Then B̃_{i−1} is nonsingular at P_{i−1} and is tangent to E_{v,i−1} at P_{i−1} of order t + 1 for some t ≥ 1. Let τ_{i+1}, ⋯, τ_{i+t} be the succession of blow-ups whose centers are infinitely near points of P_{i} = E_{v,i} ∩ E_{i,i} such that the curve B̃_{i+t} + E_{i+t} has normal crossing in the pull back of a neighborhood of P_{i}. Then we have

\begin{aligned}F_{A}(i) + ∑_{j=1}^{t} F_{B}(i + j) & ≤ & (−\frac{2}{3}n − 1 + \frac{1 + d_{i}^{2}}{3n} ) (t − 1) − \frac{1}{3}n \\ & & −d_{i} − 3 + \frac{d_{i+1}^{2} + 2d_{i}^{2} + 4}{3n} < 0. \end{aligned}

Proof Under this situation, the blow-up τ_{i+1} is of type B and the unique local branch of B̃_{i} at P_{i} satisfies the condition (I) in 6.6. Therefore the assertion follows from Lemma 6.7. Q.E.D.

Lemma 6.9. In the situation of 6.6, we further assume that τ_{w} is of type B. The point of the center P_{w−1} of τ_{w} is written as P_{w−1} = E_{v,w−1} ∩ E_{u,w−1} for some u (1 ≤ u ≤ w − 1, u ≠ v). Moreover we assume β_{w−1} ≥ 2. If C̃ satisfies the codition (I), then we have

\begin{matrix}∑_{j=0}^{t} F_{B}(w + j) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{w−1} − 2)(β_{w−1} + 2) − (β_{w−1} − 2)n \\ + (−\frac{2}{3}n − 1 + \frac{1 + d_{v}^{2}}{3n} ) (t − 1) − \frac{7}{3}n − d_{v,u} − 5 + \frac{d_{w}^{2} + 2d_{v}^{2} + d_{u}^{2} + 9}{3n} < 0. \end{matrix}

If C̃ satisfies the codition (II), then we have

\begin{matrix}∑_{j=0}^{t} F_{B}(w + j) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{w−1} − 2)(β_{w−1} + 2) − (β_{w−1} − 2)n \\ + (−\frac{2}{3}n − 1 + \frac{1 + d_{w}^{2}}{3n} ) (t − 1) − \frac{7}{3}n + d_{v} − d_{w} − d_{v,u} − 5 + \frac{d_{v}^{2} + 2d_{w}^{2} + d_{u}^{2} + 9}{3n} < 0. \end{matrix}

Proof First we assume that C̃ satifies the condition (I). We have

F_{B}(w) ≤ −\frac{1}{3} (n − \frac{1}{n}) β_{w−1}^{2} − (β_{w−1} + 1)n + 2n − 2d_{w} + d_{w,v} + d_{w,u} − d_{v,u}
−1 + \frac{d_{w}^{2} + d_{v}^{2} + d_{u}^{2}}{3n} − 4 ∑_{k=1}^{n−1} ⟨\frac{m_{w}k}{n} ⟩ ⟨\frac{m_{v}k}{n} ⟩−4 ∑_{k=1}^{n−1} ⟨\frac{m_{w}k}{n} ⟩ ⟨\frac{m_{u}k}{n} ⟩

+4 ∑_{k=1}^{n−1} ⟨\frac{m_{v}k}{n} ⟩ ⟨\frac{m_{u}k}{n} ⟩ . (6.9.1)

From (6.9.1), (6.7.1) and Lemma 5.4, we obtain the desired inequality by an easy calculation.

If the curve C̃ satisfies the condition (II), then the calculation is similar as above, and we omit it. Q.E.D.

6.10. In the situation of 6.6, we further assume that τ_{w} is of type B, and we write the point P_{w−1} := E_{v,w−1} ∩ E_{u,w−1} as in the previous Lemma. Moreover we assume that there exist two local branches C and C′ of B̃_{w−1} at P_{w−1} such that the following conditions (i) ∼ (iii) are satisfied:

(i) C is nonsingular at P_{w−1} and is tangent to E_{v,w−1} at P_{w−1} of order t + 1 for some t ≥ 1, or C defines at P_{w−1} a tangential (t, t + 1)-cusp to E_{v,w−1}, (ii) C′ is nonsingular at P_{w−1} and is tangent to E_{u,w−1} at P_{w−1} of order t′ + 1 for some t′ ≥ 1, or C′ defines at P_{w−1} a tangential (t′, t′ + 1)-cusp to E_{u,w−1}, (iii) if other local branches of B̃_{w−1} at P_{w−1} exist, then the tangent line of any component of them coincides with neither E_{v,w−1} nor E_{u,w−1}.

Let C̃ and C̃′ be the proper transform of C and C′ by the blow-up τ_{w} respectively. Then C̃ pass through P_{w} := E_{v,w} ∩ E_{w,w} , and C̃′ pass through P_{w+t} := E_{u,w} ∩ E_{w,w}. Moreover any other components of B̃_{w} except for C and C′ pass through neither P_{w} nor P_{w+t}. By changing the order of the blow-ups if necesarry, we may assume that the succession of blow-ups τ_{w+1}, ⋯, τ_{w+t} are produced at infinitely near points of P_{w} and the succession of blow-ups τ_{w+t+1}, ⋯, τ_{w+t+t′} are produced at infinitely near points of P_{w+t}. After these blow-ups, the curve B̃_{w+t+t′} + E_{w+t+t′} has normal crossing in the pull back of the union of neighborhood of P_{w} and P_{w+t}.

Lemma 6.11. In the above situation, if C and C′ are nonsingular at P_{w−1}, then we have

∑_{j=0}^{t+t′} F_{B}(w + j) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{w−1} − 2)(β_{w−1} + 2) − (β_{w−1} − 2)n

+ (−\frac{2}{3}n − 1 + \frac{1 + d_{v}^{2}}{3n} ) (t − 1) + (−\frac{2}{3}n − 1 + \frac{1 + d_{u}^{2}}{3n} ) (t′ − 1)

−\frac{11}{3}n − d_{v,u} − 7 + \frac{2d_{u}^{2} + 2d_{v}^{2} + d_{w}^{2} + 12}{3n} < 0.

If C is nonsingular at P_{w−1} and C′ defines a tangential cusp at P_{w−1}, then we have

∑_{j=0}^{t+t′} F_{B}(w + j) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{w−1} − 2)(β_{w−1} + 2) − (β_{w−1} − 2)n

+ (−\frac{2}{3}n − 1 + \frac{1 + d_{v}^{2}}{3n} ) (t − 1) + (−\frac{2}{3}n − 1 + \frac{1 + d_{w}^{2}}{3n} ) (t′ − 1)

−\frac{11}{3}n + d_{u} − d_{u,w} − d_{u,v} − 7 + \frac{d_{u}^{2} + 2d_{v}^{2} + 2d_{w}^{2} + 12}{3n} < 0.

If C and C′ also define tangential cusps at P_{w−1}, then we have

∑_{j=0}^{t+t′} F_{B}(w + j) ≤ −\frac{1}{3} (n − \frac{1}{n}) (β_{w−1} − 2)(β_{w−1} + 2) − (β_{w−1} − 2)n

+ (−\frac{2}{3}n − 1 + \frac{1 + d_{w}^{2}}{3n} ) (t + t′ − 2) − \frac{11}{3}n + d_{u} + d_{v} − 2d_{w}

−d_{v,u} − 7 + \frac{d_{u}^{2} + d_{v}^{2} + 3d_{w}^{2} + 12}{3n} < 0.

Proof Assume C is nonsingular at P_{w−1} and C′ defines a tangential cusp at P_{w−1}. Since C̃′ is nonsingular at P_{w+t} and is tangent to E_{w,w} of order t′, by the same argument as (6.7.1), we have

\begin{aligned}∑_{j=t+1}^{t+t′} F_{B}(w + j) & ≤ & (−\frac{2}{3}n − 1 + \frac{1 + d_{w}^{2}}{3n} ) (t′ − 1) + d_{u} − d_{u,w} \\ & & +4 ∑_{k=1}^{n−1} ⟨\frac{m_{w}k}{n} ⟩ ⟨\frac{m_{u}k}{n} ⟩ − \frac{4}{3}n − 2 + \frac{d_{w}^{2} + 3}{3n}. \end{aligned}

On the other hand, we have the same estimates for F_{B}(w) and ∑_{j=1}^{t} F_{B}(w + j) as (6.9.1) and (6.7.1). Therefore by using Lemma 5.4, we easily obtain the desired inequality. We omit the other two cases. Q.E.D.

Proof of Theorem 6.1. From Lemmas 6.4, 6.5, 6.7, 6.9, 6.11 and Corollary 6.8, the assertion (6.3.1) follows. Hence we complete the proof of Theorem 6.1.

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Tadashi Ashikaga Faculty of Engineering Tôhoku-Gakuin University Tagajou, Miyagi, 985 Japan