On −K^{2} for normal surface singularities II

Hao Chen

Department of Mathematics Zhongshan University Guangzhou, Guangdong 510275 People’s Republic of China

Shihoko Ishii

Department of Mathematics Tokyo Institute of Technology O-Okayama, Meguro, Tokyo 152-8551, Japan

Abstract

In this paper we continue our study [1] of the set of nonzero −K^{2} of normal surface singularities. The main results are

1) Every accumulation point of the set is the sum of −K^{2} of several normal surface singularities;

2) There are accumulation points of the set which are not −K^{2} of any one normal surface singularity, which means that the set { −K^{2}} is not closed.

1 Introduction

In our previous paper [1] we studied the set {−K^{2}|K is the numerical canonical divisor of a normal surface singularity} (for the numerical canonical divisor see Definition 2.1 below). It is proved that any nonzero −K^{2} is bigger than or equal to 1/3 (Theorem 1.4 in [1]) and that the set has no accumulation point from above (Theorem 1.10 in [1]). We know from our previous results that all accumulation points from below are rational numbers (Theorem 2.3 in [1]) and that there are indeed many accumulation points from Propositions 2.4, 2.6 and Corollary 2.6 in [1]. It is natural to ask if the set {−K^{2}} is closed, i.e., is every accumulation point the −K^{2} of a normal surface singularity? In this paper we answer this problem. Although counterexamples are given for the problem in section 4, we prove that every accumulation point is the sum of −K^{2}’s of several normal surface singularities.

2 Preliminaries

In this section we recall the basic definitions and results which are needed in the following sections.

Definition 2.1. Let A = ∑ A_{i} be a compact reduced divisor whose intersection matrix {A_{i}A_{j}}_{i,j} is negative definite on a normal Q-factorial surface Y. One defines the numerical canonical divisor K(A) as a Q-divisor with the support on A as

K(A)A_{i} = K_{Y} A_{i} (2.1)

for every i. In particular, if Y is the minimal resolution of a normal surface singularity (X, x) and A is the exceptional divisor, then K(A) is called the numerical canonical divisor of the singularity (X, x).

We call a curve C (in a surface) with C ≃ P^{1} and C^{2} = −n a (−n)-curve. The following formula in 1) is clear from the Adjunction formula, and 2) is Lemma 1.3 of [1].

Lemma 2.2. Notation as above. Suppose Y is the minimal resolution of a normal surface singularity.

1) Let p_{i} be the arithmetic genus of A_{i}, c_{i} = KA_{i} = 2p_{i} − 2 − (A_{i})^{2} and E = (A_{i}A_{j})_{1≤i,j≤n} be the intersection matrix. We have

−K^{2} = −(c_{1}, ..., c_{n})(E^{−1})^{t}(c_{1}, ..., c_{n}) (2.2)

2) −K^{2} ≥ ∑(KA_{i})^{2}/(−A_{i}^{2}).

3) (KA_{i})^{2}/(−A_{i}^{2}) = (2p_{i} − 2)^{2}/(−A_{i}^{2}) + 2(2p_{i} − 2) − A_{i}^{2} is not less than 1/3. It is 1/3 if and only if A_{i} is (−3)-curve. If it is not (−3)-curve, it is not less than 1. It is 1 if and only if A_{i} is (−4)-curve or (−1)-elliptic curve.

Lemma 2.3. Let E be a negative definite matrix of size n with non-negative integer entries at non-diagonal positions and negative integer entries at diagonal positions. Suppose that we cannot make E to be a diagonal (blocked) matrix after renumbering the order of rows and columns, respectively. Then E is the intersection matrix of exceptional divisors on a resolution of a normal surface singularity. Moreover, the genera of irreducible components of the exceptional set can be arbitrarily pre-assigned (g_{1}, .., g_{n}).

Proof. Take n smooth curves A_{1}, ..., A_{n} of genera g_{1}, ..., g_{n} and tubular neighborhoods U_{1}, ..., U_{n} of zero sections in the negative line bundle L_{1}, ..., L_{n} over these curves. Let the degree of L_{i} be the i-th diagonal entry of E. Patch U_{1}, ..., U_{n} together according to the non-diagonal entries of E, we get a complex 2-dimensional manifold Y. We know that Y is connected from our assumption of E. Thus, we get (X, x) from contracting ∪A_{i} by Grauert’s theorem. Q.E.D.

The following result is a refinement of Proposition 1.2 in [1].

Proposition 2.4. Let A be a compact connected divisor on a smooth surface whose intersection matrix is negative definite. Suppose that A does not contain (−1)-curves, and A = A_{1} + A_{2} is a nontrivial decomposition with no common components on A_{1} and A_{2}. Then we have −K(A_{1})^{2} − K(A_{2})^{2} ≤ −K(A)^{2}, and the equality holds if and only if A consists of (−2)-curves.

Proof. First we show that K(A) − K(A_{1}) − K(A_{2}) is nef on A. For an irreducible curve C ⊂ A_{1}, (K(A) − K(A_{1}) − K(A_{2}))C = (K(A) − K(A_{1}))C − K(A_{2})C. From Definition 2.1, (K(A) − K(A_{1}))C = 0. −K(A_{2})C ≥ 0 since −K(A_{2}) is an effective divisor. Similarly, we know (K(A)−K(A_{1})−K(A_{2}))C ≥ 0 for any irreducible curve C ⊂ A_{2}. Thus, (K(A) − K(A_{1}) − K(A_{2}))K(A) = K(A)^{2} − K(A_{1})^{2} − K(A_{2})^{2} ≤ 0. It is clear that the equality holds if A consists of (−2)-curves. Moreover, if A contains a non (−2)-curve, we can assume that this curve is in A_{2} without loss of generality. Thus, −K(A_{2}) is strictly bigger than 0 and −K(A_{2})C > 0 for some irreducible curves C in A_{1} and the strict inequality follows. Q.E.D.

We refer to both the concept of (−2)-insertion and the proof of the following proposition to [1].

Proposition 2.5. Let A be a (−2)-insertion of A′, then we have −K(A′)^{2} ≤ −K(A)^{2}.

The following lemma is similar to 1) of Lemma 2.2, which will be needed in the proof of our main result in section 3.

Lemma 2.6. Let X be a nonsingular surface and E = ∑_{i=1}^{r} E_{i}+∑_{j=1}^{s} F_{j} be a compact divisor on X with negative definite intersection matrix, where F_{j}, j = 1, ..., s, are (−2)-curves. Suppose φ : X → X′ is the contraction of ∑_{j=1}^{s} F_{j}, E′_{i} is the image of E_{i} under φ, E′ = ∑ E′_{i} and M′ is the intersection matrix of E′_{i}s on X′. Then −K(E)^{2} = −c(M^{′−1})^{t}c, where c = (c_{1}, ..., c_{r}), c_{i} = K(E)E_{i}.

Proof. We write K(E) = ∑ m_{i}E_{i} + ∑ n_{j}F_{j}, then K(E′) = ∑ m_{i}E′_{i}. Since all singular points of X′ are RDP’s, thus φ^{∗}(K(E′)) = K(E). Hence, −K(E)^{2} = −K(E′)^{2} = − ∑_{i=1}^{r} m_{i}K(E′)E′_{i}, where K(E′)E′_{i} = φ^{∗}(K(E′))E_{i} = K(E)E_{i} = c_{i} by projection formula. We have (M′)^{t}m = ^{t}c, where m = (m_{1}, ..., m_{r}) from Definition 2.1 of the numerical canonical divisor. So −K(E)^{2} = −m^{t}c = −c(M^{′−1})^{t}c. Q.E.D.

3 An accumulation point is the sum of −K^{2}’s

In this section we prove the following statement.

Theorem 3.1. Let W be an accumulation point of the set {−K^{2} }. Then finitely many normal surface singularities (Y_{1}, y_{1}), ...., (Y_{t}, y_{t}) exist such that

W = −K_{(Y_{1},y_{1})}^{2} − ... − K_{(Y_{t}.y_{t})}^{2} (3.1)

Proof. Let {A_{(n)}} be the sequence of exceptional divisors of minimal resolutions {X_{n}} of normal surface singularities with the corresponding sequence of {−K_{(n)}^{2}} tending to W. First, as in the proof of Theorem 2.3 of [1], we can assume that A_{(1)} → A_{(n)} is a composite of (−2)-insertions: A_{(1)} = A_{(0...0)} = ∑ E_{i} and A_{(n)} = A_{(n_{1}...n_{k})} = ∑ E′_{i} + ∑_{j=1}^{k} S_{j}, where S_{j} = ∑_{s=1}^{n_{j}} F_{js} is the insertion string with the props E′_{2j−1}, E′_{2j}. One may assume that if n → ∞, then n_{j} → ∞ for all j = 1, ..., k.

Now we get X′_{n} by contracting (−2)-curve strings S_{j}, j = 1, ..., k. Let the intersection matrix of the image of E′_{i} in X′_{n} be M_{(n_{1}...n_{k})}. This takes the following form as in the proof of Theorem 2.3 of [1].

( \begin{matrix}−\frac{n_{1}+2}{n_{1}+1} & \frac{1}{n_{1}+1} & ... & ... & 0 & 0 & ^{t}a_{1} \\ \frac{1}{n_{1}+1} & −\frac{n_{1}+2}{n_{1}+1} & ... & ... & 0 & 0 & ^{t}a_{2} \\ ... & ... & ... & ... & ... & ... & ... \\ 0 & 0 & ... & ... & −\frac{n_{k}+2}{n_{k}+1} & \frac{1}{n_{k}+1} & ^{t}a_{2k−1} \\ 0 & 0 & ... & ... & \frac{1}{n_{k}+1} & −\frac{n_{k}+2}{n_{k}+1} & ^{t}a_{2k} \\ a_{1} & a_{2} & ... & ... & a_{2k−1} & a_{2k} & N \end{matrix} ) (3.2)

and −K_{(n)}^{2} = −cM_{(n_{1}...n_{k})}^{−1}^{t}c, where c = (c_{1}, ..., c_{h}), c_{i} = K(A_{(0...0)})E_{i} from Lemma 2.6.

Since it is proved that lim M_{(n_{1}....n_{k})} = M_{(∞...∞)} (see (3.3)) is a nonsingular (thus negative definite) matrix in the proof of Theorem 2.3 of [1], we have W = −cM_{(∞....∞)}^{−1}^{t}c, where

M_{(∞...∞)} = ( \begin{matrix}−1 & 0 & ... & 0 & ^{t}a_{1} \\ 0 & −1 & ... & 0 & ^{t}a_{2} \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & −1 & ^{t}a_{2k} \\ a_{1} & a_{2} & ... & a_{2k} & N \end{matrix} ) (3.3)

Let D_{l} be the following configuration:

\begin{matrix}o_{1} − ⋯ − o_{l−3} − o_{l−2} − o_{l−1} \\ | \\ o_{l} \end{matrix} (3.4)

of (−2)-curves corresponding to RDP D_{l}. Consider the configuration A_{(∞...∞)} = ∑_{i=1}^{r} E′_{i} + ∑_{j=1}^{k}(D_{l}^{j} + D′_{l}^{j}), where D_{l}^{j}, D′_{l}^{j} for j = 1, ..., k are copies of D_{l}. Each D_{l}^{j} (respectively, D′_{l}^{j}) intersects the other part only at o_{1} (respectively, o′_{1}), which has one intersection point with E′_{2j−1} (respectively, E′_{2j}). This is equivalent to replacing each string S_{j} in A_{(n_{1}...n_{k})} with two copies of configuration D_{l}. Here we should note that A_{(∞...∞)} may not be connected. We construct a nonsingular surface X containing A_{(∞...∞)} as a compact divisor (but might be non connected) as in Lemma 2.3. Now we contract D′_{l}s in X, by φ : X → X′. Let A′_{(∞...∞)}, E′′_{2j−1}, E′′_{2j} be the images of A_{(∞...∞)}, E′_{2j−1}, E′_{2j} for j = 1, ..., k under the contraction. Since φ^{∗}(E′_{2j−1})o_{t} = 0 , for t = 1, ..., l, we have φ^{∗}(E′′_{2j−1}) = E′_{2j−1} + o_{1} + ... + o_{l−2} + (o_{l−1} + o_{l})/2. Thus, E′′_{2j−1}^{2} = −1. Similarly, E′′_{2j}^{2} = −1. Thus, the intersection matrix of A′_{(∞...∞)} in X′ is just the same as M_{(∞...∞)} in (3.3). From Lemma 2.6 we

have −K(A′_{(∞...∞)})^{2} = −cM_{(∞...∞)}^{−1}^{t}c = W. On the other hand, by contracting A′_{(∞...∞)} in X′ (since the intersection matrix M_{(∞...∞)} is negative definite) we get several normal surface singularities (Y_{1}, y_{1}), ..., (Y_{t}, y_{t}). It is clear that −K_{(Y_{1},y_{1})}^{2} − ... − K_{(Y_{t},y_{t})}^{2} = −K(A′_{(∞...∞)})^{2} = W. The conclusion is proved.

Corollary 3.2. Let W be an accumulation point of the set {−K^{2}}. Suppose {A_{(n)}} is the sequence of exceptional divisors on the minimal resolutions of normal surface singularities with {−K_{(n)}^{2}} tending to W. Moreover, the non (−2)-curve part (which can be assumed to be independent of n after choosing a suitable subsequence as showed in [1]) of {A_{(n)}} is connected. Then we can find a normal surface singularity (Y, y), such that W = −K_{(Y,y)}^{2}.

Proof. We note that in this case, except for one connected component of A_{(∞...∞)}, the other connected components are exceptional divisors of rational double points whose −K^{2}’s are zero.

4 {−K^{2}} is not closed

In this section, we give a sequence of accumulation points of {−K^{2}} which are not in this set.

Proposition 4.1. Let T_{n,m} be normal surface singularities with the following weighted dual graph (the existence of such singularities is clear from Lemma 2.3)

T_{n,m} : x − o − ⋯ − o − x − o − ⋯ − o ,

where x and o denote a (−3)-curve and a (−2)-curve respectively, there are n o′s between the two x’s and m o′s at the right side of the second x. We have

−K(T_{n,m})^{2} = (3nm + 4n + 7m + 8)/(2nm + 4n + 5m + 8) (4.1)

Proof. This is a direct computation.

The following statement is the main result of this section.

Theorem 4.2. Let

v_{m} = lim_{n→∞}(−K(T_{n,m})^{2}) = (3m + 4)/(2m + 4) (4.2)

Suppose m is a sufficiently large prime number of the form 4k + 1. Then v_{m} is not in the set {−K^{2}}.

From Theorem 4.2 we obtain the following conclusion immediately.

Corollary 4.3. The set {−K^{2}} is not closed.

Proof. We know there are infinitely many prime numbers of the form 4k +1 from Dirichlet theorem (see Chapter VI in [2]). Thus, from Theorem 4.2 there are infinitely many accumulation points of the set {−K^{2}} which are not in this set. Q.E.D.

Assume that the conclusion of Theorem 4.2 is not true, we can then get a sequence of normal surface singularities { (Y_{n}, y_{n})} with −K_{(Y_{n},y_{n})}^{2} = v_{m_{n}}, where m_{n} is a prime number of the form 4k + 1, and hence { −K_{(Y_{n},y_{n})}^{2} } tends to 3/2 from below. The main part of the proof for Theorem 4.2 is to study all possible such singularities. Let {(Y_{n}, y_{n})} be a sequence of normal surface singularities with their {−K_{(Y_{n},y_{n})}^{2}} tending to 3/2 from below, and {A_{n}} be the sequence of exceptional divisors on the minimal resolutions of {(Y_{n}, y_{n})}. As showed in [1] we can assume that the non (−2)-curve part {A} of {A_{n}} is independent of n after taking a suitable subsequence.

Lemma 4.4. Let X be a normal surface singularity whose exceptional set contains a D_{l}. We have −K_{X}^{2} ≥ 1/2. Moreover, if the D_{l} intersects at a non (−3)-curve, we have −K_{X}^{2} > 1.

Proof. Let E_{0} be a non (−2)-curve which connects to D_{l}. The exceptional set contains either the following configurations (• denotes the non (−2)-curve E_{0}):

• − o − ⋯ − o − o − o
|
o

or

\begin{matrix}o \\ | \\ • − o − o \\ | \\ o \end{matrix}

In the former case, the exceptional set contains a (−2)-insertion of the following graph H

\begin{matrix}• − o − o − o \\ | \\ o \end{matrix}

Because H is not negative definite when E_{0}^{2} = −1, we can assume that E_{0}^{2} < −1. Then −K(H)^{2} = \frac{(2p−2−E_{0}^{2})^{2}}{−1−E_{0}^{2}} = \frac{(2p−1)^{2}}{−1−E_{0}^{2}} +2(2p−1)−1−E_{0}^{2}, where p is the arithmetic genus of E_{0}. It is easy to know −K(H)^{2} > 1/2 generally and −K(H)^{2} > 1 when E_{0} is a non (−3)-curve. In the latter case, the non (−2)-curve E_{0} has E_{0}^{2} ≤ −3, because the subgraph should be negative definite. Hence, the subgraph has −K^{2} = \frac{(2p−2−E_{0}^{2})^{2}}{−2−E_{0}^{2}} = \frac{4p^{2}}{−2−E_{0}^{2}} + 4p + (−2 − E_{0}^{2})^{2} > 1, where p is the arithmetic genus of E_{0}. Then from Proposition 2.4 we obtain our conclusion.

Lemma 4.5. The A has to be one of the following possibilties:

1) one (−3)-curve;

2) the sum of two (−3)-curves.

Proof. We apply Theorem 3.1 and its proof here. There are serveral non RDP normal surface singularities X_{1}, ...X_{t}, such that 3/2 = −K_{X_{1}}^{2}... − K_{X_{t}}^{2}. Moreover, each of the corresponding exceptional divisors B_{1}, ..., B_{t} contains a divisor of the form D_{l} as showed in the proof of Theorem 3.1. From Lemma 4.4 we have t ≤ 3.

If t = 3, −K_{X_{i}}^{2} = 1/2 for each i. Again from Lemma 4.4 there is only one (−3)-curve in each B_{i}, and one B_{i} has at least two D′_{l}s from the proof of Theorem 3.1. From Propositon 2.4, −K^{2} of this B_{i} is strictly bigger than 1/2, which is a contradiction.

When t = 2, −K_{X_{i}}^{2} ≤ 1 from Lemma 4.4. If −K_{X_{1}}^{2} = 1, consider the configuration B′_{1}, which is the deletion of o_{l} and edges connecting to it from B_{1}, (where o_{l} is the irreducible component in D_{l} as in (3.4)), we have −K(B′_{1})^{2} < 1 from Proposition 2.4. Hence, B′_{1} corresponds to a rational triple point as showed in Proposition 2.8 of [1], and there is at most one (−3)-curve in B′_{1} (hence in B_{1}). On the other hand, −K_{X_{2}}^{2} = 1/2, the singularity is a rational triple point, and B_{2} contains at most one (−3)-curve. Otherwise we know −K_{X_{i}}^{2} < 1 for each i. So they are rational triple points, and there is at most one (−3)-curve in each B_{i}.

Assume t = 1, and B_{1} contains at least two non (−2)- curves T′_{i}s. By Proposition 2.4 we know −K(B′_{1})^{2} + (2p(T_{i}) − 2 − T_{i}^{2})^{2}/(−T_{i}^{2}) < 3/2, where B′_{1} = B_{1} − T_{i}. Since −K(B′_{1})^{2} ≥ 1/2 from Lemma 4.4, we have (2p(T_{i}) − 2 − T_{i}^{2})^{2}/(−T_{i}^{2}) < 1. Thus, T_{i} is a (−3)-curve. Since B_{1} is connected, we know these (−3)-curves have to be connected or connected by RDP (of fixed length) in A. It is easy to see that the configuration is obtained by inserting a string of (−2)-curves into the following configurations (x and o denote a (−3)-curve and a (−2)-curve, respectively):

x − x − x

\begin{matrix}x − o − x \\ | \\ x \end{matrix}

\begin{matrix}x − x \\ \ | \\ x \end{matrix}

We can check that the −K^{2} of the above configurations are bigger than 3/2 by direct computation. Hence, by Propositions 2.4 and 2.5 we have at most two (−3)-curves in B_{1}. We get our conclusion from the relation of A and B_{i} as described in the proof of Theorem 3.1.

In the following, the configuration D_{l} − x always means that x (where x denotes a (−3)-curve) connects to the o_{1} in the D_{l} of (3.4).

Lemma 4.6. Let X_{i}, B_{i}, i = 1, .., t be as in Lemma 4.5. If t = 1 there is at most (hence exactly) one (−3)-curve in B_{1}.

Proof. In this case, the weighted dual graph of B_{1} contains a subgraph of the form: D_{l} − x.

If there are two (−3)-curves, we have the following two possibilities:

1) The two (−3)-curves intersect.

In this case they have to intersect only at one point from the fact that its −K^{2} is not bigger than 3/2 (direct computation and Proposition 2.4). Since −K^{2} of D_{l} − x − x is less than 3/2, B_{1} has to contain one of the following subgraphs:

D_{l} − x − x − o

\begin{matrix}D_{l} − x − x \\ | \\ o \end{matrix}

or

\begin{matrix}D_{l} − x − x \\ \ | \\ o \end{matrix} .

In each case we can calculate its −K^{2} > 3/2. From Propositions 2.4 and 2.5,

1) is impossible.

2) Two x′s are connected by (−2)-curves.

Then either B_{1} contains a subgraph which is obtained by inserting a string of (−2)-curves into the three graphs as above, or B_{1} is D_{l} − x − RDP − x. In the former case we have −K^{2}(B_{1}) > 3/2 by Propositions 2.4 and 2.5. For the latter case, by a simple calculation we know the only possibility of middle RDP has to be (−2)-curve string if we want −K^{2} ≤ 3/2. However, in that situation it is always less than 3/2 (tending to 3/2 from below when the length of string tends to infinity). This contradicts to the fact −K(B_{1})^{2} = 3/2. Therefore 2) is impossible. The conclusion is proved.

Lemma 4.7. Let X_{i}, B_{i} (i = 1, ..., t) be as above. If t = 1, B_{1} and corresponding {A_{n}} are one of the following forms:

1) The B_{1} is:

D_{l} − x − o − o − o − o − o
|
o

and the A_{n} is

o − ⋯ − o − x − o − o − o − o − o
|
o

, where the left side of x is a string of (−2)-curves of length n; 2) The B_{1} is

D_{l} − x − o − o − o − o

| o and the A_{n} is

o − ⋯ − o − x − o − o − o − o
|
o

Moreover, −K(A_{n})^{2} is (3n + 3)/(2n + 5) in both 1) and 2).

Proof. From Lemma 4.6 there is only one (−3)-curve in B_{1} if t = 1. From direct computation we know that B_{1} can only have the above two possibilities from the fact −K(B_{1})^{2} = 3/2. Thus we can know A_{n} from its relation to B_{1}. Q.E.D.

Lemma 4.8. Let X_{i}, B_{i} (i = 1, .., t) be as above and t = 2. If −K(X_{i})^{2} < 1 for i = 1, 2, B_{i}, A_{n} have to be one of the following forms.

1) B_{i} :o−o−x−D_{l}, where i = 1, 2 and A_{n} : o−o−x−o−⋯−o−x−o−o, where there is a (−2)-curve string of length n between two x′s;

2) B_{1} : o − x − D_{l}; B_{2} : D_{l} − x − o − o − o − o and A_{n} : o − x − o − ⋯ − o − x − o − o − o − o.

Moreover, −K(A_{n})^{2} is 3/2 in case 1) and (27n + 67)/(18n + 45) in case 2).

Proof. If −K(B_{i})^{2} < 1 for i = 1, 2, −K(B_{i})^{2} is an accumulation point of the set {−K^{2}} (of sequence of weighted dual graphs with D_{l} in B_{i} replaced by string of (−2)-curves as showed in Theorem 3.1) as described in Proposition 2.8 in [1]. Hence, they have to be the form m/(m+1) from the conclusion of Proposition 2.8 of [1]. If we want m/(m+1)+m′/(m′ +1) = 3/2, (m, m′) has to be either (3, 3) or (2, 5). Thus −K(B_{i})^{2} = 3/4 for i = 1, 2 or −K(B_{1}) = 2/3, −K(B_{2})^{2} = 5/6. By direct computation we know that B_{1}, B_{2} have to be one of 1) and 2) from these data. The conclusion is proved.

Lemma 4.9. Let X_{i}, B_{i} (i = 1, 2) be as above. If −K(B_{1})^{2} = 1 and −K(B_{2})^{2} = 1/2, B_{2} is always of the form x − D_{l} and B_{1}, A_{n} have to be one of the following forms.

1) B_{1} : D_{l} − x − D_{l} and A_{n} : D_{l} − x − o − ⋯ − o − x;

2) B_{1} : D_{l} − x − o − ⋯ − o and A_{n} : x − o − ⋯ − o − x − o − ⋯ − o (i.e., A_{n} is of the form T_{t,w} of proposition 4.1);

3) B_{1} is of the form

o_{l−1} − o_{l−2} − ⋯ − o_{2} − o_{1}
| |
o_{l} x

and A_{n} is of the following form

x − o − ⋯ − o − o − o
|
x

4) B_{1} is of the following form

\begin{matrix}D_{l} − x − o \\ | \\ o \end{matrix}

and A_{n} is of the following form

x − o − ⋯ − o − x − o
|
o

Moreover, −K(A_{n})^{2} is (3n + 7)/(2n + 5) in case 1); (3tw + 4w + 7t + 8)/(2tw + 4w + 5t + 8) in case 2); (3n + 12)/(2n + 11) in case 3); (3n + 7)/(2n + 5) in case 4).

Proof. We can get the conclusion of B_{1}, B_{2} from the fact −K(B_{1})^{2} = 1 and −K(B_{2})^{2} = 1/2 by direct computation. The corresponding A′_{n}s can be obtained according to its relation with B_{1} and B_{2} described in the proof of Theorem 3.1. Q.E.D.

Proof of Theorem 4.2. We can easily check that if v_{m} = (3m+4)/(2m+4) is one of −K(A_{n})^{2} of Lemma 4.7, Lemma 4.8 or 1), 3), 4) of Lemma 4.9, m has to be divided by 4. This is impossible from our hypothesis on m. For 2) of Lemma 4.9, suppose v_{m} = −K(T_{t,w})^{2} for some (t, w). We have m = t − t^{2}/(4w + t + 8) from a direct computation. Suppose gcd(t, 4w + 8) = d, t = dw′ and 4w + 8 = dw′′. Then t^{2} = d^{2}w^{′2} can be divided by d(w′ + w′′), hence d can be divided by w′ + w′′. Let d = u(w′ + w′′). Then m = dw′ − uw^{′2} = uw′w′′. From our condition that m is prime we know that two of (u, w′, w′′) have to be 1. In the case u = w′′ = 1, m = w′ = 4w + 7 since d = u(w′ + w′′) = w′ + 1 = 4w + 8. This contradicts to the fact that m is of the form 4k +1. Similarly, we have contradictions in other cases. Thus, the conclusion of Theorem 4.1 follows. Q.E.D.

Acknowledgement. We express our gratitude to Professor V. Masek for the useful discussion with the second author. Proposition 2.4 is due to him. This work was started when the first author was visiting Tokyo Institute of Technology. He is grateful to the Japan Society for Promotion of Science for financial support and Tokyo Institute of Technology for hospitality. This work was completed when the second author is staying Max-Planck Institut für Mathematik.

She expresses her gratitude to Max-Plack instuitute for the support and the hospitality.

References

[1] H.Chen and S.Ishii, On −K^{2} for normal surface singularities, Inter. J. Math.,9, (1998), 653-668

[2] J-P Serre, A Course in Arithmetic, Graduate Text in Mathematics, 7, Springer-Verlag, 1973