THE MAXIMAL IDEAL CYCLES OVER COMPLETE INTERSECTION SURFACE SINGULARITIES OF BRIESKORN TYPE

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1 Kyushu J. Math. 68 (2014), doi: /kyushujm THE MAXIMAL IDEAL CYCLES OVER COMPLETE INTERSECTION SURFACE SINGULARITIES OF BRIESKORN TYPE Fan-Ning MENG and Tomohiro OKUMA (Received 12 November 2012) Abstract. For a complete intersection surface singularity of Brieskorn type, we give concrete descriptions of the maximal ideal cycle and the fundamental cycle on the minimal good resolution space. Then we provide a condition for these cycles to coincide, and a condition for the singularity to be a Kodaira singularity. The results are a natural extension of the hypersurface case obtained by Konno and Nagashima. 1. Introduction Let (V, o) be a germ of a normal complex surface singularity and f : X V a good resolution with exceptional divisor E. It is known that the topology of the singularity is determined by the weighted dual graph E of E. A divisor on X supported in E is called a cycle. The fundamental cycle Z E is by definition the smallest among the cycles F > 0 such that F is nef, i.e. FE i 0 for every irreducible component E i of E. The fundamental cycle is a topological invariant; in fact, it is determined by E. Let m be the maximal ideal of the local ring O V,o. For a non-zero function h m, let (h) E denote the exceptional part of the zero divisor div X (h). Then the smallest among the cycles (h) E, h m \{0}, is called the maximal ideal cycle and denoted by Z m. This cycle is an analytic invariant and cannot be determined by E in general. We have Z E Z m by the definition of these cycles. Therefore, it is a natural question to ask whether Z E = Z m. This equality holds on the minimal resolution for rational singularities [1], minimally elliptic singularities [10], weakly elliptic Gorenstein singularities with rational homology sphere link [12], and for hypersurfaces {z n = f(x, y)} with certain conditions [3, 22]. However, in general, it is difficult to identify the maximal ideal cycle (cf. [13, 15, 18]). In this paper, we consider a germ (V, o) (C m, o) of an isolated complete intersection singularity of Brieskorn type defined by V ={(x i ) C m q j1 x a q jmx a m m = 0, j= 3,...,m}, where a i 2 are integers. Then (V, o) is a normal surface singularity by Serre s criterion for normality. Neumann [14] proved that the universal abelian coverof a weighted homogeneous 2010 Mathematics Subject Classification: Primary 14J17; Secondary 32S25, 14B05. Keywords: surface singularities; weighted homogeneous singularities; Brieskorn complete intersections; maximal ideal cycles. c 2014 Faculty of Mathematics, Kyushu University

2 122 F.-N. Meng and T. Okuma normal surface singularity with rational homology sphere link is a complete intersection singularity of this type. The aim of this paper is to identify the maximal ideal cycle on the minimal good resolution of (V, o). We give concrete descriptions of the maximal ideal cycle and the fundamental cycle, a condition for the coincidence of these cycles, and a condition for the singularity to be a Kodaira singularity; every condition is expressed by the integers a 1,...,a m. Namely, we extend the results for Brieskorn hypersurface singularities due to Konno and Nagashima [8] to the Brieskorn complete intersections. In [8, Section 2], Konno and Nagashima constructed a good resolution using a covering method due to Tomaru [19, 23] and Fujiki [4]. We employ their method to construct a good resolution of (V, o). It is known that the resolution graph of the minimal good resolution of a weighted homogeneous surface singularity can be recovered from the Seifert invariants of the link. The Seifert invariant of the link of (V, o) is in fact obtained in [5, Section 7] (see [16] for the hypersurface case); however, the construction of the good resolution is needed for the computation of the maximal ideal cycle. This paper is organized as follows. In Section 2, we give an outline of the construction of the resolution and mention fundamental facts on cycles over a cyclic quotient singularity. In Section 3, we review a key result due to Konno and Nagashima. In Section 4, we construct a good resolution and compute the zero divisors of the pull-back of the coordinate functions. After computing the fundamental cycle in Section 5, we describe the main results of this paper in Section Preliminaries We consider a germ (V, o) (C m, o) of a Brieskorn complete intersection singularity defined by V ={(x i ) C m q j1 x a q jmx a m m = 0, j= 3,...,m}, where a i 2 are integers. We assume that (V, o) is an isolated singularity; this condition is equivalent to that every maximal minor of the matrix (q ji ) does not vanish (see [5, Section 7]). Therefore, by row operations and a diagonal linear change of coordinates, we may assume that p 3 q p 4 q (q ji ) = , (2.1) p m q m where p i,q i = 0 and p i q j = p j q i for i = j. We shall construct a good resolution of (V, o) via a tower of cyclic coverings to compute the divisors of the pull-back of the coordinate functions x 1,...,x m Outline of the construction Suppose that f : X V is the minimal good resolution and E the exceptional set. Assume that E is not a chain of rational curves. Then the dual graph of E is star-shaped (cf. Figure 3 in Section 4). Let E 0 denote the central curve of E and f : X X the morphism

3 Brieskorn complete intersections 123 which contracts the divisor E E 0 X. Then X has cyclic quotient singularities along the exceptional set E := f (E 0 ) and f is the minimal resolution of the singularities. Thus, we can read the weighted dual graph of E from the information of E X and those cyclic quotient singularities. In [8], Konno and Nagashima constructed a good resolution of the hypersurface singularity {x a xa 2 2 = xa 3 3 } via cyclic covering as an application of Tomaru s results [19, 23] (see Theorem 4.1). We adopt their method to obtain a good resolution and the information of the divisors on it. The singularity (V, o) can be obtained by a sequence of branched cyclic coverings over C 2 as follows. Let f j = p j x a q jx a 2 2 for j = 3,...,m. Put V 2 = C 2 and V k ={f k = x a k k } V k 1 C for k 3, where x k is the coordinate function of the second component C. Then we have the sequence of coverings V = V m V m 1 V 2 = C 2.Weshall construct the sequence of branched coverings π m π m 1 X m X m 1 π3 X 2, where X 2 is a partial embedded resolution of the branch locus of V 3 V 2 = C 2, and for each k 3, X k is a partial resolution of the singularity of V k with irreducible exceptional set and cyclic quotient singularities. Then we obtain that X = X m Cyclic quotient singularities Let n and μ be positive integers with μ<nand gcd(n, μ) = 1. Let n denote the primitive nth root of unity exp(2π 1/n). Then the singularity of the quotient C 2 n 0 0 n μ is called the cyclic quotient singularity of type C n,μ. A non-singular point is regarded as of type C 1,0. For integers c i 2, i = 1,...,r, we put 1 [[c 1,...,c r ]] := c 1 1 c c r If n/μ = [[c 1,...,c r ]], the weighted dual graph of the cyclic quotient singularity of type C n,μ is as in Figure 1, where all prime exceptional divisors E i are rational, H i denotes the strict transform of the image of the coordinate axis {x i = 0} C 2 by the quotient map, and (H i ) the vertex corresponding to H i. It is known that the complex structure of quotient surface singularity is determined by its resolution graph (cf. [2, 9]). In the situation above, for any positive integer λ 0, let r L(λ 0 ) := λ 0 E 0 + m i E i m 1,...,m r Z, i=1 where E 0 = H 2. Then we define a set D(λ 0 ) as follows: D(λ 0 ) := {D L(λ 0 ) DE i 0, i= 1,...,r}. We see that D(λ 0 ) is not empty and has the smallest element.

4 124 F.-N. Meng and T. Okuma (H 2 ) c 1 c r (H 1 ) E 1 E r FIGURE 1. The weighted dual graph of the cyclic quotient singularity of type C n,μ. LEMMA 2.1. (Laufer [11, Lemma 2.2]) Let D D(λ 0 ). Assume that DE i = 0 for i<r and DE r 1. Then D is the smallest element of D(λ 0 ). Proof. Suppose that D 0 D(λ 0 ) is the smallest element. Let = D D 0. Then Assume that = r i=k m i E i and m k = 0. Then E r = (D D 0 )E r 1. (2.2) E i = m i 1 c i m i + m i+1 (m k 1 = m r+1 = 0). For 1 i<r, since E i 0 and c i 2, Therefore, m i+1 >m i for k 1 i<r, and m i+1 c i m i m i 1 m i + (m i m i 1 ). E r = m r 1 c r m r <m r (1 c r ) 1. This contradicts (2.2). For any x R, we write x=min {n Z x n}. LEMMA 2.2. (Konno and Nagashima [8, Lemma 1.1]) Let e i = [[c i,...,c r ]]. Take a positive integer λ 0 and define the sequence {λ i } r i=0 by the recurrence formula λ i =λ i 1 /e i for 1 i r. Then the cycle r i=0 λ i E i is the smallest element of D(λ 0 ). COROLLARY 2.3. Let Y 0 and Y0 be the smallest element of D(λ 0) and D(λ 0 ), respectively. Then Y 0 Y0 if and only if λ 0 λ 0. LEMMA 2.4. (Konno and Nagashima [8, Lemma 1.2]) Let the sequences {e i } r i=1 and {λ i} r i=0 be as in Lemma 2.2, and for 1 i r, take relatively prime positive integers n i and μ i satisfying n i /μ i = e i. Put λ r+1 := λ r c r λ r 1. (1) If λ i 1 = λ i c i λ i+1 holds for 1 i r, then λ 1 = (μλ 0 + λ r+1 )/n. (2) If λ 0 0 (mod n), then λ i = μ i λ i 1 /n i for 1 i r. If μλ (mod n), then λ i = (μ i λ i 1 + 1)/n i for 1 i r. (3) If either λ 0 0 (mod n) or μλ (mod n), then λ i 1 = λ i c i λ i+1 holds for 1 i r. Furthermore, λ r+1 = 0 when λ 0 0 (mod n), and λ r+1 = 1 when μλ (mod n). (4) If λ 0 0 (mod n), then λ r = λ 0 /n. If μλ (mod n), then λ r =λ 0 /n.

5 3. The resolution of the branch locus Brieskorn complete intersections 125 In this section, we introduce some notation used through this paper and review the results on the resolution of the branch locus due to Konno and Nagashima [8, Section 2], which is the first step of our resolution process. Let f i = x i for i {1, 2}, and f i = p i x a q ix a 2 2 for i {3,...,m} as in Section 2. For i {1,...,m}, let C i C 2 and C C 2 be the plane curves defined by f i = 0 and m i=1 f i = 0, respectively. Then C = m i=1 C i is a reduced divisor. For 2 k m and 1 i k, we define positive integers d ik, n ik, and e ik as follows: d ik := lcm(a 1,...,â i,...,a k ), a i n ik := gcd(a i,d ik ), e ik := d ik gcd(a i,d ik ). (The symbol ˆ in the definition of d ik indicates an omitted term.) In addition, we define integers μ ik by the following condition: We also write We can easily see that e ik μ ik (mod n ik ), 0 μ ik <n ik. d k 1 := d kk, d m := lcm(a 1,...,a m ), n k := n kk, e k := e kk, μ k := μ kk. d k = d ik n ik = a i e ik, (3.1) gcd(n ik,n jk ) = 1 (1 i<j k m). (3.2) Then the next lemma follows from [8, Section 2]. LEMMA 3.1. Let φ : Y C 2 be the minimal embedded good resolution of the curve singularity (C, o) with exceptional set F. Let C i Y be the strict transform of C i. Then we have the following: (1) F is a chain of rational curves with unique ( 1)-curve; let F 0 F denote the ( 1)- curve; (2) mi=3 C i does not intersect any component of F F 0 ; (3) C 1 and C 2 intersect distinct ends of F if F is not irreducible; (4) for i 3, each C i has gcd(a 1,a 2 ) components; (5) the multiplicity of the zero divisor div Y (f i φ) along F 0 is e i2 for i {1, 2}, and d 2 for i 3; (6) for i {1, 2}, the weighted dual graph of the minimal connected chain of curves with ends F 0 and C i is as follows: (F 0 ) c i1 where n i2 /μ i2 = [[c i1,...,c isi ]]. c isi ( C i )

6 126 F.-N. Meng and T. Okuma 4. The resolution of V We use the notation of the preceding section. Let η : Y X 2 be the morphism which contracts the divisor F F 0 Y. Let D i,2 = η ( C i ) and F 2 = η (F 0 ). By Lemma 3.1, X 2 has only two singular points of types C n12,μ 12 and C n22,μ 22. Let : X 2 C 2 be the natural projection and let f j,2 = f j. Suppose that X k and {f j,k } are obtained for 2 k<m. Then we define X k+1 to be the normalization of a surface {f k+1,k = x a k+1 k+1 } X k C, where we regard x k+1 as the coordinate function of the second component C. Let π k+1 : X k+1 X k be the natural morphism and f j,k+1 = f j,k π k+1. Let F k (respectively D j,k ) denote the fiber of F 2 (respectively D j,2 ) on X k. We have the following commutative diagram (cf. Section 2.1): (X m,f m ) π m π 4 (X 3,F 3 ) π 3 (X 2,F 2 ) (V, o) (V m, o) (V 3, o) (V 2, o) C 2 THEOREM 4.1. (Tomaru [19, Section 3]; cf. [8, Theorem 2.2]) Let (U, o) be the cyclic quotient singularity of type C n,μ, m the maximal ideal of O U,o, and h m. Assume that the zero divisor of the pull-back of h on the minimal resolution of (U, o) has the weighted dual graph as in Figure 2, where n/μ = [[c 1,...,c s ]], H 0 H s+1 is the strict transform of {h = 0} with irreducible components H 0 and H s+1, and the ρ i denote multiplicities (if n = 1, then (U, o) = (C 2, o) and s = 0). Let a be a positive integer. We define integers and p as follows. Let a ā = gcd(a, lcm(ρ 0,ρ s+1 )), n = n gcd(a, ρ 0,ρ 1,...,ρ s+1 ), gcd(a, ρ 0,ρ s+1 ) and =ā n. Then p is defined by the following condition: p a gcd(a, ρ s+1 ) μβ + ρ s+1 γ (mod ), 0 p <, gcd(a, ρ s+1 ) where β and γ are integers determined by a gcd(a, ρ 0 ) β 1 (mod ρ ρ 0 0/ gcd(a, ρ 0 )), 0 β< gcd(a, ρ 0 ), ρ 0 gcd(a, ρ 0 ) γ = a gcd(a, ρ 0 ) β 1. Then the normalization W of the a-fold covering of U defined by z a = h has exactly gcd(a, ρ 0,...,ρ s+1 ) connected components. Each component W i of W has a cyclic quotient singularity of type C,p and the divisor of the function z on W i has the multiplicity ρ j /gcd(a, ρ j ) along the fiber of H j for j = 0,s+ 1. LEMMA 4.2. For 2 k m and j>k, we have the following: (1) X k and F k are non-singular outside F k ( i k D i,k); (2) each of the divisors D j,k and D k+1,k+1 has k i=2 gcd(a i,d i 1 ) components;

7 Brieskorn complete intersections 127 ρ 0 ρ 1 ρ s ρ s+1 (H 0 ) c 1 c s (H s+1 ) FIGURE 2. The weighted dual graph of the zero divisor of h. (3) every point x F k D k,k is of type C nk,μ k and the dual graph of the minimal embedded good resolution of the germ of the curve singularity (F k D k,k, x) (X k, x) is as follows: ( F k ) c 1 c s ( D k,k ) where F k and D k,k denote the strict transforms, and n k /μ k = [[c 1,...,c s ]]; (4) div Xk (x k ) = e k F k + D k,k ; (5) div Xk (f j,k ) = d k F k + D j,k ; (6) F k is irreducible. Proof. In case k = 2, the assertion follows from Lemma 3.1. Assume that it holds for some k 2. Let x be a point of F k D j,k and U X k a sufficiently small neighborhood of x. Then the point x X k is of type C 1,0 and the multiplicity of f k+1,k along F k is d k by the assumption. We apply Theorem 4.1 to the germ (U, x) and the covering x a k+1 k+1 = f k+1,k; then in the theorem is n k+1 = n k+1k+1. Let W be the normalization of the covering. If j>k+ 1, putting (s, ρ 0,ρ s+1 ) = (0,d k, 0), we obtain that W has exactly gcd(a k+1,d k ) components, which are non-singular. Thus (1) holds by induction. Suppose that j = k + 1 and put ρ 0 = d k and ρ 1 = 1. Then W is an irreducible surface with a cyclic quotient singularity of type C nj,μ j, and div W (x j ) = e j F j + D j,j. These imply (2), (3), and (4). It also follows that F k is locally irreducible. Since X k is a partial resolution of a normal surface singularity, the exceptional set F k X k is connected. Hence, (6) holds. We have a j div W (x j ) = div W (f j,j ) = (π j W ) (d k F k + D j,k ). By (3.1), we have π k+1 (d kf k ) = d k+1 F k+1. This proves (5). For 1 i m, we define integers ĝ and ĝ i as follows: ĝ i := ĝ := a 1 a m lcm(a 1,...,a m ), a 1 aˆ i a m lcm(a 1,..., aˆ i,...,a m ). Since m 1 i=2 gcd(a i,d i 1 ) =ĝ m, Lemma 4.2 immediately implies the following. LEMMA 4.3. Let X = X m, E = F m, and D i = D i,m. Then we have the following: (1) every D i is reduced and D := m i=1 D i is a disjoint union of irreducible components; (2) the set of the singular points of X is a subset of E D; (3) D m has ĝ m components;

8 128 F.-N. Meng and T. Okuma C 1,β 1 c 1,1 E 1,1,1 c 1,1 E 1,1,ĝ1 E 0 c 0 [g] c m,1 E m,1,1 c m,1 c 1,2 E 1,2,1 c 1,2 E 1,2,ĝ1 c m,2 E m,2,1 c m,2 C m,βm c 1,s1 E 1,s1,1 c 1,s1 E 1,s1,ĝ 1 c m,sm E m,sm,1 c m,sm ĝ 1 ĝ m E m,1,ĝm E m,2,ĝm E m,sm,ĝ m FIGURE 3. The weighted dual graph of the exceptional set E. (4) suppose x E D m, then x X is a cyclic quotient singularity of type C nm,μ m and the weighted dual graph of the minimal embedded good resolution of the germ (E D m, x) (X, x) is as follows: (Ē ) c 1 c s ( D m ) where Ē and D m denote the strict transforms, and n m /μ m = [[c 1,...,c s ]]; (5) div X (x m ) = e m E + D m. We introduce some notation to state the main result of this section. Let X X be the minimal resolution of X and E the fiber of E. Let E 0 E denote the strict transform of E. Let f : X V be the composite of the resolution X X and the partial resolution X = X m V m = V. Clearly f is a good resolution with exceptional set E. For any nonzero function h O V,o, we write div X (h) := div X (h f)= (h) E + H, where (h) E is supported in E and H does not contain any irreducible component of E. Let Z (i) = (x i ) E, i := n im, β i := μ im.

9 Brieskorn complete intersections 129 THEOREM 4.4. Let g and c 0 denote the genus and the self-intersection number of E 0, respectively. Then the weighted dual graph of the exceptional set E is as in Figure 3, where the invariants are as follows: c 0 = m ĝ w β w 2g 2 = (m 2)ĝ + a 1 a m dm 2, β w / = m ĝ i, i=1 [[cw,1,...,c w,sw ]] 1 if 2 0 if = 1. Furthermore Z (i) = λ (i) 0 E 0 + m s w ĝ w ν=1 ξ=1 λ (i) w,ν,ξ E w,ν,ξ (1 i m), where λ (i) 0 and the sequence {λ (i) w,ν,ξ } are determined by the following: λ (i) w,0,ξ λ (i) w,s w +1,ξ := := λ(i) 0 := e im, 1 if w = i 0 if w = i, λ (i) w,ν 1,ξ = λ(i) w,ν,ξ c w,ν λ (i) w,ν+1,ξ. The cycle Z (i) is the smallest among the cycles Z>0 such that Z is nef and the coefficient of E 0 in Z is e im. Proof. From (1) and (2) of Lemma 4.3, we see that the claims (3) (5) of Lemma 4.3 also hold for every i {1,...,m} instead of m, by taking permutations of variables. These data immediately show the dual graph except for c 0 and g. Since div X (x i ) = e im E + D i by Lemma 4.3(5), λ (i) 0 should be e im and the coefficient of the cycle Z (i) can be determined by the following: 0 = (Z (i) + D i )E w,k,ξ = λ (i) w,k 1,ξ λ(i) w,k,ξ c w,k + λ (i) w,k+1,ξ. The last assertion follows from Lemma 2.1. Recall that F 0 is the ( 1)-curve on Y (see Lemma 3.1). Let p : E 0 F 0 = P 1 be the natural map and π = π 3 π m. From the proof of Lemma 4.2, and Lemma 4.3(3), we obtain the following: π F 2 = (d m /d 2 )F m, and thus deg p = d := d 2 a 3 a m /d m ; the ramification index of a point x E 0 which corresponds to a point of E D i is d gcd(a 1,a 2 )/ĝ i for i 3, and d/ĝ i for i = 1, 2.

10 130 F.-N. Meng and T. Okuma By Hurwitz s theorem, 2g 2 = d( 2) + m i=3 = (m 2)d gcd(a 1,a 2 ) = (m 2)ĝ d gcd(a1,a 2 ) ĝ i 1 + ĝ i m ĝ i. i=1 m ĝ i i=1 By Lemma 2.4(1), we obtain (even in the case s w = 0) that λ (m) w,1,ξ = (β we m + λ (m) w,s w +1,ξ )/. Since the intersection number of E 0 and div X (x m ) is zero, c 0 e m = c 0 λ (m) m 0 = ĝ w λ (m) m 1 w,1,ξ = 2 i=1 ḓ ĝ i 1 g i ĝ w β w e m + ĝm(β m e m + 1) m. Hence, c 0 = m ĝ w β w + ĝm m e m. Since ĝ m / m e m = a 1 a m /dm 2, we obtain the assertion. LEMMA 4.5. (Konno and Nagashima [8, Lemma 4.3]) The resolution f : X V is not the minimal good resolution, i.e. E 0 is a ( 1)-curve and intersects at most two curves, if and only if m = 3 and (a 1,a 2,a 3 ) = (2, 2, 2l + 1) for an integer l>0. Proof. If f is not the minimal good resolution, V is a cyclic quotient singularity. It is well known that a Gorenstein quotient surface singularity is a rational double point, hence a hypersurface. Thus, the assertion follows from the result of hypersurface singularities [8, Lemma 4.3]. LEMMA 4.6. For 1 w m, (Z (w) ) 2 =ĝ w e wm /. Proof. Let E i,0,ξ = E 0. From Theorem 4.4, 1 if s w > 0 and (i, ν) = (w, s w ), Z (w) E i,ν,ξ = ĝ w if s w = 0 and ν = 0, 0 otherwise. Since λ (w) w,s w,ξ =e wm/ by Lemma 2.4(4), we have Z (w) Z (w) =ĝ w e wm /.

11 5. The fundamental cycle and the canonical cycle Brieskorn complete intersections 131 In this section, we compute the fundamental cycle, the canonical cycle, and the fundamental genus. Let Z E denote the fundamental cycle on E, i.e. the smallest anti-nef cycle supported on E. Since (V, o) is a Gorenstein singularity, there exists a cycle Z K such that Z K is a canonical divisor of X. We call Z K the canonical cycle on X. Let = m. Assume that a 1 a m. It follows from (3.1) that e 1m e mm = e m. THEOREM 5.1. Let w,ν = [[c w,ν,...,c w,sw ]] if s w > 0 (i.e. > 1), and let Z E = θ 0 E 0 + m s w ĝ w ν=1 ξ=1 θ w,ν,ξ E w,ν,ξ. Then θ 0 and the sequences {θ w,ν,ξ } are determined by the following: θ w,0,ξ := θ 0 := min(e m, ), θ w,ν,ξ =θ w,ν 1,ξ / w,ν (1 ν s w ). Proof. We follow the proof of [8, Theorem 1.4]. By Lemma 2.2, we only need to identify θ 0. Let u w (1 w m) be the integers determined by Then β w θ 0 + u w 0 (mod ), 0 u w <. θ w,1,ξ =θ 0 β w / = θ 0β w + u w. Note that β w = s w = 0 if = 1. The condition Z E E 0 0 is equivalent to that θ 0 c 0 m ĝ w (θ 0 β w + u w ). Using Theorem 4.4, this inequality is equivalent to the following: By (3.1), we have θ 0 a 1 a m d 2 m m ĝ w u w. ĝ w d 2 m a 1 a m = d m a w = e wm. Thus, the inequality above is equivalent to the following: θ 0 m u w e wm. Let be the set of positive integers λ satisfying the following condition: there exist integers 0 v w < for 1 w m such that λ m v w e wm, β w λ + v w 0 (mod ). (5.1)

12 132 F.-N. Meng and T. Okuma By the definition of the fundamental cycle, θ 0 = min. Let 0 ={λ (5.1) with v 1 ==v m = 0} and i ={λ (5.1) with v i = 1, v j = 0, j= i}. We see that gcd(,β w ) = gcd(, ) = 1 and min i = e im definition of these integers and (3.2). Thus, we have for 1 i m by the min 0 = and min( \ 0 ) = min m = e m. Therefore, we obtain that min = min(e m, ). LEMMA 5.2. We have Z E = Z (m) if and only if e m. Proof. This follows from Theorem 4.4, Theorem 5.1 and Lemma 2.4(3). THEOREM 5.3. Let Z 0 be the cycle which is obtained as Z E with the condition that θ 0 = in Theorem 5.1. Then Z K = E + (m 2)d m m Z 0 Z (w). Proof. Let N 0 ={w {1,...,m} = 1} and N 1 ={1,...,m}\N 0. Note that for w N 0, β w = 0 and Z (w) E 0 = ĝ w (cf. the proof of Lemma 4.6). Let B be any irreducible component of E E 0. By the adjunction formula and Theorem 4.4, ( Z K + E)E 0 = 2g 2 + = 2g 2 + w N 1 ĝ w m ĝ w + Z (w) E 0 w N 0 = (m 2)ĝ + w N 0 Z (w) E 0, ( Z K + E)B = 2 + (E B)B 1 if B = Ew,sw,ξ for some w and ξ, = 0 otherwise. It follows from Lemma 2.4(3) and (1) that Z 0 (E E 0 ) = 0 and θ w,1,ξ = β w /. By Theorem 4.4, Z 0 E 0 = c 0 m ĝ w β w = c 0 m ĝ w β w = a 1 a m dm 2 = ĝ. d m We see that for w N 0, 1 1 Z 0 = Z 0 E 0 Z (w) Z (w), (5.2) E 0

13 Brieskorn complete intersections 133 since they are numerically equivalent. From the data of the intersection numbers of Z K + E and (5.2), we obtain that Thus, the formula follows. Z K + E = ( Z K + E)E 0 Z 0 + Z (w) Z 0 E 0 w N 1 = (m 2)ĝ Z 0 + Z (w) + Z 0 E 0 w N 0 = (m 2)d m Z 0 + m Z (w). w N 1 Z (w) The arithmetic genus of the fundamental cycle, namely, 1 χ(z E ) = (1/2)Z E (K X + Z E) + 1, is called the fundamental genus. This invariant is independent of the resolution and denoted by p f (V, o). A formula of p f for weighted homogeneous surface singularities was established by Tomari (cf. [20, Theorem 3.1]). Applying the formula, Tomaru [21] obtained the following result in hypersurface case (cf. [8, Theorem 1.7]). THEOREM 5.4. If e m, then Z 2 E = 2 ĝ/d m and p f (V, o) = 1 2 ( 1)ĝ (m 2)ĝ d m If e m, then Z 2 E =ĝ me m / m and p f (V, o) = 1 2 e m (m 2)ĝ (2e m/ m 1)ĝ m e m m ĝ w w m ĝ w + 1. Proof. Assume that e m, then Z E = Z 0 and θ 0 =. By the proof of Theorem 5.3, we have Z 0 E 0 = ĝ/d m. Therefore, We see that ĝe wm = d m ĝ w /. Thus, Z 2 E = Z 0(E 0 ) = 2 ĝ/d m. 2p f (V, o) 2 = (Z K Z 0 )( Z 0 ) = E + (m 2)d m m Z 0 Z ( Z (w) 0 ) = 1 + (m 2)d m m e wm ĝ/d m = (m 2)ĝ + (1 ) ĝ m ĝ w. d m w

14 134 F.-N. Meng and T. Okuma Next, assume that e m. Then Z E = Z (m) by Lemma 5.2. It follows from Lemma 2.4(4) that λ (w) m,s m,ξ = e m/ for w = m. By Lemma 4.6 and its proof, we have ZE 2 = (Z(m) ) 2 =ĝ m e m / m and 2p f (V, o) 2 = (Z K Z (m) )( Z (m) ) = E + (m 2)d m 1 m Z 0 Z (w) 2Z ( Z (m) (m) ) =ĝ m + (m 2)d m ĝ m 1 e m d m ĝ w e m = (m 2)ĝ + ĝm m 1 (1 2e m / m ) e m ĝ w w 2ĝ m e m / m e m. 6. The maximal ideal cycle In this section, we identify the maximal ideal cycle, and mention the base points of the linear system associated with the maximal ideal cycle and Kodaira singularities. We keep the assumption that a 1 a m. Let m denote the maximal ideal of the local ring O V,o and Z m the maximal ideal cycle on X. By the definition, we have Z m = min {(L) E L = c 1 x 1 ++c m x m m, c i C, L= 0}. THEOREM 6.1. We have Z (m) Z (1). Hence, Z m = Z (m). Furthermore, the maximal ideal cycle coincides with the fundamental cycle on the minimal good resolution space and on X if and only if e m. Proof. Since e 1m e mm = e m, the first assertion follows from Theorem 4.4 and Corollary 2.3. The last assertion follows from Lemmas 4.5 and 5.2 and the fact that these two cycles coincide on any resolution of every rational surface singularity [1]. Example 6.2. Let a 1 = a 2 = 2, a 3 ==a m = 3, m 4. Then = 1 and e m = 2. Therefore, E is irreducible, Z E = E, and Z m = 2E. We also have g = (2m 7) 3 m 4 + 1, c 0 = 3 m 4. Let mult(v, o) denote the multiplicity of the local ring O V,o. LEMMA 6.3. We have mult(v, o) = m 2 i=1 a i. Proof. By row operations (without change of coordinates), the coefficient matrix (2.1) can be transformed into the following: p1 q p 2 q pm 2 qm 2 For i = 1,...,m 2, let L i denote the leading form of x a i i + pi xa m 1 m 1 + q i xa m m with respect to the weight deg x i = 1. Then L 1,...,L m 2 form a regular sequence and

15 Brieskorn complete intersections m 3 c 0 2 [g] m FIGURE 4. The weighted dual graph of E when a 1 ==a m 2 = 2, a m 1 = 3, a m = 7 and m 4. C[x 1,...,x m ]/(L 1,...,L m 2 ) is the tangent cone of (V, o). Hence, mult(v, o) = m 2 i=1 deg(l i) = m 2 i=1 a i. Let us mention the base points on E of the linear system O X ( Z m ). Let D = div X (x m ) Z m. Then the set of the base points is a subset of {t 1,...,tĝm } := E D. Let (x, y) be the local coordinates at t ξ such that E ={x = 0} and D ={y = 0}. We write l i = λ (i) m,s m,ξ and δ = l m 1 l m. Then mo X,tξ = (x l m y, x l m 1) = x l m (y, x δ ) (cf. Lemma 4.3(1)). Hence the structure of the base points is the same as that in [8, Section 3]. We obtain the following (see [8, Lemma 3.8 and Proposition 3.9] and Lemmas 4.6 and 6.3). PROPOSITION 6.4. The following conditions are equivalent: (1) mult(v, o) = Z 2 m, i.e. d m 1/a m 1 <d m 1 /a m + 1; (2) δ = 0; (3) the base points on E of the linear system O X ( Z m ) is empty. Each base point can be resolved by a succession of δ blowing-ups. Example 6.5. Let a 1 ==a m 2 = 2, a m 1 = 3, a m = 7 and m 4. Then the weighted dual graph of E is as in Figure 4; the components E m,sm,ξ correspond to the vertices with weight 7, c 0 = 2 m 4, and g = (m 6) 2 m Since = 21 > 6 = e m,wehave Z m = Z E by Theorem 6.1. We easily see that δ = l m 1 l m = 2 1 = 1. We also have d m 1 = 6 < 7 = a m. This implies that (V, o) is a Kodaira singularity (see Theorem 6.8). Definition 6.6. (Karras [6]) A normal surface singularity (W, p) is called a Kodaira singularity if there exists a pencil of curves : S such that, after a finite number of blowing-ups at non-singular points in non-multiple components of the central fiber S 0, : S S, there is a holomorphic map φ : M W from an open neighborhood M of the strict transform F of Supp(S 0 ) in S which is a resolution of (W, p) with exceptional divisor F. PROPOSITION 6.7. (Karras [6, Section 2] and [7]) Let φ : (M, F ) (W, p) be the minimal good resolution of a normal surface singularity and Z F the fundamental cycle on F. Then (W, p) is a Kodaira singularity if and only if the coefficient of F j in Z F is 1 for every component F j satisfying Z F F j < 0 and there exists an element h O W,p such that the divisor div M (h φ) is normal crossing with exceptional part (h) F = Z F.

16 136 F.-N. Meng and T. Okuma THEOREM 6.8. We have that (V, o) is a Kodaira singularity if and only if d m 1 a m. Proof. By Lemma 4.5, if the resolution f : X V is not the minimal good resolution, then the condition d m 1 a m is satisfied. On the other hand, a rational singularity with reduced fundamental cycle is a Kodaira singularity [6, Theorem 2.9]. We assume that f : X V is the minimal good resolution. We have seen that div X (x m ) is a normal crossing (cf. Lemma 4.3). Therefore, it follows from Theorem 6.1, Proposition 6.7, and the proof of Lemma 4.6 that (V, o) is a Kodaira singularity if and only if e m and e m / m =1; this condition is equivalent to that d m 1 a m. Kulikov singularities form a subclass of Kodaira singularities, which are studied in [17, 24]. Since the divisor div X (x m ) Z m is reduced, it follows from [24, Theorem 3.11] that (V, o) is a Kodaira singularity if and only if it is a Kulikov singularity. Acknowledgements. The second author is partially supported by JSPS KAKENHI Grant Number The authors would like to express sincere thanks to Professor Tadashi Tomaru for helpful advice, comments, and discussion. REFERENCES [1] M. Artin. On isolated rational singularities of surfaces. Amer. J. Math. 88 (1966), [2] E. Brieskorn. Rationale Singularitäten komplexer Flächen. Invent. Math. 4 (1967/1968), [3] D. J. Dixon. The fundamental divisor of normal double points of surfaces. Pacific J. Math. 80(1) (1979), [4] A. Fujiki. On resolutions of cyclic quotient singularities. Publ. Res. Inst. Math. Sci. 10(1) (1974/75), [5] M. Jankins and W. D. Neumann. Lectures on Seifert manifolds (Brandeis Lecture Notes, 2). Brandeis University, Waltham, MA, [6] U. Karras. On pencils of curves and deformations of minimally elliptic singularities. Math. Ann. 247 (1980), [7] U. Karras. Methoden zur Berechnung von Algebraischen Invarianten und zur Konstruktion von Deformationen Normaler Flächensingularitäten. Habilitationschrift, Dortmund, [8] K. Konno and D. Nagashima. Maximal ideal cycles over normal surface singularities of Brieskorn type. Osaka J. Math. 49(1) (2012), [9] H. Laufer. Taut two-dimensional singularities. Math. Ann. 205 (1973), [10] H. Laufer. On minimally elliptic singularities. Amer. J. Math. 99 (1977), [11] H. Laufer. Tangent cones for deformations of two-dimensional quasi-homogeneous singularities, Singularities (Iowa City, IA, 1986) (Contemporary Mathematics, 90). American Mathematical Society, Providence, RI, 1989, pp [12] A. Némethi. Weakly elliptic Gorenstein singularities of surfaces. Invent. Math. 137(1) (1999), [13] A. Némethi. Invariants of normal surface singularities, Real and complex singularities (Contemporary Mathematics, 354). American Mathematical Society, Providence, RI, 2004, pp [14] W. D. Neumann. Abelian covers of quasihomogeneous surface singularities, Singularities, Part 2 (Arcata, CA, 1981) (Proceedings of Symposia in Pure Mathematics, 40). American Mathematical Society, Providence, RI, 1983, pp [15] T. Okuma. Numerical Gorenstein elliptic singularities. Math. Z. 249 (2005), [16] P. Orlik and P. Wagreich. Isolated singularities of algebraic surface with C action. Ann. of Math. (2) 93 (1971), [17] J. Stevens. Elliptic surface singularities and smoothings of curves. Math. Ann. 267(2) (1984), [18] M. Tomari. A p g -formula and elliptic singularities. Publ. RIMS 21 (1985), [19] T. Tomaru. C -equivariant degenerations of curves and normal surface singularities with C -action. J. Math. Soc. Japan. To appear.

17 Brieskorn complete intersections 137 [20] T. Tomaru. On Gorenstein surface singularities with fundamental genus p f 2 which satisfy some minimality conditions. Pacific J. Math. 170(1) (1995), [21] T. Tomaru. A formula of the fundamental genus for hypersurface singularities of Brieskorn type. Ann. Rep. Coll. Med. Care Technol. Gunma Univ. 17 (1996), [22] T. Tomaru. On Kodaira singularities defined by z n = f(x, y). Math. Z. 236(1) (2001), [23] T. Tomaru. Pinkham-Demazure construction for two dimensional cyclic quotient singularities. Tsukuba J. Math. 25(1) (2001), [24] T. Tomaru. Pencil genus for normal surface singularities. J. Math. Soc. Japan 59(1) (2007), Fan-Ning Meng Graduate School of Science and Engineering Yamagata University Yamagata , Japan ( jpmfnfdbx@yahoo.co.jp) Tomohiro Okuma Department of Mathematical Sciences Yamagata University Yamagata , Japan ( okuma@sci.kj.yamagata-u.ac.jp)