On Yau sequence over complete intersection surface singularities of Brieskorn type

Abstract

In this paper, we study the Yau sequence concerning the minimal cycle over complete intersection surface singularities of Brieskorn type, and consider the relations between the minimal cycle A and the fundamental cycle Z. Further, we also give the coincidence between the canonical cycles and the fundamental cycles from the Yau sequence concerning the minimal cycle.

1 Introduction

After Artin’s work [1], the complex normal surface singularity theories have been researching by many mathematicians (such as Wagreich, Brieskorn, Laufer, Saito, Wahl, Neumann, Yau, etc.). It is well known that the topological type of a complex normal surface singularity is determined by its resolution graph [14]. For a given resolution graph of a complex normal surface singularity, there are various types of complex structures which realize it. We are interested in finding the relations between analytic invariants and topological invariants [7, 13, 15, 16].

Let (X, o) be the germ of a complex normal surface singularity and let $\pi :(\widetilde{X},E)\rightarrow (X,o)$ be a good resolution, where $E=\pi ^{-1}(o)$ denotes the exceptional divisor. Let $E=\bigcup _{i=1}^rE_i$ be the irreducible decomposition of E. Then $\sum _{i=1}^rE_i$ is a simple normal crossing divisor. A divisor on $\widetilde{X}$ supported in E is called a cycle. For any effective cycle $D=\sum _{i=1}^rd_iE_i\ (d_i\in \mathbb {Z}, d_i\ge 0\ \mathrm{for\ any}\ i)$ on E, $\chi (D)$ is defined by $\chi (D)=\textrm{dim}_{\mathbb {C}}\, H^0(\widetilde{X}, \mathcal {O}_D)-\textrm{dim}_{\mathbb {C}}\, H^1(\widetilde{X}, \mathcal {O}_D)$ where $\mathcal {O}_D=\mathcal {O}_{\widetilde{X}}/ \mathcal {O}_{\tilde{X}}(-D)$. From Riemann–Roch theorem, we have

$$\begin{aligned} \chi (D)=-\frac{1}{2}(D^2+K_{\widetilde{X}} D), \end{aligned}$$

(1.1)

where $K_{\widetilde{X}}$ is the canonical divisor on $\widetilde{X}$. For any irreducible component $E_i$, we have the adjunction formula

$$\begin{aligned} K_{\tilde{X}}E_i=-E_i^2+2g(E_i)-2+2\delta (E_i), \end{aligned}$$

(1.2)

where $g(E_i)$ is the geometric genus of $E_i$ and $\delta (E_i)$ is the number of nodes and cusps on $E_i$ [9]. The arithmetic genus $p_a(D)$ of D is defined by $p_a(D)=1-\chi (D)$. It follows that if B, C are cycles, we have

$$\begin{aligned} p_a(B+C)=p_a(B)+p_a(C)-1+BC. \end{aligned}$$

(1.3)

Among the non-zero effective cycles which have a non-positive intersection number with every irreducible component $E_i$ of E, there is the smallest one, which is called the fundamental cycle $Z_E$ on E. It is defined as follows (cf. [1]):

$$\begin{aligned} Z_E=\textrm{min}\left\{ D=\sum _{i=1}^r a_iE_i\Big |\ a_i>0\ \textrm{and}\ DE_i\le 0\ \mathrm{for\ any}\ i \right\} . \end{aligned}$$

Obviously, $-Z_E^2$ is one of the most important numerically invariants of (X, o) and it is independent of the choice of resolutions. The arithmetic genus of $Z_E$ is called the fundamental genus of (X, o), denoted by $p_f(X,o):=p_a(Z_E)$. This invariant is also independent of the choice of resolutions. Furthermore, there is the smallest non-zero effective cycle $(\le Z_E)$ whose arithmetic genus is equal to $p_f(X,o)$, which is called the minimal cycle A on E. It is defined as follows (cf. [9, Definition 3.1], [18, Definition 1.2]):

$$\begin{aligned} A=\textrm{min} \left\{ D>0|\ p_a(D)=p_f(X,o),\ 0<D\le Z_E \right\} . \end{aligned}$$

Yau gave the definition of Yau sequence concerning the minimal elliptic cycle for weakly elliptic singularities (cf. [19, Definition 3.3]) and showed which is important in his theories that if (X, o) is a numerically Gorenstein elliptic singularity, then

$$\begin{aligned} -K_{B_i}-(-K_{B_{i+1}})=Z_{B_i} \end{aligned}$$

for any i, where $K_{B_i}$ is the canonical cycle on $B_i$ and $\{Z_{B_i}\}$ is the Yau sequence (cf. Proof of Theorem 3.7 in [19]). As a generalization of the minimal elliptic cycle to the minimal cycle, Tomaru (cf. [18, Definition 5.1]) gave an analogue to Yau sequence concerning the minimal cycle for hypersurface singularities of Brieskorn type, and obtained a similar property for the case $p_f(X,o)\ge 2$ as Yau’s theory, i.e.,

$$\begin{aligned} -K_{B_i}-(-K_{B_{i+1}})=cZ_{B_i}, c\in \mathbb {Q}, \end{aligned}$$

(1.4)

where $\{Z_{B_i}\}$ is the Yau sequence concerning the minimal cycle. It is well known that complete intersection surface singularity of Brieskorn type is a generalization of Brieskorn hypersurface singularity. It is a natural question to ask whether the equation (1.4) also holds for Brieskorn complete intersection surface singularity.

In this paper, we consider a germ $(W,o)\subset (\mathbb {C}^m,o)$ of a Brieskorn complete intersection surface singularity defined by

$$\begin{aligned} W=\{(x_1,x_2,\dots ,x_m)\in \mathbb {C}^m|q_{j1}x_1^{a_1}+\cdots +q_{jm}x_m^{a_m}=0,\ j=3,\dots ,m\}, \end{aligned}$$

where $a_i\ge 2$ are integers. We assume that (W, o) is an isolated singularity, this condition is equivalent to that every maximal minor of the matrix $(q_{ji})$ does not vanish (cf. [4, Section 7]). By Serre’s criterion for normality, (W, o) is a normal surface singularity. Let $\pi :(\widetilde{W},E)\rightarrow (W,o)$ be the good resolution of (W, o) with exceptional divisor E, Meng-Okuma (cf. [11]) constructed a good resolution of (W, o) by employing Konno-Nagashima’s method (cf. [5]) and gave the concrete topological structure of (W, o), such as the weighted dual graph of the exceptional divisor E, the genus of the central curve $E_0$ in E, the fundamental genus $p_f(W,o)$ and the concrete description of the fundamental cycle $Z_E$ in terms of $a_1,\dots ,a_m$. Following these results, we obtain a similar equality as (1.4) for (W, o), that is,

$$\begin{aligned} -K_{B_i}-(-K_{B_{i+1}})=c\cdot Z_{B_i},c\in \mathbb {Z}. \end{aligned}$$

This paper is organized as follows. In Sect. 2, we introduce some notations and notions, and some fundamental results with respect to the minimal cycles. In Sect. 3 and 4, we give the relations between the minimal cycles and the fundamental cycles, and also consider a sequence given by Tomaru which is analogous to Yau sequence concerning the minimal cycle over Brieskorn complete intersection surface singularities, and give some new results on these singularities.

2 Preliminaries

In this section, we introduce some notations used throughout this paper, some fundamental results in terms of $a_1,a_2,\dots ,a_n$, and some fundamental facts on the minimal cycles over complex normal surface singularity.

2.1 Some fundamental results

Let $a_1,a_2,\dots ,a_m$ be positive integers. For $1\le i\le m$, we define positive integers $d_m, d_{im}, \alpha _i, e_{im}$ as follows:

$$\begin{aligned} d_{m}:&=\textrm{lcm}(a_1,\dots ,a_m),\\ d_{im}:&=\textrm{lcm}(a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_m),\\ \alpha _i:&=n_{im}:=\frac{a_i}{\gcd (a_i,d_{im})},\\ e_{im}:&=\frac{d_{im}}{{\gcd (a_i,d_{im})}}. \end{aligned}$$

In addition, we define integers $\beta _i:=\mu _{im}$ by the following condition:

$$\begin{aligned} e_{im}\mu _{im}+1\equiv 0\pmod {n_{im}},\ 0\le \mu _{im}<n_{im}. \end{aligned}$$

Let $\alpha =\prod _{i=1}^m\alpha _i$ and $\theta _0=\textrm{min}\{e_{mm},\alpha \}$. We give the following Lemma 2.2 which implies the relation between the coefficient of the central curve $E_0$ in $Z_E$ and $\alpha _i$ from the cyclic quotient singularity of type $C_{\alpha _i,\beta _i}$ for $i=1,2,\dots ,m-1$.

Remark 2.1

Let n and $\mu$ be positive integers that are relatively prime and $\mu <n$. Then the singularity of the quotient

$$\begin{aligned} \mathbb {C}^2\Big /\left<\begin{pmatrix} \epsilon _n &{} 0\\ 0 &{} \epsilon _n^{\mu } \end{pmatrix}\right> \end{aligned}$$

is called the cyclic quotient singularity of type $C_{n,\mu }$, where $\epsilon _n$ denotes the primitive n-th root $\textrm{exp}(2\pi \sqrt{-1}/n)$ of unity. If $n=1,\mu =0$, then the type $C_{1,0}$ means a non-singular point. For integers $c_i\ge 2,i=1,2,\dots ,s$, we put

$$\begin{aligned}{}[[c_1,\dots ,c_s]]:=c_1-\frac{1}{c_2-\frac{1}{\ddots - \displaystyle \frac{1}{c_s}}}. \end{aligned}$$

Suppose $n/\mu =[[c_1,\dots ,c_s]]$, it is known (cf. [3]) that if $E=\bigcup _{i=1}^sE_i$ is the exceptional divisor of the minimal resolution of $C_{n,\mu }$, then $E_i\simeq \mathbb {P}^1$ and the weighted dual graph of E is chain-shaped as in Fig. 1.

Fig. 1

The weighted dual graph of the minimal resolution of $C_{n,\mu }$

It is well known that the complex structure of quotient surface singularity is determined by its resolution graph (cf. [2, 10]).

Lemma 2.2

Suppose that $2\le a_1\le a_2\le \cdots \le a_m$. Then $\theta _0\equiv 0\pmod {\alpha _i}$ for $i\in \{1,2,\dots ,m-1\}$.

Proof

If $\alpha \le e_{mm}$, then the result is obvious following the assumption $2\le a_1\le a_2\le \cdots \le a_m$. Suppose that $e_{mm}\le \alpha$, then $\theta _0=e_{mm}$. It suffices to prove that $\frac{a_i}{\gcd (a_i,d_{im})}\Big | \frac{d_{mm}}{\gcd (a_m,d_{mm})}$ for $i\in \{1,2,\dots ,m-1\}$. We can easily see that

$$\begin{aligned} \frac{a_i\cdot d_{im}}{\gcd (a_i,d_{im})}&=\textrm{lcm}(a_i,d_{im})=d_m,\\ \frac{a_m\cdot d_{mm}}{\gcd (a_m,d_{mm})}&=\textrm{lcm}(a_m,d_{mm})=d_m. \end{aligned}$$

Since $d_{im}\equiv 0\pmod {a_m}$ for $i\in \{1,2,\dots ,m-1\}$, and

$$\begin{aligned} a_m\cdot d_m=\frac{a_m\cdot a_i\cdot d_{im}}{\gcd (a_i,d_{im})}, \quad d_{im}\cdot d_m=\frac{d_{im}\cdot d_{mm}\cdot a_m}{\gcd (a_m,d_{mm})}, \end{aligned}$$

we have

$$\begin{aligned}\frac{a_m\cdot a_i\cdot d_{im}}{\gcd (a_i,d_{im})}\Big | \frac{d_{im}\cdot d_{mm}\cdot a_m}{\gcd (a_m,d_{mm})},\end{aligned}$$

which implies that $e_{mm}\equiv 0\pmod {\alpha _i}$ for $i\in \{1,2,\dots ,m-1\}$. Thus, we obtain the assertion. $\square$

For any $x\in \mathbb {R}$, we put $\lceil x \rceil =\textrm{min}\{n\in \mathbb {Z}|n\ge x\}$ and $\lfloor x \rfloor =\textrm{max}\{n\in \mathbb {Z}|n\le x\}$. The following Lemma 2.3 is essentially following Hirzebruch resolutions for cyclic quotient singularities of type $C_{n,\mu }$ [3].

Lemma 2.3

Let $\lambda _0$ be a positive integer and let $n_i$ and $\mu _i$ be positive integers that are relatively prime with $\mu _i<n_i$. Suppose that $\epsilon _i:=n_i/\mu _i=[[c_i,\dots ,c_s]]$ with $c_i\ge 2$ and $\lambda _i=\lceil \lambda _{i-1}/ \epsilon _i\rceil$, $i=1,2,\dots ,s$, where

$$\begin{aligned}{}[[c_i,\dots ,c_s]]=c_i-\frac{1}{c_{i+1}-\frac{1}{\ddots - \displaystyle \frac{1}{c_s}}}.\end{aligned}$$

If $\lambda _0\equiv 0\pmod {n_1}$, then $\lambda _{s-1}\ge 2$ and $\lambda _{s}=\lambda _{s-1}/c_s$.

Proof

It is clear that $n_s/\mu _s=c_s$, that is $n_s=c_s\ge 2$, and

$$\begin{aligned} \frac{n_1}{\mu _1} =[[c_1,\dots ,c_s]]=c_1-\frac{1}{n_2/\mu _2}=\frac{n_2c_1-\mu _2}{n_2}. \end{aligned}$$

Since $\gcd (n_1,\mu _1)=1$ and $\lambda _0\equiv 0\pmod {n_1}$, it follows that $\mu _1=n_2$ and $\lambda _1=\lceil \lambda _0/ \epsilon _1\rceil =\lambda _0\mu _1/n_1$. Thus $\lambda _1\equiv 0\pmod {n_2}$. Further, since $\gcd (n_i,\mu _i)=1$ for $i=1,2,\dots ,s$, we have

$$\begin{aligned}&\mu _2=n_3,\dots , \mu _k=n_{k+1}, \dots , \mu _{s-1}=n_s,\\&\lambda _2=\lambda _1\mu _2/n_2,\dots , \lambda _k=\lambda _{k-1}\mu _k/n_k,\dots , \\&\lambda _s=\lambda _{s-1}\mu _s/n_s=\lambda _{s-1}/c_s, k=1,2,\dots ,s. \end{aligned}$$

It follows that $\lambda _{s-1}\equiv 0\pmod {n_s}$. Thus, we obtain the assertion following the fact $n_s\ge 2$. $\square$

From Lemma 1.2 in [5], we have the following remark.

Remark 2.4

If either $\lambda _0\equiv 0\pmod {n_1}$ or $\mu _1\lambda _0+1\equiv 0\pmod {n_1}$, then $\lambda _{i-1}+\lambda _{i+1}=\lambda _ic_i$ for $i=1,2,\dots ,s$. Furthermore, $\lambda _sc_s-\lambda _{s-1}=0$ when $\lambda _0\equiv 0\pmod {n_1}$, and $\lambda _sc_s-\lambda _{s-1}=1$ when $\mu _1\lambda _0+1\equiv 0\pmod {n_1}$.

In order to complete the proofs of our results, we also need the following results.

Lemma 2.5

([18, Lemma 5.4]) Let $\lambda ,\ell$ and d be integers satisfying $\ell \lambda +1\equiv 0 \pmod {d}$ and $0<\ell , \lambda <d$. For a non-negative integer t, let $\lambda _t$ be an integer satisfying $\ell \lambda _t+1 \equiv 0 \pmod {\ell t+d}$ and $0<\lambda _t<\ell t+d$. If $\displaystyle \frac{d}{\lambda }=[[b_1,\dots ,b_r]]$, then $\displaystyle \frac{\ell t+d}{\lambda _t}=[[b_1,\dots ,b_r, \underbrace{2, \dots ,2}_t]]$.

Corollary 2.6

Suppose that $n/\mu =[[b_1,\dots ,b_r,\underbrace{2,\dots ,2}_t]]$, where t is a non-negative integer. Let e, p be integers defined by $e\mu +1\equiv 0 \pmod {n}$ with $0< e<n$ and $ep+1\equiv 0 \pmod {n-et}$ with $0\le p <n-et$, respectively. Then $(n-et)/p=[[b_1,\dots ,b_r]]$.

Proof

When $t=0$, the assertion holds clearly. Assume that $t\ge 1$. Since $(n-e)\mu \equiv 1 \pmod {n}$, we have $n/(n-e)=[[\underbrace{2\dots ,2}_t,b_r,\dots ,b_1]]$ and

$$\begin{aligned} \frac{1}{2-\frac{1}{2-\frac{1}{\ddots \displaystyle \frac{1}{2-\frac{n}{n-e}}}}}=\frac{n-et}{n-e(t+1)}=[[b_r,\dots ,b_1]].\end{aligned}$$

Furthermore, since $((n-et)-e)p\equiv 1 \pmod {n-et}$, we have $(n-et)/p=[[b_1,\dots ,b_r]].$ $\square$

2.2 Minimal cycles over normal surface singularities

Let (X, o) be the germ of a complex normal surface singularity and $\pi :(\widetilde{X},E)\rightarrow (X,o)$ a good resolution of (X, o), where $\pi ^{-1}(o)=E=\bigcup _{i=1}^r E_i$ is the irreducible decomposition of E. Let $Z_E$ be the fundamental cycle on E and D a non-zero effective cycle with $D<Z_E$. Then we can construct a computation sequence from D to $Z_E$ as in [8]. For the relation between the arithmetic genus of D and the arithmetic genus of $Z_E$, we have the following Lemma.

Lemma 2.7

([18, Lemma 1.1]) Let D be a cycle on E such that $0\le D\le Z_E$. Then $p_a(D)\le p_a(Z_E)=p_f(X,o)$.

Among the effective cycles ($\le Z_E$), there is the smallest one whose arithmetic genus is equal to $p_a(Z_E)$, which is defined as follows.

Definition 2.8

([9, 18]) Let A be an effective cycle on E satisfying $0<A\le Z_E$. Suppose $p_f(X,o)\ge 1$. Then A is said to be a minimal cycle on E if $p_a(A)=p_f(X,o)$ and $p_a(D)<p_f(X,o)$ for any cycle D with $0\le D<A$, that is,

$$\begin{aligned} A=\textrm{min}\left\{ D>0|p_a(D)=p_f(X,o), 0<D\le Z_E \right\} . \end{aligned}$$

In 1977, Laufer showed that if (X, o) is an elliptic singularity (i.e., $p_f(X,o)=1$), then A is the minimally elliptical cycle (cf. [9]). Further, the existence and the uniqueness of the minimal cycle A can be shown as in [9]. Also, Stevens (cf. [17]) defined the minimal cycle on the minimal resolution space and called it characteristic cycle for complex normal surface singularity (X, o). In fact, it is not easy to give the concrete descriptions of the minimal cycle A when $A\ne Z_E$ for the complex normal surface singularities. For the case $A=Z_E$, Tomaru (cf. [18]) proved that $A=Z_E$ if $\textrm{lcm}(a_1,a_2)\le a_3< 2\cdot \textrm{lcm}(a_1,a_2)$ on the minimal resolution space for Brieskorn hypersurface singularity (V, o) with $p_f(V,o)\ge 1$, which is given as follows.

Theorem 2.9

([18], Theorem 4.4) Let $(\widetilde{V}, E)\rightarrow (V,o)$ be the minimal resolution with $p_f(V,o)\ge 1$, where (V, o) is the hypersurface singularity of Brieskorn type $\{(x_1,x_2,x_3)\in \mathbb {C}^3|x_1^{a_1}+x_2^{a_2}=x_3^{a_3}\}$, if $\textrm{lcm}(a_1,a_2)\le a_3<2\cdot \textrm{lcm}(a_1,a_2)$, then $A=Z_E$ on E.

Consequently, Meng et al. (cf. [12]) considered the Brieskorn complete intersection surface singularity (W, o) defined as in Sect. 1 and proved that $A=Z_E$ on the minimal resolution space if $\textrm{lcm}(a_1,\dots ,a_{m-1})\le a_m<2\cdot \textrm{lcm}(a_1,\dots ,a_{m-1})$, which is given as follows.

Theorem 2.10

([12], Theorem 3.3) Let $(\widetilde{W}, E)\rightarrow (W,o)$ be the minimal resolution, where (W, o) is the complete intersection surface singularity of Brieskorn type $\{(x_1,x_2,\dots ,x_m)\in \mathbb {C}^m|q_{j1}x_1^{a_1}+\cdots +q_{jm}x_m^{a_m}=0,\ j=3,\dots ,m\}$, if $\textrm{lcm}(a_1,\dots ,a_{m-1})\le a_m< 2\cdot \textrm{lcm}(a_1,\dots ,a_{m-1})$, then $A=Z_E$ on E.

Clearly, we always have $A\le Z_E$. Since $Z_E$ has been given the formula concretely, so in the case $A=Z_E$, they have the same status. However, for the case $A<Z_E$, it is useful to give the concrete descriptions of the minimal cycle A, which associate to (X, o) some new numerical invariants, such as the Yau cycle Y, $-Y^2$, $p_a(Y)$ and $\textrm{dim}H^1(Y,\mathcal {O}_Y)$ [6].

3 Yau sequence concerning the minimal cycle over (W, o) when $Z_E=A$

Let $\pi :(\widetilde{X},E)\rightarrow (X,o)$ be the minimal good resolution of a complex normal surface singularity (X, o), where $\pi ^{-1}(o)=E=\bigcup _{i=1}^r E_i$ is the irreducible decomposition of the exceptional divisor E. Let $Z_E$ and A be the fundamental cycle and minimal cycle on E, respectively. If $D=\sum _{i=1}^rd_iE_i$ is an effective cycle, we write $\textrm{Supp} D=\bigcup E_i, d_i\ne 0$. Suppose $p_f(X,o)\ge 2$, Tomaru (cf. [18]) defined the following sequence concerning the minimal cycle which is an analogue to the Yau sequence concerning the minimal elliptic cycle (cf. [19, Definition 3.3]).

Definition 3.1

([18, Definition 5.1]) If $Z_E A<0$, we say that the Yau sequence concerning A is $\{Z_E\}$ and the length of the Yau sequence is 1.

Suppose $Z_E A=0$. Let $B_1$ be the maximal connected subvariety of E such that $B_1\supseteq \textrm{Supp}\, A$ and $Z_E E_i=0$ for any $E_i\subseteq B_1$. Since $Z_E^2<0$, $B_1$ is properly contained in E. Let $Z_{B_1}$ be the fundamental cycle on $B_1$.

Suppose $Z_{B_1} A=0$. Let $B_2$ be the maximal connected subvariety of $B_1$ such that $B_2\supseteq \textrm{Supp}\, A$ and $Z_{B_1} E_i=0$ for any $E_i\subseteq B_2$. By the same argument as above, $B_2$ is properly contained in $B_1$.

We continue this process, if we obtain $B_t$ with $Z_{B_t} A<0$, we call $\{Z_{B_0}=Z_E,\ Z_{B_1},\dots ,Z_{B_t}\}$ the Yau sequence concerning A of (X, o) and the length of the Yau sequence is $t+1$. A connected component of $\bigcup _{E_i\nsubseteq \textrm{Supp}\, A}E_i$ is called an eliminative branch of (X, o).

From Lemma 2.7 and the definition of minimal cycle, we know that for any non-zero effective cycle D with $A\le D\le Z_E$, we have $p_a(D)=p_f(X,o)$ for a complex normal surface singularity (X, o). Thus, if $\{Z_E=Z_{B_0},Z_{B_1},\dots ,Z_{B_t}\}$ is the Yau sequence of (X, o) and $(X_{B_i},o_i)$ is the complex normal surface singularity obtained by contracting $B_i$, $i=1,\dots ,t$, we have $p_f(X_{B_1},o_1)=\cdots =p_f(X_{B_t},o_t)=p_f(X,o)$. Tomaru (cf. [18, §5]) showed that if (X, o) is a Brieskorn hypersurface singularity defined by $x_1^{a_1}+x_2^{a_2}=x_3^{a_3}\ (2\le a_1\le a_2\le a_3)$ with $p_f(X,o)\ge 2$ in a restrictive situation, then

$$\begin{aligned} -K_{B_i}-(-K_{B_{i+1}})=cZ_{B_i},\ i=0,1,\dots ,t-1, \end{aligned}$$

where $c\in \mathbb {Q}$ is a suitable positive rational number and $K_{B_i}$ is the canonical cycle on $B_i$ (A rational cycle K is called the canonical cycle if $KE_i=-K_{\widetilde{X}}E_i$ for all $E_i$, and the canonical cycle K exists such that $-K$ is a canonical divisor of $\widetilde{X}$ for Gorenstein surface singularity (cf. [19])). It is well known that the complete intersection surface singularity of Brieskorn type is the generalization of hypersurface singularity of Brieskorn type. In the following, we consider the Brieskorn complete intersection surface singularity (W, o) defined as in Section 1 with the assumption $2\le a_1\le a_2\le \cdots \le a_m$, and give some new results.

Let $\pi :(\widetilde{W},E)\rightarrow (W,o)$ be the minimal good resolution of (W, o) with exceptional divisor E. For $1\le i\le m$, we define integers $\hat{g}$ and $\hat{g}_i$ as follows:

$$\begin{aligned} \hat{g}:=\frac{a_1\cdots a_m}{\textrm{lcm}(a_1,\dots ,a_m)}, \ \hat{g}_i:=\frac{a_1\cdots a_{i-1}\cdot a_{i+1} \cdots a_m}{\textrm{lcm}(a_1,\dots ,a_{i-1}, a_{i+1},\dots ,a_m)}. \end{aligned}$$

Fig. 2

The weighted dual graph of the exceptional divisor E

Theorem 3.2

([11, 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 Fig. 2, where the invariants are as follows:

$$\begin{aligned}&2g-2=(m-2)\hat{g}-\sum _{i=1}^m \hat{g}_i,\\&c_0=\sum _{w=1}^m \frac{\hat{g}_w \beta _w}{\alpha _w}+\frac{a_1\cdots a_m}{d_m^2},\ \beta _{w}/\alpha _{w}= \left\{ \begin{array}{ll} [[c_{w,1},\dots ,c_{w,s_w}]]^{-1} &{} if \ \alpha _{w} \ge 2,\\ 0 &{} if\ \alpha _{w}=1. \end{array}\right. \end{aligned}$$

Theorem 3.3

([11, Theorem 5.1]) Let $\epsilon _{w,\nu }=[[c_{w,\nu },\dots ,c_{w,s_w}]]$ if $s_w>0$ and let

$$\begin{aligned} Z_E=\theta _0E_0+\sum _{w=1}^m\sum _{\nu =1}^{s_w}\sum _{\xi =1}^{\hat{g}_w}\theta _{w,\nu ,\xi }E_{w,\nu ,\xi }. \end{aligned}$$

Then $\theta _0$ and the sequence $\{\theta _{w,\nu ,\xi }\}$ are determined by the following:

$$\begin{aligned}\theta _{w,0,\xi }:=\theta _0:=\textrm{min}\left( e_m,\prod _{w=1}^m\alpha _w\right) , \theta _{w,\nu ,\xi }:=\lceil \theta _{w,\nu -1,\xi }/\epsilon _{w,\nu } \rceil .\end{aligned}$$

3.1 For the case $2\le a_1\le a_2\le \cdots \le a_m$

By Lemma 2.2, we know that $e_{mm}\equiv 0\pmod {\alpha _i}$ for $i\in \{1,2,\dots ,m-1\}$, and following Definition 2.8, we obtain the following Theorem.

Theorem 3.4

Suppose that $2\le a_1\le a_2\le \cdots \le a_{m-1}\le a_m$ and $\alpha _w>1$ for any $w\in \{1,2,\dots ,m-1\}$, then

$$\begin{aligned}p_a(Z_E-E_{w,s_w,\xi })<p_a(A), \xi =1,2,\dots ,\hat{g}_w, w=1,2,\dots ,m-1.\end{aligned}$$

Proof

Following Definition 2.8, we have $p_a(A)=p_a(Z_E)$, that is,

$$\begin{aligned}Z_E^2+K_{\widetilde{W}}Z_E=A^2+K_{\widetilde{W}}A,\end{aligned}$$

where $K_{\widetilde{W}}$ is the canonical divisor on $\widetilde{W}$. To prove $p_a(Z_E-E_{w,s_w,\xi })<p_a(A),w\in \{1,2,\dots ,m-1\}$, by the adjunction formula (1.2), it suffices to prove that

$$\begin{aligned} -Z_EE_{w,s_w,\xi }+E_{w,s_w,\xi }^2+1<0. \end{aligned}$$

From Theorem 3.3, we have

$$\begin{aligned} -Z_EE_{w,s_w,\xi }+E_{w,s_w,\xi }^2+1&=-\theta _{w,s_w-1,\xi }+\theta _{w,s_w,\xi }c_{w,s_w}-c_{w,s_w}+1\\&=-(\theta _{w,s_w-1,\xi }-1)+(\theta _{w,s_w,\xi }-1)c_{w,s_w}. \end{aligned}$$

If $e_{mm}\ge \prod _{w=1}^m\alpha _w$, then $\theta _0=\prod _{w=1}^m\alpha _w$. By Lemma 2.3, we have $\theta _{w,s_w-1,\xi }\ge 2$ and

$$\begin{aligned}\theta _{w,s_w,\xi }=\lceil \theta _{w,s_w-1,\xi }/c_{w,s_w} \rceil =\theta _{w,s_w-1,\xi }/c_{w,s_w}.\end{aligned}$$

Thus,

$$\begin{aligned} -Z_EE_{w,s_w,\xi }+E_{w,s_w,\xi }^2+1&=-(\theta _{w,s_w-1,\xi }-1)+(\theta _{w,s_w-1,\xi }/c_{w,s_w}-1)c_{w,s_w}\\&=1-c_{w,s_w}<0. \end{aligned}$$

Similar for the case $e_{mm}\le \prod _{w=1}^m\alpha _w$ following Lemmas 2.2 and 2.3. Thus, we complete the proof. $\square$

From Theorem 3.4, we note that the length of the Yau sequence concerning the minimal cycle A mainly depends on $e_m,\alpha _m$ and the structure of the cyclic quotient singularity $C_{\alpha _m,\beta _m}$ if we assume $2\le a_1\le a_2\le \cdots \le a_{m-1}\le a_m$. For simplicity, we may first exclude the influences of the structures of the cyclic quotient singularities $C_{\alpha _i,\beta _i}$ for $i=1,2,\dots ,m-1$.

3.2 For the case $a_{m-1}\equiv 0\pmod {\textrm{lcm}(a_1,\dots ,a_{m-2})}$

Assume that $a_{m-1}\equiv 0\pmod {\textrm{lcm}(a_1,\dots ,a_{m-2})}$. Then we have $\alpha _1=\alpha _2=\cdots =\alpha _{m-1}=1$. However, there are many cases for the relations between $e_m$ and $\alpha _m$, and the structure of the $C_{\alpha _m,\beta _m}$, where $\alpha _m/\beta _m=[[c_{m,1},\dots ,c_{m,s_m}]]$, such as $e_{mm}\le \alpha _m$ or $\alpha _m\le e_{mm}$, and $[[c_{m,k},\dots ,c_{m,s_m}]]=\frac{t+1}{t}$ or $[[c_{m,k},\dots ,c_{m,s_m}]]\ne \frac{t+1}{t}$ for some positive integer t with $1\le k\le s_m$. According to Definition 3.1, we should exclude some special cases satisfying $p_a(Z_E-E_{m,s_m,\xi })\ne p_a(A)$ for $\xi \in \{1,2,\dots ,\hat{g}_m\}$, and we obtain the following Theorem.

Theorem 3.5

Suppose that $2\le a_1\le a_2\le \cdots \le a_{m-1}\le a_m$ and $a_{m-1}\equiv 0\pmod {\textrm{lcm}(a_1,\dots ,a_{m-2})}$. If $a_m/\beta _m=[[c_{m,1},\dots ,c_{m,s_m}]]$ with $c_{m,s_m}>2$, then $p_a(Z_E-E_{m,s_m,\xi })<p_a(A)$ for $\xi =1,2,\dots ,\hat{g}_m$.

Proof

Suppose $a_{m-1}\equiv 0\pmod {\textrm{lcm}(a_1,\dots ,a_{m-2})}$, then $\alpha _1=\alpha _2=\cdots =\alpha _{m-1}=1$. Thus the weighted dual graph of the exceptional divisor E is as in Fig. 3.

Fig. 3

The weighted dual graph of E for $a_{m-1}=\textrm{lcm}(a_1,\dots ,a_{m-2})$

It is obvious that

$$\begin{aligned}\theta _0=e_{mm}=\frac{a_{m-1}}{\gcd (a_{m},a_{m-1})}\le \prod _{w=1}^m\alpha _w=\alpha _m=\frac{a_{m}}{\gcd (a_{m},a_{m-1})}.\end{aligned}$$

From Lemma 1.2 in [5], we have $Z_EE_{m,s_m,\xi }=-1$ or 0 for $\xi =1,2,\dots ,\hat{g}_m$. Since $c_{m,s_m}>2$, following the formula (1.3), we have $p_a(A)=p_a(Z_E)>p_a(Z_E-E_{m,s_m,\xi })$ for $\xi =1,2,\dots ,\hat{g}_m$ if and only if $(Z_E-E_{m,s_m,\xi })E_{m,s_m,\xi }\ge 0$, i.e., $Z_EE_{m,s_m,\xi }\ge E_{m,s_m,\xi }^2=-c_{m,s_m}$. According to the assumption $c_{m,s_m}>2$, we obtain the assertion. $\square$

Remark 3.6

From Theorems 3.4 and 3.5, we note that the length of the Yau sequence concerning the minimal cycle A is 1 if $c_{m,s_m}>2$. In other words, we have $Z_E=A$ if $c_{m,s_m}>2$ and $2\le a_1\le a_2\le \cdots \le a_{m-1}\le a_m$.

In fact, by Theorems 3.4 and 3.5, we have the following corollary.

Corollary 3.7

Suppose that $2\le a_1\le a_2\le \cdots \le a_{m-1}\le a_m$. If $a_m/\beta _m=[[c_{m,1},\dots ,c_{m,s_m}]]$ with $c_{m,s_m}>2$, then

$$\begin{aligned}p_a(Z_E-E_{m,s_m,\xi })<p_a(A), \xi =1,2,\dots ,\hat{g}_m.\end{aligned}$$

Example 3.8

Let $a_1=2,a_2=3,a_3=5$ and $a_4=\textrm{lcm}(a_1,a_2,a_3)=30, a_5=34$. Suppose that $(W,o) \subset (\mathbb {C}^5,o)$ is defined by

$$\begin{aligned} \{x_1^2+x_2^3=x_3^{5}, 2x_1^2+3x_2^3=x_4^{30},5x_1^2+7x_2^3=x_5^{34} \}. \end{aligned}$$

Then the weighted dual graph of E on the minimal good resolution of (W, o) is as in Fig. 4. Furthermore, the fundamental cycle $Z_E=15E_0+8\sum _{i=1}^{30}E_{i1}+\sum _{i=1}^{30}E_{i2}$, the fundamental genus $p_f(W,o)=856$ and $-Z_E^2=30$. However, for any $E_{k2}, k=1,2,\dots ,30$, we have $p_a(Z_E-E_{k2})=849<p_a(A)$. In fact, we have $Z_E=A$.

Fig. 4

The weighted dual graph of E for $a_1=2,a_2=3,a_3=5,a_4=30,a_5=34$

4 Yau sequence concerning the minimal cycle over (W, o) when $Z_E\ne A$

According to Theorem 3.5 and Corollary 3.7, in order to study the length of the Yau sequence concerning the minimal cycle A, it is enough to consider the case $[[c_{m,k},\dots ,c_{m,s_m}]]=[[2,2,\dots ,2]]$ for some $k\in \{1,2,\dots ,s_m\}$. Obviously, if $a_{m-1}\equiv 0\pmod {\textrm{lcm}(a_1,a_2,\dots ,a_{m-2})}$ and $a_m\equiv 0\pmod {a_{m-1}}$, then $\alpha _1=\alpha _2=\cdots =\alpha _{m-1}=\alpha _m=1$. This tells us that the length of the Yau sequence is always 1, that is $Z_E=A$. Without loss of generality, we may assume that $a_m\not \equiv 0\pmod {a_{m-1}}$.

4.1 For the case $a_{m-1}\equiv 0\pmod {\textrm{lcm}(a_1,\dots ,a_{m-2})}$ and $a_m\not \equiv 0\pmod {a_{m-1}}$

Tomaru ([18, Proposition 5.2]) condidered the Yau sequence concerning the minimal cycle A over Brieskorn hypersurface singularities under restrictive situation and we consider the Brieskorn complete intersection surface singularities (W, o) and obtained some new results. Suppose $p_f(W,o)\ge 2$, $2\le a_1\le a_2\le \cdots \le a_{m-1}\le a_m$ and $a_{m-1}\equiv 0\pmod {\textrm{lcm}(a_1,\dots ,a_{m-2})}$. Let t be a non-negative integer, and let $p_{m}$ be a non-negative integer defined by

$$\begin{aligned}p_{m}e_{m}+1\equiv 0 \pmod {(\alpha _m-te_{m})}\end{aligned}$$

with $0\le p_{m}<\alpha _m-te_{m}.$ By Theorem 3.2 and Corollary 2.6, we get the following theorem.

Theorem 4.1

Assume that the length of the Yau sequence concerning the minmimal cycle A of (W, o) is $t+1$ with $t\ge 1$, $Z_{B_t}=A$, and $E_{m,\nu ,\xi }^2=-2$ for each $E_{m,\nu ,\xi }\nsubseteq \textrm{Supp}\, A$, the coefficient of $E_{m,\nu ,\xi }$ in $Z_E$ is 1, where $1\le \nu \le s_m, 1\le \xi \le \hat{g}_m$. Then the weighted dual graph of E is given as in Fig. 5, where $s_m'=s_m-t$. Furthermore, $A=Z_E-\sum _{E_{m,\nu ,\xi }\nsubseteq \textrm{Supp}\, A} E_{m,\nu ,\xi }$ and $\displaystyle Z_E^2=-\hat{g}_m$.

Fig. 5

The weighted dual graph of E

Proof

Let $D=\sum _{E_{m,\nu ,\xi }\nsubseteq \textrm{Supp}\, A} E_{m,\nu ,\xi }$. It is easy to see that $A+D\le Z_E$ and the coefficient of any irreducible component of $\textrm{Supp}A$ in A which intersects an eliminative branch is always one. Since $Z_{B_t}=A$, $(A+D)E_i\le 0$ for each irreducible component $E_i$ of E, which implies $Z_E\le A+D$. In fact, for each irreducible component $E_i$ of $\textrm{Supp}D$, it is clear that $(A+D)E_i\le 0$. On the other hand, for every irreducible component $E_j$ of $\textrm{Supp}A=\textrm{Supp}Z_{B_t}$, since $Z_{B_{t-1}}E_j=0$, it is clear that $(A+D)E_j=AE_j+DE_j\le 0.$ Thus $Z_E=A+D$.

Since $t\ge 1$, $Z_EA=0$, which implies that $-A^2=AD$, i.e., the number of eliminative branches of (W, o). Since $a_{m-1}\equiv 0\pmod {\textrm{lcm}(a_1,\dots ,a_{m-2})}$, we have $\alpha _1=\cdots =\alpha _{m-1}=1$. Hence

$$\begin{aligned}Z_E^2=Z_E(A+D)=(A+D)D=\hat{g}_m-2\hat{g}_m=-\hat{g}_m\end{aligned}$$

following $t\ge 1$ and Fig. 2. Furthermore, any eliminative branch is a chain whose component is a rational curve with self-intersection number $-2$. Following Corollary 2.6 and Theorem 3.2, we obtain that the weighted dual graph of E is as in Fig. 5. $\square$

Theorem 4.2

([11, Thoerem 5.4]) If $e_{mm}\ge \prod _{w=1}^m\alpha _w$, then

$$\begin{aligned} p_f(W,o)=\frac{1}{2}\prod _{w=1}^m\alpha _w\left\{ (m-2)\hat{g}-\frac{\left( \prod _{w=1}^m\alpha _w-1\right) \hat{g}}{d_m}-\sum _{w=1}^m\frac{\hat{g}_w}{\alpha _w}\right\} +1. \end{aligned}$$

If $e_{mm}\le \prod _{w=1}^m\alpha _w$, then

$$\begin{aligned} p_f(W,o)=\frac{1}{2}e_{mm}\left\{ (m-2)\hat{g}-\frac{(2\lceil e_{mm}/\alpha _m\rceil -1)\hat{g}_m}{e_{mm}}-\sum _{w=1}^{m-1}\frac{\hat{g}_w}{\alpha _w} \right\} +1. \end{aligned}$$

Theorem 4.3

In the situation of Theorem 4.1, assume that $t\ge 1$ and the Yau sequence of (W, o) is $\{Z_{B_0}=Z_E, Z_{B_1},\dots , Z_{B_t}\}$. Then

$$\begin{aligned}\displaystyle -K_{B_i}-(-K_{B_{i+1}})=\frac{2p_f(W,o)-2+\hat{g}_m}{\hat{g}_m }Z_{B_i}, i=0,1,\dots ,t-1,\end{aligned}$$

where $K_{B_i}$ is the canonical cycle on $B_i$.

Proof

Since (W, o) is a Gorenstein singularity, the canonical cycle K on E exists. Thus we may write $-K$ as follows:

$$\begin{aligned} -K=\sum _{\nu =1}^{s_m-t}\sum _{\xi =1}^{\hat{g}_m}a_iE_{m,\nu ,\xi }+\sum _{\nu =s_m-t+1}^{s_m}\sum _{\xi =1}^{\hat{g}_m}x_{\nu }E_{m,\nu ,\xi }, \end{aligned}$$

where $\bigcup _{\nu =1}^{s_m-t}\bigcup _{\xi =1}^{\hat{g}_m}E_{m,\nu ,\xi }=\textrm{Supp}\, A$. Since $E_{m,\nu ,\xi }^2=-2$ for each $E_{m,\nu ,\xi }\not \subseteq \textrm{Supp}\ A$, it follows from (1.1) and (1.3) that

$$\begin{aligned}-K E_{m,s_m,\xi }=x_{s_m-1}-2x_{s_m}=0,\ -K E_{m,\nu ,\xi }=x_{\nu -1}-2x_{\nu }+x_{\nu +1}=0\end{aligned}$$

for $\nu =s_m-t+1,\dots ,s_m-1$, where $x_{s_m-t}$ is the coefficient of $E_{m,s_m-t,\xi }\subset \textrm{Supp}\, A$ in $-K$ which intersects $E_{m,s_m-t+1,\xi }$. Therefore,

$$\begin{aligned} -K=\sum _{\nu =1}^{s_m-t}\sum _{\xi =1}^{\hat{g}_m}a_iE_{m,\nu ,\xi }+c\cdot \sum _{\nu =s_m-t+1}^{s_m}\sum _{\xi =1}^{\hat{g}_m}(s_m-\nu +1)E_{m,\nu ,\xi }, \end{aligned}$$

(4.1)

where $c=x_{s_m}$. Similarly, following Definition 3.1, there is a constant $c'$ such that

$$\begin{aligned} -K_{B_1}=\sum _{\nu =1}^{s_m-t}\sum _{\xi =1}^{\hat{g}_m}b_iE_{m,\nu ,\xi }+c'\cdot \sum _{\nu =s_m-t+1}^{s_m-1}\sum _{\xi =1}^{\hat{g}_m}(s_m-\nu )E_{m,\nu ,\xi }, \end{aligned}$$

(4.2)

where $K_{B_1}$ is the canonical cycle on $B_1$. Since $t\ge 1$ and from the assumption, it is easy to see that $Z_EA=0$, $Z_EE_{m,s_m,\xi }=-1$ and $-K_{B_1}E_{m,s_m,\xi }=c'$. Then

$$\begin{aligned}(-K-(-K_{B_1})) E_j=c'Z_EE_j\end{aligned}$$

for any irreducible component $E_j$ of E, which implies that

$$\begin{aligned} -K-(-K_{B_1})=c'Z_E, \end{aligned}$$

(4.3)

and then $c'\in \mathbb {Z}$ following the definition of canonical cycle. From (4.1), (4.2) and (4.3), we have $c=c'\in \mathbb {Z}$. Hence $-K-(-K_{B_1})=cZ_E$. Since $Z_{B_i}$ is the fundamental cycle on $B_i$, the coefficient of $E_{m,\nu ,\xi }$ in $Z_{B_i}$ is also 1 for every $E_{m,\nu ,\xi }\nsubseteq \textrm{Supp}\, A$. Continuing this process, we obtain that

$$\begin{aligned}-K_{B_i}-(-K_{B_{i+1}})=cZ_{B_i},\ \ \ i=0,1,\dots ,t-1,\end{aligned}$$

where $-K_{B_0}=-K$ and $Z_{B_0}=Z_E$. Since $Z_EK_{B_1}=0$, $-KZ_E=cZ_E^2$. From Theorem 4.1, we have

$$\begin{aligned} c=\frac{KZ_E}{-Z_E^2}=\frac{2p_f(W,o)-2-Z_E^2}{-Z_E^2}=\frac{2p_f(W,o)-2+ \hat{g}_m}{\hat{g}_m}. \end{aligned}$$

From Theorem 4.2, we can obtain the integer c. $\square$

Remark 4.4

If $p_f(W,o)=2$, then $\hat{g}_m\le 2$ since $c\in \mathbb {Z}$. In fact, we have

$$\begin{aligned}\hat{g}_m|(2p_f(W,o)-2).\end{aligned}$$

Example 4.5

Let $a_1=3,a_2=4$ and $a_3=\textrm{lcm}(a_1,a_2)=12, a_4=42$. Suppose that

$$\begin{aligned}(W,o)=\left( \{x_1^3+x_2^4+x_3^{12}=0, 2x_1^3+3x_2^4+x_4^{42}=0 \},o\right) \subset (\mathbb {C}^4,o).\end{aligned}$$

Then the weighted dual graph of the minimal good resolution of (W, o) is as in Fig. 6, the fundamental cycle $Z_E=2E_0+\sum _{i=1}^{12}\sum _{j=1}^3E_{ij}$, $Z_E^2=-12$ and $p_f(W,o)=91$. The minimal cycle $A=2E_0+\sum _{i=1}^{12}E_{i1}$. It is easy to see that $A=Z_E-\sum _{i=1}^{12}\sum _{j=2}^3E_{ij}$, $Z_EA=0$, $B_1=E_0\cup (\cup _{i=1}^{12}\cup _{j=1}^2 E_{ij})$ and $B_2=E_0\cup (\cup _{i=1}^{12}E_{i1})$. Then we have $Z_{B_1}=2E_0+\sum _{i=1}^{12}\sum _{j=1}^2E_{ij}$, $Z_{B_1}A=0$ and $Z_{B_2}=A=2E_0+\sum _{i=1}^{12}E_{i1}$, $Z_{B_2}A<0$. Hence the Yau sequence is $\{Z_E,Z_{B_1},Z_{B_2}\}$ and the length of Yau sequence is 3. After computation, we have

Fig. 6

The weighted dual graph of E for $a_1=3,a_2=4,a_3=12,a_4=42$

$$\begin{aligned} -K&=111E_0+48\sum _{i=1}^{12}E_{i1}+32\sum _{i=1}^{12}E_{i2}+16\sum _{i=1}^{12}E_{i3},\\ -K_{B_1}&=79E_0+32\sum _{i=1}^{12}E_{i1}+16\sum _{i=1}^{12}E_{i2},\\ -K_{B_2}&=47E_0+16\sum _{i=1}^{12}E_{i1}. \end{aligned}$$

It is clear that $c=16$ and $-K-(-K_{B_1})=16Z_E$, $-K_{B_1}-(-K_{B_2})=16Z_{B_1}$ and $\hat{g}_4|2p_f(W,o)-2$, i.e., $2p_f(W,o)-2=15\hat{g}_4$.

Corollary 4.6

Assume that the weighted dual graph of E is given as in Fig. 7, where $s_m'=s_m-t$ and $c_{m,s_m'}>2$, and the coefficient of some $E_{m,\nu ,\xi }$ in $Z_E$ is 1 with $s_m-t+1\le \nu \le s_m, 1\le \xi \le \hat{g}_m$. Then the length of Yau sequence concerning the minimal cycle A is $t+1$ and

Fig. 7

The weighted dual graph of E

$$\begin{aligned} A=Z_E-\sum _{\nu =s_m-t+1}^{s_m} \sum _{\xi =1}^{\hat{g}_m} E_{m,\nu ,\xi }. \end{aligned}$$

Furthermore, we have $Z_E^2=-\hat{g}_m$ and

$$\begin{aligned}\displaystyle -K_{B_i}-(-K_{B_{i+1}})=\frac{2p_f(W,o)-2+\hat{g}_m}{\hat{g}_m }Z_{B_i}, i=0,1,\dots ,t-1,\end{aligned}$$

where $K_{B_i}$ is the canonical cycle on $B_i$, and $\{Z_{B_0}=Z_E,Z_{B_1},\dots ,Z_{B_t}\}$ is the Yau sequence concerning the minimal cycle A.

Fig. 8

The weighted dual graph of E

4.2 For the general case $2\le a_1\le \cdots \le a_m$

According to Lemma 1.2 in [5], we know that if $e_{mm}\beta _m+1\equiv 0\pmod {\alpha _m}$ and $a_{m-1}=\textrm{lcm}(a_1,a_2,\dots ,a_{m-2})$, then for the fundamental cycle $Z_E$,

$$\begin{aligned}\theta _0=e_{mm}=\frac{a_{m-1}}{\gcd (a_{m-1},a_m)}\le \alpha _m=\frac{a_{m}}{\gcd (a_{m-1},a_m)},\end{aligned}$$

and $\theta _{m,s_m,\xi }=\lceil \theta _0/\alpha _m\rceil =1,\xi =1,2,\dots ,\hat{g}_m$. Further, if $[[c_{m,k},\dots ,c_{m,s_m}]]=[[2,2,\dots ,2]]$ for some $k\in \{1,2,\dots ,s_m\}$, then $\theta _{m,\nu ,\xi }=1$ for $k\le \nu \le s_m,1\le \xi \le \hat{g}_m$ following Lemma 1.2 in [5]. This means that we should consider the length of Yau sequence concerning minimal cycle A without the assumption $a_{m-1}=\textrm{lcm}(a_1,a_2,\dots ,a_{m-2})$. That is, for a connected part containing the curve $E_{m,s_m,\xi }$ in the minimal resolution graph of $C_{\alpha _m,\beta _m}$ with all $E_{m,\nu ,\xi }^2=-2$ and the coefficient of $E_{m,s_m,\xi }$ in $Z_E$ is not 1 for $1\le \xi \le \hat{g}_m$, then we obtain the following theorem.

Theorem 4.7

Assume that the weighted dual graph of E is given as in Fig. 8, where $s_m'=s_m-t$ and $c_{m,s_m'}>2$, and the coefficient of $E_{m,s_m,\xi }$ in $Z_E$ is not 1 with $1\le \xi \le \hat{g}_m$. Then the length of Yau sequence concerning the minimal cycle A is $t+1$ and

$$\begin{aligned} A=Z_E-\sum _{\nu =s_m-t+1}^{s_m} \sum _{\xi =1}^{\hat{g}_m} E_{m,\nu ,\xi }. \end{aligned}$$

Proof

If $t=0$, then it is clear by Corollary 3.7. Assume that $t\ge 1$ and the coefficient of $E_{m,s_m,\xi }$ with $1\le \xi \le \hat{g}_m$ in $Z_E$ is $\theta _{s_m,\xi }:=\theta _{m,s_m,\xi }\ge 2$. Since $c_{m,s_m'}>2$, we have $Z_EE_{m,s_m,\xi }=-1$ following Lemma 1.2 in [5]. Thus, by (1.3), we have

$$\begin{aligned} p_a(Z_E-E_{m,s_m,\xi }) =p_a(Z_E)+p_a(-E_{m,s_m,\xi })-1-Z_EE_{m,s_m,\xi }=p_a(Z_E). \end{aligned}$$

Continuously, let $D=Z_E-\sum _{\nu =s_m-t+1}^{s_m} \sum _{\xi =1}^{\hat{g}_m} E_{m,\nu ,\xi }$, following Lemma 1.2 in [5] and (1.3), we have

$$\begin{aligned} p_a\left( Z_E-D\right)&=p_a(Z_E)+p_a(-D)-1-Z_ED\\&=p_a(Z_E)+\frac{1}{2}D^2-Z_ED\\&=p_a(Z_E)-\hat{g}_m+\hat{g}_m\\&=p_a(Z_E). \end{aligned}$$

Further, since $c_{m,s_m'}>2$, according to Theorem 3.5, we have

$$\begin{aligned} p_a(D-E_{m,s_m',\xi })<p_a(Z_E)=p_a(A), 1\le \xi \le \hat{g}_m. \end{aligned}$$

By Definitions 2.8 and 3.1, we have

$$\begin{aligned}Z_{B_t}=Z_E-\sum _{\nu =s_m-t+1}^{s_m} \sum _{\xi =1}^{\hat{g}_m} E_{m,\nu ,\xi }=A.\end{aligned}$$

Hence we complete the proof. $\square$

Example 4.8

Fig. 9

The weighted dual graph of E for $a_1=3,a_2=4,a_3=22,a_4=42$

Let $a_1=3,a_2=4, a_3=22$ and $a_4=42$. Assume that

$$\begin{aligned}(W,o)=\left( \{x_1^3+x_2^4=x_3^{22}, 2x_1^3+3x_2^4=x_4^{42} \},o\right) \subset (\mathbb {C}^4,o).\end{aligned}$$

Then the weighted dual graph of the minimal good resolution of (W, o) is as in Fig. 9 following Theorem 3.2. Further, by Theorems 3.3 and 4.2, we obtain that the fundamental cycle

$$\begin{aligned} Z_E =&22E_0+11\sum _{\xi =1}^6E_{1,1,\xi }+12\sum _{\xi =1}^6E_{2,1,\xi }+2\sum _{\xi =1}^6E_{2,2,\xi }+19\sum _{\xi =1}^2E_{3,1,\xi }\\&+16\sum _{\xi =1}^2E_{3,2,\xi }+13\sum _{\xi =1}^2E_{3,3,\xi }+10\sum _{\xi =1}^2E_{3,4,\xi }+7\sum _{\xi =1}^2E_{3,5,\xi }+4\sum _{\xi =1}^2E_{3,6,\xi }, \end{aligned}$$

and $p_a(Z_E)=179$. Let

$$\begin{aligned} D=Z_E-\sum _{\nu =1}^6\sum _{\xi =1}^2 E_{3,\nu ,\xi }&= 22E_0+11\sum _{\xi =1}^6E_{1,1,\xi }+12\sum _{\xi =1}^6E_{2,1,\xi }+2\sum _{\xi =1}^6E_{2,2,\xi }\\&\quad +18\sum _{\xi =1}^2E_{3,1,\xi }+15\sum _{\xi =1}^2E_{3,2,\xi }+12\sum _{\xi =1}^2E_{3,3,\xi }\\&\quad +9\sum _{\xi =1}^2E_{3,4,\xi }+6\sum _{\xi =1}^2E_{3,5,\xi }+3\sum _{\xi =1}^2E_{3,6,\xi }, \end{aligned}$$

then $p_a(D)=179$. Furthermore, for any $E_{3,\nu ,\xi }, \nu =1,2,\dots ,6, \xi =1,2$, we have $p_a(D-E_{3,\nu ,\xi })<p_a(Z_E)$. Therefore, following Theorem 3.5 and Corollary 3.7, we obtain that the minimal cycle $A=D$.