3·1. Definitions and lemmas

Definition- Lemma 3·1 . Let $(X\ni x,B)$ be a smooth lc surface germ and E a divisor over $X\ni x$ . By [ 25 , lemma 2·45], there exists a unique positive integer $n=n(E)$ and a unique sequence of smooth blow-ups

\begin{align*}X_E\;:\!=\;X_{n,E}\xrightarrow{f_{n,E}}X_{n-1,E}\xrightarrow{f_{n-1,E}}\ldots\xrightarrow{f_{1,E}}X_{0,E}\;:\!=\;X\end{align*}

such that E is on $X_E$ and each $f_{i,E}$ is the smooth blow-up at ${\rm{center}}_{X_{i-1,E}}E$ . We define $f_E\;:\!=\;f_{1,E}\circ f_{2,E}\dots\circ f_{n,E}$ , $E^i$ the exceptional divisor of $f_{i,E}$ for each i, and $E^i_j$ the strict transform of $E^i$ on $X_{j,E}$ for any $i\leq j$ . In particular, $E^i_i=E^i$ for each i and $E=E^n=E^n_n$ .

Proof. This immediately follows from [ 25 , lemma 2·29].

Proof. It immediately follows from [ Reference Hartshorne16 , excercise 5·4(a)].

Proof. We only need to show that for any positive integer n and any sequence of smooth blow-ups

\begin{align*}X_n\xrightarrow{f_n}X_{n-1}\xrightarrow{f_{n-1}}\cdots\xrightarrow{f_1}X_0\;:\!=\;X\end{align*}

over $X\ni x$ with exceptional divisors $E_k\;:\!=\;{\rm{Exc}}(f_k)$ for each k, we have $a(E_k,X,\Delta)>1-a$ .

In the following, we show that $a(E_k,X,\Delta)>1-a$ for each k by applying induction on n. When $n=1$ , by Lemmas 3·3 and 3·4,

\begin{align*}a(E_1,X,\Delta)=2-{\rm{mult}}_x(B+aC)=2-a-{\rm{mult}}_xB\geq 2-a-(B\cdot C)_x>1-a.\end{align*}

Therefore, we may assume that $n\geq 2$ , and when we blow-up at most $n-1$ times, each divisor we have extracted has log discrepancy $>1-a$ . In particular, $a(E_k,X,\Delta)>1-a$ for any $k\in\{1,2,\dots,n-1\}$ .

Let $x_1\;:\!=\;{\rm{center}}_XE_n$ , and $B_1,C_1,\Delta_1$ the strict transforms of B,C and $\Delta$ on $X_1$ respectively. Then $x_1\in E_1$ . There are three cases:

Case 1. $a+{\rm{mult}}_xB-1<0$ . By Lemma 2·9,

\begin{align*}(B_1\cdot C_1)_{x_1}\leq (B\cdot C)_x-B_1\cdot E_1=(B\cdot C)_x-{\rm{mult}}_xB<1-{\rm{mult}}_xB\leq 1.\end{align*}

By induction hypothesis for the germ $(X_1\ni x_1,\Delta_1)$ and blowing-up at most $n-1$ times, we have

\begin{align*}a(E_n,X,\Delta)=a(E_n,X_1,\Delta_1+(a+{\rm{mult}}_xB-1)E_1)\geq a(E_n,X_1,\Delta_1)>1-a\end{align*}

and finish the proof for Case 1.

Case 2. $a+{\rm{mult}}_xB-1\geq 0$ and $x_1\in E_1\cap C_1$ . Let $\tilde B_1\;:\!=\;B_1+(a+{\rm{mult}}_xB-1)E_1$ . Then

\begin{align*}K_{X_1}+\tilde B_1+aC_1=f_1^*(K_X+\Delta).\end{align*}

We have

\begin{align*}(\tilde B_1\cdot C_1)_{x_1}=(B_1\cdot C_1)_{x_1}+(a+{\rm{mult}}_xB-1)(E_1\cdot C_1)_{x_1}.\end{align*}

Since C is smooth at x, $E_1\cup C_1$ is snc at $x_1$ , so $(E_1\cdot C_1)_{x_1}=1$ . By Lemma 2·9,

\begin{align*}(B_1\cdot C_1)_{x_1}\leq (B\cdot C)_x-B_1\cdot E_1=(B\cdot C)_x-{\rm{mult}}_xB<1-{\rm{mult}}_xB.\end{align*}

Thus

\begin{align*}(\tilde B_1\cdot C_1)_{x_1}<1-{\rm{mult}}_xB+(a+{\rm{mult}}_xB-1)=a\leq 1.\end{align*}

By induction hypothesis for the germ $(X_1\ni x_1,\tilde B_1+aC_1)$ and blowing-up at most $n-1$ times,

\begin{align*}a(E_n,X,\Delta)=a(E_n,X_1,\tilde B_1+aC_1)>1-a,\end{align*}

and we finish the proof for Case 2.

Case 3. $a+{\rm{mult}}_xB-1\geq 0$ , $x_1\in E_1$ , but $x_1\not\in C_1$ . By Lemma 2·9,

\begin{align*}(B_1\cdot E_1)_{x_1}\leq B_1\cdot E_1={\rm{mult}}_xB\leq (B\cdot C)_x<1.\end{align*}

By induction hypothesis for the germ $(X_1\ni x_1,B_1+(a+{\rm{mult}}_xB-1)E_1)$ and blowing-up at most $n-1$ times and apply Lemma 3·4,

\begin{align*}a(E_n,X,\Delta)=a(E_n,X_1,B_1+(a+{\rm{mult}}_xB-1)E_1)\geq 1-(a+{\rm{mult}}_xB-1)>1-a,\end{align*}

and we finish the proof for Case 3.

Lemma 3·6. Let $X\ni x$ be a smooth germ, $B\geq 0$ an $\mathbb{R}$ -divisor on X and C a prime divisor on X such that $C\not\subset\operatorname{Supp} B$ and C is smooth at x. Let $g_1: X_1\rightarrow X$ be the smooth blow-up of x with exceptional divisor $E_1$ , $C_1\;:\!=\;(g_1^{-1})_*C$ , and $B_1\;:\!=\;(g_1^{-1})_*B$ . Let $g_2: X_2\rightarrow X_1$ be the smooth blow-up at $x_1\;:\!=\;C_1\cap E_1$ with exceptional divisor $E_2$ and $B_2\;:\!=\;(g_2^{-1})_*B_1$ . Then

\begin{align*}(B\cdot C)_x\geq 2(B_2\cdot E_2).\end{align*}

Proof. Let $C_2\;:\!=\;(g_2^{-1})_*C_1$ and $E_1'\;:\!=\;(g_2^{-1})_*E_1$ . By Lemma 2·9,

\begin{align*}B_1\cdot E_1=B_2\cdot E'_1+(E'_1\cdot E_2)(B_2\cdot E_2)\geq B_2\cdot E_2\end{align*}

and

\begin{align*}(B_1\cdot C_1)_{x_1}=\sum_{y\in g_2^{-1}(x_1)}(B_2\cdot C_2)_y+(B_2\cdot E_2)(C_2\cdot E_2)\geq (B_2\cdot E_2)(C_2\cdot E_2)= B_2\cdot E_2.\end{align*}

Thus

\begin{align*}(B\cdot C)_x=\sum_{y\in g_1^{-1}(x)}(B_1\cdot C_1)_{y}+B_1\cdot E_1\geq (B_1\cdot C_1)_{x_1}+B_1\cdot E_1\geq 2(B_2\cdot E_2).\end{align*}

3·2. Dual graph of $f_E$

Roughly speaking, this subsection shows that the dual graph of $f_E$ is almost always a chain when E is a divisor which computes the mld. We remark that [ 15 , lemma 3·18] shows that there exists such an E such that the dual graph of $f_E$ is a chain, while we show that for any such E, the dual graph of $f_E$ is a chain.

We need the following result of Kawakita. Notice that the “in particular” part of the following theorem is immediate from the construction in [ Reference Kawakita22 , remark 3].

Proof. By Theorem 3·7, there exists a weighted blow-up $f: Y\rightarrow X$ which extracts E. E contains at most two singular points of Y which are cyclic quotient singularities, and locally analytically, E is one coordinate line of each cyclic quotient singularity. Let $g: W\rightarrow Y$ be the minimal resolution of Y near E, then $f\circ g=f_E$ and $\mathcal{D}(f_E)$ is a chain.

Then any divisor E over $X\ni x$ such that $a(E,X,\Delta)={\rm{mld}}(X\ni x,\Delta)$ satisfies the following. Let $C_E\;:\!=\;(f_E^{-1})_*C$ , then $C_E\cup{\rm{Exc}}(f_E)$ is a chain and $C_E$ is one tail of $C_E\cup{\rm{Exc}}(f_E)$ . In particular, the dual graph of $C_E\cup{\rm{Exc}}(f_E)$ is the following:

Here $C_E$ is denoted by the black circle.

Proof. By the construction of $f_E$ , if $C_E\cup{\rm{Exc}}(f_E)$ is a chain, then $C_E$ is a tail of $C_E\cup{\rm{Exc}}(f_E)$ . Let $n\;:\!=\;n(E)$ and let $C_i$ be the strict transform of C on $X_{i,E}$ for each $i\in\{0,1,\dots,n\}$ . Since C is smooth at x, by the construction of $f_E$ , $C_i\cup_{k=1}^iE^k_i$ is simple normal crossing over a neighbourhood of x and its dual graph does not contain a circle for each $i\in\{0,1,\dots,n\}$ . Since $a(E,X,\Delta)={\rm{mld}}(X\ni x,\Delta)$ , $a(E^i,X,\Delta)\geq a(E,X,\Delta)$ for every $i\in\{1,2,\dots,n\}$ .

Suppose that the lemma does not hold, then there exists a positive integer $m\in\{1,2,\dots,n\}$ , such that $C_i\cup_{k=1}^iE^k_i$ is a chain, $C_i$ is a tail of $C_i\cup_{k=1}^iE^k_i$ for any $i\in\{0,1,\dots,m-1\}$ , and $C_m\cup_{k=1}^mE^k_m$ is not a chain. Since any graph without a circle that is not a chain contains at least 4 vertices, $m\geq 3$ .

By Lemma 2·8, $\cup_{k=1}^nE^k_n$ is a chain, hence $\cup_{k=1}^mE^k_m$ is a chain. Since $C_m\cup_{k=1}^mE^k_m$ is not a chain, $x_{m-1}\;:\!=\;{\rm{center}}_{X_{m-1,E}}E^m\in E^{m-1}\backslash C_{m-1}$ , and for any integer $i\in\{1,2,\dots,m-1\}$ , $f_{i,E}$ is the smooth blow-up of a point $x_{i-1}\in C_{i-1}$ , and if $i\in\{2,3,\dots,m-1\}$ , then $f_{i,E}$ is the smooth blow-up of $C_{i-1}\cap E^{i-1}$ . In particular, since $m\geq 3$ , $x_{m-3}\in C_{m-3}$ and $x_{m-2}\in C_{m-2}\cap E^{m-2}$ .

Claim 3·10. $(B_{m-1}\cdot E^{m-1})_{x_{m-1}}\geq 1$ .

Proof. Let $a_{m-1}\;:\!=\;\max\{0,1-a(E^{m-1},X,\Delta)\}$ . Since $(X\ni x,\Delta)$ is lc, $a_{m-1}\in [0,1]$ . If $(B_{m-1}\cdot E^{m-1})_{x_{m-1}}<1$ , then by Lemma 3·5,

\begin{align*} {\rm{mld}}(X\ni x,\Delta)&=a(E^n,X,\Delta)=a(E^n,X^{m-1},B_{m-1}+(1-a(E^{m-1},X,\Delta))E^{m-1})\\ &\geq a(E^n,X^{m-1},B_{m-1}+a_{m-1}E^{m-1})\\ &\geq{\rm{mld}}(X_{m-1}\ni x_{m-1},B_{m-1}+a_{m-1}E^{m-1})\\ &>1-a_{m-1}=a(E^{m-1},X,\Delta),\end{align*}

a contradiction.

Proof of Lemma 2·9 continued. Since $m\geq 3$ , by Lemmas 2·9, 3·6 and Claim 3·10,

\begin{align*}(B\cdot C)_x\geq (B_{m-3}\cdot C_{m-3})_{x_{m-3}}\geq 2(B_{m-1}\cdot E^{m-1})\geq 2(B_{m-1}\cdot E^{m-1})_{x_{m-1}}\geq 2,\end{align*}

which contradicts our assumptions.

Lemma 3·11. Let $l,r\in [0,1]$ be two real numbers. Assume that:

1. (i) $(X\ni x,\Delta\;:\!=\;B+lL+rR)$ is a smooth lc surface germ, where $B\geq 0$ is an $\mathbb{R}$ -divisor and L,R are two different prime divisors;

2. (ii) $L\not\subset\operatorname{Supp} B, R\not\subset\operatorname{Supp} B$ , and $(X\ni x,L+R)$ is log smooth; and

3. (iii) either $(B\cdot L)_x<1$ or $(B\cdot R)_x<1$ .

Then any divisor E over $X\ni x$ such that $a(E,X,\Delta)={\rm{mld}}(X\ni x,\Delta)$ satisfies the following. Let $L_E\;:\!=\;(f_E^{-1})_*L$ and $R_E\;:\!=\;(f_E^{-1})_*R$ , then $L_E\cup R_E\cup{\rm{Exc}}(f_E)$ is a chain and $L_E,R_E$ are the tails of $L_E\cup R_E\cup{\rm{Exc}}(f_E)$ . In particular, the dual graph of $L_E\cup R_E\cup{\rm{Exc}}(f_E)$ is the following:

Here $L_E$ and $R_E$ are denoted by the left black circle and the right black circle respectively.

Proof. By the construction of $f_E$ , if $L_E\cup R_E\cup{\rm{Exc}}(f_E)$ is a chain, then $L_E,R_E$ are the tails of $L_E\cup R_E\cup{\rm{Exc}}(f_E)$ . Let $n\;:\!=\;n(E)$ and let $L_i,R_i$ be the strict transforms of $L_i,R_i$ on $X_{i,E}$ for each $i\in\{0,1,\dots,n\}$ . Since $(X\ni x,L+R)$ is log smooth and by the construction of $f_E$ , $L_i\cup R_i\cup_{k=1}^iE^k_i$ is simple normal crossing over a neighbourhood of x and its dual graph does not contain a circle for each $i\in\{0,1,\dots,n\}$ . Since $a(E,X,\Delta)={\rm{mld}}(X\ni x,\Delta)$ , $a(E^i,X,\Delta)\geq a(E,X,\Delta)$ for every $i\in\{1,2,\dots,n\}$ .

Suppose that the lemma does not hold, then there exists a positive integer $m\in\{1,2,\dots,n\}$ , such that $L_i\cup R_i\cup_{k=1}^iE^k_i$ is a chain and $L_i,R_i$ are the tails of $L_i\cup R_i\cup_{k=1}^iE^k_i$ for any $i\in\{0,1,\dots,m-1\}$ , and $L_m\cup R_m\cup_{k=1}^mE^k_m$ is not a chain. Since any graph without a circle that is not a chain contains at least 4 points, $m\geq 2$ .

By Theorem 3·7, $\cup_{k=1}^nE^k_n$ is a chain, hence $\cup_{k=1}^mE^k_m$ is a chain. By the construction of $f_E$ and the choice of m, $x_{m-1}\;:\!=\;{\rm{center}}_{X_{m-1,E}}E^m\in E^{m-1}\backslash (L_{m-1}\cup R_{m-1})$ .

Claim 3·12. $(B_{m-1}\cdot E^{m-1})_{x_{m-1}}\geq 1$ .

Proof. Let $a_{m-1}\;:\!=\;\max\{0,1-a(E^{m-1},X,\Delta)\}$ . Since $(X\ni x,\Delta)$ is lc, $a_{m-1}\in [0,1]$ . If $(B_{m-1}\cdot E^{m-1})_{x_{m-1}}<1$ , then by Lemma 3·5,

\begin{align*} {\rm{mld}}(X\ni x,\Delta)&=a(E^n,X,\Delta)=a(E^n,X^{m-1},B_{m-1}+(1-a(E^{m-1},X,\Delta))E^{m-1})\\ &\geq a(E^n,X^{m-1},B_{m-1}+a_{m-1}E^{m-1})\\ &\geq{\rm{mld}}(X_{m-1}\ni x_{m-1},B_{m-1}+a_{m-1}E^{m-1})\\ &>1-a_{m-1}=a(E^{m-1},X,\Delta),\end{align*}

a contradiction.

Proof of Lemma 3·11 continued. Since $m\geq 2$ , by Lemma 2·9 and Claim 3·12,

\begin{align*}(B\cdot L)_x\geq (B_{m-2}\cdot L_{m-2})_{x_{m-2}}\geq (B_{m-1}\cdot E^{m-1})_{x_{m-1}}\geq 1\end{align*}

and

\begin{align*}(B\cdot R)_x\geq (B_{m-2}\cdot R_{m-2})_{x_{m-2}}\geq (B_{m-1}\cdot E^{m-1})_{x_{m-1}}\geq 1\end{align*}

which contradict our assumptions.

4·1. A key lemma

The following lemma is similar to [25, theorem 4·15] and plays an important role in the proof of our main theorems.

Lemma 4·1. Let m be a non-negative integer, $(X\ni x,B)$ a plt surface germ, $f: Y\rightarrow X$ the minimal resolution of $X\ni x$ , and $B_Y\;:\!=\;f^{-1}_*B$ . Then:

1. (i) for any prime divisor $F\subset{\rm{Exc}}(f)$ , $B_Y\cdot F<2$ ;

2. (ii) there exists at most one prime divisor $F\subset{\rm{Exc}}(f)$ such that $B_Y\cdot F\geq 1$ ; and

3. (iii) if $E\subset{\rm{Exc}}(f)$ is a prime divisor such that $B_Y\cdot E\geq 1$ , then $X\ni x$ is an A-type singularity and E is a tail of $\mathcal{D}(f)$ .

.

Proof. Let $F_1,\dots,F_m$ be the prime exceptional divisors of h for some positive integer m, and let $v_1,\dots,v_m$ be the vertices corresponding to $F_1,\dots,F_m$ in $\mathcal{D}(f)$ respectively. We construct an extended graph $\bar{\mathcal{D}}(f)$ in the following way:

1. (i) the vertices of $\bar{\mathcal{D}}(f)$ are $v_0,v_1,\dots,v_m$ ;

2. (ii) for any $i,j\in\{1,2,\dots,m\}$ , $v_i$ and $v_j$ are connected by a line if and only if $v_i$ and $v_j$ are connected by a line in $\mathcal{D}(f)$ ;

3. (iii) for any $i\in\{1,2,\dots,m\}$ , $v_0$ and $v_i$ are connected by $\lfloor B_Y\cdot F_i\rfloor$ lines.

Moreover, we may write

\begin{align*}K_Y+B_Y-\sum_{i=1}^ma_iF_i=f^*(K_X+B),\end{align*}

where $a_i\;:\!=\;a(F_i,X,B)-1$ . Since $(X\ni x,B)$ is plt and f is the minimal resolution of $X\ni x$ , $0\geq a_i>-1$ for each i. Let $A\;:\!=\;\sum_{i=1}^ma_iF_i$ .

If $\bar{\mathcal{D}}(f)$ is not connected, then $B_Y\cdot F_i<1$ for each i and there is nothing left to prove. Therefore, we may assume that $\bar{\mathcal{D}}(f)$ is connected.

Claim 4·2. $\bar{\mathcal{D}}(f)$ does not contain a circle.

Proof. Suppose that $\bar{\mathcal{D}}(f)$ contains a circle. We let $v_{k_0}\;:\!=\;v_0,v_{k_1},\dots,v_{k_r}$ be the vertices of this circle for some positive r such that $\bar{\mathcal{D}}(f)$ contains one of the following sub-graphs:

Let $H\;:\!=\;-\sum_{i=1}^rF_{k_i}$ . Then

\begin{align*}H\cdot F_{k_i}=-2-F_{k_i}^2=K_Y\cdot F_{k_i}\leq (K_Y+B_Y)\cdot F_{k_i}=A\cdot F_{k_i}\end{align*}

for each $i\in\{2,3,\dots,r-1\}$ ,

\begin{align*}H\cdot F_{k_i}=-1-F_{k_i}^2=K_Y\cdot F_{k_i}+1\leq (K_Y+B_Y)\cdot F_{k_i}=A\cdot F_{k_i}\end{align*}

for each $i\in\{1,r\}$ when $r\geq 2$ ,

\begin{align*}H\cdot F_{k_1}=-F_{k_1}^2=K_Y\cdot F_{k_i}+2\leq (K_Y+B_Y)\cdot F_{k_i}=A\cdot F_{k_i}\end{align*}

when $r=1$ , and

\begin{align*}H\cdot F_{i}\leq 0\leq(K_Y+B_Y)\cdot F_{i}=A\cdot F_{i}\end{align*}

for each $i\not\in\{k_1,\dots,k_r\}$ . By Lemma 2·10, $a_{k_i}\leq -1$ for each i, a contradiction.

Claim 4·3. $\bar{\mathcal{D}}(f)$ does not contain a fork.

Proof. Assume that $\bar{\mathcal{D}}(f)$ contains a fork. By Claim 4·2, $\bar{\mathcal{D}}(f)$ does not contain a circle. Therefore, $v_0$ is not a fork of $\bar{\mathcal{D}}(f)$ . In particular, there exist a positive integer r and vertices $v_{k_0}\;:\!=\;v_0,v_{k_1},\dots,v_{k_r},v_{l_1},v_{l_2}$ such that $\bar{\mathcal{D}}(f)$ contains the following sub-graph:

Let $H\;:\!=\;-\sum_{i=1}^rF_{k_i}-\frac{1}{2}F_{l_1}-\frac{1}{2}F_{l_2}$ . Then

\begin{align*}H\cdot F_{l_i}=-1-\frac{1}{2}F_{l_i}^2\leq -2-F^2_{l_i}=K_Y\cdot F_{l_i}\leq(K_Y+B_Y)\cdot F_{l_i}=A\cdot F_{l_i}\end{align*}

for each $i\in\{1,2\}$ ,

\begin{align*}H\cdot F_{k_i}=-2-F_{k_i}^2=K_Y\cdot F_{k_i}\leq (K_Y+B_Y)\cdot F_{k_i}=A\cdot F_{k_i}\end{align*}

for each $i\in\{2,3,\dots,r-1\}$ ,

\begin{align*}H\cdot F_{k_i}=-1-F_{k_i}^2=K_Y\cdot F_{k_i}+1\leq (K_Y+B_Y)\cdot F_{k_i}=A\cdot F_{k_i}\end{align*}

for $i=1$ , and

\begin{align*}H\cdot F_{i}\leq 0\leq(K_Y+B_Y)\cdot F_{i}=A\cdot F_{i}\end{align*}

for each $i\not\in\{k_1,\dots,k_r,l_1,l_2\}$ . By Lemma 2·10, $a_{k_r}\leq -1$ , a contradiction.

Proof of Lemma 4·1 continued. By Claims 4·2 and 4·3, $\bar{\mathcal{D}}(f)$ does not contain a circle or a fork. Therefore, $\bar{\mathcal{D}}(f)$ is a chain. Since $\mathcal{D}(f)$ is connected and $\bar{\mathcal{D}}(f)$ has $\mathcal{D}(f)$ as a sub-graph, $\mathcal{D}(f)$ is a chain and $v_0$ is a tail of $\bar{\mathcal{D}}(f)$ . The lemma immediately follows from the structure of $\bar{\mathcal{D}}(f)$ and $\mathcal{D}(f)$ .

4·2. A-type singularities

Lemma 4·4. Let $X\ni x$ be a surface germ of A-type, E a prime divisor on X, $f: Y\rightarrow X$ the minimal resolution of $X\ni x$ , and $E_Y\;:\!=\;f^{-1}_*E$ . Let $F_1,\dots,F_m$ be the prime exceptional divisors over $X\ni x$ . Assume that $E_Y\cup\cup_{i=1}^mF_i$ is simple normal crossing over a neighbourhood of x and the dual graph of $E_Y\cup\cup_{i=1}^mF_i$ is the following:

.

Then $(X\ni x,E)$ is plt.

Proof. We may write

\begin{align*}K_Y+E_Y-\sum_{i=1}^ma_iF_i=f^*(K_X+E),\end{align*}

where $a_i\;:\!=\;a(F_i,X,E)-1$ . Let $H\;:\!=\;-\sum_{i=1}^mF_i$ and $A\;:\!=\;\sum_{i=1}^ma_iF_i$ . Then

\begin{align*}H\cdot F_1=-F_1^2=2+K_Y\cdot F_1=1+(K_Y+E_Y)\cdot F_1=1+A\cdot F_1>A\cdot F_1\end{align*}

if $m=1$ ,

\begin{align*}H\cdot F_1=-1-F_1^2=1+K_Y\cdot F_1=(K_Y+E_Y)\cdot F_1=A\cdot F_1\end{align*}

if $m\geq 2$ ,

\begin{align*}H\cdot F_i=-2-F_i^2=K_Y\cdot F_i=(K_Y+E_Y)\cdot F_i=A\cdot F_i\end{align*}

if $m\geq 2$ and $2\leq i\leq m-1$ , and

\begin{align*}H\cdot F_m=-1-F_m^2=1+K_Y\cdot F_m=1+(K_Y+E_Y)\cdot F_m=1+A\cdot F_i>A\cdot F_m\end{align*}

if $m\geq 2$ . By Lemma 2·10, $a_i>-1$ for each i, hence $(X\ni x,E)$ is plt.

Proof. For any model X’ of X such that ${\rm{center}}_{X'}E$ is a divisor, we let $E_{X'}$ be the center of E on X’. Let $h: Y\rightarrow X$ be the minimal resolution of X and $F_1,\dots,F_m$ the prime exceptional divisors of h with the following dual graph:

Let $B_Y\;:\!=\;f^{-1}_*B$ and $a_i\;:\!=\;1-a(F_i,X,B)$ for each i. Then $a_i\in [0,1)$ for each i. There are three cases:

Case 2. E is not on Y, ${\rm{center}}_YE\;:\!=\;y\in F_i$ for some i, and ${\rm{center}}_YE\not\in F_j$ for any $j\not=i$ . In this case, since $a(E,X,B)={\rm{mld}}(X\ni x,B)$ , $a(E,X,B)\leq 1-a_i$ . By Lemma 3·5, $B_Y\cdot F_i\geq 1$ . By Lemma 4·1(iii), $i=1$ or m. We let $f_E: W\rightarrow Y$ the sequence of smooth blow-ups as in Definition-Lemma 3·1, and $g\;:\!=\;h\circ f_E$ . By Lemma 4·1(i), $B_Y\cdot F_i<2$ , hence $(B_Y\cdot F_i)_y<2$ . Since

\begin{align*}K_Y+B_Y+\sum a_iF_i=h^*(K_X+B),\end{align*}

by Lemma 3·9, $\mathcal{D}(g)$ is a chain. Moreover, by the construction of g, for any prime divisor $F\not=E_W$ in ${\rm{Exc}}(g)$ , $F^2\leq -2$ .

Case 3. E is not on Y and ${\rm{center}}_YE\;:\!=\;y\in F_i\cap F_{i+1}$ for some i. In this case, we let $f_E: W\rightarrow Y$ the sequence of smooth blow-ups as in Definition-Lemma 3·1, and $g\;:\!=\;h\circ f_E$ . By Lemma 4·1(ii), either $B_Y\cdot F_i<1$ or $B_Y\cdot F_{i+1}<1$ , hence either $(B_Y\cdot F_i)_y<1$ or $(B_Y\cdot F_{i+1})_y<1$ . Since

\begin{align*}K_Y+B_Y+\sum a_iF_i=h^*(K_X+B),\end{align*}

by Lemma 3·11, $\mathcal{D}(g)$ is a chain. Moreover, by the construction of g, for any prime divisor $F\not=E_W$ in ${\rm{Exc}}(g)$ , $F^2\leq -2$ .

By [ 25 , remark 4·9(2)], there exists a contraction $\phi: W\rightarrow Z$ over X of $\operatorname{Supp}{\rm{Exc}}(g)\backslash E_W$ . Since $\mathcal{D}(g)$ is a chain in all three cases, ${\rm{Exc}}(g)$ is snc. By Lemma 4·4, $(Z\ni z,E_Z)$ is plt for any singular point z of Z in $E_Z$ . Thus $(Z,E_Z)$ is plt near $E_Z$ , hence E is a Kollár component of $X\ni x$ .

4·3. D-type and E-type singularities

Lemma 4·6. Let $X\ni x$ be a surface germ of $D_m$ -type for some integer $m\geq 3$ , E a prime divisor on X, $f: Y\rightarrow X$ the minimal resolution of $X\ni x$ , and $E_Y\;:\!=\;f^{-1}_*E$ . Let $F_1,\dots,F_m$ be the prime exceptional divisors over $X\ni x$ . Assume that $E_Y\cup\cup_{i=1}^mF_i$ is simple normal crossing over a neighbourhood of x, $F_{m-1}^2=F_m^2=-2$ , and the dual graph of $E_Y\cup\cup_{i=1}^mF_i$ is the following:

Then $(X\ni x,E)$ is lc but not plt.

Proof. By computing intersections numbers,

\begin{align*}K_Y+E_Y+\sum_{i=1}^{m-2}F_i+\frac{1}{2}(F_{m-1}+F_m)=f^*(K_X+E).\end{align*}

Since $(Y,E_Y+\sum_{i=1}^{m-2}F_i+\frac{1}{2}(F_{m-1}+F_m))$ is log smooth over a neighbourhood of x, $(X\ni x,E)$ is lc but not plt.

Lemma 4·7. Let $0<m_1<m_2<m_3\;:\!=\;m$ be integers, $(X\ni x,B)$ a plt surface germ, $f:Y\to X$ the minimal resolution of $X\ni x$ with prime exceptional divisors $F_0,\dots,F_{m}$ with the following dual graph $\mathcal{D}(f)$ :

Let $a_i\;:\!=\;a(F_i,X,B)-1$ for each i. Then:

1. (i) $a(F_0,X,B)\leq a(F_i,X,B)$ for any $i\in\{1,2,\dots,m_1\}$ ; and

2. (ii) if $a(F_0,X,B)=a(F_l,X,B)$ for some $l\in\{1,2,\dots,m_1\}$ , then

   1. (a) $a_i=a_0$ for every $i\in\{0,1,\dots,l\}$ ,

   2. (b) $a_{m_1+1}=a_{m_2+1}=\frac{1}{2}a_0$ ,

   3. (c) $F_i^2=-2$ for any $i\in\{0,1,\dots,l-1,m_1+1,m_2+1\}$ , and

   4. (d) either $m=m_2+1=m_1+2$ , or $B=0$ and $X\ni x$ Du Val.

Proof. Let $B_Y\;:\!=\;f^{-1}_*B$ . Then

\begin{align*}K_Y+B_Y-\sum_{i=1}^ma_iF_i=f^*(K_X+B).\end{align*}

Since $(X\ni x,B)$ is plt and f is the minimal resolution of $X\ni x$ , $-1<a_i\leq 0$ for each i.

If $a_0<a_i$ for any $i\in\{1,2,\dots,m_1\}$ there is nothing to prove. Otherwise, there exists $k\in\{1,2,\dots,m_1\}$ , such that $a_k=\min\{a_i\mid 0\leq i\leq m_1\}\leq a_0$ . We define

\begin{align*}A\;:\!=\;\sum_{i=0}^{k}a_iF_i+a_{m_1+1}F_{m_1+1}+a_{m_2+1}F_{m_2+1}, \text{ and } H\;:\!=\;a_k\left(\sum_{i=0}^kF_i+\frac{1}{2}F_{m_1+1}+\frac{1}{2}F_{m_2+1}\right).\end{align*}

Then

\begin{align*}H\cdot F_i=a_k(F_i^2+2)=-a_kK_Y\cdot F_i\leq K_Y\cdot F_i\leq(K_Y+B_Y)\cdot F_i=A\cdot F_i,\end{align*}

when $0\leq i<k$ and if the equality holds then $K_Y\cdot F_i=0$ ,

\begin{align*}H\cdot F_{k}=a_k(F_k^2+1)\leq a_kF_k^2+a_{k-1}=A\cdot F_k\end{align*}

and if the equality holds then $a_k=a_{k-1}$ , and

\begin{align*}H\cdot F_{m_i+1}&=a_k(1+\frac{1}{2}F_{m_i+1}^2)=-\frac{a_k}{2}K_Y\cdot F_{m_i+1}\leq K_Y\cdot F_{m_i+1}\\&\leq (K_Y+B_Y)\cdot F_{m_i+1}\leq A\cdot F_{m_i+1}\end{align*}

for $i\in\{1,2\}$ , and if the equality holds, then:

1. (i) $K_Y\cdot F_{m_i+1}=0$ , and

2. (ii) either $m_{i+1}=m_i+1$ , or $m_{i+1}\geq m_i+2$ and $a_{m_i+2}=0$ .

Thus $H\cdot F_i\leq A\cdot F_i$ for any $i\in\{0,1,\dots,k,m_1+1,m_2+1\}$ . By Lemma 2·10, $a_i\leq a_k$ for any $i\in\{0,1,\dots,k\}$ and $a_{m_1+1}=a_{m_2+1}={1}/{2}a_k$ . Since $a_k=\min\{a_i\mid 0\leq i\leq m_1\}\leq a_0$ , $a_i=a_k=\min\{a_i\mid 0\leq i\leq m_1\}$ for any $i\in\{0,1,\dots,k\}$ . Thus for any $l\in\{1,2,\dots,m_1\}$ such that $a(F_0,X,B)=a(F_l,X,B)$ , we may pick $k=l$ , which shows (i) and (ii)(a). Moreover, we have that $H\cdot F_i=A\cdot F_i$ for any $i\in\{0,1,\dots,l,m_1+1,m_2+1\}$ , which implies that:

1. (i) $a_{m_1+1}=a_{m_2+1}={1}/{2}a_l$ , hence (ii)(b);

2. (ii) $F_i^2=K_Y\cdot F_i=-2$ for every $i\in\{0,1,\dots,l-1,m_1+1,m_2+1\}$ , hence (ii)(c); and

3. (iii) either $m=m_2+1=m_1+2$ , or there exists $i\in\{1,2\}$ such that $m_{i+1}\geq m_i+2$ and $a_{m_i+2}=0$ .

If $m=m_2+1=m_1+2$ then we get (ii)(d) and the proof is completed. Otherwise, there exists $i\in\{1,2\}$ such that $m_{i+1}\geq m_i+2$ and $a_{m_i+2}=0$ . Thus

\begin{align*}1=a(F_{m_i+2},X,B)\leq a(F_{m_i+2},X,0)\leq 1,\end{align*}

which implies that $a(F_i,X,B)=a(F_i,X,0)$ , hence $B=0$ . Moreover, since f is the minimal resolution of $X\ni x$ , $\sum_{i=1}^ma_iF_i\sim_XK_Y$ is nef over X. By Lemma 2·10, $a_i=0$ for every i, and $X\ni x$ is Du Val.

Proof. Let $F_1,\dots,F_m$ be the prime exceptional divisors of f, $a_i\;:\!=\;1-a(F_i,X,B)$ for each i, and $B_Y\;:\!=\;f^{-1}_*B$ . Then

\begin{align*}K_Y+B_Y+\sum_{j=1}^ma_jF_j=f^*(K_X+B).\end{align*}

Suppose that $E\not\subset{\rm{Exc}}(f)$ . Let $y\;:\!=\;{\rm{center}}_YE$ .

Claim 4·9. $y=F_i\cap F_k$ for some $i\not=k$ .

Proof. Then there exists an integer $i\in\{1,2,\dots,m\}$ such that $y\in F_i$ . Thus

\begin{align*} {\rm{mld}}(Y\ni y, B_Y+\sum_{j=1}^ma_jF_j)&\leq a\left(E,Y, B_Y+\sum_{j=1}^ma_jF_j\right)\\ &=a(E,X,B)={\rm{mld}}(X\ni x,B)\leq 1-a_i.\end{align*}

By Lemma 35,

\begin{align*}(B+\sum_{j\not=i}a_jF_j)\cdot F_i\geq\left(\left(B+\sum_{j\not=i}a_jF_j\right)\cdot F_i\right)_y\geq 1.\end{align*}

By Lemma 4·1(iii), $B\cdot F_i<1$ , which implies that there exists $k\not=i$ such that $y\in F_k$ . In particular, $y=F_i\cap F_k$ .

Proof of Lemma 4·8 continued. By Claim 4·9, $y=F_i\cap F_k$ for some $i\not=k$ . Possibly switching i and k, we may assume that $F_i$ is closer to the fork of $\mathcal{D}(f)$ than $F_k$ . Since $(X\ni x,B)$ is plt and f is the minimal resolution of $X\ni x$ , $0\leq a_k<1$ . Thus there exists an extraction $g: W\rightarrow X$ of $F_k$ with induced morphism $h: Y\rightarrow W$ . Let $\bar F_k$ be the center of $F_k$ on W and $w\;:\!=\;{\rm{center}}_WE$ . Then h is the minimal resolution of $W\ni w$ . Moreover, if $F_i$ is not the fork of $\mathcal{D}(f)$ , then $W\ni w$ is not an A-type singularity, and if $F_i$ is the fork of $\mathcal{D}(f)$ , then $W\ni w$ is an A-type singularity but $F_i$ is not a tail of $\mathcal{D}(h)$ . Since $(X\ni x,B)$ is plt, $(W\ni w,g^{-1}_*B+a_kF_k)$ is plt. By Lemma 4·1(iii),

\begin{align*}\left(\left(B+\sum_{j\not=i}a_jF_j\right)\cdot F_i\right)_y=((B+a_kF_k)\cdot F_i)_y\leq (B+a_kF_k)\cdot F_i<1.\end{align*}

By Lemma 3·5,

\begin{align*} {\rm{mld}}(X\ni x,B)&=a(E,X,B)=a(E,Y, B_Y+\sum_{j=1}^ma_jF_j)\\ &\geq{\rm{mld}}(Y\ni y, B_Y+\sum_{j=1}^ma_jF_j)>1-a_i\geq{\rm{mld}}(X\ni x,B),\end{align*}

a contradiction.

Proof. By Lemma 4·8, we may only consider divisors in ${\rm{Exc}}(f)$ . Let $F_1,\dots,F_m$ be the prime exceptional divisors of f such that $F_1$ is the unique fork. Since $a(F_i,X,B)\leq a(F_i,X,0)\leq 1$ for each i, there exists an extraction $g: Y_i\rightarrow X$ of $F_i$ for each i, and we let $\bar F_i$ be the strict transform of $F_i$ on $Y_i$ . By Lemmas 4·4 and 4·6, $(Y_i,\bar F_i)$ is not plt near $\bar F_i$ for any $i\in\{2,3,\dots,m\}$ and $(Y_1,\bar F_1)$ is plt near $F_1$ . By Lemma 4·7, $a(F_1,X,B)={\rm{mld}}(X\ni x,B)$ . So $E=F_1$ is the unique divisor we want.

Proof. Let $f: Y\rightarrow X$ be the minimal resolution of $X\ni x$ . By Lemma 4·8, $E\subset{\rm{Exc}}(f)$ . By Theorem 4·10, we may assume that E is not the fork of $\mathcal{D}(f)$ . Let $L_1,L_2,L_3$ be the three branches of $\mathcal{D}(f)$ and assume that E belongs to $L_1$ . By Lemma 4·7(ii)(c),(ii)(d), $X\ni x$ is a $D_m$ -type singularity, $L_2$ contains a unique curve $F_2$ and $L_3$ contains a unique curve $F_3$ respectively, such that $F_2^2=F_3^2=-2$ . Since $(X\ni x,B)$ is plt and f is the minimal resolution of $X\ni x$ , $0<a(E,X,B)\leq 1$ , so there exists an extraction $g: W\rightarrow X$ of E with the induced morphism $h: Y\rightarrow W$ . Let $E_W$ be the strict transform of E on W. By Lemmas 4·4 and 4·6, $(W,E_W)$ is lc near $E_W$ , so E is a potential lc place of $X\ni x$ .

Theorem 4·12. Let $X\ni x$ be a Du Val singularity of $D_m$ -type for some integer $m\geq 4$ or of $E_m$ -type for some integer $m\in\{6,7,8\}$ . Let $f: Y\rightarrow X$ be the minimal resolution of $X\ni x$ , and E be a divisor over $X\ni x$ such that $a(E,X,0)={\rm{mld}}(X\ni x,0)$ , then E is a potential lc place of $X\ni x$ if and only if one of the following holds:

1. (i) E is the fork of the $\mathcal{D}(f)$ ;

2. (ii) $X\ni x$ is of $D_m$ -type, the dual graph of $X\ni x$ looks like the following, and $E\in\{F_1,\dots,F_{m-2}\}$ .

Proof. Since $a(E,X,0)={\rm{mld}}(X\ni x,0)$ , $E\subset{\rm{Exc}}(f)$ . Since $X\ni x$ is Du Val, for any prime divisor $F\subset{\rm{Exc}}(f)$ , $F^2=-2$ . The if part follows from Lemmas 4·4 and 4·6.

To prove the only if part, we may assume that E is not the fork of $\mathcal{D}(f)$ . Then E is a potential lc place of $X\ni x$ if and only if there exists an extraction $g: W\rightarrow X$ of E such that $(W,E_W)$ is lc near $E_W$ , where $E_W$ is the strict transform of E on W. The theorem follows from [ 25 , theorem 4·15].

4·4. Non-plt singularities

Definition- Lemma 4·13 (25, theorem 4·7]). Let $X\ni x$ be an lc but not klt surface germ and $f: Y\rightarrow X$ the minimal resolution of $X\ni x$ . Then exactly one of the following holds:

1. (i) (B-type) ${\rm{Exc}}(f)=F$ is a smooth elliptic curve.

2. (ii) (C-type) ${\rm{Exc}}(f)=F$ is a nodal cubic curve;

3. (iii) (F-type) ${\rm{Exc}}(f)$ is a circle of smooth rational curves;

4. (iv) (H-type) ${\rm{Exc}}(f)$ has $\geq 5$ rational curves and $\mathcal{D}(f)$ has the following weighted dual graph:

Let $m\geq 5$ be an integer. For an lc singularity of H -type as above with m exceptional divisors, we call it an ${{\textbf{H}}}_m$ -type singularity.

Although the concept of Kollár component is not defined over an lc but not klt germ, the classification of surface lc singularities tells us when there exists a divisor which “looks like a Kollár component”.

Proof. E is an lc place of $(X\ni x,B)$ which implies (i). Since $(X\ni x,B)$ is lc and $X\ni x$ is not klt, $B=0$ . By the connectedness of lc places, $K_Y+E$ is plt near E if and only if E is the only lc place over $X\ni x$ , and (ii) follows from Definition-Lemma 4·13.

Now we deal with the case when $X\ni x$ is klt but $(X\ni x,B)$ is not plt:

Proof. Since $(X\ni x,B)$ is not plt, $a(E,X,B)={\rm{mld}}(X\ni x,B)=0$ , so E is an lc place of $(X\ni x,B)$ , hence a potential lc place of $(X\ni x,B)$ .

Proof. Let

\begin{align*}\delta\;:\!=\;\min\{a(E,X,B)\mid E\text{ is over }X\ni x, a(E,X,B)>0\}.\end{align*}

Then there exists $\epsilon\in (0,1)$ such that $\delta>{\rm{mld}}(X\ni x,(1-\epsilon)B)>0$ . By Theorems 4·5 and 4·10, there exists a divisor E over $X\ni x$ such that $a(E,X,(1-\epsilon)B)={\rm{mld}}(X\ni x,(1-\epsilon)B)<\delta$ and E is a Kollár component of $X\ni x$ . Since

\begin{align*}0\leq a(E,X,B)<a(E,X,(1-\epsilon)B)<\delta,\end{align*}

$a(E,X,B)={\rm{mld}}(X\ni x,B)=0$ , so E satisfies our requirements.

Finally, recall the following result:

Proof. Since $(X\ni x,B)$ is dlt but not plt, $\lfloor B\rfloor$ contains at least 2 irreducible components near x. Since X is a surface, $B=\lfloor B\rfloor$ has exactly two irreducible components near x.

There exists a divisor E over $X\ni x$ such that $a(E,X,B)=0$ . Since $(X\ni x,B)$ is dlt, $x\;:\!=\;{\rm{center}}_XE$ belongs to the log smooth strata of (X,B), so $(X\ni x,B)$ is log smooth.