Definition 2·1. We call E a prime divisor over A, if there is a proper birational morphism $\varphi:\; A' \longrightarrow A$ from a normal variety A ^′ on which E is an irreducible divisor. The generic point $P\in A$ of the image $\varphi(E)$ is called the center of E on A. In this case, we sometimes call E a prime divisor over (A, P).

Definition 2·2. For a prime divisor E over a non-singular variety A, let $\varphi:\; A'\longrightarrow A$ be a proper birational morphism with normal A ^′ such that E appears on A ^′. Let $k_E$ (or sometimes written as $k_{E/A}$ ) be the coefficient of the relative canonical divisor $K_{A^{\prime}/A}$ at E and $v_E$ the valuation defined by the prime divisor E. Here, note that $k_E$ ( $k_{E/A}$ ) does not depend on the choice of A ^′.

Definition 2·3. We say that a prime divisor E over A with the center at 0 computes ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$

if either $a(E;\, A,{\mathfrak{a}})={\textrm{mld}}(0;\, A,{\mathfrak{a}})$ (when the right-hand side is $\geq 0$ )

or $a(E;\, A,{\mathfrak{a}})<0$ (when the mld is $-\infty$ ).

Definition 3·1. Let $x_1,\ldots, x_c$ be an RSP of a regular local ring R with the algebraically closed residue field and $w_1,\ldots,w_c$ be positive integers with $\gcd(w_1,\ldots,w_c)=1$ . For $n\in {\Bbb N}$ , denote by $\mathcal I_n$ the ideal in R generated by all monomials $x_1^{s_1}\cdots x_c^{s_c}$ such that $\sum_{i=1}^c s_iw_i\geq n$ . The weighted blow-up of ${\textrm{Spec }} R$ with $wt_w(x_1,\ldots, x_c)=(w_1,\ldots, w_c)$ is the canonical projection:

\begin{align*}\textrm{Proj}_{A}(\oplus_{n\in {\Bbb N}}{\mathcal I}_n)\longrightarrow A\;:\!=\,{\textrm{Spec }} R.\end{align*}

The exceptional divisor E for the weighted blow-up is called a prime divisor obtained by a weighted blow-up of A at P.

Example 3·3. Assume ${\textrm{char }} k\neq 2,5$ . Let $A\;:\!=\,{\Bbb A}_k^3$ and ${\mathfrak{a}}=(f)^{7/10}$ , where

\begin{align*}f=(x^2+y^2+z^2)^2+x^5+y^5+z^5.\end{align*}

Then, a divisor computing ${\textrm{mld}}(0;\, A,{\mathfrak{a}})=0$ is not obtained by one weighted blow-up ([ Reference Kollár, Smith and Corti12 , exercise 6·45]).

On the other hand, there is a sequence of weighted blow-ups

\begin{align*}A_2\stackrel{\varphi_2}\longrightarrow A_1\stackrel{\varphi_1}\longrightarrow A,\end{align*}

where $\varphi_1$ is the usual blow-up at 0 and $\varphi_2$ is a weighted blow-up with weight (1, 2) at the generic point of the curve $x^2+y^2+z^2=0$ on $E_1={\Bbb P}_k^2$ . Here, $E_1$ is the exceptional divisor for $\varphi_1$ . The exceptional divisor $E_2$ for $\varphi_2$ computes ${\textrm{mld}}(0;\, A,{\mathfrak{a}})=0$

Proof. As ${\textrm{ord}}_Qg\leq {\textrm{ord}}_Q(g\mid _{\ell})$ , the first inequality is trivial. We will show the second inequality. Let $G\subset {\Bbb P}_k(r,r,s)$ be the subscheme defined by $g=0$ on ${\Bbb P}_k(r,r,s)$ . Let

\begin{align*}\pi:\; {\Bbb P}_k^2\twoheadrightarrow {\Bbb P}(r,r,s), (X_1,X_2,X_3)\mapsto (X_1^r,X_2^r,X_3^s)=(x_1,x_2,x_3)\end{align*}

be the canonical covering. Then, as $\pi^*L$ and $\pi^*G$ has no common irreducible components, Bezout’s theorem on ${\Bbb P}^2$ implies

(1) \begin{equation} \pi^*L\cdot \pi^*G=\deg\pi^*\ell\cdot\deg\pi^*g=\deg_w\ell\cdot\deg_w g=r\cdot \deg_w g , \end{equation}

Definition 4·1. Let $P\in A$ be a smooth point (not necessarily closed), K the residue field, and E a prime divisor over A with the center at P. Denote the algebraic closure of K by ${\overline{K}}$ . An RSP $\{x_1,\ldots, x_c\} $ of ${\overline{K}}\widehat{\mathcal O}_{A,P}$ at the closed point is called a squeezed system for E at P, if $v_i\;:\!=\,v_E(x_i)$ ( $i=1,\ldots, c$ ) satisfy:

1. (1) $v_1=\cdots=v_{c-1}\leq v_c$ ;

2. (2) $v_1\;:\!=\,\min\{v_E(x)\mid x\in {\mathfrak{m}}\setminus {\mathfrak{m}}^2\}$ ;

3. (3) $v_c\;:\!=\,\max\{v_E(x)\mid x\in {\mathfrak{m}}\setminus {\mathfrak{m}}^2\}$ ;

where ${\overline{K}}\widehat{\mathcal O}_{A,P}$ is the extension of the coefficient field K of the formal power series ring ${\mathcal O}_{A,P}$ to ${\overline{K}}$ , and ${\mathfrak{m}} \subset {\overline{K}}\widehat{\mathcal O}_{A,P}$ is the maximal ideal.

In this case,

\begin{align*}v'\;:\!=\,({v^{\prime}_1},\ldots, {v^{\prime}_c})=\frac{(v_1,\ldots,v_c)}{\gcd (v_1,\ldots,v_c)}\end{align*}

is called a squeezed weight for E at P.

Let E and $v'=(v^{\prime}_1,\ldots, v^{\prime}_c) $ be as above. In this case, we call E a prime divisor of squeezed type v ^′.

Example 4·3 (Theorem 1·1). For every prime divisor E over a smooth surface A with the center at 0 such that $a(E;\, A, {\mathfrak{a}})\geq 0$ for an ${\Bbb R}$ -ideal ${\mathfrak{a}}$ on A. Then, the exceptional divisor $E_1$ obtained by a squeezed blow-up for E satisfies

\begin{align*}a(E;\, A,{\mathfrak{a}})\geq a(E_1;\, A,{\mathfrak{a}}).\end{align*}

Definition 4·4. Let A, P and E as above and let $\{x_1,\ldots,x_c\}$ be a squeezed system for E and $v'=(v^{\prime}_1,\ldots,v^{\prime}_r)$ be the squeezed weight. We call the weighted blow-up of weight v ^′ with respect to the coordinate system $\{x_1,\ldots,x_c\}$ a squeezed blow-up for E.

Example 4·6. Let $A_K\;:\!=\,{\textrm{Spec }} K[[y,z]]$ and $A_{\overline{K}}\;:\!=\,{\textrm{Spec }} {\overline{K}}[[y,z]]$ , where ${\overline{K}}$ is the algebraic closure of K. Take an element $a\in {\overline{K}}\setminus K$ and let $\phi\in K[T]$ be the minimal polynomial of a. Let $\varphi_1\;:\;A_1\longrightarrow A_K$ be the usual blow-up at the closed point of $A_K$ . Then the exceptional divisor $E_1$ is the projective line ${\Bbb P}_K^1$ with the homogeneous coordinates $\{y,z\}$ . Denote the homogenised polynomial of $\phi$ by $\Phi(y,z)\;:\!=\,z^{\deg \phi}\phi(y/z)$ . Take the blow-up $\varphi_2:\; A_2\longrightarrow A_1$ with the center at the closed subscheme C defined by the ideal $(\Phi(y,z))$ on $E_1$ . As the proper transforms of any curves defined by linear forms $\ell=cy+dz=0$ ( $c,d\in K$ ) on $A_1$ do not intersect to C, it follows $v_{E_2}(\ell)=1$ . Therefore, every RSP $\{ f_1, f_2\}$ of K[[y,z]] satisfies $v_E(f_1)=v_E(f_2)=1$ .

Definition 4·7. Let E be a prime divisor over A of squeezed type $(v^{\prime}_1,v^{\prime}_2, v^{\prime}_3)$ (note that $v^{\prime}_1=v^{\prime}_2$ ) and let $E_1$ be the exceptional divisor obtained by the squeezed blow-up with respect to a squeezed system $\{x_1,x_2,x_3\}$ .

Proof. It is clear that if $E=E_1$ , then the center of E on $E_1$ is the generic point, so there is no bad curve on $E_1$ . We exclude this trivial case in the following discussions. In case the squeezed blow-up is the usual blow-up, then the exceptional divisor does not have a bad curve. Because if B is a bad curve, it is defined by linear form $\ell=\sum_i a_iX_i=0 $ with $a_3\neq 0$ , where $\{X_1,X_2,X_3\}$ is the projective coordinate system on $E_1={\Bbb P}^2$ corresponding to the squeezed system $\{x_1,x_2,x_3\}$ on ${\mathcal O}_{A,0}$ . This is a contradiction to the fact that (1, 1, 1) is the squeezed system, as we obtain another RSP $\{x_1,x_2, \ell(x_1)\}$ such that

(2) \begin{equation} v_E(x_1)<v_E(\ell(x_i)). \end{equation}

Here, we give the proof of this inequality, as this kind of discussion is used frequently in this paper.

Let $\varphi_1\;:\;A_1\longrightarrow A$ be the squeezed blow-up and $\psi :\;\widetilde A\longrightarrow A_1$ a birational morphism on which E appears. Denote the composite $\varphi_1\circ \psi $ by $\varphi$ . Let D be the proper transform of $Z(\ell(x_i))\subset A$ in $A_1$ , then $D\cap E_1 $ contains the center of E on $A_1$ by the assumption. Note that we can express

\begin{align*} (\varphi_1^*\ell(x_i))=r E_1+D,\ \ (r=v_{E_1}(\ell(x_i))). \end{align*}

Here, we remind the reader that $v_E(\ell(x_i))$ is the coefficient of the divisor $(\varphi^*\ell(x_i))=\psi^*(rE_1+D)$ at the component E. The center of E on $A_1$ is contained in D, therefore the contribution from $\psi^*(D)$ to $v_E(\ell(x_i))$ is positive. Therefore, $v_E(\ell(x_i))> r v_E(E_1)=v_{E_1}(\ell(x_i))v_E(E_1)=v_E(x_1)$ . This shows the inequality (2).

For the case where $E_1$ is an exceptional divisor of a squeezed blow-up with respect to $(v^{\prime}_1,v^{\prime}_2, v^{\prime}_3)$ with $v^{\prime}_1<v^{\prime}_3$ , if the center C of E on $E_1$ is a curve of degree $>v^{\prime}_1$ , then there is no bad curve. Because, a curve of degree $v^{\prime}_1$ cannot contain a curve of degree $>v^{\prime}_1$ . This gives the proof of “if” part of (i).

Assume a bad curve exists on $E_1$ . When the center of E on $E_1$ is a curve, then it should coincide with the bad curve by the definition, therefore the center should be of degree $v^{\prime}_1$ . When the center of E on $E_1$ is a closed point P, then a bad curve should contain P. Express the point P by the homogeneous coordinates (a,b,c) with $a,b,c\in k$ . Then a curve of degree $v^{\prime}_1$ containing P is defined by $bX_1-aX_2=0$ . Now we obtain the uniqueness of the bad curve on $E_1$ . This completes the proof of “only if” part of (i) and the proof of (ii).

Definition 4·9. Let E be a prime divisor over a smooth variety A with the center at a closed point 0. An ${\Bbb R}$ -ideal ${\mathfrak{a}}$ is called general for E if there exists a squeezed blow-up $ A_1\longrightarrow A$ for E with the exceptional divisor $E_1$ satisfying the following:

1. (1) ${\textrm{ord}}_B{\mathfrak{a}}_{A_1}{\mathcal O}_{E_1}\leq 1$ , where B is the bad curve on $E_1$ and ${\mathfrak{a}}_{A_1}$ is the weak transform of ${\mathfrak{a}}$ at $A_1$ . If there is no bad curve on $E_1$ , then we account it as the inequality automatically holds;

2. (2) in addition, if $a(E;\, A,{\mathfrak{a}})<a(E_1;\, A,{\mathfrak{a}})$ and the center P of E on $A_1$ is a smooth closed point, then there exists a squeezed blow-up $A_2\longrightarrow A_1$ for E at P. Let $E_2$ be the exceptional divisor. Then, ${\textrm{ord}}_{B^{\prime}}I_L{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}\leq 1$ , where B ^′ is the bad curve on $E_2$ , ${\mathfrak{a}}_{A_2}$ is the weak transform of ${\mathfrak{a}}$ at $A_2$ and $I_L$ is the defining ideal of the intersection $L\;:\!=\,E_2\cap E^{\prime}_1$ in $E_2$ . Here, $E^{\prime}_1$ is the proper transform of $E_1$ on $A_2$ . If there is no bad curve on $E_2$ , then we account it as the inequality automatically holds.

We say that a pair $(A,{\mathfrak{a}})$ is general if the ${\Bbb R}$ -ideal ${\mathfrak{a}}$ is general for a prime divisor computing ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ . Here, the weak transform ${\mathfrak{a}}_{iA_2}$ of an ideal ${\mathfrak{a}}_i\subset {\mathcal O}_A$ on $A_2$ is defined as

\begin{align*}{\mathfrak{a}}_i{\mathcal O}_{A_2}={\mathfrak{a}}_{iA_2}{\mathcal O}_{A_2}({-}v_{E_1}({\mathfrak{a}}_i)E_1-v_{E_2}({\mathfrak{a}}_i)E_2).\end{align*}

The weak transform ${\mathfrak{a}}_{A_2}$ of an ${\Bbb R}$ -ideal ${\mathfrak{a}}$ on A is defined as the canonical extension of the one for an ideal of ${\mathcal O}_A$ (see, for example [ Reference Kawakita9 ]).

Proof. Assume that the statement does not hold, then we may assume that P is in the hyperplane defined by $X_1=0$ in $E_1={\Bbb P}(v')$ . There exists at least one homogeneous coordinate function $X_i$ such that P does not lay in the hyperplane defined by $X_i=0$ . Then we obtain:

\begin{align*}v_E(x_i)= &\, v_{E_1}(x_i)\cdot v_E(E_1)=v^{\prime}_i\cdot v_E(E_1);\\v_E(x_1)= &\, v_{E_1}(x_1)\cdot v_E(E_1)+{\textrm{ord}}_P X_1\geq v^{\prime}_1\cdot v_E(E_1)+1.\end{align*}

This is a contradiction to the fact that

\begin{align*}v_E(x_1)\;:\;v_E(x_i)= v^{\prime}_1\;:\;v^{\prime}_i.\end{align*}

Proof. First we express the log discrepancy at E as follows:

(3) \begin{equation} \begin{array}{ll} a(E;\, A, {\mathfrak{a}})&=k_{E/A}+1-v_E({\mathfrak{a}})\\[4pt] &=k_{E/{A'}}+k_{D/A}\cdot v_E(D)+1-v_{D}({\mathfrak{a}})\cdot v_E(D) -v_E({\mathfrak{a}}_{A^{\prime}})\\[4pt] &=a(E;\, A', I_{D}\cdot{\mathfrak{a}}_{A^{\prime}})+v_E(D)\cdot a(D;\, A,{\mathfrak{a}}),\\ \end{array} \end{equation}

where $k_{E/A'}$ is the coefficient of the relative canonical divisor $K_{{\widetilde A}/{A'}}$ at E and $I_{D}$ is the defining ideal of D in A ^′. Then, by the assumption, it follows $a(E;\, A',I_{D}\cdot {\mathfrak{a}}_{A^{\prime}})<0$ and therefore we obtain

\begin{align*}{\textrm{mld}}(P;\, A', I_{D}\cdot {\mathfrak{a}}_{A^{\prime}})=-\infty.\end{align*}

By Inversion of adjunction ([ Reference Ein and Mustaţă3 , Reference Ishii and Reguera7 ]) we obtain ${\textrm{mld}}(P;\, D, {\mathfrak{a}}_{A^{\prime}}\cdot{\mathcal O}_{D})=-\infty$ . Hence, it follows ${\textrm{ord}}_P ({\mathfrak{a}}_{A^{\prime}}\cdot{\mathcal O}_{D})>1$ as claimed.

Proof. Let $f^e=f_1^{e_1}\cdots f_r^{e_r}\in {\mathfrak{a}}$ be a general element, i.e., $v_{E_1}({\mathfrak{a}})=\sum_i e_i\cdot \deg_{v^{\prime}}(\textrm{in}_{v^{\prime}} f _i)$ , where $\textrm{in}_{v^{\prime}} f$ is the initial part of f with respect to the weight v ^′.

Case 1. $\dim\overline{\{P\}}=1$ .

Let $C\;:\!=\, \overline{\{P\}}$ defined by $\ell=0$ on $E_1={\Bbb P}(v')$ , where $\ell$ is homogeneous of degree $\geq v^{\prime}_1$ with respect to the weight v ^′.

The ${\Bbb R}$ -divisor on $E_1$ induced from a general element $f^e=f_1^{e_1}\cdots f_r^{e_r}$ is expressed as follows:

\begin{align*}\left(\prod \textrm{in}_{v^{\prime}}f_i^{e_i}\right)=\alpha C +\sum_j \gamma_j C_j, \ \ \mbox{with}\ \ \alpha >1, \gamma_i \in {\Bbb R}_{>0}\end{align*}

Here, note that $\alpha>1$ follows from Lemma 5·2. As ${\mathfrak{a}}$ is general, C is not a bad curve, therefore its degree is greater than $v^{\prime}_1$ . Then, $\deg_{v^{\prime}}\ell\geq v^{\prime}_1v^{\prime}_3$ , because $\ell $ is an irreducible weighted homogeneous polynomial in $x_1,x_2,x_3$ of weight $v^{\prime}_1, v^{\prime}_1,v^{\prime}_3$ not contained in the coordinate hyperplanes in $E_1\simeq {\Bbb P}(v')$ . (Note that such a polynomial with smallest degree is in the form $ax_1^{v^{\prime}_3}+ bx_2^{v^{\prime}_3}+c{x_3}^{v^{\prime}_1}$ .) Then, we have:

\begin{align*}v_{E_1}({\mathfrak{a}})=\sum_i e_i\cdot \deg_{v^{\prime}}(\textrm{in}_{v^{\prime}} f _i)=\deg_{v^{\prime}}(\alpha C+\sum_j \gamma_j C_j)>\deg_{v^{\prime}}C=\deg_{v^{\prime}}\ell\geq v^{\prime}_1v^{\prime}_3.\end{align*}

By the assumption $a(E_1;\, A,{\mathfrak{a}})> a(E;\, A,{\mathfrak{a}})\geq 0$ , it follows

(4) \begin{equation} 0\leq a(E_1;\, A, {\mathfrak{a}})=2v^{\prime}_1+v^{\prime}_3-v_{E_1}({\mathfrak{a}})<2v^{\prime}_1+v^{\prime}_3-v^{\prime}_1\cdot v^{\prime}_3.\end{equation}

The possibilities of $(v^{\prime}_1,v^{\prime}_1,v^{\prime}_3)$ are only (1, 1, n) with $n\in {\Bbb N}$ and (2, 2, 3). In case (2, 2, 3), by (4) we have $a(E_1;\, A, {\mathfrak{a}}))<2\cdot2+3-2\cdot3=1$ . Then, in this case we have (i) and (b) of (ii).

In case (1, 1, n) for $n\in {\Bbb N}$ , we have $\deg_{v^{\prime}}\ell\geq n+1$ . Indeed, if not, we have $\deg_{v^{\prime}}\ell=n$ and $\ell=X_3+h(X_1,X_2)$ for a nonzero homogeneous polynomial h of degree n. As E has the center at the curve $\ell=0$ , in the same way as the proof of (2) we have

\begin{align*}v_E(x_3+h(x_1,x_2))>v_E(x_3),\end{align*}

and also $x_3+h(x_1,x_2)\in {\mathfrak{m}}_0\setminus {\mathfrak{m}}_0^2$ which is a contradiction to the maximality of $v_E(x_3)$ . Therefore, in this case also we have $a(E_1;\, A, {\mathfrak{a}}))<2+n-(n+1)=1$ , which shows (i) and (a) of (ii).

Case 2. $\dim \overline{\{P\}}=0$

We can take $P=(1:\; a:\; b)\in E_1={\Bbb P}(v')$ $(a,b\neq 0)$ as the homogeneous coordinate of the point P by Lemma 5·1.

First we will show that $v^{\prime}_1\neq 1$ . To see this, assume that $v^{\prime}_1=1$ . Then a curve $bX_1^{v^{\prime}_3}-X_3=0$ contains P, therefore

\begin{align*}v_E(bx_1^{v^{\prime}_3}-x_3)>v_E(x_3)=v_3,\end{align*}

and also $bx_1^{v^{\prime}_3}-x_3\in {\mathfrak{m}}_0\setminus {\mathfrak{m}}_0^2$ which is a contradiction to the maximality of $v_E(x_3)$ .

Now we may assume that $v^{\prime}_1\geq 2$ . Then, of course $v^{\prime}_1< v^{\prime}_3$ and the curve B defined by $aX_1-X_2=0$ contains P. Note that B is the bad curve.

Take a general element $f^e=f_1^{e_1}\cdots f_r^{e_r}\in {\mathfrak{a}}$ such that $v_{E_1}({\mathfrak{a}})=v_{E_1}(f^e)=\deg_{v^{\prime}}(\textrm{in}_{v^{\prime}} f^e)$ . The ${\Bbb R}$ -divisor on $E_1={\Bbb P}(v')$ induced from a general element $f^e=f_1^{e_1}\cdots f_r^{e_r}$ is expressed as follows:

(5) \begin{equation} \left(\prod \textrm{in}_{v^{\prime}}f_i^{e_i}\right)=\alpha B +\sum_j \gamma_j C_j, \ \ \mbox{with}\ \ \alpha , \gamma_i \in {\Bbb R}_{>0}.\end{equation}

By generality of ${\mathfrak{a}}$ , we have $\alpha\leq 1$ . By Lemma 5·2, we have ${\textrm{mld}}(P;\, E_1, {\mathfrak{a}}_{A_1}{\mathcal O}_{E_1})=-\infty$ . By the description (5) of the divisor defined by a general element $f^e$ , we have

\begin{align*}-\infty={\textrm{mld}}(P;\, E_1, {\mathfrak{a}}_{A_1}{\mathcal O}_{E_1}) & ={\textrm{mld}}(P;E_1,I_B^\alpha\cdot\prod_iI_{C_i}^{\gamma_i})\geq {\textrm{mld}}(P;E_1,I_B\cdot\prod_iI_{C_i}^{\gamma_i})\\& ={\textrm{mld}} (P;\, B, (\prod_iI_{C_i}^{\gamma_i}){\mathcal O}_B).\end{align*}

Hence, it follows ${\textrm{ord}}_P (\prod_iI_{C_i}^{\gamma_i}){\mathcal O}_B>1$ . Applying Lemma 3·4 to the curve B of degree $v^{\prime}_1$ , we obtain

\begin{align*}1<{\textrm{ord}}_P (\prod_iI_{C_i}^{\gamma_i}){\mathcal O}_B\leq \frac{\sum \gamma_i \deg_{v^{\prime}}C_i}{v^{\prime}_1v^{\prime}_3}\leq \frac{v_{E_1}(f^e)}{v^{\prime}_1v^{\prime}_3}\leq \frac{2v^{\prime}_1+v^{\prime}_3}{v^{\prime}_1v^{\prime}_3},\end{align*}

Here, for the third inequality, we use

\begin{align*}\sum\gamma_i\deg_{v^{\prime}}C_i\leq v_{E_1}(f^e)-\alpha v_1'.\end{align*}

Then, the only possibility of v ^′ satisfying these inequalities is (2, 2, 3) and we also have $v_{E_1}({\mathfrak{a}})=v_{E_1}(f^e)> 2\cdot 3$ which completes the proof of (i) and (ii) in case $\dim \overline{\{P\}}=0$ .

Proof. As $a(E_1;\, A,{\mathfrak{a}})\geq {\textrm{mld}}(0;\, A,{\mathfrak{a}})\geq 1$ , the inequality $a(E_1;\, A, {\mathfrak{a}})> a(E;\, A, {\mathfrak{a}})$ does not hold by (i) in Lemma 5·3.

Proof of Theorem 1·8. Let $A_1$ , $E_1$ be as in the setting above. Assuming $0\leq a(E;\, A,{\mathfrak{a}})< a(E_1;\, A,{\mathfrak{a}})$ , we will prove that $a(E;\, A,{\mathfrak{a}})\geq a(E_2;\, A,{\mathfrak{a}})$ for a divisor $E_2$ obtained by the second “blow-up” constructed below in Case 1 and Case 2.

Let $P\in E_1\subset A_1$ be the center of E. First, for every prime divisor D over $A_1$ with the center at P and with the inequality $a(D;\, A,{\mathfrak{a}})>a(E;\, A,{\mathfrak{a}})\geq0$ , we observe that

(6) \begin{equation} a(D;\, A_1, {\mathfrak{a}}_{A_1})\geq 0. \end{equation}

Indeed, we have an expression of $a(D;\, A,{\mathfrak{a}})$ as follows:

\begin{align*}a(D;\, A,{\mathfrak{a}})=a(D;\, A_1,{\mathfrak{a}}_{A_1})+v_D(E_1)(a(E_1;\, A,{\mathfrak{a}})-1).\end{align*}

As $a(D;\, A,{\mathfrak{a}})\geq0$ and $a(E_1;\, A,{\mathfrak{a}})-1<0$ (Lemma 5·3), we have $a(D;\, A_1,{\mathfrak{a}}_{A_1})\geq0$ .

Case 1. $\dim \overline{\{P\}}=1$

Let $\{y_1,y_2\}$ be a squeezed system for E on $A_1$ at P and $E_2$ the prime divisor obtained by the squeezed blow-up of $A_1$ at P with respect to $\{y_1,y_2\}$ . Let $K\;:\!=\,{\mathcal O}_{A_1,P}/{\mathfrak{m}}_{A_1,P}$ and ${\overline{K}}$ the algebraic closure of K. Let $A_{1K}\;:\!=\,{\textrm{Spec }} \widehat{\mathcal O}_{A,P}$ , $A_{1{\overline{K}}}\;:\!=\,{\textrm{Spec }} {\overline{K}}\widehat{\mathcal O}_{A,P}= {\textrm{Spec }} {\overline{K}}[[y_1,y_2]]$ . Denote the both closed points of $A_{1K}$ and of $A_{1{\overline{K}}}$ by 0. Here, we note that $\{y_1,y_2\}$ is not necessarily a squeezed system on $A_{1{\overline{K}}}$ for ${\overline{E}}$ as is shown in Example 4·6, but it does not matter. Because we are interested only in ideals which came from $A_1$ and in this case a squeezed system on $A_1$ for E works in the same way as in [ Reference Kawakita9 ] and [ Reference Ishii6 ], which one can see below:

Let $\tilde A \longrightarrow A_1$ be a log resolution of $(A_1, {\mathfrak{a}}{\mathcal O}_{A_1})$ on which E appears. Then, the base change $\tilde{\tilde A}\longrightarrow A_{1{\overline{K}}}$ by $A_{1{\overline{K}}}\longrightarrow A_1$ is also a log resolution of $(A_{1{\overline{K}}}, {\mathfrak{a}}{\mathcal O}_{A_{1{\overline{K}}}})$ on which the prime divisor ${\overline{E}}$ corresponding to E appears. Let $A_2\longrightarrow A_1$ be the squeezed blow-up with respect to the squeezed system $\{y_1,y_2\}$ and $E_2$ the exceptional divisor. By definition, it means that $A_{2{\overline{K}}}\longrightarrow A_{1{\overline{K}}}$ is squeezed weighted blow-up with respect to the squeezed system $\{y_1,y_2\}$ and ${\overline{E}}_2$ be the exceptional divisor corresponding to $E_2$ .

If ${\overline{E}}={\overline{E}}_2$ , then we have $E=E_2$ and we are done. So, we may assume that the center of ${\overline{E}}$ on $A_{2{\overline{K}}}$ is a point. Then the center $Q\in A_{2{\overline{K}}}$ is not on the proper transform of ${\overline{E}}_1$ on $A_{2{\overline{K}}}$ . This is proved as follows:

Let $w=(r,s)$ be the weight of the squeezed system $\{y_1,y_2\}$ on $A_{1}$ .

First, we show that $r=s$ does not happen. Assume $r=s$ , i.e., $w=(1,1)$ , then we can take an expression $Q=(a,b)$ of $Q\in {\overline{E}}_2={\Bbb P}_{\overline{K}}^1$ by homogeneous coordinates with $a, b\neq 0$ . Let $z\;:\!=\,by_1-ay_2\in {\mathcal O}_{A_{1{\overline{K}}}}$ . As Q is the center of ${\overline{E}}$ on ${\overline{E}}_2\subset A_{2{\overline{K}}}$ and satisfying $bY_1-aY_2=0$ ( $Y_1,Y_2$ are the homogeneous coordinates on $E_2={\Bbb P}_K^1$ corresponding to $y_1,y_2$ .), it follows

\begin{align*}z\in {\mathfrak{m}}_Q\setminus {\mathfrak{m}}_Q^2, \ \ \ \mbox{and}\ \ \ v_E(z)>v_E(y_1), v_E(y_2),\end{align*}

which is a contradiction to the fact that $\{y_1,y_2\}$ is a squeezed system. Now, we may assume that $r<s$ . Let $h=0$ be the defining equation of $E_1$ in $A_1$ around P, then ${\overline{E}}_1$ is also defined by $h=0$ and it is smooth at the closed point $0\in A_{1{\overline{K}}}$ . Therefore, we have ${\textrm{ord}}_{y_1,y_2}h=1$ . Then the initial part of h with respect to w is one of the following:

(1) $\textrm{in}_w(h)=y_1$ , (2) $\textrm{in}_w(h)=y_2$ , (3) $\textrm{in}_w(h)=y_2+a{ y_1}^d$ ( $a\in {\overline{K}}$ , $w_1d=w_2$ ). In the first two cases, ${\overline{E}}'_1\mid_{{\overline{E}}_2}$ is in the zero locus of the coordinate functions, where ${\overline{E}}'_1$ is the proper transform of ${\overline{E}}_1$ on $A_{2{\overline{K}}}$ . Therefore it does not contain the center Q of ${\overline{E}}$ by Lemma 5·1. In case (3), it follows $w=(1,d)$ . If Q is in ${\overline{E}}'_1\mid_{{\overline{E}}_2}$ , then we have $y^{\prime}_2\;:\!=\,y_2+a{ y_1}^d\in {\mathfrak{m}}_{A_{1{\overline{K}}},0}\setminus {\mathfrak{m}}_{A_{1{\overline{K}}},0}^2$ and $v_{\overline{E}}(y^{\prime}_2)>v_{\overline{E}}(y_2)$ which is a contradiction to the assumption that $\{y_1,y_2\}$ is a squeezed system. Now, in any case we obtain that $Q\not\in {\overline{E}}'_1$ .

On the other hand, $a(E;\, A,{\mathfrak{a}})$ has another expression as follows:

\begin{align*}a(E;\, A,{\mathfrak{a}})=k_{E/A_1}+k_{E_1/A}\cdot v_E(E_1)+1-v_E({\mathfrak{a}}).\end{align*}

It is sufficient to show that

\begin{align*}a({\overline{E}};\, A, {\mathfrak{a}})\geq a({\overline{E}}_2;\, A,{\mathfrak{a}}).\end{align*}

Assume contrary, then

(7) \begin{equation} 0> {\overline{a}}({\overline{E}};\, A, {\mathfrak{a}})- {\overline{a}}({\overline{E}}_2;\, A,{\mathfrak{a}})=a({\overline{E}};\, A_{2{\overline{K}}},I_{{\overline{E}}_2}\cdot{\mathfrak{a}}_{A_{2{\overline{K}}}})+(v_{{\overline{E}}}({\overline{E}}_2)-1)\cdot{\overline{a}}({\overline{E}}_2;\, A,{\mathfrak{a}}), \end{equation}

where ${\mathfrak{a}}_{A_{2{\overline{K}}}}$ is the weak transform of ${\mathfrak{a}}_{A_1}{\mathcal O}_{A_{1{\overline{K}}}}$ . For the calculation of (7), we used

1. (i) $v_{\overline{E}}({\overline{E}}_1)=v_{\overline{E}}({\overline{E}}_2)v_{{\overline{E}}_2}({\overline{E}}_1)+v_{\overline{E}}({\overline{E}}'_1)=v_{\overline{E}}({\overline{E}}_2)v_{{\overline{E}}_2}({\overline{E}}_1)$ .

Then the inequality (7) shows that $a({\overline{E}};\, A_{2{\overline{K}}},I_{{\overline{E}}_2}\cdot{\mathfrak{a}}_{A_{2{\overline{K}}}})<0$ which implies

\begin{align*}{\textrm{mld}}(Q;\, A_{2{\overline{K}}},I_{{\overline{E}}_2}\cdot{\mathfrak{a}}_{A_{2{\overline{K}}}})=-\infty.\end{align*}

Then, by Inversion of adjunction ([ Reference Ein and Mustaţă3 , Reference Ishii and Reguera7 ]), it follows

\begin{align*}{\textrm{mld}}(Q;\, {\overline{E}}_2, {\mathfrak{a}}_{A_{2{\overline{K}}}}\cdot{\mathcal O}_{{\overline{E}}_2})<0\end{align*}

which yields ${\textrm{ord}}_Q(({\mathfrak{a}}_{A_1}{\mathcal O}_{A_{1{\overline{K}}}})_{A_{2{\overline{K}}}}\cdot{\mathcal O}_{{\overline{E}}_2})={\textrm{ord}}_Q({\mathfrak{a}}_{A_{2{\overline{K}}}}\cdot{\mathcal O}_{{\overline{E}}_2})>1$ .

Let (r,s) be the squeezed weight for ${\overline{E}}$ at the closed point $0\in A_{1{\overline{K}}}$ , then

\begin{align*}a({\overline{E}}, A_{1{\overline{K}}},{\mathfrak{a}}_{A_{1{\overline{K}}}})=a(E;\, A_1, {\mathfrak{a}}_{A_1})\geq0,\end{align*}

where we the last inequality follows from (6). Now we reach the situation in Theorem 1·1 and apply the argument in ([ Reference Kawakita9 ]) for the surface pair $(A_{1{\overline{K}}},{\mathfrak{a}}_{A_{1{\overline{K}}}}) $ , we obtain

(8) \begin{equation}1<{\textrm{ord}}_Q(({\mathfrak{a}}_{A_1}{\mathcal O}_{A_{1{\overline{K}}}})_{A_{2{\overline{K}}}}\cdot{\mathcal O}_{{\overline{E}}_2})\leq \frac{v_{{\overline{E}}_2}({\mathfrak{a}}_{A_1}{\mathcal O}_{A_{1{\overline{K}}}})}{r\cdot s}\leq \frac{r+s}{r\cdot s},\end{equation}

where we note that ${\mathfrak{a}}_{A_{2{\overline{K}}}}=({\mathfrak{a}}_{A_1}{\mathcal O}_{A_{1{\overline{K}}}})_{A_{2{\overline{K}}}}$ and the third inequality follows from

\begin{align*}r+s-v_{{\overline{E}}_2}({\mathfrak{a}}_{A_1}{\mathcal O}_{A_{1{\overline{K}}}})=a({\overline{E}}_2;\, A_{1{\overline{K}}}, {\mathfrak{a}}_{A_1})= a(E_2;\, A_1, {\mathfrak{a}}_{A_1})\geq 0\end{align*}

by (6). The possible positive intergers $\{r,s\}$ satisfying (8) with $\gcd(r,s)=1$ are only $\{1,s\}$ . In this case let $z'\;:\!=\, y_1^s - cy_2$ , where $Q=(c,1)\in {\overline{E}}_2={\Bbb P}(1,s)$ , then $v_{\overline{E}}(z')> v_{\overline{E}}(y_2)$ , which is a contradiction to that $\{y_1, y_2\}$ is a squeezed system for ${\overline{E}}$ . Hence we obtain

\begin{align*}{\overline{a}}({\overline{E}};\, A, {\mathfrak{a}})\geq {\overline{a}}({\overline{E}}_2;\, A,{\mathfrak{a}}),\end{align*}

which completes the proof of the theorem for Case 1.

Case 2. $\dim\overline{\{ P\}}=0$

Since we are assuming $0\leq a(E;\, A,{\mathfrak{a}})<a(E_1;\, A,{\mathfrak{a}})$ , by Lemma 5·3 only possibility of v ^′ is (2, 2, 3) and we have $0\leq a(E_1;\, A,{\mathfrak{a}})<1$ .

Now take a squeezed blow-up $A_2 \longrightarrow A_1$ of weight $w=(w_1,w_2,w_3)$ at P and let $E_2$ be the exceptional divisor. We may assume that the condition (2) in Definition 4·9 holds. Let $Q\in E_2$ be the center of E on $A_2$ .

Let $E^{\prime}_1$ be the proper transform of $E_1$ on $A_2$ . Denote the defining ideals of $E^{\prime}_1$ and $E_2$ in $A_2$ by $I_{ E^{\prime}_1}$ and $I_{E_2}$ , respectively.

Then, we have the similar expansion of $a(E;\, A,{\mathfrak{a}})$ as in (3) as follows:

(9) \begin{equation} a(E;\, A,{\mathfrak{a}})= a(E;\, A_2, I_{E^{\prime}_1}\cdot I_{E_2}\cdot{\mathfrak{a}}_{A_2}) +v_E(E_2)a(E_2;\, A, {\mathfrak{a}})+v_E(E^{\prime}_1)a(E_1;\, A, {\mathfrak{a}}),\end{equation}

where ${\mathfrak{a}}_{A_2}$ is the weak transform of ${\mathfrak{a}}$ on $A_2$ and is also the weak transform of ${\mathfrak{a}}_{A_1}$ on $A_2$ .

Case 2·1. $\dim\overline{\{Q\}}=0$ :

We will prove $a(E_2;\, A,{\mathfrak{a}})\leq a(E;\, A,{\mathfrak{a}})$ . Assume on the contrary that $a(E_2;\, A,{\mathfrak{a}})> a(E;\, A,{\mathfrak{a}})$ . Then, by (9), we obtain

(10) \begin{equation} a(E;\, A_2, I_{E^{\prime}_1}\cdot I_{E_2}\cdot{\mathfrak{a}}_{A_2}) <0. \end{equation}

It implies that ${\textrm{mld}}(Q;\, A_2, I_{E^{\prime}_1}\cdot I_{E_2}\cdot{\mathfrak{a}}_{A_2})=-\infty$ . Let $ L\;:\!=\, E^{\prime}_1\cap E_2$ , by Inversion of adjunction, we obtain

\begin{align*}{\textrm{mld}}(Q;\, E_2, I_L{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2})<0.\end{align*}

Let B ^′ be the bad curve on $E_2$ (note that a bad curve exists in our case by Lemma 4·8). Then, we obtain

(11) \begin{equation} {\textrm{ord}}_{B^{\prime}}{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}\leq1. \end{equation}

Indeed, when $L=B'$ , then generality of ${\mathfrak{a}}$ implies that ${\textrm{ord}}_{B^{\prime}}{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}=0$ , as ${\textrm{ord}}_{B^{\prime}}I_L=1$ . On the other hand, when $L\neq B'$ , then $Q\not\in L$ and therefore generality implies ${\textrm{ord}}_{B^{\prime}}{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}\leq1$ . Now, in the same way as Case 2 in the proof of Lemma 5·3, we obtain that the weight of the second squeezed blow-up is (2, 2, 3).

We will show a contradiction under this situation. In this case, we have

(12) \begin{equation} v_{E_2}({\mathfrak{a}}_{A_1})\gt 6, \mbox{as well as }v_{E_1}({\mathfrak{a}})\gt6,\end{equation}

by applying (i) of Lemma 5·3 for $(A_1, {\mathfrak{a}}_{A_1})$ , $ E_2$ with the weight $w=(2,2,3)$ and also for $(A, {\mathfrak{a}})$ , $ E_1$ with the weight $v'=(2,2,3)$ . As the squeezed system $\{y_1,y_2,y_3\}$ at $P\in A_1$ has weight (2, 2, 3), it follows $v_{E_2}(f)\leq 3\cdot {\textrm{ord}}_P f$ for every $f\in {\mathfrak{a}}_{A_1}$ . Therefore we obtain

(13) \begin{equation} v_{E_2}({\mathfrak{a}}_{A_1})\leq 3\cdot {\textrm{ord}}_P{\mathfrak{a}}_{A_1} \leq 3\cdot {\textrm{ord}}_P{\mathfrak{a}}_{A_1}{\mathcal O}_{E_1}. \end{equation}

On the other hand, applying Lemma 3·4 to $E_1={\Bbb P}(2,2,3)$ and a general element of ${\mathfrak{a}}_{A_1}\cdot{\mathcal O}_{E_1}$ , we obtain $1<{\textrm{ord}}_P{\mathfrak{a}}_{A_1}{\mathcal O}_{E_1}\leq {v_{E_1}({\mathfrak{a}})}/{2\cdot 3}$ . Note that the first inequality follows from Lemma 5·2.

Then, it follows

(14) \begin{equation} 7=2+2+3=k_{E_1}+1\geq v_{E_1}({\mathfrak{a}})\geq 6\cdot {\textrm{ord}}_P{\mathfrak{a}}_{A_1}{\mathcal O}_{E_1}.\end{equation}

Using (12), (13) and (14) we obtain

\begin{align*}\frac{7}{2}>3\cdot {\textrm{ord}}_P{\mathfrak{a}}_{A_1}{\mathcal O}_{E_1}\geq v_{E_2}({\mathfrak{a}}_{A_1})>6\end{align*}

which is a contradiction. Therefore $a(E_2;\, A,{\mathfrak{a}})\leq a(E;\, A,{\mathfrak{a}})$ holds.

Case 2·2. $\dim \overline{\{Q\}}=1$ .

In the following, we will prove $a(E_2;\, A,{\mathfrak{a}})\leq a(E;\, A,{\mathfrak{a}})$ . Assume contrary, $a(E_2;\, A,{\mathfrak{a}})>a(E;\, A,{\mathfrak{a}})$ . The curve $\overline{\{Q\}}$ is not a bad curve, because if it is, then

\begin{align*}-\infty={\textrm{mld}}(Q;\, A_2, I_{E^{\prime}_1}\cdot I_{E_2}\cdot{\mathfrak{a}}_{A_2})={\textrm{mld}}(Q;\, E_2, I_L{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2})\end{align*}

implies ${\textrm{ord}}_QI_L{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}>1$ , while the generality of ${\mathfrak{a}}$ implies the converse inequality ${\textrm{ord}}_QI_L{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}={\textrm{ord}}_{B^{\prime}}I_L{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}\leq 1$ . We also have $\overline{\{Q\}}\neq L$ . This is proved as follows.

Let $h'\in {\mathcal O}_{A_1}$ define $E_1$ around P. As P is smooth on $E_1$ and also on $A_1$ , we have ${\textrm{ord}} h'=1$ with respect to RSP $\{y_1,y_2,y_3\}$ of ${\mathcal O}_{A_1}$ at P. Then, considering of the initial term of h ^′ with respect to the weight w, we see that one of the following holds:

1. (1) L is a coordinate axis of $E_2={\Bbb P}(w)$ ;

2. (2) L is defined by $Y_1+aY_2$ $(a\in k)$ in $E_2$ ;

3. (3) L is defined by $Y_3+f(Y_1,Y_2)$ in $E_2$ , where f is a homogeneous polynomial of degree d.

In the third case, the weight w must be (1, 1, d). In this case, if $\overline{\{Q\}}= L$ , it follows $y^{\prime}_3\;:\!=\,y_3+f(y_1,y_2)\in {\mathfrak{m}}_{A_1,P}\setminus {\mathfrak{m}}_{A_1,P}^2$ and $v_E(y^{\prime}_3)> v_E(y_3)$ , which is a contradiction to the maximality of $v_E(y_3)$ . In case (1), $\overline{\{Q\}}\neq L$ because Q is not contained in the coordinate axes (Lemma 5·1). In case (2), L becomes the bad curve, therefore $\overline{\{Q\}}\neq L$ , because $\overline{\{Q\}}$ is not the bad curve, as we saw above.

Now we obtain $Q\not\in E^{\prime}_1\cap E_2$ . By using this, we have

\begin{align*}{\textrm{mld}}(Q;\, A_2, I_{E_2}\cdot{\mathfrak{a}}_{A_2})={\textrm{mld}}(Q;\, A_2, I_{E^{\prime}_1}\cdot I_{E_2}\cdot{\mathfrak{a}}_{A_2})=-\infty.\end{align*}

By Inversion of adjunction, we have

\begin{align*}{\textrm{mld}}(Q;\,E_2,{\mathfrak{a}}_{A_2}{\mathcal O}_{E_2})=-\infty.\end{align*}

Then, we have $1< {\textrm{ord}}_Q{\mathfrak{a}}_{A_2}\cdot{\mathcal O}_{E_2}$

First we show that the squeezed weight $w=(r,r,s)$ for E at $P\in A_1$ is (1, 1, n) for $n\in {\Bbb N}$ . Let $C\;:\!=\,\overline{\{Q\}}$ be defined by $\ell=0$ in $E_2={\Bbb P}(r,r,s)$ . If $w\neq (1,1,n)$ , then the other possible weight w is (2, 2, 3). In this case the smallest possible value for the degree of $\ell$ on ${\Bbb P}(2, 2, 3)$ with respect to w is 6. Therefore, by $1< {\textrm{ord}}_Q{\mathfrak{a}}_{A_2}\cdot{\mathcal O}_{E_2}$ ,

\begin{align*}v_{E_2}({\mathfrak{a}}_{A_1})\geq \deg_w\ell\cdot{\textrm{ord}}_Q({\mathfrak{a}}_{A_1})_{A_2}\geq 6\cdot{\textrm{ord}}_Q({\mathfrak{a}}_{A_1})_{A_2}>6.\end{align*}

Now we obtain the inequality (12). The inequalities (13) and (14) also hold in the present case. Therefore, we induce a contradiction and w must be (1, 1, n). By Lemma 5·3, $deg_{w}\ell=1+n$ .

Let $\{y_1,y_2,y_3\} $ be a squeezed system at $P\in A_1$ with the weight (1, 1, n). Let $\{Y_1,Y_2,Y_3\}$ be the homogeneous coordinates of $E_2={\Bbb P}(1,1,n)$ corresponding to $\{y_1,y_2,y_3\} $ . As $\ell$ is irreducible of degree $1+n$ with respect to the weight (1, 1, n), we can express

\begin{align*}\ell=Y_1Y_3-Y_2^{n+1}.\end{align*}

For simplicity, assume ${\mathfrak{a}}={\mathfrak{a}}_1^{e_1}$ and take a general element $f\in {\mathfrak{a}}_1{\mathcal O}_{A, 0}\subset k[[x_1,x_2,x_3]]$ , where $\{x_1,x_2,x_3\}$ is a squeezed system for E at $0\in A$ of weight (2, 2, 3). Then the weak transform $f_{A_1}$ of f on $A_1$ is written as

(15) \begin{equation} f_{A_1}=(y_1\cdot y_3-y_2^{n+1})^r\cdot \ell' + g(y), \end{equation}

where $\ell'$ is weighted homogeneous and g(y) is the term with the higher weight with respect to the weight $w=(1,1,n)$ .

Here, we may assume that $P=(1,1,1)\in E_1={\Bbb P}(2,2,3)$ , then we can take a RSP at $P\in A_1$ by making use of the squeezed system $\{x_1,x_2,x_3\}$ of squeezed weight (2, 2, 3) which gives the first weighted blow-up $\varphi_1\;:\;A_1\longrightarrow A$ :

\begin{align*}z_1=\frac{x_1^3-x_3^2}{x_3^2}, \ \ \ z_2=\frac{x_2^3-x_3^2}{x_3^2}, \ \ \ z_3=x_3,\end{align*}

where $x_3 $ defines $E_1$ in the neighborhood of P. Take the minimal $m\in {\Bbb N}$ such that

(16) \begin{equation}f=x_3^m\cdot f_{A_1}\in {\mathcal O}_{A,0}\subset k[[x_1,x_2,x_3]].\end{equation}

We note that for $m\geq 2$ ,

(17) \begin{equation} {\textrm{ord}}_0 x_3^m\cdot z_i=m\ \ (i=1,2),\ \ \ \ \ \ \ {\textrm{ord}}_0 x_3^m\cdot z_3=m+1,\end{equation}

where ${\textrm{ord}}_0$ is the order with respect to the parameters $x_1,x_2,x_3 $ in ${\mathcal O}_{A,0}$ . Then, by (17),

\begin{align*}{\textrm{ord}}_0 f={\textrm{ord}}_0 (x_3^m\cdot f_A)\geq m.\end{align*}

On the other hand if $x_3^s(y_1y_3-y_2^{n+1})^r\in {\mathcal O}_{A,0}$ , it should be $s\geq 4r$ . In fact, if a quadratic monomial $z_iz_j$ $(i, j\in\{1,2\})$ appears in $y_1y_3$ which is expressed as a function of $z_1,z_2,z_3$ , then $s\geq 4r$ . If such a monomial $z_iz_j$ $(i, j\in\{1,2\})$ does not appear in $y_1y_3$ , then $z_i$ $(i<3)$ appears in $y_2$ , because $\{z_1,z_2,z_3\}$ and $\{y_1,y_2,y_3\}$ are both RSP at $P\in A_1$ . This yields $s\geq 2(n+1)r\geq4r$ .

Consider the initial part $(y_1\cdot y_3-y_2^{n+1})^r\cdot \ell'$ of $f_{A_1}$ with respect to the weight $w=(1,1,n)$ . We know that $a(E_2;\, A_1,{\mathfrak{a}}_{A_1})\geq 0$ , therefore $v_{E_2}(f_{A_1}^{e_1})=v_{E_2}({\mathfrak{a}}_{A_1}^{e_1}) \leq k_{E_2/A_1}+1=n+2$ . Then, it follows that

(18) \begin{equation} e_1(r(n+1)+\deg_{w}\ell')\leq n+2.\end{equation}

As $1<{\textrm{ord}}_Q {\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}$ , it follows $1<{\textrm{ord}}_Q (y_1y_3-y_2^{n+1})^{re_1}$ which yields $re_1>1$ . By this and (18), we have $\deg_{w}\ell'<r$ , therefore ${\textrm{ord}}_P \ell'< r$ which yields that the factor of $z_3(=x_3)$ appears in $\ell'$ at most $r-1$ times. Hence, as (16) the inclusion $x_3^m (y_1\cdot y_3-y_2^{n+1})^r\cdot \ell' \in {\mathcal O}_{A,0}$ should hold, which implies $m\geq 4r-(r-1)=3r+1$ .

Then, ${\textrm{ord}}_0 f={\textrm{ord}}_0 (x_3^m\cdot f_{A_1})\geq 3r+1,$ and therefore, taking $e_1r>1$ into account, we have

\begin{align*}{\textrm{ord}}_0{\mathfrak{a}}_1^{e_1}={\textrm{ord}}_0 f^{e_1}\geq e_1(3r+1)>3.\end{align*}

Then, for every prime divisor D over A with the center at 0 has the discrepancy $a(D;\,A,{\mathfrak{a}})<0$ , which is a contradiction to the condition that $a(E;\, A, {\mathfrak{a}})\geq 0$ .

Example 5·5. Let $f=(x_1-x_2)^2+x_3^2+x_1^4\in k[x_1,x_2,x_3]$ , $e=6/5$ and ${\mathfrak{a}}=(f)^e$ . Define E as follows:

Proof. When ${\textrm{mld}}(0;\, A,{\mathfrak{a}})\geq 0$ , then apply the theorem for a divisor E computing the mld. When ${\textrm{mld}}(0;\, A,{\mathfrak{a}})=-\infty$ , then in a similar way as in [ Reference Kawakita9 ], take a prime divisor E computing the mld. Then by taking a positive real number $t<1$ such that $a(E;\, A,{\mathfrak{a}}^t)=0$ and apply Theorem 1·8.

Proof. We can see that there is no bad curve on E. Therefore, every ${\Bbb R}$ -ideal ${\mathfrak{a}}$ is general for E.

Proof. Let $A'\longrightarrow A $ be the weighted blow-up with weight (r,s,t). By the assumption, we have $a(E;\, A,{\mathfrak{a}})< a(E';\, A,{\mathfrak{a}})$ . Then, by Lemma 5·2, we have $\alpha\;:\!=\,{\textrm{ord}}_P{\mathfrak{a}}_{A^{\prime}}{\mathcal O}_{E'}>1$ , where P is the generic point of C. Therefore, we obtain $v_{E^{\prime}}({\mathfrak{a}})=\alpha d> r+s+t-1$ , and therefore $a(E';\, A,{\mathfrak{a}})<1$ . Now, in the same way as Case 1 in the proof of Theorem 1·8, we obtain that the squeezed blow-up at P gives a divisor F satisfying $a(F;\, A,{\mathfrak{a}})\leq a(E;\, A,{\mathfrak{a}})={\textrm{mld}}(0;\, A,{\mathfrak{a}})$ .