A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds

Abstract

We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general” ${\Bbb R}$-ideal. We show that the minimal log discrepancy (“mld” for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţă–Nakamura’s conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.

1. Introduction

Throughout this paper, the base field k of varieties is an algebraically closed field of arbitrary characteristic. We study pairs $(A,{\mathfrak{a}})$ consisting of a smooth variety A of dimension $N>1$ and an “ ${\Bbb R}$ -ideal” ${\mathfrak{a}}$ which means ${\mathfrak{a}}={\mathfrak{a}}_1^{e_1}\cdots{\mathfrak{a}}_r^{e_r}$ , where ${\mathfrak{a}}_i$ ’s are non-zero coherent ideal sheaves on A and $e=(e_1,\ldots,e_r)\in {{\Bbb R}^r_{>0}}$ . We fix a closed point $0\in A$ .

The minimal log discrepancy (“mld” for short) ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ is an important invariant to measure the singularity of the pair $(A,{\mathfrak{a}})$ at 0 and plays important roles in birational geometry. We consider every prime divisor over A with the center at 0 and construct a “good model” of the divisor to approximate the mld. The prototype is as follows:

Theorem 1·1 ([ 9 , 6 ]). Assume $N=2$ . For every prime divisor E over A with the center at 0, there exists a prime divisor F obtained by one weighted blow-up with the center at 0 satisfying

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

for every ${\Bbb R}$ -ideal ${\mathfrak{a}}$ such that $a(E;\, A,{\mathfrak{a}})\geq0$ .

The inequality in the theorem implies that F is a better divisor to approximate the ${\textrm{mld}}$ . Therefore the theorem states that every prime divisor over A with the center at 0 has a better divisor which is obtained in a simple procedure. Here, we note that F is constructed from E and does not depend on the choice of an ${\Bbb R}$ -ideal ${\mathfrak{a}}$ .

Actually, in the paper [ 9 ] and [ 6 ], the main theorem is not stated in this form, but its proof shows Theorem 1·1. The paper [ 9 ] is for ${\textrm{char }} k=0$ , and the paper [ 6 ] is for ${\textrm{char }} k=p>0$ and the main statements of both papers are in the following form:

Corollary 1·2 ([ 9 , 6 ]). Assume $N=2$ . Then, for every pair $(A,{\mathfrak{a}})$ , the minimal log discrepancy ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ is computed by a prime divisor obtained by one weighted blow-up.

The corollary follows from the theorem immediately. See, for example, the proof of Corollary 1·9 in Section 5.

When we consider the case $N=3$ , we can see that one weighted blow-up is not sufficient to obtain a prime divisor computing the mld (see Example 3·3). On the other hand, in the example we can also show that the mld is computed by a prime divisor obtained by two weighted blow-ups. So it is natural to expect the following conjecture:

Conjecture 1·3. Assume $N\geq3$ . For every prime divisor E over A with the center at 0, there exists a prime divisor F centered at 0 obtained by at most $N-1$ weighted blow-ups satisfying

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

for every ${\Bbb R}$ -ideal ${\mathfrak{a}}$ such that $a(E;\, A,{\mathfrak{a}})\geq0$ .

As an immediate consequence of the conjecture, we obtain the following:

Conjecture 1·4 (Corollary of Conjecture 1·3). Assume $N\geq3$ . Then, for every pair $(A,{\mathfrak{a}})$ , the minimal log discrepancy ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ is computed by a prime divisor obtained by at most $N-1$ weighted blow-ups.

One of the motivations of the conjectures is that it is considered as a “weighted blow-up version” of Mustaţă–Nakamura Conjecture (MN-Conjecture for short):

Conjecture 1·5 (MN-Conjecture [ 13 ].) Fix N and the exponent e of ${\Bbb R}$ -ideals. Then, there exists a number $\ell_{N,e}\in {\Bbb N}$ depending only on N and e such that for any ${\Bbb R}$ -ideal ${\mathfrak{a}}$ with the exponent e the minimal log discrepancy ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ is computed by a prime divisor obtained by at most $\ell_{N,e}$ times blow-ups. Here, the blow-up means the “usual blow-up”, i.e., blow-up with the center at an irreducible reduced closed subset.

If this conjecture holds, then ACC Conjecture for these pairs holds ([ 13 ]), so it seems to be a significant conjecture. On the other hand, MN-Conjecture is equivalent to a reasonable conjecture on arc spaces ([ 5 ]), so it makes sense to study it.

Note that MN-Conjecture requires to fix an exponent e, while the weighted blow-up versions (Conjecture 1·3, 1·4) do not require it. Assume Conjecture 1·3 holds, it is also an interesting question whether the weights of the blow-ups can be bound uniformly in terms of exponents. This will strengthen the MN-Conjecture.

Another motivation of Conjecture 1·3 is for the project to bridge between positive characteristic and characteristic 0 ([ 5 ]). In [ 5 ], we have:

Lemma 1·6. Let ${\mathfrak{a}}$ be an ${\Bbb R}$ -ideal on a smooth variety $A_k$ over k ( ${\textrm{char }} k=p>0)$ and E a prime divisor over $(A_k,0_k)$ computing ${\textrm{mld}}(0_k;\, A_k,{\mathfrak{a}})$ .

If there exists an ${\Bbb R}$ -ideal $\widetilde{\mathfrak{a}}$ on a smooth variety $A_{\Bbb C}$ over ${\Bbb C}$ and a prime divisor $\widetilde{E} $ over $(A_{\Bbb C}, 0_{\Bbb C})$ , where $0_{\Bbb C}\in A_{\Bbb C}$ such that

1. 1. $\widetilde{\mathfrak{a}} ({\textrm{mod}}\ p)={\mathfrak{a}}$ (see [ 5 ] for the definition of (mod p))

2. 2. $a(\widetilde{E};\, A_{\Bbb C}, \widetilde{\mathfrak{a}})\leq a(E;\, A_k, {\mathfrak{a}})$ ,

then, ${\textrm{mld}}(0_{\Bbb C};\, A_{\Bbb C}, \widetilde{\mathfrak{a}})={\textrm{mld}}(0_k;\, A_k, {\mathfrak{a}})$ .

Remark 1·7. In particular, if such $\widetilde{\mathfrak{a}}$ and ${\widetilde{E}}$ exist for every ${\mathfrak{a}}$ and E and assume that ${\textrm{mld}}(0_k;\, A_k,{\mathfrak{a}})$ is computed by a divisor, then the set of ${\textrm{mld}}(0_k;\, A_k, {\mathfrak{a}})$ ’s is contained in the set of ${\textrm{mld}}(0_{\Bbb C};\, A_{\Bbb C}, {\mathfrak{b}})$ ’s. Therefore, if we fix the exponent e and the dimension N of $A_k$ , then the number of the values $\Lambda_e\;:\!=\,\{{\textrm{mld}}(0_k, A_k, {\mathfrak{a}}) \mid {\mathfrak{a}} \ \mbox{is} \, {a} \ $ R- $\mbox{ ideal} \, \mbox{with} \, \mbox{the exponent} \, {e}\}$ is finite for ${\textrm{char }} k>0$ , because it is proved to be finite in characteristic 0 by [ 8 ]. Similarly, if ACC holds in characteristic 0, then it also holds in positive characteristic.

Now, the problem is to construct appropriate $\widetilde{E}$ and $\widetilde{\mathfrak{a}}$ for given E and ${\mathfrak{a}}$ . If Conjecture 1·3 holds, we can reduce this problem to a divisor F of special type (i.e., obtained by at most $N-1$ weighted blow-ups), which seems easier to handle.

The main results of this paper are the following:

Theorem 1·8. Assume $N=3$ . For every prime divisor E over A with the center at 0, there exists a prime divisor F centered at 0 obtained by at most two weighted blow-ups satisfying

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

for every “general” ${\Bbb R}$ -ideal ${\mathfrak{a}}$ for E such that $a(E;\, A,{\mathfrak{a}})\geq0$ .

The terminology “general” will be defined in Definition 4·9. The weighted blow-ups will be constructed by “squeezed” blow-ups (see, Definition 4·4) depending only on E and it works for every general ideal. Here, “general” is necessary, because there exists an example of non-general ideal such that two squeezed blow-ups do not give the required divisor in the theorem (cf. Example 5·5). But it does not give a counter example for Conjecture 1·3, indeed for the example there exists another sequence of weighted blow-ups to obtain the required divisor (see, also Example 5·5).

As a corollary we obtain:

Corollary 1·9. Assume $N=3$ . Then, for every pair $(A,{\mathfrak{a}})$ with a “general” ${\Bbb R}$ -ideal ${\mathfrak{a}}$ , the minimal log discrepancy ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ is computed by a prime divisor obtained by at most two weighted blow-ups.

It is known as the Zariski’s sequence that every prime divisor E over A with the center at 0 is obtained by successive usual blow-ups from A, such that the centers of blow-ups are the center of E on each step ([ 11 , VI, 1·3]). The following corollary shows that in some cases, we obtain the two weighted blow-ups to compute the mld by just looking at the center of the second blow-up in the Zariski’s sequence.

Corollary 1·10 (Corollary 5·9). Assume $N=3$ . Let E be a prime divisor over A computing ${\textrm{mld}}(0;\, A, {\mathfrak{a}})$ for a pair $(A,{\mathfrak{a}})$ . Let $A_1\longrightarrow A$ be the first usual blow-up with the center at 0 in the Zariski’s sequence. Assume that the center $C\subset A_1$ of E is a curve of degree $\geq 2$ in the exceptional divisor $E_1\simeq {\Bbb P}^2$ . Then a weighted blow-up which is called “squeezed blow-up” at C gives a divisor computing ${\textrm{mld}}(0;\, A, {\mathfrak{a}})$ .

Note that in this case the first blow-up is also a squeezed blow-up. Example 3·3 is just in this case. In Section 5, we show a more general corollary. On the other hand, if we restrict to the case ${\textrm{mld}}\geq 1$ , then we have the following:

Theorem 1·11. Assume $N=3$ . Then, for every general pair $(A,{\mathfrak{a}})$ with ${\textrm{mld}}(0;\, A,{\mathfrak{a}})\geq 1$ , the minimal log discrepancy is computed by a prime divisor obtained by one weighted blow-up.

Corollary 1·12. Assume $N=3$ . In

\begin{align*}\Lambda=\{(A,{\mathfrak{a}}) \mid \ {\textrm{mld}}(0;\, A,{\mathfrak{a}})\geq 1\ \textit{with general }\ {\mathfrak{a}} \}\end{align*}

the Mustaţă–Nakamura Conjecture holds and also the ACC Conjecture holds for ${\textrm{char }} k\geq 0$ . Here, ACC Conjecture means that the set of ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ for the pairs in the subset $\Lambda_J\subset \Lambda$ consisting of ${\Bbb R}$ -pairs with the exponents in $J\subset {\Bbb R}_{>0}$ satisfies the Ascending Chain Condition. Here, J is a DCC set.

The corollary follows from Theorem 1·11 in the same way as in the proof of [ 6 , corollary 1·6], since the mld is computed by one weighted blow-up.

This paper is organised as follows: in Section 2 we prepare basic terminologies which will be used in this paper. In Section 3 we discuss about weighted blow-up at a (not necessarily closed) smooth point and basic formula on weighted projective space, that is the exceptional divisor appearing in a weighted blow-up. In Section 4 we construct an appropriate regular system of parameter (RSP for short) with the weight, in order to make a weighted blow-up. In Section 5 we give the proofs of the main results.

2. Preliminaries

Let A be an N-dimensional smooth variety defined over an algebraically closed field k. We fix a closed point $0\in A$ .

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 ^′.

Let ${\mathfrak{a}}$ be an ${\Bbb R}$ -ideal on A as in the beginning of the first section and $e_i$ ’s are the exponents. The log discrepancy of the pair $(A,{\mathfrak{a}})$ at E is defined as

\begin{align*}a(E;\, A, {\mathfrak{a}})\;:\!=\,k_E-\sum_i e_iv_E({\mathfrak{a}}_i)+1\end{align*}

and the minimal log discrepancy of the pair at a closed point 0 is defined as

\begin{align*}{\textrm{mld}}(0;\, A, {\mathfrak{a}})\;:\!=\,\inf \{a(E;\, A, {\mathfrak{a}})\mid E \ \mbox{prime divisor over }A \mbox{ with the center at}\ 0\}\end{align*}

It is known that for $N\geq 2$ , either ${\textrm{mld}}(0;\, A,{\mathfrak{a}})\geq 0$ or ${\textrm{mld}}(0;\, A,{\mathfrak{a}})=-\infty$ holds. For $N=1$ , we define ${\textrm{mld}}(0;\, A, {\mathfrak{a}})=-\infty$ if the left-hand side is negative, by abuse of notation, because it is convenient to describe the Inversion of adjunction.

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 $-\infty$ ).

Remark 2·4. Assume there exists a log resolution of the pair $(A, {\mathfrak{a}}{\mathfrak{m}}_0)$ , where ${\mathfrak{m}}_0$ is the maximal ideal defining $0\in A$ . If ${\textrm{mld}}(0;\, A,{\mathfrak{a}})\geq 0$ , then, on every such resolution there is a prime divisor computing ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ . If ${\textrm{mld}}(0;\, A,{\mathfrak{a}})=-\infty$ and $Z({\mathfrak{a}})\subset A$ contains an irreducible component of codimension one, there may not exist a prime divisor computing the mld among the exceptional divisors appearing in a given log resolution (cf. [ 3 , proposition 7·2]). But in this case, if we construct an appropriate log resolution of $(A, {\mathfrak{a}}{\mathfrak{m}}_0)$ by taking more blowing-ups from the given one, a prime divisor computing ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ appears on that. Therefore, for ${\textrm{char }} k=0$ or $N\leq 3$ , every pair $(A, {\mathfrak{a}})$ has a prime divisor computing ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ , since there is a log resolution for every pair.

3. Weighted blow-ups and weighted projective spaces

In this section A is always a smooth variety of dimension $N\geq2$ defined over an algebraically closed field k and $P\in A$ is a (not necessarily closed) point.

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.

More generally, let $P\in A$ be a smooth point with the not-necessarily-algebraically closed residue field K. Let ${\overline{K}}$ be the algebraic closure of the residue field of ${\mathcal O}_{A,P}$ . A weighted blow-up of A at the point P is the canonical morphism induced from a weighted blow-up $\overline{A}\longrightarrow {\textrm{Spec }} {\overline{K}}\widehat{\mathcal O}_{A,P}$ for some RSP $x_1,\ldots, x_c$ of ${\overline{K}}\widehat{\mathcal O}_{A,P}$ with $wt_w(x_1,\ldots, x_c)=(w_1,\ldots, w_c)$ for some $ (w_1,\ldots, w_c)\in {\Bbb Z}^c_{>0}$ , where ${\overline{K}}\widehat{\mathcal O}_{A,P}$ is the extension of the formal power series ring $\widehat{\mathcal O}_{A,P}$ over K to the one over ${\overline{K}}$ . Let ${\overline{E}}$ be the prime divisor obtained by the weighted blow-up $\overline{A}\longrightarrow {\textrm{Spec }} {\overline{K}}\widehat{\mathcal O}_{A,P}$ . The prime divisor E over A with the center at P corresponding to ${\overline{E}}$ is called a prime divisor obtained by a weighted blow-up of A at P. Note that if ${\overline{E}} $ gives a valuation $\overline v$ and the valuation ring ${\mathcal O}_{\overline v}$ , the prime divisor E corresponds to the valuation v whose valuation ring is $K(A)\cap {\mathcal O}_{\overline v}$ .

Note that weighted blow-ups are only defined at smooth points.

Here, we show a 3-dimensional example that the minimal log discrepancy is not computed by a divisor obtained by only one weighted blow-up, but computed by a divisor obtained by two weighted blow-ups.

The following are well known, for example see [ 10 , remark 2·6, lemma 2·7].

Remark 3·2. Let $P\in A$ be a point of a smooth variety with the residue field K.

1. (1) The set of prime divisors over A with the center at P corresponds bijectively to the set of prime divisors over $\widehat A\;:\!=\,{\textrm{Spec }} \widehat{\mathcal O}_{A,P}$ with the center at the closed point. Moreover, if prime divisors E and $\widehat E$ correspond under the above bijection, then for every ${\Bbb R}$ -ideal ${\mathfrak{a}}$ on A we have $v_E({\mathfrak{a}})=v_{\widehat E}({\mathfrak{a}})$ and also $a(E;\, A,{\mathfrak{a}})=a(\widehat E,\widehat A, {\mathfrak{a}}{\mathcal O}_{\widehat A})$ .

2. (2) Let $K'\supset K$ be a field extension and $A'\;:\!=\,{\textrm{Spec }} K'\widehat{\mathcal O}_{A,P}$ . Then, there is a surjective map from the set of prime divisors over A ^′ with the center at the closed point to the set of prime divisors over A with the center at P. If prime divisors E ^′ and E correspond by the above surjective map, then it follows $a(E';\, A', {\mathfrak{a}}{\mathcal O}_{A'})=a(E;\, A, {\mathfrak{a}})$ for every ${\Bbb R}$ -ideal ${\mathfrak{a}}$ on A.

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 ([ 12 , 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 ${\textrm{mld}}(0;\, A,{\mathfrak{a}})=0$

The following lemma for a weighted projective space with a special weight is used for our main results. The statement is easily generalised to higher dimensional case, but for simplicity of notation we state here only for 2-dimensional case.

Lemma 3·4. Let $r\leq s$ be positive integers such that $\gcd(r,s)=1$ . Let $g\in k[x_1,x_2,x_3]$ be a weighted homogeneous polynomial with respect to the weight $w=(w(x_1),w(x_2),w(x_3))=(r,r,s)$ and $Q\in {\Bbb P}_k(r,r,s)$ a closed point not contained in the coordinate planes, i.e., $Q\not\in (x_1\cdot x_2\cdot x_3=0)$ . Let $\ell\in k[x_1,x_2,x_3]$ be a weighted homogeneous polynomial of $\deg_w(\ell)=r$ such that $\ell(Q)=0$ . If $\ell \not| g$ , then it follows

\begin{align*}r\cdot s\cdot{\textrm{ord}}_Q(g)\leq r\cdot s\cdot{\textrm{ord}}_Q(g\mid _{L})\leq \deg_w g,\end{align*}

where $L\subset {\Bbb P}_k(r,r,s)$ is the divisor defined by $\ell=0$ in ${\Bbb P}_k(r,r,s)$ .

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}

In case ${\textrm{char }} k=0$ or ${\textrm{char }} k=p>0$ and $p\not| r\cdot s$ , the morphism $\pi$ is étale around Q. Therefore, $\pi^{-1}(Q)$ consists of $r^2\cdot s$ closed points $\{Q_i\mid {i=1, \ldots, r^2\cdot s}\}$ whose analytic neighbourhoods of $\pi^*G$ and $\pi^*L$ are isomorphic to those of G and L at Q, respectively. Then, by (1) we obtain

\begin{align*}r^2\cdot s\cdot{\textrm{ord}}_Q( g\mid_{L})=\sum_{i=1}^{r^2s}{\textrm{ord}}_{Q_i}(\pi^*g\mid _{\pi^*L})\leq \pi^*L\cdot \pi^*G=r\cdot \deg_w g,\end{align*}

which yields the required inequality.

In case $p | r$ , denote $r=p^e\cdot q$ ( $\gcd(p,q)=1$ ). Then, the fiber $\pi^{-1}(Q)$ consists of $q^2\cdot s$ closed points, as a topological space. For a closed point $Q_i$ ( $i=1,\ldots, q^2\cdot s$ ) in the fiber $\pi^{-1}(Q)$ we obtain

\begin{align*}{\mathfrak{m}}_{Q}{\mathcal O}_{{\Bbb P}^2}\subset {\mathfrak{m}}_{Q_i}^{p^e},\end{align*}

where ${\mathfrak{m}}_{Q}$ and ${\mathfrak{m}}_{Q_i}$ are the maximal ideals of $Q\in {\Bbb P}(r,r,s)$ and of $Q_i\in {\Bbb P}^2$ , respectively. Let $C\subset {\Bbb P}^2$ be the subscheme with the reduced structure of $\pi^*L$ . Then, we have

\begin{align*}{\mathfrak{m}}_{L, Q}{\mathcal O}_C\subset {\mathfrak{m}}_{C,Q_i}^{p^e},\end{align*}

where ${\mathfrak{m}}_{L,Q}$ and ${\mathfrak{m}}_{C,Q_i}$ are the maximal ideals of $Q\in L$ and of $Q_i\in C$ , respectively. Therefore, for every $i=1,\ldots, q^2\cdot s$ it follows

\begin{align*}p^e\cdot{\textrm{ord}}_Q (g\mid L)\leq {\textrm{ord}}_{Q_i}(\pi^*g)\mid _C.\end{align*}

Now, there are $q\cdot s$ points $Q_i$ lying on C. Then, by Bezout’s theorem on ${\Bbb P}^2$ for C and $\pi^*G$ , we obtain

\begin{align*}q\cdot s\cdot p^e{\textrm{ord}}_Q (g\mid _L)\leq q\cdot s\cdot{\textrm{ord}}_{Q_i} (\pi^*g)\mid_ C\leq C\cdot\pi^*G=\deg_w g.\end{align*}

Here noting that $q\cdot s\cdot p^e=r\cdot s$ , this is the required inequality.

In case $p | s$ , the proof is similar.

4. Squeezed systems and squeezed blow-ups

Let A be a variety of dimension $N\geq2$ over an algebraically closed field k.

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 ^′.

Note that the squeezed weight for E is determined by a prime divisor but squeezed system is not uniquely determined by the prime divisor E.

Remark 4·2. For every A, P and E as in Definition 4·1, there exists a squeezed system of ${\overline{K}}\widehat{\mathcal O}_{A,P}$ . Indeed, it is obvious that there is $x_1\in{\mathfrak{m}}\setminus {\mathfrak{m}}^2$ such that $v(x_1)$ is the minimal value among $\{v_E(x)\mid x\in {\mathfrak{m}}\setminus {\mathfrak{m}}^2\}$ . Existence of the maximal $v(x_c)$ among the set is proved by Zariski’s subspace theorem (cf. [ 1 , (10·6)]). Now, we extend $\{x_1,x_c\}$ to an RSP $\{x_1,x_2,\ldots,x_c\}$ of ${\mathcal O}_{A,P}$ . Here, if $v_E(x_i)> v_E(x_1)$ for $2\leq i \leq r-1$ , replace $x_i$ by $x_1+x_i$ . Then, we obtain a squeezed system $\{x_1,x_2,\ldots,x_c\}$ .

Actually in [ 9 ] and [ 6 ], the proofs of Theorem 1·1 show the following:

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.

Remark 4·5. As in the definitions, a squeezed system is a RSP in the local ring with extended coefficient field. A squeezed system is not in general a RSP of the original local ring ${\mathcal O}_{A,P}$ .

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 $v_E(f_1)=v_E(f_2)=1$ .

On the other hand, take the base change $\psi:\; A_{\overline{K}}\longrightarrow A_K$ by the field extension ${\overline{K}}\supset K$ . Let $z'\;:\!=\,y-az\in {\overline{K}}[[y,z]]$ . Then, the proper transform of the curve defined by $z'=0$ contains the point $(a:1)\in {\Bbb P}_{\overline{K}}^1={\overline{E}}_1$ where ${\overline{E}}_1$ is the exceptional divisor of the blow-up at the closed point of $A_{\overline{K}}$ . As $(a:1)\in {\overline{E}}_1$ satisfies $\Phi(y,z)=0$ , the proper transform of $z'=0$ intersects the center of the second blow-up induced from $\varphi_2$ . One can see that $v_E(z')>1$ , and therefore a squeezed system cannot be taken from K[[y,z]].

Now we are going to define “general” ideal.

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 $\{x_1,x_2,x_3\}$ .

An irreducible curve $B\subset E_1={\Bbb P}(v^{\prime}_1,v^{\prime}_2, v^{\prime}_3)$ with the following properties is called a bad curve for E on $E_1$ .

1. (1) B is a curve of degree $v^{\prime}_1$ with respect to $(v^{\prime}_1,v^{\prime}_2, v^{\prime}_3)$ . (In the discussions on a weighted projective space, “degree” always means degree with respect to $(v^{\prime}_1,v^{\prime}_2, v^{\prime}_3)$ , and it is sometimes denoted by $\deg_{v^{\prime}}$ .)

2. (2) B contains the center of E.

Lemma 4·8. Under the setting of Definition 4·7, the following hold:

1. (i) A bad curve does not always exist. More precisely a bad curve does not exist if and only if one of the following holds;

   1. (a) the squeezed weight is (1, 1, 1); or

   2. (b) the squeezed weight $(v^{\prime}_1,v^{\prime}_2,v^{\prime}_3)$ satisfies $v^{\prime}_1<v^{\prime}_3$ and the center of E on $A_1$ is a curve of $\deg_{v^{\prime}}\ >v^{\prime}_1$ on $E_1\simeq {\Bbb P}(v^{\prime}_1,v^{\prime}_2,v^{\prime}_3)$ ; or

   3. (c) $E=E_1$ .

2. (ii) If a bad curve exists, then it is unique in $E_1$ .

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 [ 9 ]).

Remark 4·10. In (2), we assume smoothness of the center P of E on $A_1$ . But it turns out that it always holds by Lemma 5·1.

Remark 4·11. The definition of generality of an ${\Bbb R}$ -ideal is rather complicated. However, one can see that under a fixed exponent, the inequalities of orders at specific curves of $E_1$ and $E_2$ are open conditions in the space of regular functions of A, which is the reason why we call the ideal ${\mathfrak{a}}$ “general”. The following gives a sufficient condition for generality of the ideal.

Under the same symbols as in Definition 4·9, the ${\Bbb R}$ -ideal ${\mathfrak{a}}$ is general for E if one of the following hold:

1. (1) there is no bad curve on $E_1$ or $E_2$ ;

2. (2) assume the bad curves $B\subset E_1$ and $B'\subset E_2$ exist. ${\textrm{ord}}_B{\mathfrak{a}}_{A_1}{\mathcal O}_{E_1}=0$ , and ${\textrm{ord}}_{B^{\prime}}$ ${\mathfrak{a}}_{A_2}{\mathcal O}_{E_2}=0$ .

5. Proofs of the main results

For the proofs of the main theorems we need the following lemma which guarantees that the second weighted blow-up is possible.

Lemma 5·1. Let E be a prime divisor over a smooth N-fold A ( $N\geq 2$ ) with the center at the closed point 0. Let $\{x_1,\ldots, x_N\}$ be a RSP at 0. Let $v_i\;:\!=\,v_E(x_i)$ , $v\;:\!=\,(v_1,\ldots,v_N)$ and define

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

Let $\varphi_1:\; A_1\longrightarrow A$ be the weighted blow-up with respect to $\{x_1,\ldots, x_N\}$ with weight v’. Denote the exceptional divisor of $\varphi_1$ by $E_1$ . Assume $E\neq E_1$ and let C be the center of E on $A_1$ and $P\in C$ the generic point of C.

Then,

\begin{align*}P\in E_1 \setminus \left\{\bigcup (X_i=0)\right\}\subset E_1={\Bbb P}(v^{\prime}_1, \ldots, v^{\prime}_N),\end{align*}

where $X_i$ is a homogeneous coordinate function corresponding to $x_i$ . In particular, P is smooth on $A_1$ and also on $E_1$ .

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*}

The following lemma is a basic idea appeared in [ 9 ].

Lemma 5·2. Let ${\mathfrak{a}}$ be an ${\Bbb R}$ -ideal on A with $a(E;\, A, {\mathfrak{a}})\geq 0$ . Let $A'\longrightarrow A$ be a proper birational morphism with normal A^′, and D an irreducible divisor on A^′ with the same center on A as that of E. Assume $a(D;\, A, {\mathfrak{a}}) > a(E;\, A, {\mathfrak{a}})$ and the generic point P of the center of E on A^′ is smooth and not contained in the other exceptional divisors for $A'\longrightarrow A$ .

Then, we have

\begin{align*}{\textrm{mld}}(P;\, D, {\mathfrak{a}}_{A^{\prime}}{\mathcal O}_{D})<0, \ \ \mbox{in particular}\end{align*}

\begin{align*}{\textrm{ord}}_P{\mathfrak{a}}_{A^{\prime}}{\mathcal O}_{D}>1,\end{align*}

where ${\mathfrak{a}}_{A^{\prime}}$ is a weak transform of ${\mathfrak{a}}$ on A ^′.

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 ([ 3 , 7 ]) 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.

Setting for the proof of Theorem 1·8.

Let E be a prime divisor over a smooth 3-fold A with the center at a closed point 0. Let ${\mathfrak{a}}$ be a general ${\Bbb R}$ -ideal on A such that $a(E;\, A, {\mathfrak{a}})\geq0$ . Let

\begin{align*}\varphi_1\;:\; A_1 \longrightarrow A\end{align*}

be a squeezed blow-up for E satisfying the condition (1) in Definition 4·9. Let the squeezed system $\{x_1,x_2,x_3\}$ and the weight $v'=(v_1',v^{\prime}_2,v^{\prime}_3)$ correspond to the squeezed blow-up $\varphi$ (note that $v^{\prime}_1=v^{\prime}_{2}$ ). Denote the exceptional divisor for $\varphi$ by $E_1$ . If $a(E_1;\, A, {\mathfrak{a}})\leq a(E;\, A, {\mathfrak{a}})$ , then $E_1$ is the required prime divisor F in the theorem. Therefore, from now on, we assume that the inequalities $a(E_1;\, A, {\mathfrak{a}}) \gt a(E;\, A, {\mathfrak{a}})\geq 0$ hold.

Lemma 5·3. Let A, E and $ E_1$ be as above. If ${\mathfrak{a}}$ is general for E and the inequalities $a(E_1;\, A, {\mathfrak{a}})> a(E;\, A, {\mathfrak{a}})\geq 0$ hold, then we obtain the following:

1. (i) $0<a(E_1;\, A,{\mathfrak{a}})<1$ ;

2. (ii) $v'=(1,1,n) $ with $n\geq 1$ or $v'=(2,2,3)$ .

   1. (a) In case (1, 1, n) the center of E on $A_1$ is a curve in $E_1={\Bbb P}(1,1,n)$ of degree $n+1$ .

   2. (b) In case (2, 2, 3) the center of E on $A_1$ is either a curve of degree 6 or a closed point in $E_1={\Bbb P}(2, 2, 3)$ .

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 ^′.

We divide the proof into two cases according to the dimension of the center of E on $A_1$ . Let $P\in A_1$ be the generic point of the center of E on $A_1$ .

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

Corollary 5·4 (Theorem 1·11). Let A be a smooth variety of dimension 3 over an algebraically closed field k. For any general pair $(A,{\mathfrak{a}})$ with ${\textrm{mld}}(0;\, A, {\mathfrak{a}})\geq 1$ the minimal log discrepancy is computed by a prime divisor obtained by one weighted blow-up.

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 [ 9 ] and [ 6 ], 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 ([ 3 , 7 ]), 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 ([ 9 ]) 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 $a(E;\, A, {\mathfrak{a}})\geq 0$ .

The condition “general” is necessary as far as we use “squeezed” blow-ups to construct a required divisor in Theorem 1·8. Actually, we have a non-general ideal such that two squeezed blow-ups do not give the required divisor.

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:

g $\varphi_1\;:\;A_1\longrightarrow A$ be the weighted blow-up with weight (1, 1, 2) with respect to the coordinates $\{x_1,x_2,x_3\}$ . Let $E_1$ be the exceptional divisor of $\varphi_1$ . Let $\varphi_2\;:\;A_2\longrightarrow A_1$ be the (usual) blow-up with the center at $E_1\cap (f_{A_1}=0)$ , where $(f_{A_1})$ is the weak transform of (f) on $A_1$ . Let $E_2$ be the exceptional divisor of $\varphi_2$ . Let $\varphi_3\;:\;\widetilde A\longrightarrow A_2$ be the (usual) blow-up with the center at $E_2\cap (f_{A_2}=0)$ , where $(f_{A_2})$ is the weak transform of (f) on $A_2$ . Let E be the exceptional divisor of $\varphi_3$ . Then, $\varphi_1$ and $\varphi_2$ are squeezed blow-ups for E, ${\mathfrak{a}}$ is not general for E and the following hold:

\begin{align*}0=a(E;\, A,{\mathfrak{a}})<a(E_2;\, A,{\mathfrak{a}})=\frac{1}{5}<a(E_1;\, A,{\mathfrak{a}})=\frac{3}{5}.\end{align*}

So, we can see that the squeezed blow-ups do not work for this ideal. But if we do not stick to squeezed blow-up, we can find two weighted blow-ups to obtain the required F in the theorem. Let $\{x^{\prime}_1, x^{\prime}_2,x^{\prime}_3\}$ be another RSP defined by $x^{\prime}_i=x_i$ $(i=1,3)$ and $x^{\prime}_2=x_1-x_2$ . Then, $v_E(x^{\prime}_1)=1,v_E( x^{\prime}_2)=2,v_E(x^{\prime}_3)=2$ . (We can see that this RSP is not squeezed.) Now, let $\psi_1\;:\;A^{\prime}_1\longrightarrow A$ be the weighted blow-up with weight (1, 2, 2) with respect to $\{x^{\prime}_1, x^{\prime}_2,x^{\prime}_3\}$ . Let $E^{\prime}_1$ be the exceptional divisor of $\psi_1$ . Let $\psi_2\;:\;A^{\prime}_2\longrightarrow A^{\prime}_1$ be the (usual) blow-up with the center at $E^{\prime}_1\cap (f_{A^{\prime}_1}=0)$ . Let $E^{\prime}_2$ be the exceptional divisor of $\psi_2$ . Then, we can see that $E=E_2$ at the generic points. So, E itself is obtained by two weighted blow-ups.

The example suggests us that we may take an appropriate weighted blow-up to obtain the required F in the theorem, if ${\mathfrak{a}}$ is not general.

Corollary 5·6 (Corollary 1·9). Assume $N=3$ . Then, for every “general” pair $(A,{\mathfrak{a}})$ , the minimal log discrepancy ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ is computed by a prime divisor E obtained by at most two weighted blow-ups. More concretely, the blow-ups are squeezed blow-ups for E.

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 [ 9 ], 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.

Corollary 5·7. Let E be a prime divisor over A with the center at 0 and $E_1={\Bbb P}(r,r,s)$ $(r,s\geq 1)$ the exceptional divisor of a squeezed blow-up for E. Assume that $a(E;\, A,{\mathfrak{a}})\geq 0$ and the center of E on $E_1$ is a curve of degree $>r$ , then there is a prime divisor F such that

\begin{align*}a(F;\, A,{\mathfrak{a}})\leq a(E;\, A,{\mathfrak{a}})\end{align*}

holds for every ${\Bbb R}$ -ideal ${\mathfrak{a}}$ and F is obtained by at most two weighted blow-ups.

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

The proof of the theorem shows also the following corollary.

Corollary 5·8. Let E be a prime divisor over A with the center at 0 computing ${\textrm{mld}}(0;\, A,{\mathfrak{a}})\geq 0$ . Let E^′ be the exceptional divisor of a weighted blow-up with weight $v\;:\!=\,(r,s,t)$ , where $\gcd(r,s,t)=1$ . Assume that the center C of E on E^′ is a curve of degree $d\geq r+s+t-1$ If ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ is not computed by E^′, then the mld is computed by the divisor obtained by one additional weighted blow-up at C.

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 $a(F;\, A,{\mathfrak{a}})\leq a(E;\, A,{\mathfrak{a}})={\textrm{mld}}(0;\, A,{\mathfrak{a}})$ .

The following is a special case of the corollary above. Example 3·3 is in this case.

Corollary 5·9 (Corollary 1·10). Let E be a prime divisor over A with the center at 0 computing ${\textrm{mld}}(0;\, A,{\mathfrak{a}})\geq 0$ . Let E^′ be the exceptional divisor of the usual blow-up with the center at 0. Assume that the center C of E on E^′ is a curve of degree $d\geq 2$ Then, ${\textrm{mld}}(0;\, A,{\mathfrak{a}})$ is computed by the divisor obtained by one additional weighted blow-up at C.