数理解析研究所講究録 1233 巻 2001 年 95-101

95
Certain associated graded ringsof 3-dimensional regular local rings are regular
昌孝 (Masataka Tomari)

金沢大学理学部 泊This note is a preliminary version.

Introduction. The study of various blowing-ups is very important in the theory of
singularities. In many case some blowing-up appears as the blowing-down of divisorsof algebraic variety , and is understood naturally as afiltered blowing-up. From thispoint of view, one of most interesting results in this field is M. Kawakita’s classificationof aspecial divisorial contraction of dimension three [2]. In [2], Kawakita proved thatevery divisorial contraction to asmooth 3-dimensional point is aweighted blowing-up
induced by certain weighting on aregular system of parameters of 3-dimensional regular
local ring. It is natural to study his theorem from the theory of filtered blowing-ups,
and this is my motivation for this talk.
In this paper, Iwill discuss the filtered blowing-up of singularities, and, by using
special equi-singular deformation induced from afiltration on local ring, Ishow thefollowing simple assertion,

and $F=\{F^{k}\}_{k\geq 0}$ be a filtration on such that$gr_{F}A=\oplus_{k\geq 0}F^{k}/F^{k+1}$ is an integral domain with isolated singularity.
Then $gr_{F}A$ is$i.e.$.
,
regular,
Theorem. 1Let$A$

$A\cong \mathrm{C}\{x_{1}, x_{2}, x_{3}\}$$gr_{F}A\cong \mathrm{C}[y_{1}, y_{2}, y_{3}]$In this paper, afiltration $F$ on the local ring $(A, m)$ is; $F=\{F^{k}\}$ ;adecreasing, $F^{k}=A(k\leq-1)$ ,sequence of ideals $F^{k}\subset A$ such that $F^{0}(A)=A$ ,and $7?=\oplus_{k\geq 0}F^{k}T^{k}\subset A[T]$ is afinitely generated -algebra, where $T$ isfor
an indeterminate. There is an integer $N$ such that the relation $F^{kN}=F^{N}\cdots$$G=gr_{F}(A)$$k\geq0$$m$andremarkdenote-primary.
Weis, and we assume thatall
, where $72’=\oplus_{k\in Z}F^{k}T^{k}$ is the extended Rees algebra.
that$m\supset F^{1}$$F^{k}F^{l}\subset$$A$$F^{k+l}(\forall k, l)$$F^{N}$$F^{N}$$G=\mathcal{R}’/T^{-1}\mathcal{R}’$

Theorem 1is shown as aspecial case of the following more general results.
Theorem 2. Let $(V,p)$ be a normal -dimensional isolated terminal singularity ofindex (resp. canonical, resp. $log$ terminal, resp. $log$ canonical), and $F=\{F^{k}\}$be a filtration on $A=O_{V,p}$ such that $G=grFA$ is an integral domain with isolated
singularity. Then
$d$

$r$

(1) $G$ is normal and terminal singularity of index
terminal, resp. $log$ canonical).

$r$(resp. canonical, resp.

$log$(2) There is a filtration $F_{B}=\{F_{B}^{k}\}$ on the canonical cover (the index one cover)and there exsuch that $G_{B}=grFB\cong$ the canonical cover$F_{B}^{kM}\cap
A=F^{k}\subset
A$$M\geq1$
relationssuch that theists an integer
for $k\geq 0$ and$(gr_{F_{B}\cap A}(A))^{(M)}=gr_{F}(A)$ hold.

$B=\oplus_{m=0}^{k-1}\omega_{A}^{[m]}$$0\beta$(3) If $d=3$ and $(V,p)$ is teminal, then the relation $e(m_{B}, B)=e((G_{B})_{+}, G_{B})(=$$1,2)$ hold.

96
We have acorollary as follows:
Corollary 3. $(V,p):3$ -dimensional cyclic $te$ minal and $F$ :as above, then$A^{\wedge}$$gr_{F}(A)^{\wedge}\cong$.

As the case of index one, we obtain Theorem 1from Corollary 3. Here recall thatevery isolated quotient singularity of dimension not less than three is rigid.
In general, if we consider afiltration induced from adivisorial contraction, theassociated graded ring is not necessary an integral domain with isolated singularity$($$[1,3])$.

\S 1. Sketch of proof of Theorem 2.
We assume that there is no N
such that $G^{(N)}=G$ , where$\geq 2$$G^{(N)}$.

is defined by$G^{(N)}=\oplus_{k\geq 0}G_{kN}\subset Ci$

Step 1. Let: X
be the filtered blowing-up by F with$F^{k}=\phi_{*}(O_{X}(-kE))$
E. We obtain the relationfor k. (cf[6,$\S 2])$ .
$\psi$$=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(\mathrm{f}\mathrm{t})$$arrow V=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A$$=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(G)$$\in \mathrm{Z}$Proof. Since G is an integral domain and V

72’$=\oplus_{k\in}zF^{k}T^{k}\subset A[T,T^{-1}]$$=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A$is anormal domain.

is normal, we can easily see that

This claim is shown as follows: We have
, where, thensatisfies the conditions
is normal. Further, if $P\not\in V(u)$ , then we obtain the relations$A[T, T^{-1}]_{P}$ which is normal.

$u=T^{-1}\ni \mathcal{R}_{-1}’$$G=\mathcal{R}’/u\mathcal{R}’$$P\in V(u)\subset \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{R}’)$$G_{P}\cong \mathcal{R}_{P}’/u\mathcal{R}_{P}’$$\mathcal{R}_{P}’$$R_{0}$

and$S_{1}$. If, hence$\mathcal{R}_{P}’=(\mathcal{R}_{T}’)_{P}=$By the assumption thatis afinitely generated -algebra, there is apositive$F^{kN}=F^{N}\cdots$
integer $N>0$ such that, for $k\geq 0$ , i.e., $\mathcal{R}^{(N)}=A[F^{N}T^{N}]$ . Here is
the blowing-up with center
and $F^{kN}=\psi_{*}(F^{kN}O_{X})=\psi_{*}(O_{X}(kN))$ . Since $Q(G)$has ahomogeneous element of degree 1, we have $O_{X}(k)=(O_{X}(1)^{\emptyset k})^{**}$ for. We$G$$(-E)$$O_{X}(1)=O_{X}$,$O_{X}(N)=O_{X}$$(-NE)$
havehence. Since is an integral domain,
defines avaluation on $Q(A)$ such that $F^{k}=\{x\in Q(A)|V(x)\geq k\}$ . Further$\{F^{kN}\}$ defines the valuation $V’$ on $Q(A)$ as $F^{kN}=\{x\in Q(A)|V_{E}(x)\geq kN\}$
where$V_{E}(x)=ord_{E}(x)$ on $X$ . Therefore $F^{k}=\{x\in Q(A)|V_{E}(x)\geq k\}$ for.
$A$$\mathcal{R}$$F^{N}$$\psi$$F^{N}$$\forall k\in \mathrm{Z}$$V$$\{F^{k}\}$$\forall k\in \mathrm{Z}$Step 2. X has only cyclic quotient singularities, in particular X has only
singularities.

Proof. (cf \S 5 [6]). For$\log$

terminal, there exists $f\in F^{d}-F^{d+1}$ ,with $P\in V_{+}(f^{*})$ , where. Here we denotes. Now
is aregular ring. This is shown as follows: We see that
$A_{f}[T,T^{-1}]$ is regular and thatis regular. Hence
so is.
$P\in E=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(G)$$\subset X=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(\mathrm{f}\mathrm{t})$$f^{*}=fT^{d}\in \mathcal{R}_{d}$$\overline{f}T^{d}\in G_{d}$$\oplus_{k\in}z(\mathcal{R}f^{*)_{k}}$$\mathcal{R}_{f^{*}}=$$(\mathcal{R}_{f}*)_{\mathrm{t}\tau-1})^{-1}=$$n_{f}*/T^{-1}\mathcal{R}f*$$=\mathcal{R}_{f}’./T^{-1}n_{f^{\mathrm{r}}}’=G_{\overline{[}}$$\mathcal{R}_{f^{\mathrm{r}}}$Now let $B=(n_{f*})_{P}=\oplus_{k\in Z}((n_{f*})_{P})_{k}$ and $t\in B$ be ahomogeneous unit of theminimal degree $N(P)$ . Let $C=B/t-1$ . Then, by [6,\S 5], $C$ is aregular local ring.
Hereis afinite direct summand of $C$ .
$((\mathcal{R}f*)_{P})_{0}$Step 3. (The

Proof. Let$\log$

canonical condition of A implies that ) G is normal.

($v_{0}\in\omega_{A}^{[r]}$bbee aa ggeenneerraattoorr aatt p aass$\omega_{A}^{[r]}=A$.

$\omega_{0}$. We define the integer

97
by the relation $divx(\omega_{0})=-(r+a’)E$ on $X$ . That is $\omega_{X}^{[r]}\cong O_{X}(-(r+a’)E)$ or$K_{X}= \psi^{*}(K_{V})-(1+\frac{a’}{r})E$ . Since
canonical, we have $a’\leq 0$ . We will showisthe following.
$a’$

$A$

Claim. $R^{1}\psi_{*}(O_{X}(-mE))=0$ for m$\log$$\geq 1$, (m$\in \mathrm{Z})$.

Proof of the claim. We have the relation$O_{X}(-mE)$$\cong\omega_{X}((r-1)K_{X} +(r+a’)E-mE))$$\cong\omega x((r-1)(K_{X}+E)-(m-1-a’)E)$ .

Further $(r-1)(K_{X}+E)-(m-1-a’)E$ is relatively numerically equivalent$(m-1-a’)E=-(m-1- \frac{a’}{r})E$ with respect to . Since $-E$ is relatively -ample,$(r-1)(K_{X}+E)-(m-1-a’)E$ is. Hence by the vanishing theorem of Grauert-Viehweg, we obtain the claim.
$\mathrm{t}\mathrm{o}-\frac{r-1}{r}a’E-$$\psi$$\psi$$\psi- \mathrm{n}\mathrm{e}\mathrm{f}$$\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{r},\mathrm{K}\mathrm{a}\mathrm{w}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{a}$Here we have the exact sequence$0arrow O_{X}(-(k+1)E)arrow O_{X}(-kE)arrow O_{E}(k)arrow \mathrm{O}$for$k\in \mathrm{Z}$. By the claim, we obtain the following exact sequence$0arrow F^{k+1}=\psi_{*}(O_{X}(-(k+1)E))arrow F^{k}=\psi_{*}(O_{X}(-kE))arrow H^{0}(O_{E}(k))$$arrow R^{1}\psi_{*}(O_{X}(-(k+1)E))=0$

for $k\geq 0$ .

We have$0arrow H_{c_{+}}^{0}(G)arrow Garrow\oplus_{k\in}\mathrm{z}H^{0}(O_{E}(k))arrow H_{c_{+}}^{1}(G)arrow 0$

.

Since
is an integral domain, $H_{G_{+}}^{0}(G)=0$ . Furtherisnormal. This is shown as follows: Letbe the normalization of in $Q(G)$ . Since$(G)$ , we have
has only isolated singularity , $G-/G$ has finite length. Hence on. By Demazure, with, there exists
the relation
as follows: $\overline{G}(k)=O_{E}(kD)T^{k}$ , for. Hence $\Gamma_{*}(G)=R(E, D)=\overline{G}$ .
$G$$\oplus_{k\in \mathrm{Z}}H^{0}(O_{E}(k))=\Gamma_{*}(G)$$G$$\overline{G}$$G$$E=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}$$\overline{G}(k)=G(k)\sim\sim$$T\in Q(\overline{G})_{1}$$D\in D\mathrm{i}\mathrm{t}\mathrm{t}(E)(3\mathrm{Q}$$k\in \mathrm{Z}$Therefore, we obtain the relation $H_{G_{+}}^{1}(G)_{k}=0$ for $k\leq-1$ . And the relation$H_{c_{+}}^{1}(G)=0$ follows.
Step 4. We will discuss theterminal property of $G=R(E, D)$ under the assumption that is
terminal of index .
$\log$$A$$\log$$r$

We have the following.
Lemma [8]. Let us assume the conditions that $G$ is an integral domain where Spec(G)$V(G_{+})$ is normal Gorenstein and that Spec(A)$V(m)$ is Gorenstein. Then the follorving relations hold.
–$\frac{\omega_{X}^{[m]}(mE-\alpha E)}{\omega_{X}^{[m]}(mE-(\alpha+1)E)}\cong\omega_{E}^{[m]}(mD’+\alpha D)$

Here $O_{E}(k)--O_{E}(kD)T^{k}$ as before, with

for

$m$

,

$ce\in \mathrm{Z}$$D= \sum_{V\in trr^{1}(X)}\frac{p_{V}}{q_{V}}V$.

with $(p_{V}, q_{V})=1$ ,98$q_{V}\geq 1$

and $D’= \sum_{V\in Irr^{1}(X)}\frac{q_{V}-1}{q_{V}}V$.

By the relation$\omega_{X}^{[r]}(rE-\alpha E)\cong O_{X}(-(a’+\alpha)E)$, for all$\alpha\in \mathrm{Z}$

we obtain, for all$\omega_{E}^{[r]}(rD’+\alpha D)\cong O_{E}((\alpha+a’)D)$$\alpha\in \mathrm{Z}$by Lemma. Hence $K_{R}^{[r]}=R(a’)$ follows.

Here
is regular, $G=R(E, D)$ is$a’<0$(resp.
(resp. $a’\leq 0$ ) by Theorem (2.5)
canonical)if
if
terminal
and only
and Theorem (2.8) of [7].

$\log$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(R)-V(R_{+})=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(G)-V(G_{+})$$\log$

We will discuss the index of R. By Lemma, we have the following exact sequence.
$0 arrow\frac{T^{m}\omega_{R}^{[m]}}{T^{m-1}\omega_{\mathcal{R}}^{[m]}},arrow K_{R(E,D)}^{[m]}arrow$$\oplus_{k\in \mathrm{Z}}\mathrm{K}\mathrm{e}\mathrm{r}\{H^{1}(\omega_{X}^{[m]}(mE-(k+1)E))arrow H^{1}(\omega_{X}^{[m]}(mE-kE))\}arrow 0$

for$m\in \mathrm{Z}$.

If there exist $r’>1$ where the relation $K_{R(E,D)}^{[r’]}=R(a$ ” is satisfied for some integer$)$$a”\in \mathrm{Z}$, we have the relation$\frac{a’}{r’},=\frac{a’}{r}$Here $(K_{R}^{[r’]})_{k}=R_{k+a},,$ , hence. We obtain $a”<0$ .

$(K_{R}^{[r’]}.)_{k}=0$

if $k\leq-1$ .

For $k\geq 0$ , we set $m=r’\geq 1$ and obtain the relations$\omega_{X}^{[m]}(mE-(k+1)E)=\omega x$$((m-1)(K_{X}+E)-kE)$ ,and$(m-1)(K_{X}+E)-kE \equiv-(-\frac{m-1}{r}a’+k)E$ .

This is, hence the following vanishing hold$\psi- \mathrm{n}\mathrm{e}\mathrm{f}$$H^{1}(\omega_{X}^{[m]}(mE-(k+1)E))=0$

Hence$\frac{T^{m}\omega_{\mathcal{R}’}^{[m]}}{T^{m-1}\omega_{R}^{[m]}},\cong K_{R(E,D)}^{[m]}$with $m=r’$ . Hence. For$V(T^{-1})=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(R(E, D))\subset \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{R}’)$$c$.

$\geq 0$.

,is locally principal along$T^{m}\omega_{\mathcal{R}}^{[m]}$$c\neq 0\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{C}[T^{-1}]$$\omega_{\mathcal{R}}^{[m]},/(T^{-1}-c)\omega_{R}^{[m]},=\cup k\in \mathrm{Z}\psi_{*}(\omega_{X}^{[m]}(-kE))=\omega_{A}^{[m]}$module for samefor k

, it follows thatis aprincipal $\mathcal{R}’/(T^{-1}-c)\mathcal{R}’=A-$

Step 5. We will show: The condition that A is a canonical (resp. terminal) singularity
implies that G is also a canonical (resp. terminal) singularity.

Proof. Let ci :$\mathcal{V}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{R}’)arrow \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{C}[T^{-1}]\cong \mathrm{C}$with

for $c\neq 0$ . Let us introduce the filtration of idealson, whereway:for$[6]\S 5$ , we obtain the following diagram after the blowing-up of$F^{l}(\mathcal{R}’)=\mathcal{R}’|\iota.\mathcal{R}’\subset \mathcal{R}’$, andby the following. As is shown inby this$\mathcal{V}_{0}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}G$$\{F^{l}(\mathcal{R}’)\}$$72’=\oplus_{k\geq l}F^{l}T^{l}\subset \mathcal{R}’$$\mathcal{R}’$$l\in \mathrm{Z}$$\mathcal{V}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{R}’)$$\mathcal{V}_{c}\cong V$

99$111\iota 1\mathrm{d}\iota 1\cup 11$.
$\frac{\xi_{\iota}}{r}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{R}’)=\mathcal{V}$$\mathrm{Y}"=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(\mathcal{R}_{F}(\mathcal{R}’))$$\swarrow\omega$$\omega"[searrow]$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{C}[T^{-1}]$

where ”gives the filtered blowing-up for each fiber as follows:
is nothingand ” : ”
is nothing but the graded blowing-up of$F$. By J. Wahl [9], ”is alocallyforby
but the blowing-up ofis regular. Here is antrivial family under the assumption that
relation
following
$d+1$
have
the
we
and
-dimensional scheme-Gorenstein$\omega$$\omega$$\mathrm{Y}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(G)$” $0:\mathrm{Y}$ ”$carrow \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{V}_{c})$$\omega_{c}$$c\neq 0\in \mathrm{C}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A)$$0arrow \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathcal{V}_{0})$$\omega$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(G)-V(G_{+})$$\mathcal{V}$$r$$K_{R}^{[r]},$$\cong \mathcal{R}’(a’+r)$of
There is ameromorphic -ple $d+1$ -form
an isomorphism. This induces the isomorphism$\Omega\sim 0$$r$$\omega_{\mathrm{Y}’}^{[r]},$$=O_{\mathrm{Y}},$$\mathcal{R}’$.
such that,;$1arrow\tilde{\Omega}_{0}$$\mathcal{R}’arrow K_{R}^{[r]}$gives, $(r+a’)\xi^{*}(\tilde{\Omega}_{0})$ ,, where the relation Proj
that is, we have the relation.
is holomorphic on. Since $a’\leq-r$ ,. Here
on
not
does
which
vanishes
is aholomorphic -ple -form on
Hence”$c\neq0$
for$)_{c}-E$ . Here,and(72) for,for.
gives agenerator of
the case $c=0$ . Here$gr_{F}(\mathcal{R}’)=$$div_{\mathrm{Y}’}’\tilde{\Omega}_{0}=-(r+a’)\mathrm{E}$$\mathrm{E}\cong E\cross \mathrm{C}$$r$${\rm Res}_{(\mathrm{Y}’)_{\mathrm{c}}},(\tilde{\Omega}_{0})$$\mathrm{d}$$(\mathrm{Y}" )_{c}$$(\mathrm{Y}" )_{c}=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(G^{\mathfrak{h}})=C(E, D)$$(\mathrm{Y}" )_{c}=X=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}$$(\mathrm{Y}$$\mathrm{Y}"$$\xi^{*}(\tilde{\Omega}_{0})$$E=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(G)$${\rm Res}_{(Z’)_{c}}(\tilde{\Omega}_{0})$$c\in \mathrm{C}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{C}[T^{-1}])$$\omega_{(Z)_{c}}^{[r]}$We state the following claim.
: ””such that the natural
Claim. There is aresolution of singularities”:
is locally trivial along the fiber over $\{0\}=V(T^{-1})$ :induced map$\tilde{\mathrm{Y}}$$\beta$$\tilde{\omega}$$arrow \mathrm{Y}$$\tilde{\mathrm{Y}}"arrow \mathrm{C}$$\tilde{\mathrm{Y}}$”$arrow\beta$$\mathrm{Y}"=\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(\mathcal{R}_{F}(\mathcal{R}’))$$\tilde{\omega}"[searrow]$$\swarrow\omega$”

SpecC $[T^{-1}]$
be the horizontal divisor ofLet$c\neq0$, we have the relation:For$\mathrm{F}\subset\tilde{\mathrm{Y}}"arrow C$$\tilde{\mathrm{Y}}$”which is exceptional for${\rm Res}|_{\overline{\mathrm{Y}}},,c(\beta^{*}(\tilde{\Omega}_{0}))=\beta^{*}({\rm Res}|_{\mathrm{Y}_{c}},,\tilde{\Omega}_{0})$$\beta$$\tilde{\mathrm{Y}}"arrow \mathrm{Y}"$.

.

Since $(A, m)$ has only canonical singularities, this is holomorphic. Henceis holomorphic.
. Therefore
morphic on$\tilde{\mathrm{Y}}"$:$\tilde{\Omega}_{0}$is hol0-${\rm Res}|_{\overline{\mathrm{Y}}’ 0},(\beta^{*}(\tilde{\Omega}_{0})$Q.E.D. for the claim.

on the local ring
Step 6. Here we will introduce afiltration
has the desired properties as is claimed in Theorem 2.
$F_{B}$

By atentative way, we set$F_{B}^{k}(\omega_{A}^{[m]})\subset\omega_{A}^{[m]}$$B=\oplus_{k=0}^{r-1}\omega_{A}^{[k]}$as follows:$F_{B}^{k}( \omega_{A}^{[m]})=\sum_{ma’+rh\geq k\cdot gcd(a’,r)}\psi_{*}(\omega_{X}^{[m]}(mE-kE)))\subset\omega_{A}^{[m]}$and$F_{B}^{k}(B)=\oplus_{m=0}^{r-1}F_{B}^{k}(\omega_{A}^{[m]})U^{m}\subset B=\oplus_{m=0}^{r-1}\omega_{A}^{[m]}U^{m}$

.

,

which

100
The main point which we have to check here is the assertion that the associated gradedring of grpB B is nothing but the graded canonical cover G $R\{E$ , D). We can show
this asserition by the following formula about graded cyclic covers which we will recall
in the below.
$\ovalbox{\tt\small REJECT}$Now,$K_{G}$

is a$\mathrm{Q}$-Cartier divisor of index r and there exists $\varphi\in k(X)$ such that

$rK_{E}-a’D=div_{X}(\varphi)$.

Corollary (1.7.1) of [7]. Let $S=S(R, Kr, \varphi T^{a’})$ be the normal graded cyclic r-coveras described in [7]. Then the Pinkham-Demazure construction withofrespect towith $\alpha a’+\beta r=s(=(r, \mathrm{a}’))$ is given by) as follows:(1) $F$ is the cyclic cover of $E$ given by$R=\mathrm{k}(\mathrm{X})D)$$S$$\tilde{T}=T^{\beta}u^{a}$$S=\mathrm{R}\{\mathrm{F},\tilde{D}$$\rho:F=Spec_{E}(\bigoplus_{l=0}^{\epsilon-1}O_{E}(l(\frac{r}{s}(K_{X}+D’)-\frac{a’}{s}D)))arrow E$.

(2) $\tilde{D}=\rho^{*}\{\alpha(K_{X}+D’)+\beta D\}$ .

(3) We obtain the relation$K_{S}=S( \frac{a’}{s})$.

By using Lemma B and the above theorem we can check the assertion. The details
are left to the readers.

Further we obtain the following relations;$F_{B}^{k} \cap A=F_{B}^{k}(\omega_{Z}^{[0]})=\psi_{*}(O_{X}(-hE))=F^{[k\frac{g\mathrm{c}d(a’,r)}{r}]}h\geq k\frac{\sum_{g\mathrm{c}d(a’,r)}}{r}$.

Step 7. Now we assume that
$4=3$ and that $(V,p)$ is aterminal singularity$D$ ).
of index . Then so is
Since $gr_{F_{B}}(B)$ is the graded canonical$B)$
cover of $gr_{F}(A)$ , grpB { is aterminal 3-dimensional singularity of index one, hence
is regular or conpund Du Val singularity. In particular, $gr_{F_{B}}(B)$ is ahypersurface
isolated singularity by M. Reid [4].
$d=\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{n}$$gr_{F}(A)=\mathrm{R}\{\mathrm{F},$$r$

We have the following results on multiplicities of filtered rings;
Lemma [5]. Let, $x_{x}\in(G_{B})_{+}$ be aand ,homogeneous minimal generator with $\deg x_{1}\leq\deg x_{2}\leq\ldots.\leq\deg x_{s}$ . Then we havethe followings.
$P(G_{B}, \lambda)=\sum_{k\geq 0}l((G_{B})_{k})\lambda^{k}\in \mathrm{Z}[[\mathrm{A}]]$$x_{1}$$\ldots$$\deg x_{2}\cdots$ $\deg x_{d}\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G_{B}, \lambda)\leq e(m_{B}, B)\leq e((G_{B})_{+}, G_{B})$(1).
$e((G_{B})_{+},
Gb)$
Hence, ifequals the round up of the rational number$\deg x_{1}\cdot\deg x_{2}\cdots$, then we have the equality $e(m_{B}, B)=$$\deg x_{1}\cdot$$\deg x_{d}\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G_{B}, \lambda)$$e((G_{B})_{+}, G_{B})$.

(2) Ifis ahypersurface isolated singularity which is defined by aquasi-homogeneouspolynomial of type $(\deg x_{1}, \ldots, \deg x_{d+1}; h)$ , then $\deg x_{1}\cdot\deg x_{2}\cdots\deg x_{d}\lim_{\lambdaarrow 1}(1-$$G_{B}$$\lambda)^{d}P(G_{B}, \lambda)=\frac{h}{\deg x_{d+1}}$

number$\frac{h}{\deg x_{d+1}}$

and $e((G_{B})_{+}, G_{B})$ equals to the round up of the rational.

Hence we obtain the relation $e(m_{B}, B)=e((G_{B})_{+}, G_{B})(=1,$or2).

101This completes the proof of Theorem 2.

References. Inst. Math. Sci.
1. T. Hayakawa: Blowing ups of 3-dimensional terminal singularities. II. Publ.
36 (2000), no. 3, 423-456.
2. M. Kawakita: Divisorial contractions in dimension three which contract divisors to smooth points.
Invent, math.145, 105-119 (2001).
3. Y. Kawamata: Divisorial contractions to 3-dimensional terminal quotient singularities. in ”Higherdimensional complex varieties (de Gruyter, 1996)” 241-245
Algebraic Geometry,
4. M. Reid: Young person’s guide to canonical singulariteis. In: Block,Bowdoin 1985, Part I(Proceedings of Symposia in Pure Mathematics, vol. 46 , 354-414) Amer.
Math. Soc. (1987)
5. M. Tomari: Multiplicity of filtered rings and simple K3 singularities of multiplicity two. preprint
6. M. Tomari, Kei-ichi Watanabe: Filtered rings, filtered blowing-ups and normal two dimensional
singularities with “star-shaped ”resolution. Publ. Res.Inst.Math.Sci. Kyoto Univ.25-5(1989),681-740.
7. M. Tomari, Kei-ichi Watanabe: Cyclic covers of normal graded rings, to appear in Kodai Math. J.
8. M. Tomari: Papers which include the proof of Lemma in Step 4of \S 2, in preparation.
9. J. Wahl: Deformations of quasi-homogeneous surface singularities, Math. Ann. 280, 103-12${\rm Res}$$\mathrm{S}\mathrm{J}.(\mathrm{e}\mathrm{d})$