E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE s a Noethean ng fo any Wel dvso D on X (e.g., Elzondo and Snvas [ES0]). uthemoe, H 0 (X O X

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1 THE TOTAL COORDINATE RING O A NORAL PROJECTIVE VARIETY E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE. Intoducton The total coodnate ng TC(X) of a vaety s a genealzaton of the ng ntoduced and studed by Cox [Cox95] n connecton wth a toc vaety. Consde a nomal pojectve vaety X wth dvso class goup Cl(X), and let us assume that t s a ntely geneated fee abelan goup. We dene the total coodnate ng of X to be TC(X) = D H 0 (X O X (D)) whee the sum as above s taken ove all Wel dvsos of X contaned n a xed complete system of epesentatves of Cl(X). We efe to Denton. fo a pecse denton of TC(X). Such ngs gew out of an old poblem of classcal algebac geomety, whch we descbe as follows. Let p, :::, p m be dstnct m ponts n the pojectve space P ove C, and let : X! P be the blow upofp at fp ::: p m g. Put E = ; (p ) fo = ::: m. Let H be a hypeplane n P and put A = ; (H). Then Cl(X) s a fee abelan goup wth bass E, :::, E m, A. Then, we may egad H 0 (X O X ( P m = n E + n m+ A)) as the lnea system on P of degee n m+ passng though p wth multplcty at least ;n fo each. The total coodnate ng TC(X) = n ::: n m+ Z X m H 0 X O X n E + n m+ A = facltates the computaton of H 0 (X O P m X ( nteges. In fact, f TC(X) s a Noethean ng, then the dmenson of H 0 (X O P m X ( n = E + n m+ A)), whee Z denotes the ng of n = E + n m+ A)) s a atonal functon n n, :::, n m+. In the event that TC(X) s Noethean fo a nomal pojectve vaety wth functon eld K whose dvso class goup s nte fee, the ng R(D) = nz H 0 (X O X (nd))t n K[t t ; ] Patally suppoted by JSPS-CONACYT, and CONCYT

2 E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE s a Noethean ng fo any Wel dvso D on X (e.g., Elzondo and Snvas [ES0]). uthemoe, H 0 (X O X (E + nd)) nz s a ntely geneated R(D)-module fo any Wel dvsos D and E. The total coodnate ng sometmes appeas as an nvaant subng. uka exhbted numeous examples whee the total coodnate ngs ae not Noethean, ths beng elated to the Hlbet's fouteenth poblem. See [uk0] and [uk0]. In the pesent pape, we shall pove that the total coodnate ng s a unque factozaton doman fo any connected nomal Noethean scheme whose dvso class goup s ntely geneated fee abelan goup n Coollay.. The man theoem of the pape s the followng: Theoem.. Let X be a connected nomal Noethean scheme wth functon eld K. Let D ::: D be Wel dvsos on X, and let t ::: t be vaables ove K. We set X R = H 0 X O X n D t n t n S = K[t ::: t ]: n ::: n Z. Then, R s a Kull doman.. Assume that Q(R) =Q(S), whee Q( ) stands fo the eld of factons. Then thee s a natual sujecton ' :Cl(X) hd ::: D ;! Cl(R) whee D s the mage n Cl(X) of a Wel dvso D on X. 3. If thee exst nteges n, :::, n such that P n D s an ample dvso, then the map ' as above s an somophsm. The followng s an mmedate consequence (ef. Remak.) of the above theoem. Coollay.. Wth the notaton as n Theoem., R s a unque factozaton doman f the set of the mages of D ::: D geneate Cl(X). In patcula, the total coodnate ng s a unque factozaton doman fo a connected nomal Noethean scheme whose dvso class goup s a ntely geneated fee abelan goup. We should menton that ndependently Bechtold and Hausen [BH0] poved that the total coodnate ng s a unque factozaton doman fo a locally factoal vaety ove an algebacally closed eld whose Pcad goup s ntely geneated fee abelan goup. Ou method s puely algebac, smple, and totally deent fom thes. Hee, we gve a pecse statement of the esult of Bechtold and Hausen [BH0]. Let X be a nomal vaety ove an algebacally closed eld whose Pcad goup s ntely geneated fee abelan goup. Let A(X) = DPc(X) H 0 (X O X (D)) be the ng dened n the same way as n Denton. usng all Cate dvsos. Obseve that f the dvso class goup of X s ntely geneated fee abelan goup, ths s the

3 THE TOTAL COORDINATE RING O A NORAL PROJECTIVE VARIETY 3 subng of TC(X) [see () n Denton.] consstng of those gaded peces coespondng to Cate dvsos. Then, Poposton 8.4 n [BH0] mples that the followng fou condtons ae equvalent () A(X) s a unque factozaton doman, () X s locally factoal, (3) Cl(X) = Pc(X), (4) A(X) = TC(X). Let X eg be the open subscheme consstng of nonsngula ponts of X. uthemoe, assume that the dvso class goup of X s ntely geneated fee abelan goup. Snce the codmenson of X n X eg n X s at least, TC(X) = TC(X eg ) s satsed. Then, Poposton 8.4 n [BH0] mples that TC(X eg ) s a unque factozaton doman snce X eg s locally factoal. Theefoe we know that TC(X) s a unque factozaton doman by the esult of Bechtold and Hausen. Hee, we emak that A(X eg ) concdes wth TC(X eg ) snce X eg s locally factoal. On the othe hand, A(X) = TC(X) f and only f X s locally factoal. We emak that we do not assume that a scheme s of nte type ove an algebacally closed eld n Coollay.. We shall pove Theoem. n the next secton. In the nal secton, we gve some examples of total coodnate ngs. We pove that the total coodnate ng of the blow up of a pojectve space at a nte numbe of ponts on a lne s ntely geneated, see Example 3.3. Next we gve an example of a nomal pojectve vaety X that has a non-ntely geneated total coodnate ng n Example 3.4. Acknowledgement. The st autho would lke to thank the hosptalty of the Tokyo etopoltan Unvesty, dung hs vst n Octobe 00, whee the poof of the man theoems wee dscussed. The second and thd authos vsted exco n 000. We would lke to thank the exchange pogam between exco and Japan, JSPS-Conacyt whch suppoted ou vst. We also would lke to thank James Lews fo many coectons, to Paul Robets and Vasudevan Snvas fo eadng the st daft of ths atcle, and to loan Bechtold, Jugen Hausen and the efeee fo many valuable comments.. The total coodnate ng In ths secton we dene the total coodnate ng and pove Theoem.. Denton.. Let X be a connected nomal Noethean scheme wth functon eld K. We assume that the dvso class goup Cl(X) of X s somophc to Z s. Let D, :::, D s be Wel dvsos on X such that the set of the mages geneate Cl(X). Let t, :::, t s be vaables ove K. We set X R(X D ::: D s )= H 0 X O X n D t n t ns s K[t ::: t s ] n ::: n sz whee we egad H 0 X X O X n D = fa K j dv X (a)+ X n D 0g[f0g as an addtve subgoup of K. It s easly seen that R(X D ::: D s ) s a subng of K[t ::: t ]. The ng R(X s D ::: D s ) s unquely detemned by X up to somophsm, that s, t s ndependent of the choce of D, :::, D s up to somophsm. We call t

4 4 E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE the total coodnate ng TC(X) of X. We may egad the total coodnate ng of X as a gaded ng wth the gadng gven by Cl(X). We sometmes wte t as () TC(X) = DCl(X) H 0 (X O X (D)) whee D denotes the class n Cl(X) epesented by awel dvso D. Befoe povng the man theoem, we gve a emak. Remak.. Wth the notaton gven n Theoem., t s easly seen that Q(R) concdes wth Q(S) f one of the followng two condtons ae satsed () the set of the mages of D, :::, D s geneate Cl(X), () thee exst nteges n, :::, n such that P n D s an ample dvso. Even f Q(R) = Q(S) s satsed, the map ' n Theoem. s not an somophsm n geneal, as can be seen n the followng example. Let : X! P be the blow up of P at a pont p. Put E = ; (p). Let H be ahypeplane n P and put A = ; (H). Let K be the functon eld of X and let t be avaable ove K. Set R = nz H 0 (X O X (na))t n S = K[t ]: Usng the pojecton fomula, we have H 0 (X O X (na)) = H 0 (P O P (nh)). Theefoe Q(R) = Q(S) s satsed. In ths case, Cl(X) s a Z-fee module of ank wth bass E and A. Snce R s a polynomal ng ove the base eld, we have Cl(R) = 0. Theefoe the map ' n Theoem. s not an somophsm n ths case. o the emande of ths secton, we pove Theoem.. Let H be the set of educed and educble closed subschemes of X of codmenson. Put D = X H m fo = :::. Let O X be the local ng of X at, and let X be the coespondng maxmal deal. o H, we denote by v the nomalzed valuaton of the dscete valuaton ng O X. Then we have R = Hee we set n ::: n Z R = fa K j v (a)+ P n m 0 fo each H gt n t n n ::: n Z fo H. It s easy to see that R fa K j v (a)+ P n m 0gt n t n s also a subng of S = K[t R R S fo any H, and that R = \ H R s satsed. ::: t : ] such that

5 THE TOTAL COORDINATE RING O A NORAL PROJECTIVE VARIETY 5 Snce the local ng (O X X ) s a dscete valuaton ng, thee exsts an element X such that X = O X. By the equalty we have fa K j P v (a)+ n m 0g = ;P n m O X R = = n ::: n Z n ::: n Z = O X h( ;m ;P n m O X t n t n O X ( ;m t ) n (;m t ) n t ) ::: ( ;m t ) : Theefoe, R s a Noethean nomal doman. We emak that R s the unque homogeneous pme deal of heght of R. Let fq j g be the set of non-homogeneous pme deals of heght of R. Snce R s a Kull doman, R =(R ) R \ \ (R ) Q! s satsed (see Theoem.3 n atsumua [at90]). Snce S = R [ ; ], we have and () (3) S = \ (R ) Q R = \ H R = \! (R ) R \ S: H Hee fo H,we set P = R \ R. Then we have P = = n ::: n Z a K n ::: n ZH 0 (X O X ( X v (a)+ P n m > 0, and v G (a)+ P n m G 0foeach G H n D ; ))t n t n : t n t n We shall pove that R s a Kull doman. By equaton (), t s enough to show that fo any a R nf0g, a s a unt n (R ) R except fo ntely many 's. (Hee, we emak that Q(R) does not have to concdes wth Q(S).) We note that (R ) R \ R = R \ R = P : By equaton (3) as above, t s easy to see that thee exst only ntely many 's such that a s contaned n P. We have thus poven that R s a Kull doman. By Theoem.3 n atsumua [at90], the set fp j H g ncludes the set of homogeneous pme deals of heght of R. Set H 0 = f H j ht R P =g:

6 6 E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE Thus fp j H 0 g s the set of homogeneous pme deals of heght ofr. (Let E be the dvso gven n Remak.. Then, t s easy to see that ht R P E =. Theefoe E H n H 0 n ths case.) By Theoem.3 n [at90], we have R = \ H 0 R P A \ S: We denote by Dv(X) the goup of Wel dvsos on X. It s the fee abelan goup geneated by H. Let P(X) Dv(X) be the subgoup of pncpal dvsos whee fdv X (a) j a K g dv X (a) = X H v (a): By denton, Cl(X) s the quotent goup Dv(X)=P(X). Let v 0 be the nomalzed valuaton of the dscete valuaton ng R P fo H 0. o the emande of the poof, we assume that Q(R) concdes wth Q(S). We emak that R P = (R ) R s satsed fo H 0 snce Q(R) = Q(S). We denote by HDv(R) the set of homogeneous Wel dvsos on R. That s, t s the fee abelan goup geneated by fp j H 0 g. Let HP(R) HDv(R) denote the subgoup geneated by whee fdv R (b) j b s a non-zeo homogeneous element of Rg Then, t s known that the natual map dv R (b) = X H 0 v 0 (b)p : HDv(R)=HP(R)! Cl(R) s an somophsm (e.g., see Samuel [Sam64]). Let : Dv(X)! HDv(R) be the sujectve homomophsm gven by ( ) = P H 0 and ( )=0fo H n H 0. Consde the followng dagam: 0 ;! P(X) ;! Dv(X) ;! Cl(X) ;! 0 # 0 ;! HP(R) ;! HDv(R) ;! Cl(R) ;! 0 o H 0 and a K, v (a) = v 0 (a) s satsed snce R P a K, (dv X (a)) = dv R (a) s satsed. Hence we have (P(X)) HP(R). By denton, the addtve goup HP(R) s geneated by fdv R (a) j a K g and fdv R (t ) j = ::: g fo = (R ) R. Theefoe fo

7 THE TOTAL COORDINATE RING O A NORAL PROJECTIVE VARIETY 7 snce Q(R) =Q(S). o H 0, snce R P =(R ) R and R = O X h( ;m t ) ::: ( ;m t ) v 0 (;m t )=0s satsed. Theefoe X we have v 0 (t )=m. Hence fo each, (D )=( v 0 (t )P =dv R (t ) H m )= X s satsed. Snce HP(R) s geneated by (P(X)) and f(d ) j = ::: g, we have an somophsm Cl(R) H 0 ' Dv(X)/P(X)+hD ::: D + h j H n H 0 = Cl(X) hd ::: D + h j H n H 0 : By ths somophsm, we ave at the natual sujecton ' : Cl(X) hd ::: D ;! Cl(R): o the est of ths secton, we shall pove that ' s an somophsm f the subgoup hd ::: D of Dv(X) contans an ample dvso. We emak that Q(R) =Q(S) s satsed n ths case. It s enough to show H = H. 0 In ode to show ths, we need only show that R P =(R ) R fo each H. It s sucent to show R R P fo each H. s a gaded ng such that R R Let f be a homogeneous element of R. Snce R Q(R), t s easy to see that thee exst K and nteges p, :::, p, q, :::, q such that Thus we have t p t p tq t q R and f = tq t p dv X ()+ X t q t p p D 0 R : dv X ()+ X q D 0 v (=)+ X (q ; p )m 0: We want to show f R P. If v ()+ P p m =0,then t p t p have f R P. R n P s satsed. Theefoe n ths case, we Assume P P P that v () + p m > 0. Put p = v () + p m. We emak that v ()+ q m p. Snce hd ::: D contans an ample dvso, thee exst K and nteges s, :::, s such that dv X ()+ X s D + p 0 v ()+ X s m + p = 0:

8 8 E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE Then we have dv X ()+ X dv X ()+ X (q + s )D 0 (p + s )D 0 v ()+ X (p + s )m = 0: Theefoe we have t q +s t q+s R and t p +s t p+s R n P. Snce f = tq t q t p t p we have f R P. Ths completes the poof of Theoem.. = tq +s t q+s t p +s t p+s Remak.3. Y Hu and Sean Keel [HKe00, Theoem.9] poved the followng. Let X be a Q-factoal pojectve vaety suchthatpc(x)q = N (X). Then X s a o deam space f and only f TC(X) s ntely geneated. If X s a o deam space then X s agitquotent of V = spec(tc(x)) by the tous G = Hom(N G m ) Hee a o deam space s a vaety wth nce geometc popetes. o example, the nef cone Nef(X) s the ane hull of ntely many sem-ample lne bundles, and thee exst small Q-factoal modcatons of X. 3. Some examples We gve some examples of total coodnate ngs n the secton. Example 3.. It s well known that the dvso class goup s a ntely geneated fee abelan goup fo a smooth complete toc vaety (e.g., see 63p n [ul93]). uthemoe n ths case, Cox [Cox95] poved that TC(X) s a homogeneous polynomal ng. He called TC(X) the homogeneous coodnate ng of X. Remak 3.. Total coodnate ngs have a deep elaton wth nvaant theoy as uka [uk0] has shown. Hee we pesent some of hs esults. Let P be the pojectve space of dmenson ove the eld of complex numbes C. Let X be the blow upofp at m dstnct ponts. Then we have Cl(X) ' Z m+. Assume that the m ponts ae not on any hypeplane n P. Then wth a sutable lnea acton of G = Ga m;; on a polynomal ng S ove C wth m vaables, the nvaant subng S G s somophc to TC(X). On the othe hand, Nagata [Nag6] poved that the eectve cone E(X) ncl(x) s not ntely geneated as a sem-goup f X s a blow up of P at 9 geneal ponts. Theefoe, TC(X) s not Noethean n ths case. It s a counteexample of Hlbet's 4-th poblem. uka also genealzed n [uk0] a esult of Dolgachev and ealzed the oot system T p q n the cohomology goup of a cetan atonal vaety of Pcad numbe p + q + ;. As

9 THE TOTAL COORDINATE RING O A NORAL PROJECTIVE VARIETY 9 an applcaton he poved that the nvaant subng of a tenso poduct wth an actons of Nagata type s nntely geneated f the Weyl goup of the coespondng oot system T p q s nnte. Example 3.3. Let p ::: p m be m ponts on a pojectve lne contaned n the pojectve -space P ove an algebacally closed eld k. Then the total coodnate ng TC(X) of the blow upx = Bl p ::: p m (P ) of P at fp ::: p m g s a Noethean ng. We gve apoof: Let p, :::, p m be m dstnct ponts n P. Let : X! P be the blow up of P at fp ::: p m g. Put E = ; (p ) fo = ::: m. Let H be a hypeplane n P and put A = ; (H). Then Cl(X) s a fee abelan goup wth bass E, :::, E m, A. Let B = k[z 0 ::: Z ] be the homogeneous coodnate ng of P. We denote by I the homogeneous pme deal of B coespondng to the pont p. o = ::: m and s Z, we dene Then we have I s s = f s 0 B f s<0: [ b \\ mbm ] a = H 0 (X O X (aa ; b E ;;b m E m )) whee [] a denotes the homogeneous component of degee a. Theefoe we have TC(X) = = a b ::: b mz b ::: b mz B[T ::: T m m ] H 0 (X O X (aa ; b E ;;b m E m )) ( b \\ mbm )T b T bm m o the est of the poof, we assume that fp ::: p m g le on a lne n P. We may assume I = (f Z ::: Z ) I = (f Z ::: Z ). I m = (f m Z ::: Z ) whee f f m k[z 0 Z ] ae lnea foms such thatanytwo elements n ff f m g ae lnealy ndependent ove k. Hee, fo = ::: m and s Z, we put f s = f s f s 0 f s<0:

10 0 E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE Let g be a polynomal n B. Put g = X c ::: c 0 g c c Z c Z c whee g c c k[z 0 Z ]. Then g s contaned n b f and only f g c c f b ;c ;;c k[z 0 Z ] fo any nteges c ::: c 0. Hee, b o b ; c ;;c ae possbly negatve. Then g s contaned n b \\ mbm f and only f g c c f b ;c ;;c k[z 0 Z ] fo any nteges c ::: c 0 and = ::: m. uthemoe, ths s equvalent to g c c f b ;c ;;c f m bm;c ;;c k[z 0 Z ] fo anynteges c ::: c 0. In patcula, g b \\ mbm f and only f g c c Z c Z c b \\ mbm fo any c ::: c 0. Hee we clam that (4) TC(X) =B[T ; ::: T ; m Z T T m ::: Z T T m f T ::: f m T m ]: It s easy to see that the ght-hand sde s ncluded n the left sde. Assume that g b \\ mbm. Then gt b Tm bm s contaned n the left sde. We want to show that the ght sde also contans t. We may assume that g = g Z c Z c, whee g k[z 0 Z ] and c ::: c ae non-negatve nteges. Snce g Z c Z c b \\ mbm,we may assume that whee g k[z 0 Z ]. Then g = g f b ;c ;;c f m bm;c ;;c gt b T bm m = g f b ;c ;;c f m bm;c ;;c Z c Z c T b T bm m = g (Z T T m ) c (Z T T m ) c Hee, f b ; c ;;c < 0, then If b ; c ;;c 0, then (f b ;c ;;c T b ;c ;;c ) (f m bm;c ;;c T bm;c ;;c m ): f b ;c ;;c T b ;c ;;c =(T ; ) c ++c ;b : f b ;c ;;c T b ;c ;;c =(f T ) b ;c ;;c : Thus gt b Tm bm s contaned n the ng on the ght-hand sde n (4), and ths completes the poof. Obseve that f m, then we obtan B[T ; ::: T ; m Z T T m ::: Z T T m f T ::: f m T m ] = k[t ; ::: T ; m Z T T m ::: Z T T m f T ::: f m T m ]: We shall gve an example of a nomal pojectve vaety wth nntely geneated total coodnate ng.

11 THE TOTAL COORDINATE RING O A NORAL PROJECTIVE VARIETY Example 3.4. Let us consde the weghted polynomal ng B k ove a eld k n thee vaables x y z of degee a b c, espectvely. Take the weghted pojectve plane P k(a b c) =Poj(B k ) and consde the blow up : X k (a b c)! P k (a b c) at the smooth pont p k (a b c) :=Ke ; k[x y z] ' ;! k[t] whee ' s the homomophsm of k-algebas dened by '(x) =t a, '(y) =t b and '(z) =t c. We denote X k (a b c) and p k (a b c) smply by X k and p k, espectvely. Usng ntesecton theoy on X k, Cutkosky [Cut9] studed the nte geneaton of the symbolc Rees ng R s (p k ) = n0 p (n) k T n B k [T ]. We emak that the total coodnate ng TC(X k ) s equal to R s (p k )[T ; ]. uthemoe, TC(X k ) s Noethean ng f and only f R s (p k )s. Assume that a =7n ; 3 b= n(5n ; ) c=8n ; 3wthn 4 and (n 3) =. In ths case, the total coodnate ng TC(X k ) s a Noethean ng f and only f the chaactestc of k s postve (see Goto, Nshda and Watanabe [GNW94]). Refeences [BH0] loan Bechtold and Jugen Hausen, Homogeneous coodnates fo algebac vaetes, AG.043. [Cox95] Davd A. Cox, The homogeneous coodnate ng of a toc vaety, J. Algebac Geom. 4 (995), no. 3, 7{50. [Cut9] S. Dale Cutkosky, Symbolc algebas of monomal pmes, J. ene angew. ath. 46 (99), 7{89. [El94] Jave Elzondo, The Eule sees of estcted Chow vaetes., Composto ath. 94 (994), no.3, 97{30. [EL98] E. Jave Elzondo and Paulo Lma-lho, Eule-chow sees and pojectve bundles fomulas, J. Algebac Geom. 7 (998), 695{79. [ES0] E. Jave Elzondo and V. Snvas, Some emaks on Chow vaetes and Eule-Chow sees, J. Pue Appl. Algeba 66 (00), no. -, 67{8. [ul93] Wllam ulton, Intoducton to toc vaetes, st. ed., Annals of athematcs Studes, vol. 3, Pnceton Unvesty Pess, Pnceton, NJ, 993. [GNW94] Sho Goto, Koj Nshda, and Ke-ch Watanabe, Non-Cohen-acaulay symbolc blow-ups fo space monomal cuves and counteexamples to Cowsk's queston, Poc. Ame. ath. Soc. 0 (994), no.. [Ha77] Robn Hatshone, Algebac geomety, Gaduate Texts n athematcs, vol. 5, Spnge-Velag, New Yok, 977. [HKe00] Y Hu and Sean Keel, o deam spaces and GIT, chgan ath. J. 48 (000), 33{348, Dedcated to Wllam ulton on the occason of hs 60th bthday. [at90] Hdeyuk atsumua, Commutatve ng theoy, Cambdge Unvesty Pess, 990. [uk0] [uk0] Shgeu uka, Counteexample to Hlbet's fouteenth poblem fo the 3-dmensonal addtve goup, pepnt No. 343, Novembe 00. Shgeu uka, Geometc ealzaton of T-shaped oot system and counteexamples to Hlbet's fouteenth poblem., pepnt No. 37, August 00. [Nag6] asayosh Nagata, On atonal sufaces II., em. Coll. Sc. Unv. Kyoto Se. A ath. 33 (960/96), 7{93. [Sam64] Pee Samuel, Lectues on unque factozaton domans, Tata Inst

12 E. JAVIER ELIZONDO, KAZUHIKO KURANO, AND KEI-ICHI WATANABE Insttuto de atematcas, Cudad Unvestaa, UNA, exco, D 0450, exco E-mal addess: jave@math.unam.mx Depatment of athematcs, ej Unvesty, Hgash-ta --, Tama-ku, Kawasak 4-857, Japan E-mal addess: kuano@math.mej.ac.jp Depatment of athematcs, College of Humantes and Scences, Nhon Unvesty, Setagaya-ku, Tokyo , Japan E-mal addess: watanabe@math.chs.nhon-u.ac.jp