THE DIVISOR CLASS GROUPS AND THE GRADED CANONICAL MODULES OF MULTI-SECTION RINGS

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1 THE DIVISOR CLASS GROUPS AND THE GRADED CANONICAL MODULES OF MULTI-SECTION RINGS KAZUHIKO KURANO Abstract. We shall descrbe the dvsor class group and the graded canoncal module of the mult-secton rng T (X; D 1,..., D s ) defned n (1.1) below for a normal projectve varety X and Wel dvsors D 1,..., D s on X under a mld condton. In the proof, we use the theory of Krull doman and the equvarant twsted nverse functor due to Hashmoto [4]. 1. Introducton We shall descrbe the dvsor class groups and the graded canoncal modules of mult-secton rngs assocated wth a normal projectve varety. Suppose that Z, N 0 and N are the set of ntegers, non-negatve ntegers and postve ntegers, respectvely. Let X be a normal projectve varety over a feld k wth the functon feld k(x). We always assume dm X > 0. We denote by C 1 (X) the set of closed subvaretes of X of codmenson 1. For V C 1 (X) and a k(x), we defne as ord V (a) l OX,V (O X,V /αo X,V ) l OX,V (O X,V /βo X,V ) dv X (a) ord V (a) V Dv(X) Z V, where α and β are elements n O X,V such that a α/β, and l OX,V ( ) denotes the length as an O X,V -module. We call an element n Dv(X) a Wel dvsor on X. For a Wel dvsor D n V V, we say that D s effectve, and wrte D 0, f n V 0 for any V C 1 (X). For a Wel dvsor D on X, we put H 0 (X, O X (D)) {a k(x) dv X (a) + D 0} {0}. Here we note that H 0 (X, O X (D)) s a k-vector subspace of k(x) Mathematcs Subject Classfcaton. Prmary: 14C20, Secondary: 13C20. Ths work was supported by KAKENHI

2 Let D 1,..., D s be Wel dvsors on X. We defne the mult-secton rngs T (X; D 1,..., D s ) and R(X; D 1,..., D s ) assocated wth D 1,..., D s as follows: (1.1) T (X; D 1,..., D s ) (n 1,...,n s) N 0 s H 0 (X, O X ( n D ))t n 1 1 t ns s k(x)[t 1,..., t s ] R(X; D 1,..., D s ) H 0 (X, O X ( n D ))t n 1 1 t n s s k(x)[t ±1 1,..., t ±1 s ] (n 1,...,n s ) Z s We want to descrbe the dvsor class groups and the graded canoncal modules of the above rngs. For a Wel dvsor F on X, we set M F H 0 (X, O X ( (n 1,...,n s ) Z s n D + F ))t n 1 1 t n s s k(x)[t ±1 1,..., t ±1 s ], that s, M F s a Z s -graded reflexve R(X; D 1,..., D s )-module wth [M F ] (n1,...,n s) H 0 (X, O X ( n D + F ))t n 1 1 t n s s. We denote by M F the somorphsm class of the reflexve module M F n Cl(R(X; D 1,..., D s )). For a normal varety X, we denote by Cl(X) the class group of X, and for a Wel dvsor F on X, we denote by F the resdue class represented by the Wel dvsor F n Cl(X). In the case where Cl(X) s freely generated by D 1,..., D s, the rng R(X; D 1,..., D s ) s usually called the Cox rng of X and denoted by Cox(X). Remark 1.1. Assume that D s an ample dvsor on X. In ths case, T (X; D) concdes wth R(X; D), and t s a Noetheran normal doman by a famous result of Zarsk (see Lemma 2.8 n [6]). It s well-known that Cl(T (X; D)) s somorphc to Cl(X)/ZD. Mor [8] constructed a lot of examples of non-cohen Macaulay factoral domans usng ths somorphsm. It s well-known that the canoncal module of T (X; D) s somorphc to M KX, and the canoncal sheaf ω X concdes wth M KX. Watanabe proved a more general result n Theorem (2.8) n [12]. We want to establsh the same type of the above results for mult-secton rngs. For R(X; D 1,..., D s ), we had already proven the followng: Theorem 1.2 (Elzondo-Kurano-Watanabe [2], Hashmoto-Kurano [5]). Let X be a normal projectve varety over a feld such that dm X > 0. Assume that D 1,..., D s are Wel dvsors on X such that ZD ZD s contans an ample Carter dvsor. Then, we have the followng: (1) R(X; D 1,..., D s ) s a Krull doman. (2) The set {P V V C 1 (X)} concdes wth the set of homogeneous prme deals of R(X; D 1,..., D s ) of heght 1, where P V M V. 2

3 (3) We have an exact sequence 0 ZD Cl(X) p Cl(R(X; D 1,..., D s )) 0 such that p(f ) M F. (4) Assume that R(X; D 1,..., D s ) s Noetheran. Then ω R(X;D1,...,D s ) s somorphc to M KX as a Z s -graded module. Therefore, ω R(X;D1,...,D s) s R(X; D 1,..., D s )- free f and only f K X ZD n Cl(X). Suppose that Cl(X) s fntely generated free Z-module generated by D 1,..., D s. By the above theorem, the Cox rng Cox(X) s factoral and ω Cox(X) M KX Cox(X)(K X ), where we regard Cox(X) as a Cl(X)-graded rng. The man result of ths paper s the followng: Theorem 1.3. Let X be a normal projectve varety over a feld k such that d dm X > 0. Assume that D 1,..., D s are Wel dvsors on X such that ND ND s contans an ample Carter dvsor. Put U {j tr.deg k T (X; D 1,..., D j 1, D j+1,..., D s ) d + s 1}. Then, we have the followng: (1) T (X; D 1,..., D s ) s a Krull doman. (2) The set {Q V V C 1 (X)} {Q j j U} concdes wth the set of homogeneous prme deals of T (X; D 1,..., D s ) of heght 1, where and Q j Q V P V T (X; D 1,..., D s ) n 1,...,n s N 0 n j >0 (3) We have an exact sequence 0 j U ZD j Cl(X) T (X; D 1,..., D s ) (n1,...,n s). q Cl(T (X; D 1,..., D s )) 0 such that q(f ) M F k(x)[t 1,..., t s, {t 1 j j U}]. (4) Assume that T (X; D 1,..., D s ) s Noetheran. Then ω T (X;D1,...,D s) s somorphc to M KX t 1 t s k(x)[t 1,..., t s, {t 1 j j U}] as a Z s -graded module. Further, we have q(k X + D ) ω T (X;D1,...,D s). 3

4 Therefore, ω T (X;D1,...,D s ) s T (X; D 1,..., D s )-free f and only f K X + D j U ZD j n Cl(X). Here, tr.deg k T denotes the transcendence degree of the fractonal feld of T over a feld k. Remark 1.4. Wth notaton as n the prevous theorem, ht(q j ) 1 f and only f j U. Ths wll be proven n Lemma 3.3. Snce ND ND s contans an ample Carter dvsor, Q j (0) for any j. Therefore, ht(q j ) 2 f and only f j U. 2. Examples Example 2.1. Let X be a normal projectve varety wth dm X > 0. Assume that all of D s are ample Carter dvsors on X. Then, T (X; D 1,..., D s ) s Noetheran by a famous result of Zarsk (see Lemma 2.8 n [6]). Assume that s 1. By defnton, U snce dm X > 0. By Theorem 1.3 (3), Cl(T (X; D 1 )) s somorphc to Cl(X)/ZD 1. By Theorem 1.3 (4), ω T (X;D1 ) s T (X; D 1 )-free module f and only f K X ZD 1 n Cl(X) (see Remark 1.1). Next, assume that s 2. In ths case, U {1, 2,..., s}. By Theorem 1.3 (3), Cl(X) s somorphc to Cl(T (X; D 1,..., D s )). By Theorem 1.3 (4), ω T (X;D1,...,D s) s T (X; D 1,..., D s )-free module f and only f K X D 1 D s n Cl(X). When ths s the case, K X s ample, that s, X s a Fano varety. Example 2.2. Set X P m P n. Let p 1 (resp. p 2 ) be the frst (resp. second) projecton. Let H 1 be a hyperplane of P m, and H 2 a hyperplane of P n. Put A p 1 (H ) for 1, 2. In ths case, Cl(X) ZA 1 +ZA 2 Z 2, and K X (m+1)a 1 (n+1)a 2. We have Cox(X) R(X; A 1, A 2 ) k[x 0, x 1,..., x m, y 0, y 1,..., y n ]. Cox(X) s a Z 2 -graded rng such that x s (resp. y j s) are of degree (1, 0) (resp. (0, 1)). Let a, b, c, d be postve ntegers such that ad bc 0. Put D 1 aa 1 + ba 2 and D 2 ca 1 + da 2. Then, both D 1 and D 2 are ample dvsors. Consder the mult-secton rngs: R(X; D 1, D 2 ) p,q Z Cox(X) p(a,b)+q(c,d) T (X; D 1, D 2 ) p,q 0 Cox(X) p(a,b)+q(c,d) Here, both R(X; D 1, D 2 ) and T (X; D 1, D 2 ) are Cohen-Macaulay rngs. 4

5 By Theorem 1.2 (4), we know R(X; D 1, D 2 ) s a Gorensten rng K X ZD 1 + ZD 2 n Cl(X) (m + 1, n + 1) Z(a, b) + Z(c, d). In ths case, we have U {1, 2} snce all of a, b, c and d are postve. Theorem 1.3 (4), we have By T (X; D 1, D 2 ) s a Gorensten rng K X + D 1 + D 2 0 n Cl(X) m + 1 a + c and n + 1 b + d. Example 2.3. Let a, b, c be parwse coprme postve ntegers. Let p be the kernel of the k-algebra map S k[x, y, z] k[t ] gven by x T a, y T b, z T c. Let π : X P Proj(k[x, y, z]) be the blow-up at V + (p), where a deg(x), b deg(y), c deg(z). Put E π 1 (V + (p)). Let A be a Wel dvsor on X satsfyng π O P (1) O X (A). In ths case, we have Cl(X) ZE + ZA Z 2, and K X E (a + b + c)a. Then, we have Cox(X) R(X; E, A) R s(p) : S[t 1, pt, p (2) t 2, p (3) t 3,...] S[t ±1 ], T (X; E, A) R s (p) : S[pt, p (2) t 2, p (3) t 3,...] S[t]. By Theorem 1.2 (4), we have ω R s (p) M KX In ths case, U {1}. By Theorem 1.3 (4), R s(p)(k X ) R s(p)( 1, a b c). ω Rs(p) M KX t 1 t 2 k(x)[t 1, t ±1 2 ] ω R s (p) t 1 t 2 k(x)[t 1, t ±1 2 ] R s(p)( 1, a b c) t 1 t 2 k(x)[t 1, t ±1 2 ] R s (p)( 1, a b c). Therefore, both of R s(p) and R s (p) are quas-gorensten rngs, that were frst proven by Sms-Trung (Corollary 3.4 n [11]). Cohen-Macaulayness of such rngs are deeply studed by Goto-Nshda-Shmoda [3]. Dvsor class groups of ordnary and symbolc Rees rngs were studed by Shmoda [10], Sms-Trung [11], etc. 3. Proof of Theorem 1.3 In ths secton, we shall prove Theorem 1.3. Throughout ths secton, we assume that X s a normal projectve varety over a feld k such that d dm X > 0, and D 1,..., D s are Wel dvsors on X such that ND ND s contans an ample Carter dvsor. We need the followng lemmta. They are well-known, but the author could not fnd a reference. 5

6 Lemma 3.1. Let G be an ntegral doman contanng a feld k. Let P be a prme deal of G. Assume that both tr.deg k G and tr.deg k G/P are fnte. Then, the heght of P s less than or equal to tr.deg k G tr.deg k G/P. Proof. Assume the contrary. Then there exsts a rng G whch satsfes the followng fve condtons: k G G. G s fntely generated (as a rng) over k. tr.deg k G tr.deg k G. tr.deg k G/P tr.deg k G /(G P ). tr.deg k G tr.deg k G/P < ht(g P ). However, usng the dmenson formula (e.g. 119p n [7]), we have ht(g P ) tr.deg k G tr.deg k G /(G P ) tr.deg k G tr.deg k G/P. It s a contradcton. q.e.d. Lemma 3.2. Let r be a postve nteger. Let F 1,..., F r be Wel dvsors on X. Let S be the set of all non-zero homogeneous elements of T (X; F 1,..., F r ). Then the followng condtons are equvalent: (1) There exst non-negatve ntegers q 1,..., q r such that r 1 q F s lnearly equvalent to a sum of an ample Carter dvsor and an effectve Wel dvsor. (2) There exst postve ntegers q 1,..., q r such that r 1 q F s lnearly equvalent to a sum of an ample Carter dvsor and an effectve Wel dvsor. (3) S 1 (T (X; F 1,..., F r )) k(x)[t ±1 1,..., t ±1 r ]. (4) Q(T (X; F 1,..., F r )) k(x)(t 1,..., t r ), where Q( ) denotes the feld of fractons. (5) tr.deg k T (X; F 1,..., F r ) dm X + r. Usng Theorem n [1], t s easy to see that T (X; F 1,..., F r ) s Noetheran f and only f T (X; F 1,..., F r ) s fntely generated (as a rng) over the feld H 0 (X, O X ). Therefore, f T (X; F 1,..., F r ) s Noetheran, then the condton (5) s equvalent to that the Krull dmenson of T (X; F 1,..., F r ) s dm X + r. Proof. (2) (1), and (3) (4) (5) are trval. Frst we shall prove (1) (3). Suppose r q F D + F, 1 where q s are non-negatve ntegers, D s a very ample Carter dvsor and F s an effectve dvsor. We put C H 0 (X, O X ( n F + md))t n 1 (3.1) 1 t nr r t m r+1 m Z r (n 1,...,n r ) N 0 k(x)[t 1,..., t r, t ±1 r+1]. 6

7 We regard C as a Z r+1 -graded rng wth C (n1,...,n r,m) H 0 (X, O X ( n F + md))t n 1 1 t n r r t m r+1. Then, we have T (X; F 1,..., F r ) (n 1,...,n r) N 0 r C (n1,...,n r,0), so T (X; F 1,..., F r ) s a subrng of C. Thus, S 1 C s a Z r+1 -graded rng such that S 1 T (X; F 1,..., F r ) (S 1 C) (n1,...,n r,0). (n 1,...,n r ) N 0 r Snce r 1 q F D s lnearly equvalent to an effectve dvsor F, there exsts a non-zero element a n H 0 (X, O X ( q F D)). For any 0 b H 0 (X, O X (D)), (at q 1 1 t qr r t 1 r+1)(bt r+1 ) s contaned n S. Therefore, S 1 C contans (bt r+1 ) 1. Hence, k(x) s contaned n S 1 C. Snce k(x) (S 1 C) (0,...,0), k(x) s contaned n S 1 T (X; F 1,..., F r ). By the assumpton of (1), there exsts a postve nteger l such that and (S 1 C) (lq1,...,lq r,0) 0 (S 1 C) (lq1 +1,lq 2,...,lq r,0) 0. Then, t s easy to see that t 1 S 1 C. Therefore, S 1 C contans k(x)[t ±1 1,..., t ±1 r ]. Hence S 1 T (X; F 1,..., F r ) concdes wth k(x)[t ± 1,..., t ± r ]. Next, we shall prove (5) (2). Let D be a very ample dvsor. Consder the rng R(X; F 1,..., F r, D). Frst, assume that H 0 (X, O X ( u F vd)) 0 for some ntegers u 1,..., u r, v such that v > 0. By the assumpton (5), there exsts postve ntegers u 1,..., u r such that H 0 (X, O X ( u F )) 0. Therefore we may assume that there exsts postve ntegers u 1,..., u r and v such that H 0 (X, O X ( u F vd)) 0. Here, we have u F vd + ( 7 u F vd).

8 Therefore u F s the sum of an ample dvsor vd and the dvsor u F vd whch s lnearly equvalent to an effectve dvsor. Next, assume that for any ntegers u 1,..., u r and v, (3.2) H 0 (X, O X ( u F vd)) 0 f v > 0. We put P (n 1,...,n r,m) Z r+1 m>0 R(X; F 1,..., F r, D) (n1,...,n r,m). By the assumpton (5), P s a prme deal of R(X; F 1,..., F r, D) of heght 1 by Lemma 3.1. (Here, snce D s an ample dvsor, tr.deg k R(X; F 1,..., F r, D) dm X + r + 1. Note that P s an deal of R(X; F 1,..., F r, D) by (3.2) above. By (5), tr.deg k R(X; F 1,..., F r, D)/P dm X + r.) However R(X; F 1,..., F r, D) has no homogeneous prme deal of heght 1 that contans H 0 (X, O X (D))t r+1 by Theorem 1.2 (2). Ths s a contradcton. q.e.d. Put A k(x)[t ±1 1,..., t ±1 s ] and B k(x)[t 1,..., t s ]. Recall that D 1,..., D s are Wel dvsors on a normal projectve varety X such that ND ND s contans an ample Carter dvsor. We denote T (X; D 1,..., D s ) and R(X; D 1,..., D s ) smply by T and R, respectvely. Snce T R B, T s a Krull doman. We have proven Theorem 1.3 (1). By Theorem 1.2 (2), we have R A A P NHP 1 (R) where NHP 1 (R) s the set of non-homogeneous prme deals of R of heght 1. It s easy to see R P T P T for P NHP 1 (R). Therefore, we have A T P T. P NHP 1 (R) Snce T P T s a dscrete valuaton rng, P T s a non-homogeneous prme deal of T of heght 1. For V C 1 (X), put Q V P V T. Then, R PV T QV, snce ND contans an ample dvsor. Therefore Q V s a homogeneous prme deal of T of heght 1. 8 R PV R P,

9 On the other hand, we have Q T t B (t ) and T Q B (t ). Note that s B A ( B (tj )). Then, we have (3.3) Put T j T R B R PV T QV j1 A B P NHP 1 (R) (n 1,...,n j 1,n j+1,...,n s ) N 0 s 1 H 0 (X, O X ( j We need the followng lemma. T P T ( s j1 B (tj ) ) n D ))t n 1 1 t n j 1 j 1 tn j+1 j+1 tns s. Lemma 3.3. Wth notaton as above, the followng condtons are equvalent: (1) T Qj B (tj ). (2) The heght of Q j s 1. (3) The heght of Q j s less than 2. (4) j U, that s, tr.deg k T j d + s 1. Proof. By Lemma 3.2, we have Q(T ) Q(B). It s easy to see that B (tj ) s a dscrete valuaton rng. Snce Q j s a non-zero prme deal of a Krull doman T, the equvalence of (1), (2) and (3) are easy. Here, we shall prove (1) (4). Note that T/Q j T j. Then, we have Q(T j ) T Qj /Q j T Qj B (t )/(t )B (t ) k(x)(t 1,..., t j 1, t j+1,..., t s ). The mplcaton (4) (3) mmedately follows from ht(q j ) tr.deg k T tr.deg k (T j ) 1. Ths nequalty follows from Lemma 3.1 and the fact T j T/Q j. By (3.3), Lemma 3.3 and Theorem 12.3 n [7], we know that {Q V V C 1 (X)} {Q j j U} s the set of homogeneous prme deals of T of heght 1, and {P T P NHP 1 (R)} s the set of non-homogeneous prme deals of T of heght 1. Further we obtan T ( ) T Qj. T QV P NHP 1 (R) 9 T P T. q.e.d.

10 The proof of Theorem 1.3 (2) s completed. Let Dv(X) Z V be the set of Wel dvsors on X. Let HDv(T ) ( ) Z Spec(T/Q V ) Z Spec(T/Q j ) be the set of homogeneous Wel dvsors of Spec(T ). Here, we defne ϕ : Dv(X) HDv(T ) by ϕ(v ) Spec(T/Q V ) for each V C 1 (X). Then, t satsfes the followng: For each a k(x), we have ϕ(dv X (a)) dv T (a) Z Spec(T/Q V ) HDv(T ). If j U, then dv T (t j ) Spec(T/Q j ) + ϕ(d j ). If j U, then dv T (t j ) ϕ(d j ). They are proven essentally n the same way as n pp n [2]. Then, we have an exact sequence 0 j U ZD j Cl(X) q Cl(T ) 0 such that q(f ) ϕ(f ) n Cl(T ). Here, remember that Cl(T ) concdes wth HDv(T ) dvded by the group of homogeneous prncpal dvsors (e.g., Proposton 7.1 n Samuel [9]). It s easy to see that the class of the Wel dvsor q(f ) corresponds to the somorphsm class of the reflexve module ( ) ( ) M F T Qj M F A T Qj M F k(x)[t 1,..., t s, {t 1 j The proof of Theorem 1.3 (3) s completed. Remark 3.4. It s easy to see j U}]. t d 1 1 t d s s M F + d D M F for any ntegers d 1,..., d s. Therefore, we have M F t d 1 1 t d s s k(x)[t 1,..., t s, {t 1 j U}] t d 1 1 t ds s ( MF + j d D k(x)[t 1,..., t s, {t 1 j j U}] ). 10

11 Hence, s somorphc to M F t d 1 1 ts ds k(x)[t 1,..., t s, {t 1 j j U}] (3.4) M F + d D k(x)[t 1,..., t s, {t 1 j j U}] as a T -module. Note that ths s not an somorphsm as Z s -graded modules. The somorphsm class whch the module (3.4) belongs to concdes wth q(f + d D ). In the rest, we assume that T s Noetheran. We shall prove that ω T s somorphc to M KX t 1 t s k(x)[t 1,..., t s, {t 1 j j U}] as a Z s -graded module. (Suppose that t s true. If we forget the gradng, t s somorphc to M KX + D k(x)[t 1,..., t s, {t 1 j j U}] by Remark 3.4, that s correspondng to q(k X + D ) n Cl(T ). Therefore, we know that ω T s T -free f and only f K X + D j U ZD j n Cl(X).) Put X X \ Sng(X). We choose postve ntegers a 1,..., a s and sectons f 1,..., f t H 0 (X, a D ) such that a D s an ample Carter dvsor, X k D + (f k ), and all of the D s are prncpal Carter dvsors on D + (f k ) for k 1,..., t. Put W {n Z s n 0 f U}. Put D D X for 1,..., s. Consder the morphsm Y Spec X O X ( n W Further, we have the natural map ξ : Y Spec(T ). n D )t n 1 1 t n s π X. The group G s m naturally acts on Spec(T ) and Y, and trvally acts on X. Both π and ξ are equvarant morphsms. Clam 3.5. There exsts an equvarant open subscheme Z of both Y and Spec(T ) such that the codmenson of Y \ Z n Y s bgger than or equal to 2, and the codmenson of Spec(T ) \ Z n Spec(T ) s bgger than or equal to 2. Proof. For u U, there exst ntegers c 1u,..., c su such that H 0 (X, O X ( c ud )) 0, c uu a u, and c u > 0 f u. 11 s

12 In fact, f u U, there exst postve ntegers q 1,..., q u 1, q u+1,..., q s such that q D u s a sum of an ample dvsor D and a Wel dvsor F whch s lnearly equvalent to an effectve dvsor by Lemma 3.2. Then, H 0 (X, O X (q( u for q 0. For each u U, we set q D ) a u D u ) H 0 (X, O X (q(d + F ) a u D u ) 0 (b 1u,..., b su ) (c 1u,..., c su ) + (a 1,..., a s ). Here, note that b uu 0 and b u > 0 f u. We choose 0 g u H 0 (X, O X ( c u D )) for each u U. Consder the closed set of Spec(T ) defned by the deal J generated by and {f k t a 1 1 t a s s k 1,..., t} { gu f k t b 1u 1 t b su s k 1,..., t; u U }. By Theorem 1.3 (2), we know that the heght of J s bgger than or equal to 2 snce there s no prme deal of T of heght one whch contans J. We choose d k k(x) satsfyng H 0 (D + (f k ), O X (D )) d k H 0 (D + (f k ), O X ) for each k and. Then t (3.5) Y π 1 (D + (f k )) and π 1 (D + (f k )) Spec(C k ), k1 k1 where C k H 0 (D + (f k ), O X )[d k1 t 1,..., d ks t s, {(d kj t j ) 1 j U}]. We put Z Spec(T ) \ V (J). Then we have [ t (3.6) Z Spec ( T [(f k t a 1 1 t a s s ) 1 ] ) { Spec ( T [(g u f k t b 1u 1 t b su s ) 1 ] )}]. Here, we have (3.7) T [(f k t a 1 1 t as s ) 1 ] H 0 (D + (f k ), O X )[(d k1 t 1 ) ±1,..., (d ks t s ) ±1 ] ( 1 C k [ (d kj t j )) ]. 12 u U

13 On the other hand, T [(g u f k t b 1u 1 t b su s ) 1 ] 1 t bsu s ) 1 ] (n1,...,n s ) T [(g u f k t b1u (n) Z s (n) Z s n u 0 (n) Z s n u 0 R[(g u f k t b 1u 1 t bsu s ) 1 ] (n1,...,n s ) R[(f k t a 1 1 t a s s ) 1, (g u f k t b 1u 1 t b su s ) 1 ] (n1,...,n s) C k [{(d kj t j ) 1 j u}, (g u f k t b 1u 1 t bsu s ) 1 ]. Let β ku be an element n H 0 (D + (f k ), O X ) such that g u f k t b 1u 1 t b su s β ku (d k1 t 1 ) b 1u (d ks t s ) b su for k 1,..., t and u U. Then, (3.8) C k [{(d kj t j ) 1 j u}, (g u f k t b 1u 1 t b su s ) 1 ] C k [ β ku (d kj t j ) By (3.5), (3.6), (3.7) and (3.8), we know that Z s an open subscheme of Y. The deal of C k generated by (d kj t j ) and β ku (d kj t j ) u U s the unt deal or of heght two. (If U, then Z Y by the constructon. If U {u} and f β ku s a unt element, then ths deal s the unt. In other cases, ths deal s of heght 2.) Therefore, the codmenson of Y \ Z n Y s bgger than or equal to two. q.e.d. We can defne the graded canoncal module as n Defnton 3.1 n [5] usng the theory of the equvarant twsted nverse functor [4]. By Clam 3.5 above and Remark 3.2 n [5], we have ω T H 0 (Y, ω Y ). On the other hand, we have j u j u 1 ]. ω Y s ΩY/X π O X (K X ) π O X ( π O X ( D )( 1,..., 1) OY π O X (K X ) D + K X )( 1,..., 1), where ( 1,..., 1) denotes the shft of degree (Theorem n [4]). 13

14 Then, we have H 0 (Y, ω Y ) H 0 (X, π π O X ( D + K X )( 1,..., 1)). By the projecton formula (Lemma 26.4 n [4]), π π O X ( D + K X )( 1,..., 1) ( O X ( ) D + K X ) π O Y ( 1,..., 1) O X ( D + K X ) O X ( n D ) ( 1,..., 1) n W n W O X ( (n + 1)D + K X ) ( 1,..., 1) Therefore, we have n W +(1,...,1) O X ( H 0 (Y, ω Y ) H 0 (X, n D + K X ). O ( X n W +(1,...,1) n D + K X )) n D + K X )) H 0 (X, O ( X n W +(1,...,1) n D + K X )) n W +(1,...,1) H 0 (X, O X ( M KX t 1 t s k(x)[t 1,..., t s, {t 1 j We have completed the proof of Theorem 1.3. References j U}]. [1] W. Bruns and J. Herzog, Cohen-Macaulay rngs, Cambrdge Studes n Advanced Mathematcs 39, Cambrdge Unversty Press, [2] E. J. Elzondo, K. Kurano and K.-. Watanabe, The total coordnate rng of a normal projectve varety, J. Algebra 276 (2004), [3] S. Goto, K. Nshda and Y. Shmoda, The Gorenstenness of symbolc Rees algebras for space curves, J. Math. Soc. Japan 43 (1991), [4] M. Hashmoto, Equvarant Twsted Inverses, n Foundatons of Grothendeck Dualty for Dagrams of Schemes (J. Lpman, M. Hashmoto, eds.), Lecture Notes n Math. 1960, Sprnger (2009), pp [5] M. Hashmoto and K. Kurano, The canoncal module of a Cox rng, Kyoto J. Math. 51 (2011), [6] Y. Hu and S. Keel, Mor dream spaces and GIT, Mchgan Math J. 48 (2000), [7] H. Matsumura, Commutatve rng theory, Cambrdge Unversty Press,

15 [8] S. Mor, On affne cones assocated wth polarzed varetes, Japan J. Math. 1 (1975), [9] P. Samuel, Lectures on unque factorzaton domans, Tata Inst. Fund. Res., Bombay, [10] Y. Shmoda, The class group of the Rees algebras over polynomal rngs, Tokyo J. Math. 2 (1979), [11] A. Sms and N. V. Trung, The dvsor class group of ordnary and symbolc blow-ups, Math. Z. 198 (1988), [12] K.-. Watanabe, Some remarks concernng Demazure s constructon of normal graded rngs, Nagoya Math. J. 83 (1981), Department of Mathematcs School of Scence and Technology Mej Unversty Hgashmta 1-1-1, Tama-ku Kawasak , Japan kurano@sc.mej.ac.jp 15