Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. XXXVI, 159-174 (1990)

On Monomial k-Buchsbaum Curves in pr. MARIO F I O R E N T I N I - L E T U A N HOA(*)

1. -

Introduction.

In this paper a curve means an one-dimensional closed subscheme C o f P r which is locally Cohen-Macaulay and equidimensional. L e t R = k[xo,..., xr] and n = (x0, ...xr) c R, where k is an algebraically closed field. The module

M(C) = ( ~ H 1 ( p r , ~C (n)), n

where ~c is the ideal sheaf of C, is called the Hartshorne-Rao module of C (cf. [23]). We will say that C is k-Buchsbaum if n k M(C) = 0. It is clear that if C is k-Buchsbaum then C is k'-Buchsbaum for any k' > k. C is called strictly k-Buchsbaum if C is k-Buchsbaum and it is not (k - 1)-Buchsbaum. It follows from [11] that C is arithmetically Cohen-Macaulay (resp. Buchsbaum) iff C is 0-Buchsbaum, i.e. M(C) = 0, (resp. 1-Buchsbaum). In this sense, the concept of k-Buchsbaum curves is a natural generalization of that of arithmetically Buchsbaum curves. This notion was introduced recently in [8] and [9]. F o r any curve C c P ~ , there is a unique integer kc such that C is strictly kcBuchsbaum. This is an important invariant of C. F o r example, knowing kc, one can give upper bounds for the Castelnuovo's regularity or for the degrees of the defining equations of C (see [20], [21] and [18]). In [10] Gaeta gives a geometrical characterization of k-Buchsbaum curves. The aim of this paper is to estimate the number kc of a monomial curve. By a monomial curve we mean a curve of P~ with generic zero (84, 8 d - a i r al , ..., s d - a r - l t at-1 , t d } , where 0 < al < ... < at-1 < d are integers with g.c.d. (al, ...,ar-l,d)= 1. Such a curve is also called a projection of one-dimensional Veronese variety, see [27]. On the other hand, studying Veronese varieties and their projections,

(*) Indirizzo degli Autori: MARIOFIORENTINI: Depart. Math. Univ. Ferrara, via Machiavelli 35, 44100 Ferrara, Italia; LE TUAN HOA: Institute of Math., Box 631, Boho, Hanoi, Vietnam.

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MARIO F I O R E N T I N I - LE T U A N HOA

Gr6bner has posed in [14] the problem of classifying these projections according to when they are arithmetically Cohen-Macaulay or not (see [16] and [27] for more informations on this problem). Hence, one may consider the problem of finding kc as an extended GrSbne~s problem for projections of one-dimensional Veronese varieties. A precise description of those monomial curves C of p3 for which kc = 1 has been given by Bresinsky, Schenzel and Vogel in [6]. Using the theory of affine semigroup rings one has already found some criteria for kc = 0 or kc = 1 ([2], [16] and [27]). Following these works we will give an effective method to give bounds for kc or to compute kc explicitly. Now, let us briefly describe the content of the paper. In Section 2 we first recall some basic facts on affine semigroup rings. Then we give a criterion for a monomial curve C to be k-Buchsbaum in terms of the associated affme semigroup of C. Using this result, we obtain, in Section 3, lower and upper bounds for kc. In Section 4 we can determine explicitly the number kc for the case al = 1 and a~_ ~ = d - 1. In Section 5 we give a practical method to compute kc for monomial curves in p 3 . This method combines the theory of affine semigroup rings and the theory of liaison of curves in P~. 2. -

Preliminaries.

Throughout this paper, we set R = k[x0, ..., x~] and n = (x0, ..., x0. Let C be a curve in p 3 . Let I(C) denote the defining ideal of C. Set A = R/I(C) and m = m4. It is known that riM(C) = 0 iff mHlm (A) = 0 (see [11], proof of Theorem 1.3). Using the proof of[24], Corollary 4.7 we have the following generalization: LEMMA 2.1.

C is a k-Buchsbaum curve iff m kHim (A) = 0.

More generally, if A = ~ An is a d-dimensional graded k-algebra with n_>0

maximal homogeneous ideal m(d >- 1), then A is called k-Buchsbaum ring if mkH~ (A) = 0 for i = 0, ..., d - 1. Lemma 2.1 says that a curve C is k-Buchsbaum iff the homogeneous coordinate ring of C is k-Buchsbaum. Now, let C be a monomial curve in p r given parametrically by (8 d, 8d-ch ta, , ..., S d - a , - , ta~-l , t d } ,

where 0 =: ao < al < a2 < ... < ar_l < a~ :-- d and g.c.d. (a~, ..., a~) = 1. Let e~ denote the tuple (d - a, a). Let N denote the set of non-negative integers. We denote by S the semigroup in N 2 generated by the set I = {e~; i = 0 , . . . , r ) .

ON MONOMIAL k-BUCHSBAUM CURVES IN p r

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Then one can identify the homogeneous coordinate ring of C with the naturally graded semigroup ring k[S]. Hence, C is k-Buchsbaum iff k[S] is a k-Buchsbaum ring. In [13] and [25] one has already given conditions for k[S] to be a Cohen-Macaulay or Buchsbaum ring in terms of the underlying semigroup S. The aim of this section is to give an extension of these results for k-Buchsbaum affine semigroup rings. For that we need some more notations. Let H denote the following semigroup in N2: H = {(~, fl) 9 N 2; ~ + fl --- 0 mod d}.

If A, B c_N z , we denote by A + B the set of elements a + b with a 9 A and b 9 B. If x 9 N 2 , [x]o and [X]l denote the first and the second components of x, respectively. Further, we denote by So = (d, d - a l , . . . , d - a t - 1) (resp. $1 = = (ao, ..., at)) the numerical semigroup in N generated by the first (resp. the second) components of all elements in the generating set I of S. For i = 0,1 the set Ai = {0, coi(1), ..., ~ i ( d - 1)}, where

~,i(3")=min{aeSi; a---jmodd},

j= l,...,d-1,

is called the Apery sequence of Si (see [2]). We set

S'={e 9149

and e + n e d 9

for some m , n 9

Then we have: LEMMA 2.2 ([28],

Corollary 3.4).

HI(k[S])=k[S'\S],

where m =

= k[S\{0}].

LEMMA 2.3 ([2], Lemma 2.1). LEMMA 2.4. e 9215

S'=Hn(So•

Every element of S' has the form e + mo eo + ml ed for some and mo,ml 9

PROOF. Let u e S ' . By Lemma 2.3, [u]i 9 Si(i = O, 1). By the definition of Ai there exist a i 9 and m i 9 such that [u]i=ai+mid. Then e=(ao,al)eHn(Ao• and we have u=([U]o,[U]l)=e+moeo+ + m~ ea, q.e.d. The following easy, but key proposition extends results of[13] and [25]. PROPOSITION 2.5. k[S] is a k-Buchsbaum ring iff one of the following equivalent conditions is satisfied: (i) S ' + k [ S \ ( 0 ) ] _cS, (ii) Let J = H (~ (Ao • A 1 ) \ S . Then J + kI c_S, where kA denotes the set A + . . . + A (k times).

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PROOF. The equivalence of (i) and (ii) follows from Lemma 2.4 and the fact that S is generated by I. The equivalence of k-Buchsbaum property of k[S] and (i) follows from Lemma 2.2, q.e.d. REMARK 1. i) Proposition 2.5 says that k[S] is a Cohen-Macaulay ring (*> k = 0) iff J = 0. ii) Lemma 2.2 is valid only for 2-dimensional affine semigroup rings ([28], Corollary 3.4). But analyzing the proof of[28], Lemma 4.11 one can give an another proof of Proposition 2.5 without using Lemma 2.2. Moreover, with the last mentioned proof one can also obtain a criterion, similar as in Proposition 2.5(i), for an arbitrary affine semigroup ring to be k-Buchsbaum (under the assumption that the corresponding ring k[S'] is a CohenMacaulay ring). For an element e e H, the number 8(e) = ([e]0 + [e]l)/d is called the degree of e. Let U= 0. If u e U, we define the degree of u with respect to U and to (ul,...,u~) as follows: 8U(u) = min (Pl + -.. + Ps; u = Pl ul + ... + p~ us and Pi e N}. This definition is dependent on the choice of a set of generators (ul, ...,us). In this paper, a numerical semigroup will be always given by a explicitely indicated set of generators. Hence, the notation 8~(u) can not arise any confusion. For short we shall write 8i(x) for 8s~(x), where i - - 0 or 1. LEMMA 2.6. (cfr. [2], Lemma 3.1 and [19], Lemma 1). Then the following three conditions are equivalent:

Let e e S' \ (0).

(i) e e S ,

(ii) 8(e) >_ 80 ([e]o), (iii) 8(e) > 81([e]1). REMARK 2. A simple method for computing the degree 8u(U) of u e U with respect to U = (ul, ..., us) is the following: {1 8u(0) := 0,

By(l) =

ifleU, ff 1 ~ U,

8u(U) = 1 + min {Su(U-Ui);u~ <-u, l <_i<_n}. With this remark, Proposition 2.5 and Lemma 2.6 we have an algorithm to check whether k[S] is a k-Buchsbaum ring or not. But this algorithm is, of

ON MONOMIAL k-BUCHSBAUM CURVES IN P ~

163

course, still very complicated. F o r k = 1, there is an effective algorithm for some subclasses of monomial curves in [16].

3. -

General

case.

In this section we will give some estimations for the number k c . REMARK 3. Denote by C ( a l , . . . , a t - 1, d) the monomial curve given parametrically by { S d , s d - = l t = l , . . . , s g - ~ - l t ~ ' - l , t d } . If we interchange s and t then we obtain the curve C(d - a r - 1 , ..., d - a l , d). Hence kc(al ..... ~,_~,~) = = kc(d_a,_~,...,d_al,d

).

We need the following notations. L e t M be a graded A-module. We define (i) a(M) = inf {n; [M]. ~ 0}, (ii) e(M)= sup {n; [M]. r 0}, (see[24], p. 36). PROPOSITION 3.1.

Assume that J r 0. Set

= max (~o ([e]o) - d(e), dl ([e]l) - d(e)}, eeJ

and = max{do ([e]o) + dl ([ell) - d(e)} - min ~(e) - 1. eeJ

eeJ

Then 2 < k c -<~. PROOF. F o r short, we set k = k c . F i r s t we will prove that k-> Z. L e t e 9 J be an arbitrary element. By Proposition 2.5, we have e + keo 9 S and e + keg 9 S (recall t h a t eo = (d, 0) and ed = (0, d)). By L e m m a 2.6 this follows that do([e]o) = do([e + keg]o) <- d(e + keg) = ~(e) + k , and d1 ([e]l) = dl ([e + keo]l) -< d(e + keo) = d(e) + k . Hence k -> 2. To prove the second inequality it suffices to show that k[S] is an h-Buchs-

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baum ring for some h-< tz. Set , ' = m a x {8o ([e]o) + 81 ([e]l) - 8(e)). Let f e S' \ S be an arbitrary element. By L e m m a 2.4, f = e + mo eo + mled for some e e J (because f ~ S) and too, ml 9 N. Since f ~ S, we must have eo + moeo ~ S and e + mled ~ S. Again, by L e m m a 2.6, this gives 8(e) +mo = 8(e + moeo) < 8! ([e + mo eo]l) = 8! ([e]O,

and 8(e) + ml = 8(e +

m 1 ea)

< 8o ([e +

m 1 ed]o)

= 8o ([e]o).

F r o m this it follows that 8 ( f ) = 8(e) + mo + ml - 8o ([e]o) + 81 ([ell) - ~(e) - 2 ---ix'- 2. On the other hand, it is obvious that 8(f) ___rain 8(e). Hence, by L e m m a s 2.2 eEJ

and 2.4 it follows that e(H~ (k[S])) _
The following example and Example 2 show that both bounds of Proposition 3.1 are sharp. EXAMPLE 1. Let C = C ( 1 , 4 , 9 ) . In this case S 0 = ( 9 , 8 , 5 ) , S1 = = (1, 4, 9) = N, Ao = (0, 5, 8, 10, 13, 15, 16, 20, 21)andA1 = {0, 1,..., 8}. Hence, using Lemma 2.6, we get easily that J = {(15, 3), (20, 7)). We have 8o(15) = = 81(3) = 3, 8((15, 3)) = 2 and 8o (20) = 81 (7) = 4, 8((20, 7)) = 3. Hence ~ = 1, tz = 2. Also 1 <- kc <- 2. Using [16], Theorem 3.2 one can easily check that k[S] isn't a Buchsbaum ring. Hence kc = 2. PROPOSITION 3.2. sition. Set hi =

Suppose that J r 0. Let ~ be as in the previous propomax

eeJ 0,
{8~([e]i + md) - 8(e)),

and oJ = maxe~j8(e) - miE'n8(e).

i = 0, 1,

ON MONOMIAL k-BUCHSBAUM CURVES IN p r

165

Then max{~o, ~} - k c <- m i n {~o, ~1} + co. In particular, if ~ = 0, i.e. if J consists of elements of the same degree, then k c = ~c = ~1.

PROOF.

Set k = k c . Without loss of generality, we can assume that

~o --- 21. F i r s t we shall show that ~o + o~_ k, or equivalent, that k[S] is a (~o + oJ)Buchsbaum ring. L e t e 9 J and f 9 (2o + ~)I. Then 8(f) = ~o + o~. B y Proposition 2.5 we have to show that e + f e S. Since e + f 9 S', by L e m m a 2.4 it follows that e + f = u + meo + ned for some u 9 H n (Ao X A1) and m , n e N . If u 9 S, there is nothing to prove. Hence one can assume that u 9 J. Moreover, since u + ~eo does belong already to S, it suffices to consider the case m -< ~ - 1. By the definition of ~ we get t h a t 8(u) - 8(e). Then, from the formula of ~o it follows that 8o ([e + f ] o ) = 8o ([u + meo + ned]o) = 8o ([u]0 + rod) -< 8(u) + ~o -_ 8(u) + ~ + ;(o = 8(e) + 8 ( f ) = 8(e + f ) . Hence, b y L e m m a 2.6, e + f e S. Now, we will show that k -> lb. Take an element e 9 J and 0 <- m -- 2 - 1 such that 21 = 81([e]1 - md) - 8(e). Note that ~1 = max(~o, ~1} -> ~. Hence ~1 - 1 - m -> 0. Consider the element f = meg + ()~1- 1 - m) eo 9 (~1 - 1) I. Then we have 81 ( [ f + e]l) = 81 ([e + reed + (~1 - 1 - m) eo]l) = 81 ([e]l + md) = = ~1 + 8(e) > ~1 - 1 + 8(e) = 8 ( f ) + 8(e) = 8(e + f ) .

B y L e m m a 2.6, e + f ~ S. This means k[S] isn't a (~1- 1)-Buchsbaum ring which follows t h a t k>-~l, q.e.d. REMARK 4.

L e t C be an a r b i t r a r y curve in P ~ . The n u m b e r

diam M ( C ) = e ( M ( C ) ) - a ( M ( C ) ) + 1 =

= e ( g ~ (R/I(C))) - a(H~ (R/I(C))) + 1 is called d i a m e t e r of M ( C ) . We obviously have k c <- diatoM(C).

Note that we have used this fact in the proof for the u p p e r bound of Proposition 3.1. T h e r e w e r e given in [1], [12] and [20] some classes of curves in p a ,

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MARIO FIORENTINI - LE TUAN HOA

for which kc -- diatoM(C). This result is also valid for an a r b i t r a r y monomi-,d curve in P " with ~ = 0. Indeed, in this case 8(e) = n is a constant for all e e J. L e t f b e an element o f S ' \ S with the maximal degree. Then, b y L e m m a 2.2, it follows immediately that diam M(C) = ~ f ) - n + 1. On the other side, b y L e m m a 2.4, f = u + n o e o + m ~ e d for some u e J . Hence m o + m i = ~ ( f ) - 8(u) = 8 ( f ) - n = diam M(C) - 1. Since u + mo eo + ml ed = f ~t S, we t h e n get kc >- diam M(C). Also kc = diamM(C). In general, this is not true. As examples we can take the Buchsbaum monomial curves constructed in[19], Satz 4. EXAMPLE 2 (cf. also [2A], p. 171 and [9], R e m a r k 9). L e t us consider a monomial curve C in p 3 lying on a smooth quadric, i.e. C is given by {sa+b,satb,sbta,t"+b}, where a > b > 0 and g.e.d.(a,b)=l. Then we have A0 = Al = (0, b, 2b,..., (a - 1) b, a, 2 a , . . . , (b - 1) a, ba}. Hence J = ((ib, (d - i)b; 1 + b _< i <_ a - 1 }. All elements of J have the same d e g r e e b. Hence, by Proposition 3.2, kc = a - b - 1. Applying Proposition 3.1 we get aL~o the same result (~ = ~ = b - a - 1).

4. - C a s e

al = 1 and a~_ 1 = d-

I~

In this case, So = $1 = N; Ao = Al = {0, 1, ..., d - 1}. Propositions 3.1 and 3.2 become much simpler as follows

PROPOSITION 4.1. (i)

Assume that al = 1, at-1 = d - 1

max {80(m), 8 1 ( m ) } - 1 =: 2' <-kc -<

O
(ii) k c = PROOF.

max

0
8o(m)-1=

max

0
max

O
and r < d . T h e n

{8o(m)+Sl(d-m)}-3,

81(m)-1.

U n d e r the assumption of the proposition we have

J = {eL; 0 < a < d

and a C a o , ...,at} r

Hence ~ = 0 and 8(e) = 1 for all e e J. The proposition follows now immediately from Propositions 3.1 and 3.2, q.e.d. COROLLARY 4.2 ([2], Corollary 4.8 and [27], T h e o r e m 4.1). Assume that a~ = 1 and a, _~ = d - 1 . Then C is arithmetically Buchsbaum iff for all 0
ON MONOMIAL k-BUCHSBAUM CURVES IN p r

167

consider the case r < d. Then we have max {80 (m) + 81(d - m)} - 3 -< 2 ~ ' - 1.

0
Hence ~' -< kc - 2 ~ ' - 1. F r o m this it follows t h a t kc = 1 iff ~' = 1, i.e. iff 8 0 ( m ) - 2 and 81(m)-<2 for all 0 < m < d , q.e.d. Under the assumption of Proposition 4.1 one can easily deduce from [28], Corollary 3.8 that e(H 2 (k[S])) < 0. By R e m a r k 4 we have e(H 1 (k[S])) = kc (because a(H~(k[S])) = 1). Hence 1 + kc = reg (k[S]) = index of regularity of the Hilbert function of C = n c , where reg(k[S]) denotes the Castelnuovo's regularity of k[S] (see [7] and [18] for the definition and some basic facts on Castelnuovo's regularity of kBuchsbaum algebras) and nc is defined in [26]. So, the equality (ii) of Proposition 4.1 gives us an upper bound for the defining equations of C, namely: COROLLARY4.3.

mc<-kc+2 =

max

0
8o(m)+1=

max

0
81(m)+1,

where m c denotes the maximal degree of a minimal basis of I(C). The inequality m c <-kc + 2 extends also[26], Corollary 1 which states t h a t for a monomial Buchsbaum curve with al = 1 and a t - 1 = d - 1 there is a basis for the defining prime ideal of C consisting of binomials of degrees - 3. F o r integers a,b r 0 we set [a/b] = max {n e N; n <_a / b ) , ]a/b[= n'fin {n e N; n >- a / b } .

Then we have the following result which was stated without proof in [18], Example 3.8. COROLLARY4.4 (cf.[26], where 1 <- a <- b <- d-1. Then

Corollary

2).

Let

C = C(1, ..., a, b, ..., d),

a

PROOF. Without restriction, we m a y assume that a + b <-d. Then one can easily check that [(b - 2)/a] -> [(d - a - 2)/(d - b)]. L e t 2' be the number defined in Proposition 4.1. Since ] m i n i = [(m - 1) In] + 1 for every integers

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m, n_> 1, we have

'

a 1[}1

d-b

= @ "

On the other hand, for m <- b we have (d - m ) / ( d - b) <- (b + a - m ) / a . Hence l(d - m ) / ( d - b)[ <- ](b + a - m)/a[. Also, for 1 <- m ~ b we have

e.(m)+el(d-m)----g

+

<_

+ b+a

g

m

B

(because [~1 + ~] ~ ~ +/~). Hence m a x . { 8 o ( m ) + ~ l ( d - m ) } - 3 --

0
max

a+l-m";b-1

{~o(m)+~1(d-m)} -3 < -

<- [(b - 2)/a] = ~'. By Proposition 4.1(i) we get kc = ~'= [ ( b - 2)/a],

q.e.d.

REMARK 5. Corollary 4.4 shows that for this special case the lower bound of Proposition 4.1 is sharp. The authors coldn't find an example for which kc > ~' in this proposition. EXAMPLE 3. Let C = C ( 1 , . . . , a , d - l , d ) , where l < - a < - d - 2 . Then, by Corollary 4.4, C is a strictly ( d - a - 2)-Buchsbaum curve. This example shows t h a t for every h, h <- d - 3, there exists a projective monomial curve of a given degree d which is strictly h-Buchsbaum. Note that, by a result of[15] on Castelnuovo's regularity, a reduced irreducible curve of degree d in p r is always (d - r)-Buchsbaum (see [18], Corollary 3.1.6).

5. - k - B u c h s b a u m

monomial

curves in

p3.

We need the following technical, but useful lemma. It is an improvement of the lower bound in Proposition 3.1. LEMMA 5.1. Assume that for an element e 9 we have $/([e]i)> $(e) (i = 0, or 1), and there exist f 9 S ' , f l , -.-,fh 9 I such that e = f + f l + ... +fh. Then kc >- h + $i ([e]i) - $(e). PROOF. Without restriction, we may assume that i = 0 . L e t n = -- ~o([e]0) - $(e) - 1 _>0. Then, by L e m m a 2.6, e + ned ~ S. H e n c e f + f l + . . . +

169

ON MONOMIAL k-BUCHSBAUM CURVES IN p r

+fh + ned ~ S. A l s o , b y P r o p o s i t i o n 2.5, w e m u s t h a v e kc>h+n=h+$o([e]o)-~(e)-l,

q.e.d.

a) C u r v e s g i v e n by {s d, s ~- 1 t, s d - ~ t ~, tg}. I n t h i s s e c t i o n w e s h a l l c o n s i d e r t h e m o n o m i a l c u r v e s C = C(1, a , d). I f a = d - 1, b y C o r o l l a r y 4.4 o r E x a m p l e 2 w e k n o w t h a t k c = d - 3. O n t h e o t h e r h a n d , f r o m [27], T h e o r e m 3.5 i t f o l l o w s t h a t C is a r i t h m e t i c a l l y C o h e n M a c a u l a y iff [(d - 1 ) / ( a - 1)] > [(d - 1 ) / a ] . I n g e n e r a l , w e h a v e t h e f o l l o w i n g estimation: LEMMA 5.2.

L e t C = C(1, a, d). A s s u m e

[(d - 1)/(a

that (r

- 1)] = [ ( d - 1 ) / a ]

k c -> 1 ) .

T h e n kc >- a - 2. PROOF. L e t r = [(d - 1 ) / a ] + 1. T h e n w e h a v e d / a <- r <- (d - 1 ) / a + 1. From this it follows that ( r - 1) d - ( a - 1) -< r ( d - a) <- (r - 1) d . Hence, if we consider the element

e = ( r ( d - a), (r - 1) d - r ( d - a)), t h e n 0<- [ e l ] - > a - 1. S i n c e So = ( d , d e e S ' b y L e m m a 2.3. L e t

1, d - a ) ,

$1 = ( 1 , a , d )

= N, we have

f = e + ( a - 1 - [e]l) el + ([d/a] - 1) e~. Then [ f ] l = [e]l + ( a - 1 - [e]l) + ([d/a] - 1) a = a [ d / a ] - 1 < d . H e n c e r ( [ f ] l ) = a - 1 + ( [ d / a ] - 1). T h e r e f o r e 81 ( [ f ] l ) - ~ ( f ) = ( a - 1 + [d/a] - 1) - ( r - 1 + a - 1 - [e]l + [d/a] - 1) = = [ e ] l - ( r - 1) = ( r - 1)

d -

r(d

-

a)

( b e c a u s e [(d - 1 ) / ( a - 1)] = [(d - 1 ) / a ] = r -

-

(r -

1)

=

r(a

1). B y L e m m a

-

1) - ( d - 1) > 0 ,

3.1 w e t h e n g e t

k c -> r ( [ f ] l ) - ~ ( f ) + ( a - 1 - [e]l) + ([d/a] - 1) = [e]l - ( r - 1) + + ( a - 1 - [ell) + [ d / a ] - 1 = a - 2 - [(d - 1 ) / a ] + [ d / a ] . O n t h e o t h e r h a n d , f r o m t h e a s s u m p t i o n [(d - 1 ) / ( a - 1)] = [ ( d l o w s t h a t [(d - 1 ) / a ] = [d/a]. H e n c e k c -> a - 2, q.e.d.

1 ) / a ] , i t fol-

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REMARK 6. L e t d = p a + q with 0 - q -< a - 1. Then a, d satisfy the condition of L e m m a 5.2 iff q > 0 and p + q ~< a - 1. EXAMPLE 4. F r o m L e m m a 5.2, Proposition 2.5(ii) and R e m a r k 6 one can find all strictly 2-Buchsbaum curves of the t y p e C(1, a, d). T h e y are C(1, 4, 5), C(1, 4, 6) and C(1, 4, 9). In fact, one can show t h a t the set of monomial curves in p 3 linked to one of these curves gives all strictly 2-Buchsbaum monomial curves in p 8 (see [17]). T h e y will be listed in Section 5b. REMARK 7. kc>a-2.

We could not find any example of C = C ( 1 , a , d )

with

Note t h a t for these curves we have S1 = N, A1 = (0, 1, ..., d - 1} and So = = (d, d - 1, d - a ) . Hence, in m a n y cases, one can easily compute the set J and the u p p e r bound ~ for k c defined in Proposition 3.1. In all example (e.g. a = d - 1, d - 2, d - 3 or d = a + q with a - 1 = 0 mod q) what we have computed, it is always tz = a - 2. Hence, by L e m m a 5.2 we get k c = a - 2. The following lemma can be also used for showing kc = a - 2 in other cases: LEMMA 5.3.

L e t d = p a + q w i t h p, p > O a n d p + q < _ a - 1 .

Then

a(H~m (k[S])) = p.

PROOF.

Recall that a(H~m (k[S])) = inf (n; [H~ (k[S])]. ~ 0).

By L e m m a s 2.2, 2.4 and R e m a r k 6 we then get a(H~m (k[S])) = rain ~(e) eeJ

and

J r 0.

L e t e e J be an a r b i t r a r y element. Then, by the definition of J, t h e r e exist n u m b e r s m , n such that [e]o = m ( d - a) + n ( d - 1)

and

8o([e]o) = m + n .

Moreover, given [e]o, [e]l is uniquely determined as the n u m b e r 0 <- [e]l < d with [ e ] o + [ e ] l - m o d d . Since [ e ] o e A o , [e]o can not be the form m ' ( d - a) + n ' ( d - 1) + l ' d with l' > 0. F r o m this it follows that n - a - 1. Suppose that q _< n_< a - 1. I f m - p then [e]o = ( m - p)(d - a) + (n - q)(d - 1) + p ( d - a) + q(d - 1) = = (m - p)(d - a) + (n - q)(d -

1) + (p + q - 1) d ~ A o ,

a contradiction. Hence m -< p - 1. But in this case we have 0 -< m a + n -< d.

ON MONOMIAL k-BUCHSBAUM CURVES IN p r

171

Therefore [e]l = m a + n. Then e = ( m ( d - a) + n ( d - 1), m a + n ) = he1 + mea ~t J ,

a contradiction. Thus the case n >_ q doesn't occur. Analogously, the case n < _ q - 1 and m < _ p also doesn't occur. Summing up, we g e t : n - < q - 1 and m _ p + l . Then [e]0 >-(p + 1 ) ( d - a ) = ( p - 1 ) d + ( p - 1 ) a + 2 q > Hence ~(e) > p - 1,

( p - 1)d.

q.e.d.

Using the algorithms of[3] and[5] we get the following lemma. LEMMA 5.4. Suppose that d = p a + q with p, q > 0, p + q - a - 1 and a - [(a - 2)/(p + q - 1)] q + q. Set 1 =](a - 1)/(p + q - 1)[. Then C has the following minimal free resolution: l-1

0 ---> ( ~ R ( ( q - 1) t - a - p - q) --* t=l

l-1

t__~ {R((q - 1) t - a - q) ~ R ( ( q - 1) t + 1 - a - p - q)} 9 ~R(-(lp

+ q + 1)) @ R ( ( q - 1 ) ( / - 1) - a - p) --, l-1

---, R ( - p

- q) @ ~ t~O

R ( t q - a - t) ~ R ( - I p

- 1) --~ R ~ R / I ( C ) ~ O,

where R = k[x0, x l , x2, x.~]. We drop the proof because of tedious calculations. F o r example, if q = 1,2 then d, a always satisfy the condition of the lemma. As a consequence of this lemma, we get reg(k[S]) = a + p - 2. Hence e ( H ~ k[S])) <_ a + p - 3. Combining Lemmas 5.2, 5.3 and R e m a r k 4, we also obtain kc = a - 2 for a , d which satisfy the a.~sumptions of L e m m a 5.4 EXAMPLE 5.1. We consider strictly 3-Buchsbaum curves of the type C (1, a, d ). By L e m m a 5.2. we must have a = 5 and d = 6,7,8,11,12,16. Using the afore-mentioned fact that k c = a - 2 for a -< d - 3 or for a , d satisfying the assumptions of L e m m a 5.4, we can conclude that these curves have k c = 3. Hence, all strictly 3-Buchsbaum curves of this type are: C(1, 5, 6), C(1, 5, 7), C(1, 5, 8), C(1, 5, 11), C(1, 5, 12) and C(1, 5, 16). All they but C(1, 5, 8) are quasi complete intersection, see[9] and [21].

172

MARIO FIORENTINI

- LE TUAN

HOA

QUESTION 1. Give the complete description (in terms of a, b,d) of all monomial curves which are strictly 3-Buchsbaum. EXAMPLE 6. Analogously, we can find all strictly 4-Buchsbaum curves of the type C(1,a,d). They are C(1,6,d) with d = 7 , 8 , 9 , 1 0 , 1 3 , 1 4 , 15, 19, 20, 25. Th e unique case which can not be decided by the method in Example 5 is the curve C(1, 6, 10). For this curve we can, however, use the algorithms of [3] and [5] to compute its minimal free resolution and then to verify that kc = 4.

QUESTION2. Give the complete description (in terms of a, b, d) of all monomial curves which are strictly 4-Buchsbaum. b) Some other classes. In this subsection, using the theory of liaison, we shall compute kc for some classes of monomial curves in pS. DEFINITION 5.5 [22]. Two c u r v e s C 1 and C~ in p a are algebraically directly linked by a complete intersection X containing C1, C2 if i) I( C2) / I (X) = Home~ (Ocl , Ox ), ii) I(C1)/I(X) ~- Homp8 (Ocz, Ox). We say C and C' are linked ff there exist curves C = Co, C1 ..., C. = C' such that Ci, Ci + 1 are directly linked for i = 0,..., n - 1. This is an equivalence relation and will be called 1/a/son. It follows from[23] that the Hartshorne-Rao module M(C) determines the liaison equivalence class of C. Hence, we have COROLLARY 5.6.

If C1 and C2 are linked, then kcl = kc2.

From this corollary we can now use results in Section 5a) and the beautiful paper of Bresinsky and Huneke [4] to compute kc for some classes of C. 1) By[4], Example 2.10 all the curves C ( p , p + 3 , 2 p + 3 ) with p_>l and g.c.d. ( p , 3 ) = 1 lie in the liaison class of C(1, 4, 5). Since kc(1,4,5)--2 (see w5a), Case a = d - 1), we get C(p, p + 3, 2p + 3) = 2. Note that this example belongs to the class of monomial curves in Example 2.

ON MONOMIAL koBUCHSBAUM CURVES IN p r

173

2) Analogously, from[4], Example 4.28 it follows t h a t kc(p,p+8,~+3) = 2

and

kc(2p+s,2p+r

= 2,

where p - > l and g.c.d. (p, 3) = 1. 3) Finally, from[4], Example on p. 66 and w5a), Case a = d - 2, we have kc(a_~.,a+~,~_~)=13, where a is an odd number ->13 and a ~ 5 mod 7. Acknowledgement. A part of this paper was done while the second author was visiting the Department of Mathematics of F e r r a r a University. F o r the friendly atmosphere there he expresses his appreciation. Thanks are also due to N. V. Trung for his help during the preparation of this paper.

Pervenuto in Redazione il 20 giugno 1990.

ABSTRACT This paper is devoted to the study of projective monomial k-Buchsbaum curves C. First, using the theory of affine semigroup rings, we give a criterion for C to be kBuchsbaum. Then we give some lower and upper bounds for the number kc such that C is strictly kc-Buchsbaum. For some classes of monomial curves we can compute kc explicitly.

REFERENCES [1] E. BALLICO,On the order of projective curves, Rend. Cir. Mat. Palermo, Serie II, 38 (1989), pp. 155-160. [2] H. BRESINSKY, Monomial Buchsbaum ideals in P~, Manuscripta Math., 47 (1984), pp. 105-132. [3] H. BRESINSKY,Minimal free resolutions of monomial curves in p3, Linear Alg. Appl., 59 (1984), pp. 121-129. [4] H. BRESINSKY- C. HUNEKE, Liaison ofmonomial curves in p3, j. Reine Angew. Math., 365 (1986), pp. 33-66. [5] H. BRESINSKY- R E N S C H U C H , Basisbestimmung VeronesescherProjektionsideale mit allgemciner NuUstelle (tom , tom-"t~ , to'n-st~ , t~), Math. Nachr., 96 (1980), pp. 257-269. [6] H. BRESINSKY- P. SCHENZEL- W. VOGEL,On liaison, arithmetically Buchsbaum curves and monomial curves in pS, j. Alg., 86 (1984), pp. 283-301. [7] D. EISENBUD - S. GOTO, Linear free resolutions and minimal multiplicity, J. Alg., 88 (1984), pp. 89-133.

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[8] PH. ELLIA, Ordres et cohomogie des fibres de rang dcux sur l'espace projectif, Preprint, Nice, 170 (1987). [9] M. FXORENTISI - A. T. LASCU, Projective embeddings and linkage, Rend. Sem. Mat. Fis. Milano, 57 (1987), pp. 161-182. [10] F . GAETA, A geometrical characterization of smooth k-Buchebaum curves, to appear in Springer Lecture Notes in Math. [11] A. V. GERAMITA- P. MAROSCIA-W. VOGEL,A note on arithmetically Buchsbaum curves in P~, Matematiche (Catania), 40, fasc. I-II (1985), pp. 21-28. [12] S. GmFFRIDA - R. MAGGIONI,On the Rao module of a curve lying on a smooth cubic surface in p3, -H, preprint CIMPA, Nice (1990). [13] S. GOTO- N. SUZUKI- K. WATANABE, On affine semigroup rings, Japan J. Math, 2 (1976), pp. 1-12. [14] W. GROBNER, Uber Veronesesche Variet~ten und deren Projectionen, Arch. Math., 16 (1965), pp. 257-264. [15] L. GRUSON - L. LAZARFELD - C. PESI~NE, On a theorem of Castelnuovo and the equations defining space curves, Invent. Math., 72 (1983), pp. 491-506. [16] L. T. HOA, Algorithmetical aspects of the problem of classifying multi-projections of Veronesian varieties, Manuscripta Math., 63 (1989), pp. 317-331. [17] L. T. HOA, k-Buchsbaum monomial curves in p3. Manuscripta Math. (to appear). [18] L.T. HOA - R. M. MIRO-ROIG- W. VOGEL, On numerical invariants of locally Cohen-Macaulay schemes in P~, preprint M.P.I. (1990). [19] J. KASTNER, Zu einem Problem yon H. Bresinsky fiber monomiale Buchebaum Kurven, Manuscripta Math., 54 (1985), pp. 197-204. [20] J. MIGLIORE- R. M. Mmo-RoxG, On k-Buchsbaum curves in P 8, Commun. Alg., 18(8) (1990), pp. 2403-2422. [21] R. M. MIRO-ROm, Some remarks on quasi-complete intersection space curves, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., 34 (1988), pp. 237-245. [22] C. PESKINE - L. SzPmo, Liaison des variet~s alg$briques, Invent. Math., 26 (1973), pp. 271-302. [23] R. PAO, Liaison among curves in p3, Invent. Math., 50 (1979), pp. 205-217. [24] J. STOCKRAD- W. VOGEL, Buchsbaum rings and applications, Springer, Berlin, Heidelberg, New-York (1986). [25] N. V. TRUNG, Classification of the double projections of Veronese varieties, J. Math. Kyoto Univ., 22 (1983), pp. 567-581. [26] N. V. TRUNG, Degree bound for the defining equations of projective monomial curves, Acta Math. Vietnam, 9 (1984), pp. 157-163. [27] N. V. TRUNG, Projections of one-dimensional Veronese Varieties, Math. Nachr., 118 (1984), pp. 47-67. [28] N. V. TRUNG - L. T. HOA, Affine semigroups and Cohen-Macaulay rings generated by monomials, Trans. Amer. Math. Soc., 298 (1986), pp. 145-167.