Projective closures of affine monomial curves
aa r X i v : . [ m a t h . A C ] J a n PROJECTIVE CLOSURES OF AFFINE MONOMIAL CURVES
JOYDIP SAHA, INDRANATH SENGUPTA, AND PRANJAL SRIVASTAVAA
BSTRACT . Our aim in this paper is to study the projective closures ofthree important families of affine monomial curves in dimension , in or-der to explore possible connections between syzygies and the arithmeticCohen-Macaulay property.
1. I
NTRODUCTION
Let r ≥ and n = ( n , . . . , n r ) be a sequence of r distinct positiveintegers with gcd( n ) = 1 . Let us assume that the numbers n , . . . , n r gen-erate the numerical semigroup Γ( n , . . . , n r ) in the set of all non-negativeintegers N : Γ( n , . . . , n r ) = { r X j =1 z j n j | z j nonnegative integers } minimally, that is, if n i = P rj =1 z j n j for some non-negative integers z j ,then z j = 0 for all j = i and z i = 1 . Let η : k [ x , . . . , x r ] → k [ t ] bethe mapping defined by η ( x i ) = t n i , ≤ i ≤ r . Let p ( n , . . . , n r ) =ker( η ) . Let β i ( p ( n , . . . , n r )) denote the i -th Betti number of the ideal p ( n , . . . , n r ) . Therefore, β ( p ( n , . . . , n r )) denotes the minimal number ofgenerators p ( n , . . . , n r ) . For a given r ≥ , let β i ( r ) = lub( β i ( p ( n , . . . , n r )) ,where is lub is taken over all the sequences of positive integers n , . . . , n r .Herzog [9] proved that β (3) is and it follows easily that β (3) is a finiteinteger as well. Bresinsky [5] (and [6]) defined a class of monomial curvesin A and proved that β (4) = ∞ . He used this observation to prove that β ( r ) = ∞ , for every r ≥ . We call this family the Bresinsky curves .Subsequently, it has been proved in [10] that all the higher Betti numbersof Bresinsky curves are also not bounded above by a fixed integer. Sub-sequent to Bresinsky curves came the examples of
Backelin curves [2] and
Mathematics Subject Classification.
Primary 13D02, 13F055, 13P10, 13P20.
Key words and phrases.
Monomial curves, Gr¨obner bases, Betti numbers.The first author thanks SERB, Government of India for NPDF at ISI kolkata, throughthe research project PDF/2019/001074.The second author is the corresponding author. the examples of
Arslan curves [1]. All three families have one strong re-semblance that all are monomial curves in the affine space and there is noupper bound on the minimal generating set of the defining ideals of thesecurves.Let us assume that n r > n i for all i < r , for the sequence n = ( n , . . . , n r ) .We fix n = 0 . We define the semigroup Γ( n , . . . n r ) ⊂ N , ( N isthe set of all non negetive integers) generated by { ( n i , n r − n i ) | ≤ i ≤ r } . Let us denote p ( n , . . . n r ) be the kernel of k -algebra map η H : k [ x , . . . , x r ] −→ k [ s, t ] , η H ( x i ) = t n i s n r − n i , ≤ r . Then homogeniza-tion of the ideal p ( n , . . . , n r ) w.r.t. the variable x is p ( n , . . . n r ) . Thusthe projective curve { [( a n r : a n r − n b n : · · · : b n r )] ∈ P nk | a, b ∈ k } isthe projective closure of the affine curve C ( n , . . . , n r ) := { ( b n , . . . b n r ) ∈ A nk | b ∈ k } and we denote it by C ( n , . . . , n r ) . We say that the Projec-tive curve C ( n , . . . , n r ) is arithmetically Cohen-Macaulay if the vanishingideal p ( n , . . . n r ) is a Cohen-Macaulay ideal.This paper is devoted to a study of the three families Backelin, Bresin-sky and Arslan curves and also their projective closures. First we studyBackelin’s curves and its projective closure and show that such curves arearithmetically Cohen-Macaulay and all betti numbers are unbounded. Wealso find their syzygies. Next we compute the syzygies of the projective clo-sure of Bresinsky curves and finally the syzygies of the projective closureof Arslan curves. Our aim in this paper is to find suitable conditions on theaffine monomial curves which ensure the arithmetic Cohen-Macaulaynessof its projective closure and we suspect that there may be some criterionthrough their syzygies. This paper is a case study with the help of threevery important and interesting classes of curves.2. B ACKELIN ’ S E XAMPLES
Let us first begin with a description of Bakelin’s example of monomialcurves in A . Beckelin defines the following semigroups h s, s + 3 , s + 3 n +1 , s + 3 n + 2 i , for n ≥ , r ≥ n + 2 and s = r (3 n + 2) + 3 . In [2], it hasbeen shown that type of such semigroups are not bounded by embeddingdimension. Notations 2.1.
We fix some notations. For n ≥ , r ≥ n + 2 and s = r (3 n + 2) + 3 , • Γ nr := Γ( s, s + 3 , s + 3 n + 1 , s + 3 n + 2) and we denote Γ nr :=Γ( s, s + 3 , s + 3 n + 1 , s + 3 n + 2) • We define P nr := p ( s, s + 3 , s + 3 n + 1 , s + 3 n + 2) . ROJECTIVE CLOSURES OF AFFINE CURVES 3 • We denote P nr := p ( s, s + 3 , s + 3 n + 1 , s + 3 n + 2) , therefore P nr = ( P nr ) H w.r.t the variable x . • B nr will denote affine Bakelin’s curves C ( s, s +3 , s +3 n +1 , s +3 n +2) and the projective closure of Backelin’s curves will be denoted by B nr . • For any monomial ideal I in polynomial ring over a field, G ( I ) denotes the unique minimal generating set of I .
3. G
ENERATING SET FOR B AKELIN ’ S CURVE
Theorem 3.1. (Gastinger)
Let A = k [ x , . . . , x r ] be a polynomial ring, I ⊂ A the defining ideal of a monomial curve defined by natural numbers a , . . . , a r whose greatest common divisor is . Let J ⊂ I be a subideal.Then J = I if and only if dim k A/ h J + ( x i ) i = a i for some i . (Note that theabove conditions are also equivalent to dim k A/ h J + ( x i ) i = a i for any i .)Proof. See in [3]. (cid:3)
Let I ⊂ k [ x , . . . , x r ] be a monomial ideal, then it has unique minimalgenerating set. We denote the minimal generating set by G ( I ) . Theorem 3.2.
The defining ideal P nr of monomial curve associated to Γ nr is minimally generated by following binomials f = x x − x x f i = x n − i x i − − x n − i +12 x i − , i nf j = x r − n +3+ j x n − − j − x j x r − − j , ≤ j ≤ n − f j = x r − n +3+ j x n − j − x j +13 x r +1 − j , j n − f = x r − n +21 x n x − x r +24 f = x n +12 x − x n x f = x n +12 − x n − x x Proof.
We proceed by Gastinger’s theorem stated as in 3.1.Let J nr = h{ f , f i , f j , f j , f , f | ≤ i ≤ n, ≤ j ≤ n − }i and A nr = { x x , x n − i +12 x i − , x r − n +3+ j x n − − j , x r − n +3+ j x n − j , x r +24 , x n x , x n +12 | ≤ i ≤ n, ≤ j ≤ n − } . Then J nr + ( x ) = h A nr i . At first we note that J nr ⊂ P nr . A/J nr + h x i is the vector space over k whose basis consists of the imagesof monomials are listed bellow • S = { x α : 0 ≤ α ≤ r + 1 }• S = { x β : 0 ≤ β ≤ n }• S = { x γ : 0 ≤ γ ≤ r + 1 }• S = { x α x β : 1 ≤ α ≤ r − β + 1 , ≤ β ≤ n − }• S = { x α x β : 1 ≤ α ≤ r − β + 2 , n + 1 ≤ β ≤ n }• S = { x α x : 1 ≤ α ≤ r + 1 } JOYDIP SAHA, INDRANATH SENGUPTA, AND PRANJAL SRIVASTAVA • S = { x α x : 1 ≤ α ≤ n − }• S = { x β x γ : 1 ≤ β ≤ n − , ≤ γ ≤ n − β ) }• S = { x α x β x γ : 1 ≤ α ≤ n − , ≤ β ≤ n − , ≤ γ ≤ }• S = { x α x β x : n ≤ α ≤ r − β + 1 , ≤ β ≤ n − }• S = { x α x n : 1 ≤ α ≤ r − n + 2 } Then cardinality of this basis is, X i =1 | S i | = 2 n + 2 r + 3 + ( n − r − n + 2) + n r − n + 3) + ( r + 1)+ ( n −
1) + 32 n ( n −
1) + 2( n − + ( r − n + 2) + n −
12 (2 r − n + 4)= 3 nr + 3 n + 2 r + 4 . Hence dim k ( A/J nr + h x i ) = 3 nr + 3 n + 2 r + 4 = s + 3 n + 1 . So byGastinger Theorem we have J nr = P nr . (cid:3) Let us denote the above generating set of P nr by S nr , thus S nr = { f , f i , f j , f j , f , f | ≤ i ≤ n, ≤ j ≤ n − } .4. G R ¨ OBNER BASIS OF B AKELIN ’ S CURVE
Definition 1.
Let G = ( g , . . . , g t ) ⊂ k [ x , . . . , x r ] and = f ∈ k [ x , . . . , x r ] . We say that f reduces to zero modulo G , denoted by f → G , if f can bewritten as f = P ti =1 a i g i , such that whenever a i g i = 0 , we have Lm ( f ) ≥ Lm ( a i g i ) . Lemma 4.1. (Buchberger’s Criterion)
Let I be an ideal in A = k [ x , . . . , x r ] and G = { g l , . . . , g t ) } be a generating set for I . Then, G is a Gr¨obner basisfor I if and only if S ( g i , g j ) −→ G , for every i = j Lemma 4.2.
Let G = ( g , . . . , g t ) ⊂ A = k [ x , . . . , x r ] and let f, g ∈ G benon-zero with Lc ( f ) = Lc ( f ) = 1 , and gcd ( Lm ( f ) , Lm ( g )) = 1 . Then (i) S ( f, g ) = Lm ( g ) · f − Lm ( f ) · g (ii) S ( f, g ) = − ( g − Lm ( g )) · f + ( f − Lm ( f )) · g −→ G Proof. (i) We have lcm ( Lm ( f ) , Lm ( g )) = Lm ( f ) · Lm ( g ) . The restis clear from the definition of S -polynomial.(ii) Suppose f = Lm ( f ) + f ′ and g = Lm ( g ) + g ′ , for f ′ , g ′ ∈ k [ x , . . . , x r ] . Then we have,
ROJECTIVE CLOSURES OF AFFINE CURVES 5 S ( f, g ) = Lm ( g ) · f − Lm ( f ) · g = ( g − g ′ ) · f − ( f − f ′ ) · g = − g ′ · f + f ′ · g = − ( g − Lm ( g )) · f + ( f − Lm ( f )) · g Suppose Lm ( g ′ .f ) = Lm ( f ′ .g ) . Then Lm ( g ′ ) Lm ( f ) = Lm ( f ′ ) Lm ( g ) , which implies that Lm ( g ) divides Lm ( g ′ ) .Lm ( f ) . Since Lm ( g ) and Lm ( f ) are coprime, we get Lm ( g ) divides Lm ( g ′ ) , which isa contradiction. So Lm ( S ( f, g )) = max ( Lm ( g ′ .f ) , Lm ( f ′ .g )) (cid:3) Theorem 4.3.
Let g = x r +21 − x x r be a polynomial in k [ x , . . . , x ] . Sup-pose G nr = ( S nr \ { f n − } ) ∪ { g } . Then G nr also a generating set forthe defining ideal P nr .Proof. Follows from the relation g = f , ( n − + x r − n +24 · f ,n . (cid:3) Theorem 4.4.
Let us consider the degree reverse lexicographic monomialorder on k [ x , . . . , x ] induced by x > x > x > x then G nr is Gr¨obnerbasis of defining ideal P nr with respect to this order.Proof. We proceed by Buchberger’s algorithm,(1) S ( f , f (2 ,i ) ) = x n +12 x x − x n x = x f , for i = 1= x n − i +22 x i − − x n − i +11 x i − x = x f (2 ,i − , for 2 ≤ i ≤ n. (2) S ( f , f (3 ,i ) ) = x n − x r − n +54 − x r +21 x = f (2 , x r − n +54 − gx , for i = n − x i x r − − i − x r − n +4+ i x n − − i x = − f (3 ,i +1) x , for 2 ≤ i ≤ n − . (3) S ( f , f (4 ,i ) ) = x n +13 x r +1 − n +34 − x r − n +31 x n x = f (2 ,n ) x x r − n +44 − f (3 , x x for i = n − x i +43 x r +1 − i − x r − n +4+ i x n − i − x = − f (4 ,i +1) x , for 0 ≤ i ≤ n − . (4) S ( f , g ) = x x x r − x r +31 . Since gcd( Lm ( f ) , Lm ( g )) = 1 , so bylemma 4.2 S ( f , g ) −→ G nr (5) S ( f , f ) = − x r − n +31 x n − x + x r +24 x = − f (3 , ( x ) (6) S ( f , f ) = x n x x − x x n x = f (2 , ( x x ) (7) S ( f , f ) = − x x n x + x n − x x = f (2 , ( x n x x + x x x n ) JOYDIP SAHA, INDRANATH SENGUPTA, AND PRANJAL SRIVASTAVA (8) for i < j , we have S ( f (2 ,i ) , f (2 ,j ) ) = x j − i x n − j +12 x j − − x n − i +12 x j − i )3 x i − = f ( j − i − X l =0 ( x l x ( n − i ) − l x j − i ) − l +1)3 x i +(3 l − ) (9) S ( f (2 ,i ) , f (3 ,j ) ) = x i + j )+13 x r − − j − x r − n +3+ i + j x n − i − j x i − , Nowwe have, S ( f i , f j ) = f (3 , ( − x x i − + f n ( x x r − − j ) , for i + j = n = f (3 , ( i + j ) − n − ( − x x i − ) + f (2 ,n ) ( x i + j ) − n +23 x ( r − − j )4 )+ f ( x i + j ) − n − x r +3 n − j − ) , for i + j > n = f (4 ,i + j ) x i − ) , for i + j < n (10) S ( g, f (2 ,i ) ) = x r − n +2+ i x n − i +12 x i − − x x i − x r = f (3 ,i − x x i − for ≤ i ≤ n − for i = n , Since gcd ( Lm ( g ) , Lm ( f (2 ,n ) ) = 1 ,we have S ( f (2 ,n ) , g ) −→ G nr (11) S ( f (2 ,i ) , f (4 ,j ) ) = x i + j )3 x r +1 − j − x r − n +3+ i + j x n − i − j +12 x i − , Wehave, S ( f (2 ,i ) , f (4 ,j ) ) = f (3 ,l − ( − x x i − ) + f ( − x r − n + l +31 x n − l x i − ) l = i + j ≤ n − f ( − x r − n +21 x x i − ) + f (3 ,n − ( − x x i − ) l = i + j = n − g ( − x x i − ) + f ( − x r − n +31 x i − ) + f (2 ,n ) ( x x r +1 − j ) i + j = n = f (4 , ( i + j ) − n − ( − x x i − ) + f (2 ,n ) ( x i + j ) − n +13 x r +1 − j )+ f ( x l − n − x r − − j +3 n ) i + j ≥ n + 1 (12) S ( f (2 ,i ) , f ) = x i − x r +24 − x r − n +2+ i x n − i +12 x i − = − f (4 ,i − x i +14 , ≤ i ≤ n. (13) S ( f (2 ,i ) , f ) = x n − i +22 x i − − x n − i x x i − = f (2 , ( − x n − i +11 x x i − − x n − i +22 x i − ) − ( x x − x x )( i − X l =1 ( x n − i + l x ( n − l )2 x i − (7+3( l − x l ) , ≤ i ≤ n for i = 2 , S ( f (2 , , f ) = f (2 , ( x n − i +11 x x i − + x n − i +22 x i − for i = 1 , S ( f (2 , , f ) = f x (14) S ( f (2 ,i ) , f ) = − x n − i x n − i +22 x i − + x n − i − x x i , i n . ROJECTIVE CLOSURES OF AFFINE CURVES 7
Since gcd( Lm ( f (2 ,i ) ) , Lm ( f )) = 1 , we have S ( f (2 ,i ) , f ) −→ G nr (15) S ( f (3 ,i ) , f (3 ,j ) ) = − x i x r − − i x j − i + x j − i x j x r − − j = x x − x x ( j − i − X l =0 ( x l x ( j − i ) − ( l +1)2 x j − (1+3 l )3 x r − j +( − l )4 ) , i < j (16) S ( f (3 ,i ) ,g ) = x n − i x r − x n − − i x i x r − − i = − f (2 ,i +1) x r − − i (17) S ( f (3 ,i ) , f (4 ,j ) ) = x n +1 − j x j +13 x r +1 − j − x n − j +1+ i x i x r − − i , Nowwe have, S ( f (3 ,i ) , f (4 ,j ) ) = f ( − x n x i − x r − − i + ( x n − x − x n x ) x x i − x r − i +14 , i = j = f (2 ,l ) x i +23 x r +1 − j , i < j, l = j − i = f (2 , x ( i − j )+11 x j − x r +1 − j + ( − f ) i − j X l =0 x l x n − j + i − l x i − − l x r − − i +3 l ) , i > j (18) S ( f (3 ,i ) , f ) = x i x r +24 − x i x i x r − − i = − f ( P il =0 x l x i − l x i − l x r − i +3 l − ) , ≤ i ≤ n − (19) S ( f (3 ,i ) , f ) = x r +3+ i x − x i x i x r − − i = gx i x − f ( i X l =0 x l x i − l )2 x i − l )3 x r − i +3 l − ) (20) S ( f (3 ,i ) , f ) = x r + n +2+ i x x − x n + i +22 x i x r − − i = g ( x n + i x x ) + f ( − x i x x x r − ) − f ( i − X l =0 ( x l x n + i +(1 − l )2 x i − (3 l +1)3 x r − i − l ) , ≤ i < n − (21) S ( g, f (4 ,i ) ) = − f x r , i = 0= f (2 , ( x n − i x i − x r +1 − i + x x r − ) , i = 1= f (2 , ( x n − i x i − x r +1 − i + x n − i +12 x r − )+ f ( i − X l =0 x n − i + l x n − − l x i − − l x r +2 − i +3 l ) , i ≥ . (22) S ( g, f ) = x n x r +24 − x n +12 x x r = f ( − x r ) JOYDIP SAHA, INDRANATH SENGUPTA, AND PRANJAL SRIVASTAVA (23) S ( g, f ) = − x n +22 x x r + x r + n +21 x Since gcd( Lm ( g ) , Lm ( f )) = 1 , therefore S ( g, f ) −→ G (24) Since gcd( Lm ( g ) , Lm ( f )) = 1 , therefore S ( g, f ) −→ G (25) S ( f (4 ,i ) , f (4 ,j ) ) = x j − i x j +13 x r +1 − j − x j − i x i +13 x r +1 − i = f ( j − i − X l =0 x l x ( j − i ) − ( l +1)2 x j − (2+3 l )3 x r − j +(1+3 l )4 ) , i < j (26) S ( f (4 ,i ) , f ) = x n − i x r +24 − x n − − i x i +23 x r +1 − i = f (2 ,i +1) ( − x r +1 − i ) , ≤ i ≤ n − (27) S ( f (4 ,i ) , f ) = x r − n +3+ i x n − i − x − x i +23 x r +1 − i = x ( f (3 ,i ) ) , i < n − for i = n − , S ( f (4 ,i ) , f ) = gx − f (2 ,n ) x r − n +44 (28) S ( f (4 ,i ) , f ) = x r +2+ i x x − x i x i +13 x r +1 − i = g ( x i x x ) − f ( i − X l =0 x l x i − l x i +( − l − x r − i +1+3 l ) , i n − . (29) S ( f , f ) = x r +21 x − x x r +24 = gx (30) S ( f , f ) = − x n +12 x r +24 + x r + n +11 x x = f (2 , ( x r +21 x + x x r +14 + f (3 , x n − x x (31) S ( f , f ) = x n − x x − x n x n x = f (2 , x n x By Buchberger’s Criterion S-polynomials of all generator reduces to zero,so G nr is Gr¨obner basis of P nr with respect to degree reverse lexicographicmonomial order > induced by x > x > x > x (cid:3) Corollary 4.5.
Let us consider the degree reverse lexicographic monomialorder on k [ x , . . . , x ] induced by x > x > x > x then with respect tothis order G (in < ( P nr )) is the set { x x , x n − i x i − , x r − n +3+ j x n − − j , x r − n +3+ j x n − j , x r − n +21 x n x , x n +12 x , x n +12 | ≤ i ≤ n, ≤ j ≤ n − } .Proof. Follows from the theorem 4.4. (cid:3)
We will take two theorems from [4].
Lemma 4.6.
Let I be an ideal in A = k [ x , . . . , x r ] and I H ⊂ A [ x ] itshomogenization w.r.t. the variable x . Let < be any reverse lexicographicmonomial order on A and < the reverse lexicographic monomial order on A [ x ] extended from A such that x i > x is the least variable. ROJECTIVE CLOSURES OF AFFINE CURVES 9 If { f , . . . , f n } is the reduced Gr¨obner basis for I w.r.t < , then { f H , . . . , f Hn } is the reduced Gr¨obner basis for I H w.r.t < , and in < ( I H ) = (in < ( I )) A [ x ] .Proof. See lemma 2.1 in [4]. (cid:3)
Theorem 4.7.
Let n : n , . . . , n r be a sequence of positive integers with n r > n i for all i < n . Let < any reverse lexicographic order on A = k [ x , . . . , x r ] such that x i > x r for all ≤ i < r . and < the inducedreverse lexicographic order on A [ x ] , where x n > x . Then the followingconditions are equivalent:(i) The projective monomial curve C ( n , . . . , n r ) is arithmetically Cohen-Macaulay.(ii) in < (( p ( n , . . . , n r )) H ) (homogenization w.r.t. x ) is a Cohen-Macaulayideal.(iii) in < ( p ( n , . . . , n r )) is a Cohen-Macaulay ideal.(iv) x r does not divide any element of G (in < ( p ( n , . . . , n r ))) .Proof. See theorem 2.2 in [4]. (cid:3)
Theorem 4.8.
The projective closure B nr of Bakelin’s curve is arithmeti-cally Cohen-Macaulay.Proof. From the corollary 4.5, we see that x does not divide any elementof G (in < ( P nr ) . The proof follows from the theorem 4.7. (cid:3) Lemma 4.9.
The reduced Gr¨obner basis of the ideal P nr w.r.t. reverselexicographic monomial order on k [ x , . . . , x ] induced by x > x > x >x > x is G nr = { f H | f ∈ G nr } . Where f H is the homogenization of f w.r.t. the variable x and G nr is same as in 4.3.Proof. Follows from the lemma 4.6 (cid:3)
5. H
ILBERT S ERIES OF B AKELIN ’ S CURVE
Lemma 5.1. I ⊂ S be a graded ideal and < a monomial order on S .Then S/I and
S/in < ( I ) have the same Hilbert function, i.e. H ( S/I, i ) = H ( S/in < ( I ) , i ) for all i . Lemma 5.2.
Let I = ( x A , ..., x A r ) ⊂ K [ x , ..., x n ] be a monomial ideal.Let p ( I ) denote the numerator of the Hilbert series of K [ x , x , ..., x n ] /I ,and let | A | denote the total degree of the monomial x A . Then p ( I ) = p ( x A ) − P i = ni =2 t | A i | p ( x A , ..., x A i − : x A i )) . Theorem 5.3.
The numerator of the Hilbert series of the B nr is − nt r +2 − t r +3 + (3 n + 4) t r +4 − (2 n + 2) t r +5 − t n +1 − t n +2 + 2 t n +2 − t n +1 + t n +3 + t n +4 − t n +3 − P ni =2 ( t n +2 i − + t n +2 i +1 − t n +2 i ) Proof. : From Corollary 3.5, in < ( P nr ) is generated by the set { x x , x n − i x i − , x r − n +3+ j x n − − j , x r − n +3+ j x n − j , x r − n +21 x n x , x n +12 x , x n +12 | ≤ i ≤ n, ≤ j ≤ n − } . We will use algorithm 2.6 of [11]. Rear-range in < ( P nr ) so that they are in ascending lexicographic order on thereversed set of variables x > x > x > x . We have in < ( P nr ) = { x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n , x n +12 , x r − n +21 x n x , x n +12 x , x n − x , x x , x n − x , ..., x n − } . By Lemma 3.10, Hilbert function of k [ x , x , x , x ] / P nr is equal to theHilbert function of k [ x , x , x , x ] /in < ( P nr ) , it is sufficient to computethe Hilbert function of the latter. Let I denote the monomial ideal in < ( P nr ) .We apply Lemma 5.2 to the ideal I,then ROJECTIVE CLOSURES OF AFFINE CURVES 11 p ( I ) = p ( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n , x n +12 , x r − n +21 x n x , x n +12 x , x n − x , x x , x n − x , ..., x n − )= p ( x r +21 ) − t r +2 p (( x r +21 : x r +11 x )) − · · · − t r +2 p (( x r +21 , x r +11 x , ..., x r − n +41 x n − : x r − n +31 x n − ) − t r +3 p (( x r +21 , x r +11 x , ..., x r − n +31 x n − : x r − n +21 x n +12 )) − · · · − t r +3 p (( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +41 x n − : x r − n +31 x n )) − t n +1 p ((( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n : x n +12 )) − t r +3 p (( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n , x n +12 : x r − n +21 x n x )) − t n +2 p ((( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n , x n +12 ,x r − n +21 x n x : x n +12 x ) − t n +1 p ((( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n , x n +12 , x r − n +21 x n x ,x n +12 x : x n − x )) − t p ((( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n , x n +12 , x r − n +21 x n x , x n +12 x , x n − x : x x )) − t n +3 p (( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n , x n +12 , x r − n +21 x n x ,x n +12 x , x n − x , x x : x n − x )) − · · · − t n − p (( x r +21 , x r +11 x , ..., x r − n +31 x n − , x r − n +21 x n +12 , ..., x r − n +31 x n , x n +12 , x r − n +21 x n x ,x n +12 x , x n − x , x x , x n − x , ...x x n − : x n − ))= (1 − t r +2 ) − ( n − t r +2 (1 − t ) − nt r +3 (1 − t ) − t n +1 (1 − t r − n +3 − t r +3 (1 + t − t ) − t n +2 (1 − t n − nt r − n +2 + nt r − n +3 ) − t n +1 (1 − t n +1 ) − ( n + 1) t r − n +3 + ( n + 1) t r − n +4 ) − t (1 − t n − − t n + t n − ) − (1 + t − t )( n X i =2 t n +2 i − )= 1 − t r +2 − ( n − t r +2 + ( n − t r +3 − nt r +3 + nt r +4 − t n +1 + t r +4 − t r +3 − t r +5 + 2 t r +4 − t n +2 + t n +2 + nt r +4 − nt r +5 − t n +1 + t n +2 + ( n + 1) t r +4 − ( n + 1) t r +5 − t + t n +3 + t n +4 − t n +3 − n X i =2 ( t n +2 i − + t n +2 i +1 t n +2 i )= 1 − nt r +2 − t r +3 + (3 n + 4) t r +4 − (2 n + 2) t r +5 − t n +3 + 2 t n +2 − t n +1 + t n +4 + t n +3 − t n +2 − t n +1 − t − n X i =2 ( t n +2 i − + t n +2 i +1 − t n +2 i ) . Hence the Hilbert Series of the ring k [ x , x , x , x ] /I is p ( I )(1 − t ) . (cid:3)
6. S
YZYGIES OF P ROJECTIVE CLOSURE OF B RESINSKY ’ S CURVE
In [5], Bresinsky defined the following family of curves , for h ≥ , C ((2 h − h, (2 h − h + 1) , h (2 h + 1) , h (2 h + 1) + 2 h − . He showed that minimal number of generators of the defining ideal, thatis µ ( p ((2 h − h, (2 h − h + 1) , h (2 h + 1) , h (2 h + 1) + 2 h − is an unbounded function of h . Gr¨obner basis and syzygies of Bersinsky’scurves has been studied in [10]. It has been proved in [10] that all bettinumbers are unbounded function of h . Recently Herzog and Stamate provedin [4] that Bresinsky’s curves are not arithmetically Cohen-Macaulay usingGr¨obner basis criterion. Here we will study syzygies of projecive closure ofBresinsky’s curves and will show that all betti numbers are not unboundedfunction of h , infact type is . Notations 6.1.
For h ≥ , we define the following notations, • let B h = C ((2 h − h, (2 h − h + 1) , h (2 h + 1) , h (2 h + 1) +2 h − • Q h = p ((2 h − h, (2 h − h + 1) , h (2 h + 1) , h (2 h + 1) +2 h − . Theorem 6.1.
Consider the degree reverse lexicographic monomial orderinduced by x > x > x > x in the polynomial ring k [ x , x , x , x ] . For h ≥ , let us consider the following polynomials • p = x x − x x • p = x h − x h − • p (3 ,j ) = x j +11 x h − l − x j x h − l , ≤ j ≤ h − • p (4 ,i ) = x i +11 x h − i − x i − x h − i , ≤ i ≤ h • p = x h − x h − x h +14 • p (6 ,i ) = x i x h − − i x i − x h +1+ i , ≤ i ≤ h − • p = x x h − x − x h • p = x h x h − x h − .Let G h = { p , p , p , p , p } ∪ { p (3 ,j ) | ≤ j ≤ h − } ∪ { p (4 ,i ) | ≤ i ≤ h } , then G h is Gr¨obner basis of Q h with respect to the given monomialorder.Proof. We proceed by Buchberger’s algorithm 4.1. We compute S -polynomialsand show that it reduces to zero upon division by G h .(1) S ( p , p ) = x h − x x x h − x = − p (2) S ( p , p (3 ,j ) ) = x j +12 x h − l − x j x x h − j − = − x p (3 ,l +1) , for ≤ j < h − x h x − x h +11 x = x p − x p (4 , h ) , when i = 2 h − . ROJECTIVE CLOSURES OF AFFINE CURVES 13 (3) S ( p , p (4 ,i ) ) = x i x h − i − x i +21 x h − i − x = − x p (4 ,i +1) , for ≤ i < hS ( p , p (4 , h ) ) = x x h − x h +21 x , since gcd(Lt( p ) , Lt( p (4 , h ) )) =1 , S ( p , p (4 , h ) ) −→ G h (4) S ( p , p ) = x h x h +14 − x x h − x = p x h +14 − p (3 , x h − x (5) S ( p , p (6 ,i ) ) = x h +2+ i − x i x h − − i x i = − p (6 ,i +1) , ≤ i ≤ h − − p , i = 2 h − (6) S ( p , p ) = x h +13 − x x h − x = − p (6 , (7) S ( p , p ) = x x h − x h +11 x h +14 = x h − p + p (3 , x h − x − x h +14 p (4 , h ) (8) since gcd(Lt( p ) , Lt( p (3 ,l ) )) = 1 , S ( p , p (3 ,l ) −→ G h (9) S ( p , p (4 ,i ) ) = x i x i − x h − − x i +11 x h − = p (3 , ( − x i − ) + p ( i − X l =0 x l x i − − l x i − − l x h − i + l ) , ≤ i ≤ h = − p (3 , , i = 1 . (10) Since gcd(Lt( p ) , Lt( p (5) )) = 1 , S ( p , p ) −→ G h (11) S ( p , p (6 ,i ) ) = x i x h +1+ i − x i x h − x i = − p ( i +1 X l =0 x l x i − l x h + i − l x l ) , ≤ i ≤ h − . (12) S ( p , p ) = x x h − x x h − x = p x h − (13) S ( p , p ) = x h x h − − x h x h − x h = p ( P h − l =0 x l x h − − l x h − − l x l ) (14) S ( p (3 ,i ) , p (3 ,j ) ) = x j x j − i x h − j − x j − i x i x h − i = p ( j − i − X r =0 x r x j − − r x j − i − − r x h − j + r ) . (15) S ( p (3 ,j ) , p (4 ,i ) ) = − x h x h − i + x h − x h − i = p ( − x h − i ) , for j = i = x h − j + i − x h − i − x i − j x h − i + j x h − j = − x h − i p (6 ,i − j − , for j < i = x j − i x h + i − j − x h − i − x h − i + j x h − i = p ( − x j − i x h − j )+ p ( − j − i − X k =0 x k x j − i − − k x h − − k x h − j + k ) , for j > i. (16) S ( p (3 ,j ) , p ) = x j x h − x h +14 − x j x h + j x h − j = p x j x h − p ( j − X k =0 x k x j − − k x h + j − − k x h − j + k ) (17) S ( p (3 ,j ) , p (6 ,i ) ) = x h − x x h − x h +24 = p x + p x h − x h +14 for j = i = x j − i − x h +1+ i − j − x h − − i + j x h − j + i − = p for j = i + 1= p (3 , ( x j − i − x h +1+ i − j ) − p ( x j − i − x h − j + i +24 ) − p ( j − i − X k =0 x k x j − i − − k x h − − k x h − j + i +2+ k ) , for j > i + 1= p x i − j + p ( i − j X k =0 ( x k x h − − k x i − j − k x h +1+ k ) , for j < i. (18) S ( p (3 ,j ) , p ) = − x h + j − x h − j +14 + x j x h − j = p (3 , x j − x h − l + p ( − x j − x h − j +14 ) − p ( j − X k =0 x k x j − − k x h − − k x h − j +1+ k ) , for 0 < j ≤ h − p , for j = 0 (19) S ( p (3 ,j ) , p ) = x h − j − − x h − j − x j x h − l = p x h − j − + p ( h − j − X k =0 x k x h − − k x h − j − − k x h +1+ k ) . ROJECTIVE CLOSURES OF AFFINE CURVES 15 (20) S ( p (4 ,i ) , p (4 ,j ) ) = x j − i x j − i x h − j − x j − i x i − x h − i = p ( P j − i − k = o x k x j − i − − k x j − − k x h − j + k ) .(21) Since gcd(Lt( p (4 ,i ) ) , Lt( p )) = 1 , S ( p (4 ,i ) , p ) −→ G h .(22) S ( p (4 ,i ) , p (6 ,j ) ) = x x h +1+ i − x x i x i − x h − i = p ( x x h + i + x x h + i − x + p (3 , x x i − x , for i = j = x j +2 − i x h +1+ j − x j − i x i − x h − i + j +24 = p ( j − i +1 X l =0 x l x j +1 − i − l x h + j − l x l ) + p (3 , x j − i x i − x j − i +24 , for i < j = x i − j − x h +1+ j − x i − j − x i − x h − i + j +24 = p (3 , x i − j − x j − p ( i − j − X l =0 x l x i − j − − l x i − − l x h − i + j +2+ l ) , for i ≥ j + 2= x x h +1+ j − x j x h +14 = p x h + j − p (3 , x j x , for i = j + 1 . (23) S ( p (4 ,i ) , p ) = x h x i − x i − x i − x h − i +14 = p (3 , , for i = 1= p (3 , x i − + p ( − i − X l =0 x l x i − − l x i − − l x h − i +1+ l ) , for 1 < i ≤ h. (24) S ( p (4 ,i ) , p ) = x h − i x h − − x h − i − x i − x h − i = p ( h − i − X l =0 x l x h − i − − l x h − − l x l ) + p (3 , x h − i − x i − x h − i , for 1 ≤ i < h = x x h − − x h − x h = p (3 , x h − , for i = 2 h. (25) S ( p (6 ,i ) , p (6 ,j ) ) = x j − i x h +1+ j − x j − i x h +1+ i x j − i = p ( j − i − X l =0 x l x j − i − − l x h + j − l x l ) , for i < j. (26) S ( p (6 ,i ) , p ) = x i x i x h − x i x h +1+ i = − p ( P il =0 x l x i − l x h + i − l x l ) (27) S ( p (6 ,i ) , p ) = x h − − i x h − − x h − − i x h − − i x h +1+ i = p ( h − − i X l =0 x l x h − − i − l x h − − l x l ) . (28) S ( p , p ) = x h − x h − − x h − x h x h − = p ( P h − l =0 x l x h − − l x h − − l x l ) We see that all S-polynomials reduce to zero upon division by G h , there-fore G h is Gr¨obner basis of Q h with respect to degree reverse lexicographicmonomial order on k [ x , x , x , x ] induced by x > x > x > x . (cid:3) Suppose for any g ∈ k [ x , x , x , x ] , we denote g H ∈ k [ x , x , x , x , x ] the homogenization of g w.r.t. the indeterminates x . Lemma 6.2.
For h ≥ , let G h := { g H | g ∈ G h } . Then G h is a Gr¨obnerbasis of the homogenized ideal Q h with respect to degree reverse lexico-graphic monomial order on k [ x , x , x , x , x ] induced by x > x > x >x > x .Proof. Follows from theorems 6.1 and 4.6. (cid:3)
Lemma 6.3.
The ideal Q h , for h ≥ is generated by following polynomials • p H = x x − x x • p H = x h − x x h − • p H (3 ,j ) = x j +11 x h − l − x x j x h − j ; ≤ j ≤ h − • p H (4 ,i ) = x i +11 x h − i − x x i − x h − i ; ≤ i ≤ h • p H = x h − x h − x h +14 Proof.
Let T h = { p H , p H , p H (3 ,j ) , p H (4 ,i ) , p H | ≤ j ≤ h − , ≤ i ≤ h } ,for h ≥ . Then T h ⊂ G h , for h ≥ . We have (cid:3) Notations 6.2.
For h ≥ we define the following matrices,(1) B h := (cid:0) p H p H p H (3 , · · · p h (3 , h − p h (4 , · · · p H (4 , h ) p H (cid:1) × (4 h +3) (2) B h := (cid:0) Ω ij (cid:1) (4 h +3) × (8 h +4) , where Ω ij , ≤ i ≤ h + 3 , ≤ j ≤ h + 4 are defined as follows, • Ω ((2 , = x x , Ω (2 h +2 , = x , Ω (4 h +1 , = − x and Ω ( i, = 0 ,for i ∈ [4 h + 3] \ { , h + 2 , h + 1 } . • Ω (2 , = x , Ω (4 , = x , Ω (2 h +3 , = − x and Ω ( i, = 0 , for i ∈ [4 h + 3] \ { , , h + 3 } . ROJECTIVE CLOSURES OF AFFINE CURVES 17 • Ω (1 , l ) = x x l x h − − l , Ω (3+ l, l ) = − x , Ω (4+ l, l ) = x and Ω ( i, l ) = 0 , for i ∈ [4 h + 3] \ { , l, l } , ≤ l ≤ h − . • Ω (1 , h +1+ l ) = x x l − x h − − l , Ω (2 h +2+ l, h +1+ l ) = − x , Ω (2 h +3+ l, h +1+ l ) = x and Ω ( i, h +1+ l ) = 0 , for i ∈ [4 h + 3] \ { , h + 2 + l, h +3 + l } , ≤ l ≤ h − . • Ω (1 , h +1+ l ) = x l +11 x h − − l , Ω (3+ l, h +1+ l ) = − x , Ω (4+ l, h + l +1) = x and Ω ( i, h +1+ l ) = 0 , for i ∈ [4 h + 3] \ { , l, l } , ≤ l ≤ h − . • Ω (1 , h − l ) = x l +11 x h − − l , Ω (2 h +2+ l, h − l ) = − x , Ω (2 h +3+ l, h − l ) = x and Ω ( i, h − l ) = 0 , for i ∈ [4 h + 3] \ { , h + 2 + l, h +3 + l } , ≤ l ≤ h − . • Ω (1 , h − = x h − x x h − , Ω (2 , h − = − x x + x x , and Ω ( i, h − = 0 , for i ∈ [4 h + 3] \ { , } , . • Ω (1 , h ) = x x h − , Ω (2 , h ) = − x x , Ω (3 , h ) = − x , Ω (2 h +3 , h = x , and Ω ( i, h ) = 0 , for i ∈ [4 h + 3] \ { , , , h + 3 } , . • Ω (1 , h +1) = x x h − , Ω (2 , h +1) = − x x , Ω (2 h +2 , h +1) = − x , Ω (4 h +2 , h +1) = x , and Ω ( i, h +1) = 0 , for i ∈ [4 h + 3] \ { , , h + 2 , h + 2 } , . • Ω (1 , h +2) = x h − x h , Ω (2 , h +2) = − x x , Ω (3 , h +2) = x h , Ω (4 h +3 , h +2) = − x , and Ω ( i, h +2) = 0 , for i ∈ [4 h + 3] \ { , , , h + 3 } , . • Ω (1 , h +3) = x h − , Ω (2 , h +3) = − x h +14 , Ω (3 , h +3) = x h − x , Ω (4 h +3 , h +3) = − x , and Ω ( i, h +3) = 0 , for i ∈ [4 h + 3] \ { , , , h + 3 } , . • Ω (1 , h +4) = x h − x h , Ω (2 , h +4) = − x h +13 , Ω (3 , h +4) = x h − x , Ω (4 h +3 , h +3) = − x , and Ω ( i, h +3) = 0 , for i ∈ [4 h + 3] \ { , , , h + 3 } , .(3) B h := (cid:0) ∆ ij (cid:1) (8 h +4) × (4 h +3) , where ∆ ij , ≤ i ≤ h + 4 , ≤ j ≤ h + 3 are defined as follows • ∆ (4 h + l,l +1) = x , ∆ (3+ l,l +1) = − x , ∆ (4 h + l +1 ,l +1) = − x , ∆ (4+ l,l +1) = x , and ∆ ( i,l +1) = 0 , for i ∈ [8 h + 4] \ { h + l, l, h + l +1 , l } , ≤ l ≤ h − . • ∆ (6 h − , h − = x , ∆ (2 h +1 , h − = − x , ∆ (8 h − , h − = − x , ∆ (1 , h − = x , ∆ (8 h +1 , h − = x and ∆ ( i, h − = 0 , for i ∈ [8 h + 4] \ { h − , h + 1 , h − , , h + 1 }• ∆ (6 h − l, h − l ) = x , ∆ (6 h + l, h − l ) = − x , ∆ (2 h +1+ l, h − l ) = − x , ∆ (2 h +2+ l, h − l ) = x , and ∆ ( i, h − l ) = 0 , for i ∈ [8 h +4] \ { h − l, h + l, h + 1 + l, h + 2 + l } ≤ l ≤ h − • ∆ (3 , h − = x , ∆ (8 h, h − = − x , ∆ (6 h, h − = x , ∆ (2 , h − = − x , ∆ (2 h +2 , h − = − x and ∆ ( i, h − = 0 , for i ∈ [8 h + 4] \{ , h, h, , h + 2 } • ∆ (4 h +1 , h − = x , ∆ (8 h − , h − = x , ∆ (8 h, h − = − x , ∆ (2 , h − = − x , and ∆ ( i, h − = 0 , for i ∈ [8 h + 4] \ { h + 1 , h − , h, }• ∆ (8 h − , h ) = x , ∆ (1 , h ) = − x , ∆ (8 h +1 , h ) = − x , ∆ (4 h, h ) = x , and ∆ ( i, h ) = 0 , for i ∈ [8 h + 4] \ { , h − , h + 1 , h }• ∆ (4 h +1 , h +1) = x h , ∆ (3 , h +1) = − x h − x , ∆ (8 h − , h +1) = − x h , ∆ (8 h +3 , h +1) = − x , ∆ (8 h +2 , h +1) = x and ∆ ( i, h +1) =0 , for i ∈ [8 h + 4] \ { h + 1 , , h − , h + 3 , h + 2 }• ∆ (4 h +2 , h − = x h , ∆ (4 h +1+ i, h +2) = − x h − − i x i +14 , ∆ (8 h, h +2) = x h − x , ∆ (6 h − j, h +2) = x (2 h − − j x j +14 , ∆ (1 , h +2) = x h , ∆ (8 h +3 , h +2) = x , ∆ (8 h +4 , h +2) = − x , and ∆ ( i, h +2) = 0 , for i ∈ [8 h + 4] \{ h − , h + 1 + i, h, h − j, , h + 3 , h + 4 } , ≤ i ≤ h − , ≤ j ≤ h − • ∆ (8 h, h +3) = x h , ∆ (3 , h +3) = − x h − x , ∆ (4 h +1+ i, h +3) = − x h − − i x x i , ∆ (6 h − i, h +3) = x h − i x i , ∆ (1 , h +3) = x x h − , ∆ (8 h +2 , h +3) = x , ∆ (8 h +4 , h +3) = − x , and ∆ ( i, h +3) = 0 , for i ∈ [8 h +4] \{ h, , , h +1+ i, h − i, , h + 2 , h + 4 } (4) B h := (cid:0) Γ ij (cid:1) (4 h +3) × , where Γ i , ≤ i ≤ h + 3 are defined asfollows • Γ (4 h − , = x h , • Γ ( m, = − x h − m x m ,for ≤ m ≤ h − • Γ (4 h − , = − x (2 h − x , • Γ (2 h − n, = x h − − n x n +14 ,for ≤ n ≤ h − • Γ (4 h, = − x h , • Γ (4 h +1 , = − x , • Γ (4 h +2 , = − x , • Γ (4 h +3 , = x , • Γ ( i, h +1) = 0 , for i ∈ { , . . . , h + 4 } \ { h − , i, h − , h − j, h, h +1 , h +2 , h +3 } , ≤ i ≤ h − , ≤ j ≤ h − . Theorem 6.4.
For h ≥ , we consider the homogenized ideal Q h (defin-ing ideal of projective closure of Bresinsky’s curve) in the polynomial ring k [ x , x , x , x , x ] . Then a minimal graded free resolution of Q h is givenby, B h : 0 −→ R B h −→ R h +3 B h −→ R h +4 B h −→ R h +3 B h −→ R −→ R/ Q h −→ Where matrices B i , ≤ i ≤ are defined in 6.2. ROJECTIVE CLOSURES OF AFFINE CURVES 19
Proof.
One can check that B h : 0 −→ R B h −→ R h +3 B h −→ R h +4 B h −→ R h +3 B h −→ R −→ R/ Q h −→ is a chain complex by calculating B ih B i +1 h = 0 , ≤ i ≤ . We proceedby Buchsbaum Eisenbud acyclicity criterion. Let r i be the i th expectedrank of B h . Then r = (4 h + 3) − (8 h + 4) + (4 h + 3) − , r =8 h + 4 − (4 h + 3) + 1 = 4 h + 2 , r = 4 h + 3 − h + 2 , r = 1 . Weneed to show that grade( I r i ( B ih )) ≥ i, ≤ i ≤ (where I r i ( B ih ) denotesthe ideal generated by r i × r i minors of the matrix B ih )(1) We take p H ∈ I r ( B ) and we have I r ( B ) ≥ .(2) We take • L [21] h := [134 . . . (4 h + 3) | (4 h + 1) . . . (8 h − h (8 h + 2)(8 h +3)(8 h + 4)] = x h − ( − x h + x h − x )( x h − x h − x h +14 ) • L [22] h := [2 . . . (4 h + 3) | . . . h (8 h + 1)(8 h + 2)(8 h + 4)] = − x h − ( − x x + x x )( − x x h − x + x h x )( x x h ) .Since gcd of any two prime factors of L [21] h and L [22] h is 1, L [21] h , L [22] h forms a regular sequence. Hence grade( I r ( B h )) ≥ (3) We take • L [31] h := [1(4 h ) . . . (6 h − . . . (8 h − h − h )(8 h +3)(8 h + 4) | . . . (4 h + 1)(4 h + 3)] = x h +21 , • L [32] h := [1 . . . h (8 h + 1)(8 h + 2) | . . . (4 h + 2)] = x h ( − x h + x h − x ) • L [33] h := [3 . . . (2 h + 1)(4 h + 1)(6 h ) . . . h (8 h + 3) | . . . (4 h + 2)]= x x h − x − x ( h − X i =0 x i x h − − i x h − i x i ) − x x h − x h − x x h − + x x h − x h − x x h + x x h − x h +13 x h − Let us consider the ideal h L [31] h , L [32] h i . Then primary decomposi-tion of this ideal is h x h +21 , x h i ∩ h x h +21 , ( − x h + x h − x ) i . Henceassociated primes are h x , x i , h x , ( − x h + x h − x ) i . We ob-serve that L [33] h / ∈ h x , x i and L [33] h / ∈ h x , ( − x h + x h − x ) i .If L [33] h ∈ h x , x i then x x h − x h ∈ h x , x i which is a contra-diction, if L [33] h ∈ h x , ( − x h + x h − x ) i then agin x x h − x h ∈h x , ( − x h + x h − x ) i a contradiction. Therefore { L [31] h , L [32] h , L [33] h } forms a regular sequence hence grade( I r ( B h )) ≥ .(4) We take • Γ (4 h, = − x h , • Γ (4 h +1 , = − x , • Γ (4 h +2 , = − x , • Γ (4 h +3 , = x .Hence grade( I r ( B h )) ≥ . Therefore the complex B h is exact, minimality follows from the fact, allentries of matrices B ih , ≤ i ≤ are lies in homogenious maximal ideal h x , . . . , x i . (cid:3)
7. S
YZYGIES OF PROJECTIVE CLOSURE OF A RSLAN C URVES
In [1] Arslan gave a necessary and sufficient criterion for Cohen-macaulaynessof tangent cone of a monomial curve at origin using Gr¨obner basis. He in-troduced the following curves, for h ≥ , A h := C ( h ( h + 1) , h ( h + 1) +1 , ( h + 1) , ( h + 1) + 1) and showed that tangent cone of this curves atorigin are Cohen-Macaulay. It has been shown in [4] that Arslan curvesare arithmetically Cohen-Macaulay using Gr¨obner basis. Here we will findsyzygies of projective closure of Arslan curves, i.e A h and will show thatall betti numbers are unbounded function of h . Lemma 7.1.
Let us consider the polynomials, for h ≥ , • w = x x − x x • g i = x i x h − i +13 − x i +12 x h − i , for ≤ i ≤ h − • g h = x h +12 − x h x • q j = x j +11 x h − j − x j x h − j x , for ≤ j ≤ h .Suppose U h = { w, g i , q j | ≤ i, j ≤ h } Then U h is a Gr¨obner basis ofdefining ideal of projective closure of the Arslan’s curve A h w.r.t. degree re-verse lexicographic monomial order on the polynomial ring k [ x , x , x , x , x ] induced by x > x > x > x > x .Proof. We use proposition 5.2 in [4] and theorem 4.6.
Notations 7.1.
For h ≥ , let I h := p ( h ( h + 1) , h ( h + 1) + 1 , ( h + 1) , ( h +1) + 1) and I h := p ( h ( h + 1) , h ( h + 1) + 1 , ( h + 1) , ( h + 1) + 1) . Alsofor h ≥ , we define the following matrices.(1) A h := (cid:0) w g ... g h q ... q h (cid:1) (2) A h := (cid:0) Φ ij (cid:1) (2 h +3) × (4 h +1) where Φ ij are defined as follows, • Φ ( h +2 , = x , Φ ( h +3 , = − x , Φ (2 h +3 , = x , Φ (2 , = x , Φ ( i, =0 , for i ∈ { , . . . , h + 1 } \ { , h + 2 , h + 3 , h + 3 } . • Φ (1 , = x i x h − i − x , Φ ( h +3+ i, = − x , Φ ( h +4+ i, = x , Φ ( i, =0 , for i ∈ { , . . . , h +1 }\{ , h +3+ i, h +4+ i } , ≤ i ≤ h − . ROJECTIVE CLOSURES OF AFFINE CURVES 21 • Φ (1 ,h +2+ i ) = x i +12 x h − i − , Φ (2+ i,h +2+ i ) = − x , Φ (3+ i,h +2+ i ) = x , Φ ( j,h +2+ i ) = 0 , ,for j ∈ { , . . . , h +1 }\{ , i, i } , ≤ i ≤ h − • Φ (1 , h +1) = x h , Φ ( h +1 , h +1) = − x , Φ ( h +2 , h +1) = − x , , Φ ( j, h +1) =0 ,for j ∈ { , . . . , h + 1 } \ { , h + 1 , h + 2 }• Φ (1 , h +2+ i ) = x i x h − i , Φ (2+ i, h +2+ i ) = − x , Φ (3+ i, h +2+ i ) = x , Φ ( j, h +2+ i ) = 0 , ,for j ∈ { , . . . , h +1 }\{ , i, i } , ≤ i ≤ h − • Φ (1 , h +1) = x h − x , Φ ( h +1 , h +1) = − x , Φ ( h +2 , h +1) = − x , Φ ( j, h +2+ i ) =0 , for j ∈ { , . . . , h + 1 } \ { , h + 1 , h + 2 }• Φ (1 , h +2+ i ) = x i +11 x h − i − , Φ ( h +3+ i, h +2+ i ) = − x , Φ ( h +4+ i, h +2+ i ) = x , Φ ( j, h +2+ i ) = 0 ,for j ∈ { , . . . , h + 1 } \ { , h + 3 + i, h +4 + i } , ≤ i ≤ h − (3) A h := (cid:0) Ψ ij (cid:1) (4 h +1) × (2 h − where Ψ ij are defined as follows, • Ψ (2 h +2+ i, i ) = x , Ψ ( h +2+ i, i ) = − x , Ψ (2 h +3+ i, i ) = − x , Ψ ( h +3+ i, i ) = x , Ψ ( j, i ) = 0 , for ,for j ∈ { , . . . , h − } \ { h + 2 + i, h +2 + i, h + 3 + i, h + 3 + i } , ≤ i ≤ h − • Ψ (3 h +2+ i,h + i ) = x , Ψ (3 h +3+ i,h + i ) = − x , Ψ (2+ i,h + i ) = − x , Ψ (3+ i,h + i ) = x , Ψ ( j,h + i ) = 0 , for ,for j ∈ { , . . . , h − } \ { h + 2 + i, h +3 + i, i, i } , ≤ i ≤ h − • Ψ (3 h +1 , h − = x , Ψ (2 h +1 , h − = − x x , Ψ (3 h +2 , h − = x , Ψ (4 h +1 , h − = − x x , Ψ (1 , h − = − x x , Ψ ( h +1 , h − = x , Ψ (1 , h − = x x , Ψ (2 , h − = − x x , Ψ (2 h +2 , h − = − x x , Ψ ( h +2 , h − = x x Theorem 7.2.
For h ≥ , a graded free minimal resolution of I h (the defin-ing ideal of the projective closure of Arslan’s curve A h ) is M h : 0 −→ R h − A h −→ R h +1 A h −→ R h +3 A h −→ R −→ R/ I h −→ . Where the matrices A ih , ≤ i ≤ has defined in 7.1.Proof. We proceed exactly same way as in the proof of theorem 6.4 that iswe use Buchsbaum-Eisenbud acyclicity criterion.Let r i be the i th expectedrank of A h . Then r = (2 h + 3) − (4 h + 1) + (2 h − − , r = 4 h +1 − (2 h −
1) = 2 h + 2 , r = 2 h − . We need to show that grade( I r i ( A ih )) ≥ i, ≤ i ≤ .(1) We take following minors from A h • D [21] h := [13 · · · (2 h +3) | · · · (2 h +2)] = ( x h +12 − x h x )( − x h +13 + x x h ) x h − • D [22] h := [2 · · · (2 h + 3) | h + 2) · · · (4 h + 1)] = ( x x − x x )( − x x h + x h x ) x h − ) . It is clear from factorization that D [21] h , D [22] h forms a regular se-quence. Hence grade( I r ( A h )) ≥ .(2) We take following minors from A h • D [31] h := [2 · · · h (2 h +2) · · · (3 h +1) | · · · (2 h − x h +11 x h − − x h − x = x h − ( x h +11 − x h x ) • D [32] h := [2 · · · h | · · · (2 h − x h − x h +13 − x h x h = x h − ( x h +13 − x h +12 x h ) • D [33] h := [( h + 2) · · · (2 h + 1)(3 h + 2) · · · h | · · · (2 h − x h x h − x h − x h x = x h − ( x x h − x h x ) From factorization of polynomials, primary decomposition of theideal h D [31] h , D [32] h i is h x h − , x h − i∩h x h − , x h +13 − x h +12 x h i∩h x h +11 − x h x , x h − i∩h x h +11 − x h x , x h +13 − x h +12 x h i . Therefore associated primes of the ideal h D [31] h , D [32] h i are, • h x , x i , • h x , x h +13 − x h +12 x h i , • h x h +11 − x h x , x i , • h x h +11 − x h x , x h +13 − x h +12 x h i . One can easily check that the polynomial D [33] h does not lie in anyassociated primes of the ideal h D [31] h , D [32] h i . Therefore { D [31] h , D [32] h , D [33] h } forms a regular sequence. Hence grade( I r ( A h )) ≥ . Minimalityfollows from the fact, all entries of matrices A ih , ≤ i ≤ are liesin homogenious maximal ideal h x , . . . , x i . (cid:3) R EFERENCES [1] F.Arslan,
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On Prime Ideals with Generic Zero x i = t n i , Proceedings of the Amer-ican Mathematical Society, Vol.47, No.2, february 1975.[6] H. Bresinsky and L. T. Hoa, Minimal generating sets for a family of monomial curvesin A , Commutative algebra and algebraic geometry (Ferrara), Lecture Notes in Pureand Appl. Math. 206 (Dekker, New York, 1999) 5–14. ROJECTIVE CLOSURES OF AFFINE CURVES 23 [7] Decker, W.; Greuel, G.-M.; Pfister, G.; Sch¨onemann, H.: S
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Email address : [email protected] Discipline of Mathematics, IIT Gandhinagar, Palaj, Gandhinagar, Gujarat 382355,INDIA.
Email address : [email protected] Discipline of Mathematics, IIT Gandhinagar, Palaj, Gandhinagar, Gujarat 382355,INDIA.
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