Symbolic blowup algebras and invariants associated to certain monomial curves in P 3
aa r X i v : . [ m a t h . A C ] M a r SYMBOLIC BLOWUP ALGEBRAS AND INVARIANTS ASSOCIATED TOCERTAIN MONOMIAL CURVES IN P CLARE D’CRUZ ∗ AND MOUSUMI MANDAL † Abstract.
In this paper we explicitly describe the symbolic powers of the ideal defining the curve C ( q, m ) in P parametrized by ( x d +2 m , x d + m y m , x d y m , y d +2 m ), where q, m are positive integers, d = 2 q + 1 and gcd( d, m ) = 1. We show that the symbolic blowup algebra is Noetherian andGorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal p := p C ( q,m ) defining the curve C ( q, m ) is computed. We also give a formula for the Castelnuovo-Mumford regularity of the symbolic powers p ( n ) for all n ≥ Introduction
Let k be a field, A = k [ x , . . . , x t ] a polynomial ring and I a homogeneous ideal in A withno embedded components. For every n ≥
1, the n -th symbolic power of I is defined as I ( n ) := \ p ∈ Ass(
A/I ) ( I n A p ∩ A ). By a classical result of Zariski and Nagata n -th symbolic power of a givenprime ideal consists of the elements that vanish up to order n on the corresponding variety. However,describing the generators of symbolic powers is not easy. One can verify that I n ⊆ I ( n ) and in factfor 0 = I ⊂ A, I r ⊆ I ( n ) holds if and only if r ≥ n . It is a challenging problem to determine forwhich n and r the containment I ( n ) ⊆ I r holds true. The results in [10] and [14] show that I ( n ) ⊆ I r for n ≥ ( t − r . In the direction of comparing the symbolic powers and ordinary powers of ideals,B. Harbourne raised the following conjecture in [1, Conjecture 8.4.3]: For any homogeneous ideal I ⊂ A , I ( n ) ⊆ I r if n ≥ r ( t − − ( t − n for which I ( n ) ⊆ I r holds for a given ideal I and for an integer r . To answer this question C. Bocci and B. Harbournedefined an asymptotic quantity called resurgence which is defined as ρ ( I ) = sup { m/r | I ( m ) I r } (see [3]). From the results in [10] and [14] it follows that this quantity exists for radical ideals.In fact, 1 ≤ ρ ( I ) ≤ t − γ ( I ) in [3]. This invariant isdefined as γ ( I ) = lim n →∞ α ( I ( n ) ) n , where α ( I ) denotes the least degree of a homogeneous generator of Mathematics Subject Classification.
Primary: 13A30, 1305, 13H15, 13P10.
Key words and phrases.
Symbolic Rees algebra, resurgence, Waldschmidt constant, Castelnuovo-Mumfordregularity. ∗ partially supported by a grant from Infosys Foundation. † Supported by ISIRD(IIT KGP) grant No.: IIT/SRIC/MA/ANL/2017-18/45, India. ∗ AND MOUSUMI MANDAL I . They showed that if I is an homogeneous ideal, then α ( I ) /γ ( I ) ≤ ρ ( I ) and in addition if I isa zero dimensional subscheme in a projective space, then α ( I ) /γ ( I ) ≤ ρ ( I ) ≤ reg( I ) /γ ( I ), wherereg( I ) denotes the Castelnuovo-Mumford regularity [3, Theorem 1.2.1]. Hence, if α ( I ) = reg( I ),then ρ ( I ) = α ( I ) /γ ( I ). Later, in [13, Conjecture 2.1] Harbourne and Huneke raised the followingConjecture: Let I be an ideal of fat points in A and m = ( x , . . . , x t ). Then I ( n ( t − ⊆ m n ( t − I n holds true for all I and n . In the same paper they showed that the conjecture is true for fat pointideals arising as symbolic powers of radical ideals generated in a single degree in P .The resurgence and the Waldschmidt constant has been studied in a few cases: for certain generalpoints in P [2], smooth subschemes [12], fat linear subspaces [11], special point configurations [8]and monomial ideals [4]. The behaviour of Castelnuovo-Mumford regularity of symbolic powers isnot easy to predict. From a result of Cutkosky, Herzog and Trung, it follows that if I is an ideal ofpoints in a projective space and the symbolic Rees algebra M n ≥ I ( n ) is Noetherian, then reg( I ( n ) ) is aquasi-polynomial ([6, Theorem 4.3]). Moreover, lim n →∞ reg( I ( n ) ) n ! exists and can even be irrational[7].Though there are several results available for the the resurgence, the Waldschmidt constant andthe Castelnuovo-Mumford regularity of symbolic powers, there is no precise result for monomialcurves in a projective space. Though it is well known that p (2) / p is a cyclic module (for examplesee [19, Lemma 2.1]), the explicit description of the generator has been crucial in our study of thevarious invariants. In this paper we focus on the ideal defining the monomial curve C ( q, m ) in P parametrized by ( x d +2 m , x d + m y m , x d y m , y d +2 m ), where q, m are positive integers, d = 2 q + 1 withgcd( d, m ) = 1.Another topic of interest is the Gorenstein property of the symbolic blowup algebras. If q = 1, i.e., d = 3, then the monomial curves we consider coincide with the curves in [19, Theorem 3.2(v)(2)(a)].However, our proof here is different. We use properties of monomial ideals to obtain our results.These computations are also useful in computing the Castelnuovo-Mumford regularity of the sym-bolic powers. The Gorenstein property of monomial curves in P have also been studied by Schenzelin [20]. However, their curves do not overlap with the monomial curves we consider in this paper.We now briefly describe the contents of our paper. Let φ : R = k [ x , x , x , x ] −→ S = k [ x, y ] bea homomorphism given by φ ( x ) = x d +2 m , φ ( x ) = x d + m y m , φ ( x ) = x d y m and φ ( x ) = y d +2 m . Forthe rest of this paper p := p C ( q,m ) := ker φ . In Section 2, we prove a few preliminary results. InSection 3, we prove some results on monomial ideals which will be used in the subsequent sections.In Section 4, we explicitly describe the generators of p ( n ) (Theorem 4.6) and show that the symbolicRees algebra M n ≥ p ( n ) is Noetherian. As computing symbolic powers is not easy we use a simpletrick. We consider the ring T = R/ ( x , x ). Then p T is a monomial ideal and eventually for all n ≥ p ( n ) T is a monomial ideal (Proposition 4.4). In Section 5, we show that R s ( p ) is Gorenstein. Moreover, the symbolic fiber cone F s ( p ) := R ( p ) ⊗ R R/ m = M n ≥ p ( n ) / mp ( n ) is Cohen-Macaulay(Theorem 5.5).Section 6 is devoted to study certain invariants associated to p , namely the resurgence, the Wald-schmidt constant and the Castelnuovo-Mumford regularity. We verify that Conjecture 8.4.3 in[1] and Conjecture 2.1 in [13] is true for p (Corollary 6.8, Corollary 6.8). We express the resur-gence for the monomial curve C ( q, m ) in terms of the degree of the curve C ( q, m ). In particularwe show that ρ ( p ) = e ( R/ p ) − e ( R/ p ) − e ( R/ p ) is the degree of C ( q, m ) (Theorem 6.10). TheWaldschmidt constant is calculated for the same (Theorem 6.13). We next give an explicit for-mula for the Castelnuovo-Mumford regularity for all the symbolic powers p ( n ) and show that it isa quasi-polynomial (Theorem 6.28). As a consequence we show that lim n −→∞ reg R p ( n ) ! = e ( R/ p )2(Corollary 6.29). We end this paper by comparing all these invariants and show that there existmonomial curves for which Theorem 1.2.1(b) in [3] may not hold true (Lemma 6.31).2. Basic results
In this section we prove several results which are probably well known in literature. We provideproofs for the sake of convenience. For the rest of this paper q , m and d are as in the introductionand p := p C ( q,m ) ⊆ R .It is well known that the generators for p are the 2 × x x x q + m x x x q x m ! [18]. In particular, if g = x q x x m − x q + m +13 , g = x q +11 x m − x x q + m and g = x x − x (2.1)then p = ( g , g , g ). Lemma 2.2. (1) R/ p is Cohen-Macaulay. In particular x , x is a regular sequence in R/ p .(2) e ( x , x ); R p ! = 2( q + m ) + 1 = d + 2 m. Proof. (1) From the Hilbert-Burch theorem it follows that the minimal free resolution of p is of theform 0 −→ R [ − ( q + m + 2)] φ −→ R [ − ( q + m + 1)] ⊕ R [ − ψ −→ p −→ φ = x x − x − x x q x m x q + m and ψ = (cid:16) g g g (cid:17) . Hence depth( R/ p ) = 2 = dim( R/ p ) = 2. Thisimplies that R/ p is Cohen-Macaulay. As x , x is a system of parameters for R/ p , it is a regularsequence by [15, Corollary 11.12]. CLARE D’CRUZ ∗ AND MOUSUMI MANDAL (2) Since x , x is a regular sequence, e ( x , x ); R p ! = ℓ R ( p + ( x , x )) ! = ℓ R ( x , x , x , x q + m )+13 ) ! = 2( q + m ) + 1 . (cid:3) Put f := x q + m g − x q x m g + x q − x q + m − x x m g (2.4)= − x q + m )+13 − x q − x x q + m − x m + 3 x q x x q + m x m − x q +11 x m . (2.5) Lemma 2.6. (1) For i = 1 , , , x i f ∈ p .(2) f ∈ p (2) .(3) For all n = 1 , . . . , q + m + 1 , f n ∈ p n − .Proof. (1) As g i ∈ p for i = 1 , , x f = x q + m − g g − g ∈ p x f = − x q − x q + m − x m g − g g ∈ p x f = − x q − x m g g − g ∈ p . (2.7)(2) From (1) it follows that x f ∈ p ⊆ p (2) . As x p , f ∈ p (2) .(3) Let 1 ≤ n ≤ q + m + 1. By the definition of f , f n = ( x q + m g − x q x m g + x q − x q + m − x x m g ) f n − = ( x q + m f n − ) g − ( x q x m f n − ) g + ( x q − x q + m − x x m f n − ) g ∈ p n − p [from (1)]= p n − . (2.8) (cid:3) Computations with monomial ideals
In general, symbolic powers are not easy to compute. Hence, we first consider the ring T := R/ ( x , x ) ∼ = k [ x , x ]. Since p T is a monomial ideal, p n T is also. Consider p T = ( x , x x q + m , x q + m +13 ) , ( f ) T = ( x m + q )+13 ) , I n := X n +2 n = n ( f n T )( p T ) n ⊆ p ( n ) T. (3.1)Our aim in this section is to compute ℓ ( T /I n ). For this, we fist need to show that ( I n : x q + m ) ⊆ I n − .Next we will compute ℓ ( I n − / ( I n : x q + m )). Lemma 3.2.
For all n ≥ , ( I n : x q + m ) ⊆ I n − . Proof.
From the definition of I n we get( I n : x q + m ) = X a +2 a = n ; a =0 ( f a ( p T ) a : x q + m ) + (( x , x x q + m , x q + m +13 ) n : x q + m )= X a +2 a = n ; a =0 ( x q + m +13 ) f a − ( p T ) a + ( x n ) + n X i =1 ( x ( q + m )( i − ( x , x ) i − x n − i )2 )( x , x ) ⊆ I n − . (cid:3) Our next step is to describe the generating set of I n modulo ( I n : x q + m ). Lemma 3.3.
The minimal set of generators of I n − / ( I n : x q + m ) form a vector space basis over k .Proof. Put M = I n − / ( I n : x q + m ) and m ′ = ( x , x ). Since x q + m m ′ I n − ⊆ ( p T ) I n − ⊆ I n , we get m ′ I n − ⊆ ( I n : x q + m ). Hence m ′ M = 0 which implies that M/ m ′ M ∼ = M . By graded Nakayama’sLemma the generators of M form a vector space basis over T / m ′ ∼ = k . (cid:3) In Lemma 3.7 we explicitly describe the generating set of I n − modulo ( I n : x q + m ). We state aresult on monomial ideals which follows from [9, Proposition 1.14] and will be consistently used inall the proofs which involve monomial ideals. Proposition 3.4.
Let I = ( u , . . . , u r ) and J = ( v ) be monomial ideals in a polynomial ring overa field k . Then I : J = ( { u i /gcd ( u i , v ) : i = 1 , ..., r } ) . Lemma 3.5.
For all n ≥ , p n T ⊆ x n − ( x , x q + m ) + ( I n +1 : x q + m ) Proof.
We prove by induction on n . If n = 1, then p T = x ( x , x q + m ) + ( x q + m +13 ) ⊆ x ( x , x q + m ) + ( I : x q + m ) . Hence the claim is true for n = 1. Let n >
1. Then p n T = ( p T )( p n − T ) ⊆ (( x , x x q + m ) , x q + m +13 ) (cid:16) x n − ( x , x q + m ) + ( I n : x q + m ) (cid:17) [by induction hypothesis]= x n − ( x , x q + m ) + ( x n − x q + m )3 ) + ( x , x x q + m )( I n : x q + m )+( f T ) (cid:16) x n − ( x , x q + m ) + ( I n : x q + m ) (cid:17) (3.6) CLARE D’CRUZ ∗ AND MOUSUMI MANDAL
We now verify that all the terms except x n − ( x , x q + m ) are in ( I n +1 : x q + m ). x q + m (cid:16) ( x n − ) x q + m )3 (cid:17) = ( x n − )( x q + m )+13 x q + m − ) ⊆ f ( p n − T ) ⊆ I n − = I n +1 ( x , x x q + m )( I n : x q + m ) ⊆ ( p T )( I n : x q + m ) ⊆ ( I n +1 : x q + m ) x q + m (cid:16) x q + m +13 x n − ( x , x q + m ) (cid:17) = x n − · x ( x , x q + m ) · x q + m )+13 ⊆ f ( p n − T ) ⊆ I n − = I n +1 f ( I n : x q + m ) ⊆ ( I n +1 : x q + m ) (cid:3) We are now ready to describe the generators of I n − modulo ( I n : x q + m ). Lemma 3.7.
For all n ≥ , I n − = n − X a =0 x (2( q + m )+1) a x n − − a ) − ( x , x q + m ) + ( I n : x q + m ) if ( n − (cid:18) x (2( q + m )+1) ( n − ) (cid:19) + n − X a =0 x (2( q + m )+1) a x n − − a ) − ( x , x q + m ) + ( I n : x q + m ) if | ( n − . Proof.
From (3.1) we get I n − = n − X a =0 f a ( p T ) n − − a if 2 ( n − f ( n − T ) + n − X a =0 f a ( p T ) n − − a if 2 | ( n − ⊆ n − X a =0 f a (cid:16) x n − − a ) − ( x , x q + m ) + ( I n − a : x q + m ) (cid:17) if 2 ( n − x (2( q + m )+1)) ( n − ) + n − X a =0 ( f a T ) (cid:16) x n − − a ) − ( x , x q + m ) + ( I n − a : x q + m ) (cid:17) if 2 | ( n −
1) [by Lemma 3.5] ⊆ n − X a =0 x n − − a ) − x (2( q + m )+1) a ( x , x q + m ) + ( I n : x q + m ) if 2 ( n − x (2( q + m )+1) ( n − ) + n − X a =0 x n − − a ) − x (2( q + m )+1) a ( x , x q + m ) + ( I n : x q + m ) if 2 | ( n − . This implies that I n − ⊆ RHS . The other inclusion follows from Lemma 3.2 and checking element-wise. (cid:3)
Proposition 3.8.
For all n ≥ , ℓ I n − ( I n : x q + m ) ! = n. Proof.
From Lemma 3.3, ℓ (cid:18) I n − ( I n : x q + m ) (cid:19) = dim k (cid:18) I n − ( I n : x q + m ) (cid:19) , which is the number of minimal set ofgenerators of I n − / ( I n : x q + m ). From Lemma 3.7, we observe that in the generators of I n − modulo( I n : x q + m ), the terms which are of even degree in x are x n − − a )2 x (2( q + m )+1) a where a ≤ ( n − / x are all distinct and form a linearly independent set. Hencedim k I n − ( I n : x q + m ) ! = n/
2) if 2 n −
11 + [2( n − /
2] if 2 | n − n. (3.10) (cid:3) Proposition 3.11.
For all n ≥ , ℓ (cid:18) TI n (cid:19) = (2( q + m ) + 1) n + 12 ! . Proof.
We prove by induction on n . If n = 1, then ℓ (cid:18) TI n (cid:19) = ℓ k [ x , x ]( x , x x m + q , x q + m +13 ) ! = 1 + 2( q + m ) . Now let n >
1. From the exact sequence0 −→ T ( I n : x q + m ) .x q + m −→ TI n −→ TI n + ( x q + m ) −→ ℓ (cid:18) TI n (cid:19) = ℓ TI n + ( x q + m ) ! + ℓ T ( I n : x q + m ) ! = ℓ T ( x q + m , x n ) ! + ℓ TI n − ! + ℓ I n − ( I n : x q + m ) ! [Lemma 3.2]= 2( q + m ) n + (2( q + m ) + 1) n ! + n [by induction hypothesis and Proposition 3.8]= (2( q + m ) + 1) n + 12 ! . (cid:3) CLARE D’CRUZ ∗ AND MOUSUMI MANDAL The symbolic powers
In this section we explicitly describe the symbolic powers p ( n ) . Using the fact x , x is a regularsequence in R , we get the results we are interested in for the symbolic powers p ( n ) (Proposition 4.4,Theorem 4.6). Let I n := X n +2 n = n f n p n ⊆ p ( n ) . (4.1) Proposition 4.2.
Let n ≥ . Then(1) I n ⊆ p ( n ) .(2) Let m = ( x , x , x , x ) . Then ( I n + ( x , x )) is an m -primary ideal.Proof. (1) As ( f ) ⊆ p (2) (Lemma 2.6(3)), X n +2 n = n f n p n ⊆ X n +2 n = n p n + n = p ( n ) . (4.3)(2) By (4.1), p n ⊆ I n and ( p n + ( x , x )) = (( x , x x q + m , x q + m +13 ) n , x , x ) which implies that m = ( q p n + ( x , x )) ⊆ ( q I n + ( x , x )) ⊆ m . (cid:3) Proposition 4.4.
For all n ≥ , e ( x , x ); R p ( n ) ! = ℓ R p ( n ) + ( x , x ) ! = ℓ R R ( I n , x , x ) ! = ℓ (cid:18) TI n (cid:19) = (2( q + m ) + 1) n + 12 ! . Proof.
From Proposition 4.2(1), I n ⊆ p ( n ) . Hence, e ( x , x ); R p ( n ) ! = ℓ R R p ( n ) + ( x , x ) ! [as R/ p ( n ) is Cohen-Macaulay] ≤ ℓ R R ( I n , x , x ) ! = ℓ R (cid:18) TI n (cid:19) = ℓ R/ ( x ,x ) (cid:18) TI n (cid:19) [as ( x , x ) ⊆ Ann(
T /I n )]= ℓ T (cid:18) TI n (cid:19) [(3.1)]= (2( q + m ) + 1) n + 12 ! [Proposition 3.11]= e ( x , x ); R p ! ℓ R p R p p n R p ! [Lemma 2.2(2)]= e ( x , x ); R p ! ℓ R p R p p ( n ) R p ! [since p ( n ) R p = p n R p ]= e ( x , x ); R p ( n ) ! [by [15, 1.8]] . (4.5)Thus equality holds in (4.5) which proves the theorem. (cid:3) We end this section by explicitly describing the generators of p ( n ) for all n ≥ Theorem 4.6.
For all n ≥ ,(1) p ( n ) = I n .(2) p (2 n ) = ( p (2) ) n and p (2 n +1) = p ( p (2) ) n ,Proof. (1) By Proposition 4.4 we get p ( n ) + ( x , x ) = I n + ( x , x ) . Localizing at m we get ( p ( n ) +( x , x )) R m = ( I n + ( x , x )) R m . From Lemma 2.2(1), we conclude that x R m , x R m is a regularsequence on R m / p ( n ) R m . Hence( p ( n ) , x ) R m = ( I n , x ) R m + x (( p ( n ) , x ) : x ) R m = ( I n , x ) R m + x ( p ( n ) , x ) R m . By Nakayama’s Lemma, ( p ( n ) , x ) R m = ( I n , x ) R m . This implies that p ( n ) R m = I n R m + x ( p ( n ) : x ) R m = I n R m + x p ( n ) R m . Once again by Nakayama’s lemma, p ( n ) R m = I n R m . This implies that p ( n ) R m / I n R m = (0) R m . Asthis is a graded module, p ( n ) / I n = (0) which implies that p ( n ) = I n . ∗ AND MOUSUMI MANDAL (2) For all n ≥
3, applying Proposition 4.2(1) we get p ( n ) = I n = X a +2 a = n f a p a ⊆ X a +2 a = n p a ( p (2) ) a ⊆ p ( n ) . Hence equality holds and p ( n ) = X a +2 a = n p a ( p (2) ) a . Thus p (2 n ) = X a +2 a =2 n p a ( p (2) ) a = n X a =0 p n − a ( p (2) ) a ⊆ n X a =0 ( p (2) ) n = p (2 n ) p (2 n +1) = X a +2 a =2 n +1 p a ( p (2) ) a = n X a =0 p n +1 − a ( p (2) ) a ⊆ n X a =0 p ( p (2) ) n = pp (2 n ) ⊆ p (2 n +1) . (cid:3) Corollary 4.7.
For all n ≥ , R/ p ( n ) is Cohen-MacaulayProof. As x , x is a system of parameters in R/ p ( n ) and e (( x , x ); R p ( n ) ) = ℓ (cid:16) R/ p ( n )+( x ,x ) (cid:17) (Theo-rem 4.4), R/ p ( n ) is Cohen-Macaulay. (cid:3) Gorenstein property of symbolic blowup algebras
In this section we discuss the Gorenstein property of symbolic blowup algebras. If q = 1, thenthe curves we are interested in has been studied in [19]. Our proof here is different.Throughout this section U := k [ x , x , x , x , u , u , u , v ] and K := ( w , w , z , z , z ) where w = x u − x u + x q + m u w = x u − x u + x q x m u z = x v − x q + m − u u + u ,z = x v + x q − x q + m − x m u + u u ,z = x v + x q − x m u u + u . Before we prove our main result we prove some preliminary results.
Lemma 5.1.
U/K is Gorenstein.Proof.
Using the Buchsbaum-Eisenbud criterion one can check that minimal free resolution of
U/K is 0 −→ U φ −→ U φ −→ U φ −→ U −→ UK −→ where φ = ( w w z z z ) ,φ = v − x q − x m u − u u − v − u u − x q + m − u x q − x m u u x − x u − u − x x − u x q + m − u x − x , φ = w w z z z . Moreover, K is generated by the Pfaffians of order 4 of the anti-symmetic matrix φ and R/K isGorenstein. (cid:3)
Proposition 5.3.
Let τ : k [ x , x , x , x , u , u , u , v ] → R s ( p ) be an homomorphism given by τ ( x i ) = x i ( ≤ i ≤ ), τ ( u i ) = g i t ≤ j ≤ and τ ( v ) = f t . Then ker τ = K .Proof. By Lemma 5.1 all associated primes of K are minimal primes. As K ⊆ ker τ and ht( K ) =ht(ker τ ) = 3, ker τ is a minimal prime of K .We claim that there exists a , a , a ∈ k , ( a , a , a ) = (0 , ,
0) such that α = a x + a x + a x [ P ∈ Ass( K ) P . Otherwise ( x , x , x ) ⊆ [ P ∈ Ass( K ) P which implies that ( x , x , x ) + K ⊆ [ P ∈ Ass( K ) P andhence ( x , x , x , u , u ) ⊆ [ P ∈ Ass( K ) P . Consequently, ( x , x , x , u , u ) ⊆ Q for some Q ∈ Ass( K ).This implies that ht( Q ) ≥ α = a x + a x + a x [ P ∈ Ass( K ) P . Then αv = a x v + a x v + a x v = a z + a z + a z − [ a ( − x q + m − u u + u ) + a ( x q − x q + m − x m u + u u ) + a ( x q − x m u u + u )]= a z + a z + a z − β where β = a ( − x q + m − u u + u ) + a ( x q − x q + m − x m u + u u ) + a ( x q − x m u u + u ). Then U [1 /α ]( w , w , v + β/α ) ∼ = k [ x , x , x , x , u , u , u , v ][1 /α ]( w , w , v + β/α ) ∼ = k [ x , x , x , x , u , u , u ][1 /α ]( w , w ) ∼ = R ( p )[1 /α ] . Recall that p = ( g , g , g ) and g , g , g form a d -sequence [17] and R ( p ) = M n ≥ p n t n ∼ = k [ x , x , x , x , u , u , u ]( w , w ) . Moreover R ( p ) is a domain [16, Theorem 3.1] and dim R ( p ) = 5. Hence ( w , w ) is a prime idealof height 2. Since α ( w , w ), ht( w , w ) U [1 /α ] = 2. This implies that ( w , w , v + β/α ) U [1 /α ] is ∗ AND MOUSUMI MANDAL a prime ideal and ht( w , w , v + β/α ) U [1 /α ] = 3. In the ring U [1 /α ],( w , w , v + β/α ) U [1 /α ] = ( w , w , ( a z + a z + a z ) /α ) U [1 /α ] ⊆ KU [1 /α ] ⊆ (ker τ ) U [1 /α ] . Since α ker τ , ht( w , w , v + β/α ) U [1 /α ] = ht((ker τ ) U [1 /α ]) = 3 and( w , w , v + β/α ) U [1 /α ] = KU [1 /α ] = (ker τ ) U [1 /α ] . (5.4)By our choice of α and (5.4) K = KU [1 /α ] ∩ U = (ker τ ) U [1 /α ] ∩ U = ker τ. (cid:3) Theorem 5.5. (1) R s ( p ) = R [ p t, f t ] .(2) R s ( p ) is Cohen-Macaulay.(3) R s ( p ) is Gorenstein.(4) The symbolic fiber cone F s ( p ) = M n ≥ p ( n ) / mp ( n ) is Cohen-Macaulay.Proof. (1) The proof follows from Theorem 4.6.(2) and (3) follows from Lemma 5.1 and Proposition 5.3.(4) Let m = ( x , x , x , x ). Then F s ( p ) ∼ = U/ ( K + m ) ∼ = k [ u , u , u , v ] / ( u , u u , u ). Sincedim( F s ( p )) = 2 and the images of u and v form a regular sequence in F s ( p ), F s ( p ) is Cohen-Macaulay. (cid:3) Invariants associated to symbolic powers
In this section we compute certain invariants namely the resurgence, the Waldschmidt constant,regularity associated to the symbolic powers of p . Finally we compare these invariants.6.1. Containment.
In order to compare the symbolic powers and ordinary powers C. Bocci and B. Harbourne in [3]defined the resurgence of an ideal I in R as ρ ( I ) := sup (cid:26) nr : I ( n ) * I r (cid:27) . We can also compute the resurgence in the following way. For any ideal I ⊆ R let ρ n ( I ) := min { r : I ( n ) * I r } . Then ρ ( I ) := sup ( nρ n ( I ) : n ≥ ) . Lemma 6.1.
For all k ≥ and j = 0 , , p ( k (2 q +2 m )+ j ) ⊆ p k (2 q +2 m − j and p ( k (2 q +2 m )+ j ) p k (2 q +2 m − j +1 .Proof. For all k ≥ j = 0 ,
1, by Lemma 2.6(3) and Theorem 4.6 we get p ( k (2 q +2 m )) p j = (( p + f ) q + m ) k p j = ( q + m X i =0 f q + m − i p i ) k p j ⊆ ( p q + m − i ) − p i ) k p j = p k (2 q +2 m − j . Let j = 0. We will show that p ( k (2 q +2 m )) p k (2 q +2 m − . By Lemma 2.6(2) and Theorem 4.6 weget f k ( q + m ) ∈ p ( k (2 q +2 m )) . Observe that, f k ( q + m ) ∼ = ( x q +11 x m ) k ( q + m ) mod ( x ). Also p k (2 q +2 m − = ( x q x x m , x q +11 x m , x ) k (2 q +2 m − mod ( x )and hence ( x q +11 x m ) k (2 q +2 m − is a minimal generator of p k (2 q +2 m − mod ( x ). Since k (2 q + 1)( q + m ) = k (2 q + 2 qm + q + m ) < k (2 q + 2 qm − q + 2 q + 2 m −
1) + q + 1= k ( q + 1)(2 q + 2 m −
1) + q + 1 , comparing the powers of x we get f k ( q + m ) p k (2 q +2 m − . Hence the lemma is true for j = 0.Using the similar argument we can show that f k ( q + m ) g ∈ p (( k (2 q +2 m ))+1) \ p k (2 q +2 m − . This showsthat the lemma is true for j = 1. (cid:3) Lemma 6.2.
For all k ≥ and j = 2 , . . . , q + 2 m − , p ( k (2 q +2 m )+ j ) ⊆ p k (2 q +2 m − j − and p ( k (2 q +2 m )+ j ) p k (2 q +2 m − j .Proof. Let k = 0. If j = 2 j ′ and j ′ = 1 . . . q + m −
1, then by Lemma 2.6(3) and Theorem 4.6 weget p (2 j ′ ) = ( p + f ) j ′ = j ′ X i =0 f i p j ′ − i ) ⊆ p i − j ′ − i = p j ′ − = p j − . (6.3)If j = 2 j ′ + 1 and j ′ = 1 , . . . , q + m −
1, then from Theorem 4.6(2) and (6.3) we get p (2 j ′ +1) = p (2 j ′ ) p ⊆ p j ′ − p = p j ′ = p j − . (6.4)Hence the lemma is true for k = 0.Now let k ≥
1. Then by Theorem 4.6(2), induction hypothesis and (6.3) we get p ( k (2 q +2 m )+ j ) = ( p (2 q +2 m ) ) k p ( j ) ⊆ p k (2 q +2 m − p j − = p k (2 q +2 m − j − . We now show that p ( k (2 q +2 m )+ j ) p k (2 q +2 m − j . Let j = 2 j ′ where j ′ = 1 , . . . , q + m −
1. ByLemma 2.6(2) and Theorem 4.6(2) we get f k ( q + m )+ j ′ ∈ ( p (2) ) ( k ( q + m )+ j ′ ) = p ( k (2 q +2 m )+2 j ′ ) ∗ AND MOUSUMI MANDAL and f k ( q + m )+ j ′ = ( x q +11 x m ) k ( q + m )+ j ′ mod ( x ) . (6.5)Since p = ( x q x x m , x q +11 x m , x ) mod ( x ),( x q +11 x m ) k (2 q +2 m − j ′ ∈ p k (2 q +2 m − j ′ mod ( x ) (6.6)and is a minimal generator. Comparing the power of x in (6.5) and (6.6) we get f k ( q + m )+ j ′ p k (2 q +2 m − j ′ mod ( x ) since(2 q + 1)( k ( q + m ) + j ′ ) − ( q + 1)( k (2 q + 2 m −
1) + 2 j ′ ) = − k ( q + m ) + ( q + 1) − − j ′ < f k ( q + m )+ j ′ p k (2 q +2 m − j ′ .Using the above argument we can show that if k > j = 2 j ′ + 1 where j ′ = 1 , . . . q + m − f k ( q + m )+ j ′ g ∈ p ( k (2 q +2 m )+ j ) \ p k (2 q +2 m − j . (cid:3) As a consequence of Lemma 6.1 and Lemma 6.2 we verify Conjecture 8.4.3 on [1] for p . Lemma 6.7.
Let r ≥ . Then for all n ≥ r − , p ( n ) ⊆ p r .Proof. If n ≥ r −
1, then p ( n ) ⊆ p (2 r − . Hence it is enough to show that p (2 r − ⊆ p r . By Theorem4.6 we have p (2 r − = p ( p (2) ) r − ⊆ p r . (cid:3) We now verify that Conjecture 2.1 in [13] holds true in our case.
Corollary 6.8. p (3 n ) ⊆ m n p n .Proof. Let r ≥
1. If n = 2 r , then by repeatedly applying Theorem 4.6 we get p (3(2 r )) = ( p (2) ) r = p (2 r ) p (4 r ) ⊆ m r p r = m n p n as p (2) ⊆ m and p (4 r ) = ( p (2) ) r ⊆ p r .If n = 2 r −
1, then p (3 n ) = p (6( r − p (3) . Using the even case argument, p (6( r − ⊆ m r − p r − .Hence p (3 n ) = p (6( r − p (3) ⊆ m r − p r − p (2) p ⊆ m r − p r − = m n p n . (cid:3) We have an improved version of Corollary 6.8.
Corollary 6.9.
For all n ≥ , p (2 n ) ⊆ m n p n and p (2 n +1) ⊆ m n p n .Proof. Let r ≥
1. Then by applying Theorem 4.6(2) and (2.4) we get p (2 n ) = ( p (2) ) n = ( p + ( f )) n ⊆ ( mp ) n = m n p n p (2 n +1) = pp (2 n ) ⊆ m n p n +1 ⊆ m n p n . (cid:3) As a consequence of Lemma 6.1 and Lemma 6.2 we give the exact value for the resurgence ρ ( p ). Theorem 6.10.
For all q, m ≥ , ρ ( p ) = e ( R/ p ) − e ( R/ p ) − . Proof.
Let j = 0 , k ≥ n k,j = k (2 q + 2 m ) + j . Then by Lemma 6.1, for all k ≥ ρ k (2 q +2 m )+ j ( p ) = k (2 q + 2 m −
1) + j + 1. Hencesup k ( n k,j ρ n k,j ( p ) ) = sup ( k (2 q + 2 m ) + jk (2 q + 2 m −
1) + j + 1 ) = 2 q + 2 m q + 2 m − . (6.11)Let j = 2 , . . . , q + 2 m − k ≥ n k,j = k (2 q + 2 m ) + j . Then by Lemma 6.2, for all k ≥ ρ k (2 q +2 m )+ j ( p ) = k (2 q + 2 m −
1) + j . Hencesup k ( n k,j ρ n k,j ( p ) ) = sup k ( k (2 q + 2 m ) + jk (2 q + 2 m −
1) + j ) = 2 q + 2 m q + 2 m − . (6.12)From (6.11) and (6.12) we get ρ ( p ) = sup k ( n k,j ρ n k,j ( p ) : j = 0 , . . . , q + m − ) = 2 q + 2 m q + 2 m − . As e ( R/ p ) = 2( q + m ) + 1, the result follows. (cid:3) Waldschmidt Constant.
For a homogeneous ideal I ⊂ R , let α ( I ) denote the least generating degree of I . The Waldschmidtconstant of I is defined as γ ( I ) = lim s −→∞ α ( I ( s ) ) s . Here we will compute the Waldschmidt constant for p . Theorem 6.13. γ ( p ) = α ( p ) = 2 .Proof. As deg( g ) = deg( g ) ≥ deg( g ) = 2, α ( p ) = 2. By Theorem 4.6(2), p (2 n ) = ( p (2) ) n and p (2 n +1) = p ( p (2) ) n . Since p (2) = ( p + f ) and deg( f ) = 2( q + m ) + 1 ≥ α ( p ) = 2 · α ( p ) = 4. Thus α ( p (2 n ) )2 n = 4 n n = 2 and p (2 n +1) n + 1 = 4 n + 22 n + 1 = 2. Hence γ ( p ) = 2. (cid:3) Regularity.
In this subsection we compute the regularity of the symbolic powers of p . Let p T , f T and I n asin (3.1). We first prove a preliminary result (Lemma 6.14) which indicates that it is enough tocompute the regularity of T /I n . Lemma 6.14. reg( R/ p ( n ) ) = reg( T /I n ) . ∗ AND MOUSUMI MANDAL
Proof. As x , x is a regular sequence in R/ p , by [5, Remark 4.1],reg R p ( n ) ! = reg R p ( n ) + ( x ) ! = reg R p ( n ) + ( x , x ) ! = reg (cid:18) TI n (cid:19) . (cid:3) Let G ( F ) := M n ≥ I n /I n +1 be the associated graded ring corresponding to the filtration F := { I n } n ≥ . We show G ( F ) is Cohen-Macaulay and this result is very useful in computing the regularity.We first prove a preliminary lemma. Lemma 6.15.
For all n ≥ ,(1) p n T : ( x ) ⊆ p n − T .(2) ( p n T : x q + m )+13 ) ⊆ p n − T .Proof. (1) By Proposition 3.4,( p n T : x ) = n − X i =0 (cid:16) x n − i )2 x ( q + m ) i ( x , x ) i : x (cid:17) + ( x ( q + m ) n ( x , x ) n : x )= n − X i =0 (cid:16) x n − i − x ( q + m ) i ( x , x ) i (cid:17) + ( x ( q + m ) n ( x , x ) n − ) ⊆ p n − T, (6.16)( p n T : x q + m )+13 )= X i =0 ( x n − i )2 x ( q + m ) i ( x , x ) i : x q + m )+13 ) + n X i =3 ( x n − i )2 x ( q + m ) i ( x , x ) i : x q + m )+13 )= x n ( x , x ) + n +2 X i =3 ( x n +2 − i )2 x ( q + m )( i − ( x , x ) i − )( x x q + m − , x x q + m , x x q + m +13 , x q + m +23 ) ⊆ ( x n ) + n X i =3 ( x n +2 − i )2 x ( q + m )( i − ( x , x ) i − )( x , x x q + m , x q + m +13 ) ⊆ p n − T. (6.17) (cid:3) For any element r ∈ T , let r ⋆ denote the image in G ( F ). Theorem 6.18. G ( F ) is Cohen-Macaulay. Proof.
We first show that ( x ) ⋆ is a regular element in G ( F ). We claim that ( I n : x ) = I n − for all n ≥
1. Clearly x I n − ⊆ ( p T ) I n − ⊆ I n . For the other inclusion, from Lemma 6.15(1) we get( I n : x ) = X a +2 a = n ( f a T ) (cid:16) p a T : x (cid:17) ⊆ X a +2 a = n ( f a T )( p a − T ) ⊆ I n − . Let denote the image in
T / ( x ). Then G ( F )( x ) ⋆ ∼ = M n ≥ I n I n +1 + x I n − = G ( F ) . To show that x q + m )+13 is a regular element in G ( F ), we need to verify that(( I n +2 + x I n ) : ( x q + m )+13 )) = I n + x I n − . (6.19)One can verify that x q + m )+13 ( I n +2 + x I n ) ⊆ f T ( I n + x I n − ) ⊆ I n +2 . For the other inclusion, forall n ≥ I n +2 + x I n ) : ( x q + m )+13 ))= X a +2 a = n +2 ( f a p a T : x q + m )+13 ) + X a +2 a = n ( x f a p a T : ( x q + m )+13 ))= X a +2 a = n +2; a =0 ( f a − p a T ) + ( p n +2 T : x q + m )+13 )+ X a +2 a = n ; a =0 ( x f a − p a T ) + ( x p n T : x q + m )+13 ) ⊆ X a +2 a = n +2; a =0 ( f a − p a T ) + p n T + X a +2 a = n ; a =0 ( x f a − p a T ) + x p n − T [by (6.17)] ⊆ I n + x I n − . (6.20) (cid:3) As x ⋆ ∈ [ G ( F )] is a regular element, we can use it to determine reg( T / ( I n + x )). Lemma 6.21.
Let n ≥ . Then reg TI n + ( x ) ! = r (cid:16) q + m + (cid:17) if n = 2 r, (2 r − (cid:16) q + m + (cid:17) − if n = 2 r − . Proof.
By Theorem 4.6, I r + ( x ) = ( I + ( x )) r + ( x ) = ( x , x q + m )+13 ) r + ( x ) = ( x , x r (2( q + m )+1)3 ) . ∗ AND MOUSUMI MANDAL the minimal free resolution of
T / ( I r + ( x )) is / / T [ − ( r (2( q + m ) + 1) + 2)] x r (2( q + m )+1)3 − x ! / / T [ − ⊕ T [ − ( r (2( q + m ) + 1))] / / T / / TI r + ( x ) / / This implies thatreg TI r + ( x ) ! = r (2( q + m ) + 1) = 2 r (cid:18) q + m + 12 (cid:19) . Let n = 2 r −
1. Then by Theorem 4.6, I r − + ( x ) = ( I , x )( I r − , x ) + ( x ) = ( x , x x ( q + m )3 , x q + m +13 )( x , x ( r − q + m )+1)3 ) + ( x )= ( x , x x (2 r − q + m )+ r − , x (2 r − q + m )+ r )= ( x , x x n ( q + m )+ r − , x n ( q + m )+ r ) . By Hilbert-Burch Theorem, the minimal free resolution of
T /I n is / / T [ − n ( q + m ) − r − x n ( q + m )+( r − − x − x x / / T [ − ⊕ T [ − n ( q + m ) − r ] / / T / / TI n + ( x ) / / . (6.23) Hence reg(
T /I n + ( x )) = n ( q + m + 1) + r − n ( q + m + 1) − . (cid:3) Lemma 6.24.
For all n ≥ , reg( T /I n + ( x q + m )+13 )) = 2 n + 2( q + m ) − .Proof. As ( f T ) = ( x q + m )+13 ), from Theorem 4.6, for all n ≥ I n + ( x q + m )+13 ) = ( p T ) n + ( x q + m )+13 ) = ( x n , x n − x q + m , x n − x q + m +13 , x q + m )+13 ) . By Hilbert-Burch Theorem the minimal free resolution of I n + ( x q + m )+13 ) is / / T [ − (2 n + q + m )] ⊕ T [ − (2 n − q + m ))] x q + m − x − x x − x q + m x n − / / T [ − (2 n )] ⊕ T [ − (2 n − q + m )] ⊕ T [ − (2( q + m ) + 1)] / / T / / TI n + ( x q + m )+13 ) / / . Hence reg(
T /I n + ( x q + m )+13 )) = 2 n + 2( q + m ) − (cid:3) We now use the fact that ( x q + m )+13 ) ⋆ ∈ [ G ( F )] is a regular element to compute reg( T /I n ). Proposition 6.25.
Let n ≥ . Then reg (cid:18) TI n (cid:19) = r (cid:16) q + m + (cid:17) if n = 2 r, (2 r − (cid:16) q + m + (cid:17) − if n = 2 r − . Proof.
Let n = 2 r . We prove the Proposition by induction on r . If r = 1, then the result followsfrom Lemma 6.24. Let r >
1. By Theorem 6.18, ( x q + m )+13 ) ⋆ is a regular element in G ( F ). Hence,we have the exact sequence,0 / / TI r − [ − q + m ) − .x q + m )+13 / / TI r / / TI r + ( x q + m )+13 ) / / . (6.26)Then from the exact sequence (6.26) we getreg (cid:18) TI r (cid:19) = max ( reg TI r − ! + 2( q + m ) + 1 , reg TI r + ( x q + m )+13 ) !) = max (cid:26) (2 r − (cid:18) q + m + 12 (cid:19) + 2( q + m + 12 ) , r + 2( q + m ) − (cid:27) [Lemma 6.24]= max (cid:26) r (cid:18) q + m + 12 (cid:19) , r + 2( q + m ) − (cid:27) = 2 r (cid:18) q + m + 12 (cid:19) . Let n = 2 r − r ≥
1. If r = 1, then the result follows from Corollary 6.21. Let r >
1. As( x ) ⋆ is a nonzerodivisor in G ( F ), we have the exact sequence0 −→ TI r − [ − .x −→ TI r − −→ TI r − + ( x ) −→ . (6.27)As all the modules in (6.27) are Artinian,reg( T /I r − ) = max { reg( T / ( I r − )[ − , reg( T / ( I r − + ( x ))) } = max { (2 r − (cid:18) q + m + 12 (cid:19) + 2 , (2 r − (cid:18) q + m + 12 (cid:19) − } = (2 r − (cid:18) q + m + 12 (cid:19) − . (cid:3) Theorem 6.28.
Let n ≥ . Then reg R p ( n ) ! = n ( e ( R/ p ) /
2) + θ where θ = if n is even − / if n is odd . ∗ AND MOUSUMI MANDAL
Proof.
From Lemma 6.14 and Proposition 6.25 we getreg R p ( n ) ! = r (cid:16) q + m + (cid:17) if n = 2 r, (2 r − (cid:16) q + m + (cid:17) − if n = 2 r − . Since e ( R/ p ) = 2 q + 1 + 2 m , the result follows. (cid:3) As an immediate corollary we have:
Corollary 6.29. lim n −→∞ reg (cid:16) R p ( n ) (cid:17) n = e ( R/ p )2 . Comparing invariants.
In this subsection we compare the various invariants. We verify that Theorem 1.2.1(b) does notalways hold true if the scheme defined by ideal I is not zero-dimensional. Lemma 6.30. ρ ( p ) ≥ α ( p ) γ ( p ) Proof. As α ( p ) /γ ( p ) = 1 and ρ ( p ) ≥ (cid:3) Lemma 6.31. If q = m = 1 , then ρ ( p ) ≥ reg( p ) /γ ( p ) . If either q > or m > , then ρ ( p ) < reg( p ) /γ ( p ) .Proof. From (2.3) it follows that reg( R/ p ) = q + m . Hence ρ ( p ) − reg( p ) γ ( p ) = 2 q + 2 m q + 2 m − − q + m q + m )(5 − q − m )2(2 q + 2 m − . If q = m = 1, then 5 − q − m = 1. If either q > m >
1, then 5 − q − m < (cid:3) References [1] T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. Knutsen, W. Syzdek and T. Szemberg,
A primer onSeshadri constants.
Interactions of classical and numerical algebraic geometry, 33-70, Contemp. Math., ,Amer. Math. Soc., Providence, RI, 2009. 1, 3, 14[2] C. Bocci and B. Harbourne,
The resurgence of ideals of points and the containment problem.
Proc. Amer. Math.Soc. (2010), no. 4, 1175-1190. 2[3] C. Bocci and B. Harbourne,
Comparing powers and symbolic powers of ideals , J. Algebraic Geom. (2010),no. 3, 399-417. 1, 2, 3, 12[4] C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Seceleanu, A. Van Tuyl and ThanhVu, The Waldschmidt constant for squarefree monomial ideals.
J. Algebraic Combin. (2016), no. 4, 875-904.2[5] M. Chardin, Some results and questions on Castelnuovo-Mumford regularity.
Syzygies and Hilbert functions,1-40, Lect. Notes Pure Appl. Math., , Chapman & Hall/CRC, Boca Raton, FL, 2007. 16[6] D. Cutkosky, J. Herzog and N. V. Trung,
Asymptotic behaviour of the Castelnuovo-Mumford regularity.
Com-positio Math. (1999), no. 3, 243-261. 2 [7] D. Cutkosky, Irrational asymptotic behaviour of Castelnuovo-Mumford regularity.
J. Reine Angew. Math. (2000), 93-103. 2[8] M. Dumnicki, B. Harbourne, U. Nagel, A. Seceleanu, T. Szemberg and H. Tutaj-Gasi´ n ska, Resurgences forideals of special point configurations in P N coming from hyperplane arrangements. J. Algebra (2015),383-394. 2[9] V. Ene and J. Herzog,
Gröbner bases in commutative algebra.
Graduate Studies in Mathematics,
AmericanMathematical Society, (2012). 5[10] L. Ein, R. Lazarsfeld, and K. E. Smith,
Uniform bounds and symbolic powers on smooth varieties . Invent. Math. (2001), no. 2, 241-252. 1[11] G. Fatabbi, Giuliana, B. Harbourne and A. Lorenzini,
Inductively computable unions of fat linear subspaces.
J.Pure Appl. Algebra (2015), no. 12, 5413-5425. 2[12] E. Guardo, B. Harbourne and A. Van Tuyl,
Asymptotic resurgences for ideals of positive dimensional subschemesof projective space.
Adv. Math. 246 (2013), 114-127. 2[13] B. Harbourne and C. Huneke,
Are symbolic powers highly evolved?
J. Ramanujan Math. Soc. (2013),247-266. 2, 3, 14[14] M. Hochster and C. Huneke,
Comparison of symbolic and ordinary powers of ideals . Invent. Math. (2002),no. 2, 349-369. 1[15] M. Herrmann, S. Ikeda and U. Orbanz,
Equimultiplicity and blowing up. An algebraic study. With an appendixby B. Moonen. (1988 ) Springer-Verlag, Berlin. 3, 9[16] C. Huneke,
On the Symmetric and Rees Algebra of an Ideal Generated by a d -sequence. J. Algebra (1980),no. 2, 268-275. 11[17] C. Huneke, The theory of d-sequences and powers of ideals.
Adv. in Math. (1982), no. 3, 249-279. 11[18] M. Morales, Syzygies of monomial curves and a linear diophantine problem of Frobenius . Max-Planck-Institutf¨ u r Mathematik, 1987. 3[19] M. Morales and A. Simis, Arithmetically Cohen-Macaulay monomial curves in P . Comm. Algebra (1993),no. 3, 951-961. 2, 10[20] P. Schenzel, Examples of Gorenstein domains and symbolic powers of monomial space curves. J. Pure Appl.Algebra 71 (1991), no. 2-3, 297-311. 2[21] M. Waldschmidt, Propri ´ e t ´ e s arithm ´ e tiques de fonctions de plusieurs variables. II. (French) S´ e minaire PierreLelong (Analyse) ann´ e e 1975/76, pp. 108?135. Lecture Notes in Math., Vol. 578, Springer, Berlin, 1977 1 Chennai Mathematical Institute, Plot H1 SIPCOT IT Park, Siruseri, Kelambakkam 603103, TamilNadu, India
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