criterion for complete intersection of certain monomial

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form a minimal arithmetic sequence and find some criterion for complete intersection of C. Keywords : Monomial Curves, Complete Intersection. 1. Introduction.
CRITERION FOR COMPLETE INTERSECTION OF CERTAIN MONOMIAL CURVES Alok Kumar Maloo and Indranath Sengupta Abstract. Let k be a field and m0 , . . . , me−1 (e ≥ 3) be a sequence of positive integers with gcd(m0 , . . . , me−1 ) = 1. Let C be the affine monomial curve in the e-space Aek defined parametrically by X0 = T m0 , . . . , Xe−1 = T me−1 . In this article, we assume that m0 , . . . , me−1 form a minimal arithmetic sequence and find some criterion for complete intersection of C. Keywords : Monomial Curves, Complete Intersection.

1. Introduction Let e ≥ 3 be an integer and m0 , . . . , me−1 be a sequence of positive integers with gcd(m0 , . . . , me−1 ) = 1, such that m0 , . . . , me−1 generate the P + semigroup Γ := e−1 i=0 Z mi minimally. Let k be a field. In general it is not known when the affine monomial curve C = C(m0 , . . . , me−1 ) in the e-space Aek , defined parametrically by X0 = T m0 , . . . , Xe−1 = T me−1 , is a complete intersection. It was shown in [H] that if e = 3 then C is a complete intersection if and only if Γ is symmetric. Also it is known that if m0 , . . . , me−1 form an almost arithmetic sequence, that is if some e − 1 terms of m0 , . . . , me−1 form an arithmetic sequence, then C is a set-theoretic complete intersection ( see [P1] ). In this article our aim is to study the complete intersection property of the affine monomial curves when m0 , . . . , me−1 form an arithmetic sequence. We show that THEOREM 3.5. Let m0 < m1 < m2 be a minimal arithmetic sequence of positive integers. Let k be a field. Then the monomial curve C(m0 , m1 , m2 ) in the affine space A3k is a complete intersection if and only if m0 is even. THEOREM 3.7. Let e ≥ 4 be an integer and m0 , . . . , me−1 be a minimal arithmetic sequence of positive integers. Let k be a field. Then the affine monomial curve C(m0 , . . . , me−1 ) in the e-space Aek is never a complete intersection. This work was done when the second author was a CSIR senior research fellow at Indian Institute of Science, Bangalore. AMS Classification Number : 13P10. 1

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For an almost arithmetic sequence of positive integers, an explicit description of a standard basis of Γ is given in [PS] and this has been used in [P2] to construct a minimal set of generators for the relation ideal of C. We prove our results as an application of some results from [P2] and [PS]. 2. Preliminaries We shall use the notations and terminology from [P2] and [PS]. We recall a few of them. Let e ≥ 3 be an integer and let p = e − 2. Let m0 , . . . , mp+1 be positive integers such that gcd(m0 , . . . , mp+1 ) = 1. Let Γ denote the numerical semiP + + group generated by m0 , . . . , mp+1 , i.e., Γ := p+1 i=0 Z mi , where Z denotes the set of non-negative integers. The set {m0 , . . . , mp+1 } is called minimal if it forms a minimal set of generators for Γ. The integers m0 , . . . , mp+1 are said to be an almost arithmetic sequence if some p+1 of them form an arithmetic sequence. For the rest of the note we make the following assumptions: 0 < m0 < . . . < mp form an arithmetic sequence P with common difference d and mp+1 is arbitrary. Put n := mp+1 , Γ0 := pi=0 Z+ mi and Γ := Γ0 +Z+ n. Let S denote the standard basis of Γ with respect to m0 , i.e. S = {γ ∈ Γ | γ − m0 6∈ Γ}. For integers a, b let [a, b] denote the set {i ∈ Z | a ≤ i ≤ b}. For i ≥ 0 there exist unique integers qi , ri , such that ri ∈ [1, p] and i = qi p + ri . Put gi := qi mp + mri . Then 0 = g0 < g1 < g2 < . . . and gi ∈ Γ0 for all i ≥ 0. With the above notations we recall the relevant part of [PS, §3]: Lemma 2.1. (1) gi + gj = εm0 + gi+j , where ε = 1 or 0 according as ri + rj ≤ p or ri + rj > p . (2) Let u := min{i ≥ 0 | gi 6∈ S} and v := min{b ≥ 1 | bn ∈ Γ0 } . Then there exist unique integers w ∈ [0, v−1], z ∈ [0, u−1], λ ≥ 1, µ ≥ 0 and ν ≥ 1 such that (a) gu = λm0 + wn; (b) vn = µm0 + gz ; (c) gu−z + (v − w)n = νm0 . Moreover ν = λ + µ +  where  = 1 or 0 according as ru−z < ru or ru−z ≥ ru . (3) Let V := [0, u − 1] × [0, v − 1] and W := [u − z, u − 1] × [v − w, v − 1] and U := V \ W . Then S = {gi + bn | (i, b) ∈ U }. If, in particular, (i, b), (j, c) ∈ U such that gi + bn ≡ gj + cn (mod m0 ), then (i, b) = (j, c). (4) Every element of Γ can be expressed uniquely in the form am0 + gi + bn for some a ∈ Z+ and (i, b) ∈ U . Continuing as in [PS], put q = qu , r = ru , q 0 = qu−z and r0 = ru−z . For the the rest of this note q, q 0 , r, r0 , u, v, w, z, λ, µ, ν, W and U will have the meaning assigned to them by Lemma (2.1).

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Let k be a field and X0 , . . . , Xp , Y, T be indeterminates. Let p denote the kernel of the k-algebra homomorphism η : k[X0 , . . . , Xp , Y ] → k[T ], defined by η(Xi ) = T mi , η(Y ) = T n . Let C := k[T m0 , T m1 , . . . , T mp , T n ] be the affine monomial curve in the e-space Aek defined by (p + 2)-integers. We also denote this curve by C(m0 , . . . , mp , n). A minimal set of generators for p, the relation ideal of C, is explicitly constructed in [P2]. We recall the construction and the result: For i, j ∈ [1, p − 1], let ξij := Xi Xj − X01−ε Xi+j−εp Xpε , where  = 0 or 1 according as i + j ≤ p or i + j > p . For i ∈ [0, p − r], let φi := Xr+i Xpq − X0λ−1 Xi Y w . 0

For j ∈ [0, p − r0 ], let ψj := Xr0 +j Xpq Y v−w − X0ν−1 Xj . ( 0 Y v − X0µ Xr−r0 Xpq−q if r0 < r, Let θ := 0 Y v − X0µ Xp+r−r0 Xpq−q −1 if r0 ≥ r. ( [0, p − r] if µ 6= 0 or W = ∅, Let I := [max(rz − r + 1, 0), p − r] if µ = 0 and W 6= ∅ ( ∅ if W = ∅, and J := 0 [0, min(z − 1, p − r )] if W 6= ∅. Then by [P2, (4.5)] the set {ξij | 1 ≤ i ≤ j ≤ p − 1} ∪ {θ} ∪ {φi | i ∈ I} ∪ {ψj | j ∈ J} forms a minimal set of generators for p. 3. Results Rings are assumed to be commutative with unity. Let A be a ring. For an A-module M let µ(M ) denote the minimum number of elements required to generate M as an A-module. Let C denote the affine monomial curve defined by (p + 2)-integers which form an almost arithmetic sequence. Let p denote the relation ideal of C. Then by [P2, (4.6)], µ(p) is either p(p−1)/2+p−r+2 or p(p−1)/2+p−r0 +2 or p(p − 1)/2 + 2p − r − r0 + 3. Hence µ(p) ≥ p(p − 1)/2 + 2. Therefore a necessary condition for C to be a complete intersection is p ≤ 2. This means that a monomial curve in affine e-space defined by an almost arithmetic sequence is never a complete intersection if e ≥ 5. In this section we look at the case when e = 3 and e = 4 and C is defined by an arithmetic sequence. Now we deduce the following result from Lemma (2.1): Lemma 3.1. In the notation of Lemma (2.1), every element of Γ0 is of the form αm0 + gi for some integers α ≥ 0 and 0 ≤ i ≤ m0 − 1. Proof. Let γ ∈ Γ0 . Then by the Lemma (2.1) we may write γ = αm0 + gi , with α ≥ 0, i ≥ 0. Choose this expression with i minimal. We claim that i ≤ m0 − 1. Suppose i ≥ m0 . Then again by Lemma (2.1) we may write γ = αm0 + gi−m0 + gm0 − εm0 , where ε = 1 or 0. Therefore γ = (α − ε + qm0 + 1 + d)m0 + gi−m0 , which contradicts the minimality of i. Hence i ≤ m0 − 1. 

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Lemma 3.2. Assume that m0 < m1 < . . . < mp+1 = n form an arithmetic sequence. Write m0 = a(p + 1) + b, where a and b are unique integers such that a ≥ 0 and b ∈ [1, p + 1]. Then (1) u = p + 1, q = 1, r = 1. (2) λ = 1, w = 1. (3) v = a + 1. ( a+d if z 6= 0, (4) z = p + 1 − b and µ = a + d + 1 if z = 0. ( ( 0 if 1 ≤ b ≤ p, b if 1 ≤ b ≤ p, 0 0 (5) q = and r = 1 if b = p + 1 1 if b = p + 1. ( a + d + 1 if z 6= 0, (6) ν = a + d + 2 if z = 0. Proof. (1) Since g0 = 0 and gi = mi for 1 ≤ i ≤ p, by the minimality of m0 , . . . , mp , n it follows that u ≥ p + 1. Now gp+1 − m0 = mp + m1 − m0 = n ∈ Γ. Hence gp+1 6∈ S, which shows that u = p + 1. (2) We have gp+1 = mp + m1 = m0 + n , hence λ = 1 and w = 1. (3) Suppose βn ∈ Γ0 , for some β ≥ 1. Then by the Lemma (3.1) we may write βn = αm0 + gi for some α ≥ 0 and 0 ≤ i ≤ m0 − 1. Therefore β(m0 + (p + 1)d) = αm0 + (qi + 1)m0 + id, and so (3.3)

(β − α − qi − 1)m0 = (i − β(p + 1))d.

Since gcd(m0 , . . . , mp , n) = 1, so gcd(m0 , d) = 1, and hence m0 | β(p+1)−i. Therefore (3.4)

β(p + 1) − i = tm0 , for some t ∈ Z.

We claim that t ≥ 1. From equation (3.4) we see that tm0 ≥ β(p + 1) − m0 + 1, since i ≤ m0 − 1. So (t + 1)m0 ≥ β(p + 1) + 1 ≥ 1, and hence t + 1 ≥ 1, i.e., t ≥ 0. Now if t = 0 then from equations (3.3) and (3.4) we get β − α − qi − 1 = 0 and β(p + 1) = i, which implies that ri = i − qi p = (α + 1)p + β > p, which is absurd. Hence t ≥ 1. From equation (2) we get β(p + 1) − i = ta(p + 1) + tb. So β(p + 1) ≥ ta(p+1) + tb > ta(p+1) ≥ a(p+1), as t ≥ 1 and b ≥ 1. Therefore β > a and hence β ≥ a+1. Furthermore we note that (a+1)n = (a+1)(m0 +(p+1)d) = (a + d)m0 + m0 + (p + 1 − b)d = (a + d)m0 + mp+1−b ∈ Γ0 . Hence v = a + 1. (4) Follows from the equation (a + 1)n = (a + d)m0 + mp+1−b obtained in part (3). (5) We see that u − z = b and hence the result follows. (6) We always have r0 ≥ r = 1, so ν = λ + µ, and hence the result.



Theorem 3.5. Let m0 < m1 < m2 be a minimal arithmetic sequence of positive integers. Let k be a field. Then the monomial curve C(m0 , m1 , m2 ) in the affine space A3k is a complete intersection if and only if m0 is even.

CRITERION FOR COMPLETE INTERSECTION OF CERTAIN MONOMIAL CURVES 5

Proof. First suppose that m0 is even. Then m0 = 2a + 2 for some integer a ≥ 0. Now by Lemma (3.2), u = 2, λ = 1, w = 1, v = a + 1, z = 0, µ = a + d + 1, q = q 0 = 1, r = r0 = 1. In this case W = ∅ and so we have I = {0} and J = ∅. This shows that a minimal generating set for the relation ideal p is {X12 − X0 X2 , X2a+1 − X0a+d+1 } and hence µ(p) = 2. Conversely suppose that m0 is odd. Then m0 = 2a + 1 for some integer a ≥ 0. By Lemma (3.2) we see that u = 2, λ = 1, w = 1, v = a + 1, z = 1, µ = a + d, q = 1, q 0 = 0, r = r0 = 1, ν = a + d + 1. Since W 6= ∅ and µ 6= 0, we therefore have I = {0} and J = {0}. Hence {X12 − X0 X2 , X2a+1 − X0a+d X1 , X1 X2a − X0a+d+1 } is a minimal generating set for the relation ideal p. This shows that µ(p) = 3.  Proposition 3.6. Let p ∈ {1, 2} and m0 , . . . , mp+1 be a minimal almost arithmetic sequence of positive integers. Then the following statements are equivalent: (1) The monomial curve C(m0 , . . . , mp+1 ) is a complete intersection. (2) card(I) + card(J) = 1. (3) Either r = p, W = ∅ or r = rz = p, W 6= ∅, µ = 0. Proof. Follows directly from [P2, (4.5)].



Theorem 3.7. Let e ≥ 4 be an integer and let m0 , . . . , me−1 be a minimal arithmetic sequence of positive integers. Let k be a field. Then the affine monomial curve C(m0 , . . . , me−1 ) in the e-space Aek is never a complete intersection. Proof. We need to consider only the case e = 4. By Lemma (3.2) we get r = 1 in this case. Hence by Proposition (3.6), it can never be a complete intersection.  Remark 3.8. If the sequence is not an arithmetic sequence then either case is possible, as indicated by Proposition (3.6). Let us consider a few examples to illustrate this. Examples 3.9. (1) Let m0 = 10, m1 = 15 and n = 7. Then W = ∅ and hence C(10, 15, 7) is a complete intersection. (2) Let m0 = 5, m1 = 9 and n = 6. Then W 6= ∅, µ = 0. Hence C(5, 9, 6) is a complete intersection. (3) Let m0 = 16, m1 = 21 and n = 13. Then W 6= ∅, µ = 6. Hence C(16, 21, 13) is not a complete intersection. (4) Let m0 = 8, m1 = 10, m2 = 12 and n = 15. Then W = ∅, r = 2. Hence C(8, 10, 12, 15) is a complete intersection. (5) Let m0 = 7, m1 = 12, m2 = 17 and n = 11. Then W 6= ∅, r = 2, rz = 1, µ = 3. Hence C(7, 12, 17, 11) is not a complete intersection.

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Acknowledgements This work was done when the second author visited Department of Mathematics, IIT Kanpur in December 1998. He expresses his sincere gratitude to Prof.Ashok Sengupta, Department of Mechanical Engineering, IIT Kanpur, for his kind help in arranging this visit. References [H] J. Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Mathematica 3 (1970), 175-193. MR 42:4657 [P1] D.P. Patil, Certain monomial curves are set-theoretic complete intersections, Manuscripta Mathematica 68 (1990), 399-404. [P2] D.P. Patil, Minimal sets of generators for the relation ideal of certain monomial curves, Manuscripta Mathematica 80 (1993), 239-248. [PS] D.P. Patil and B. Singh, Generators for the derivation modules and the relation ideals of certain curves, Manuscripta Mathematica 68 (1990), 327-335. Department of Mathematics, Indian Institute of Technology, Kanpur 208016, INDIA. E-mail address: [email protected] Department of Mathematics, Jadavpur University, Kolkata 700 032, INDIA. E-mail address: [email protected]