Monomial curves in affine four space Kazufumi Eto Nov. 8, 2011 We will give a theorem for a question whether every affine monomial curve is a set-theoretic complete intersection. Let k be a field, N > 2 and n1 , . . . , nN natural numbers whose gcd is one. The curve C = {(tn1 , tn2 , . . . , tnN ) : t ∈ k} is called a monomial curve in affine N -space defined by n1 , n2 , . . . , nN . And the kernel of the ring homomorphism k[X1 , . . . , XN ] −→ k[t],

Xi 7−→ tni

is called the defining ideal of the monomial curve C. In general, for an ideal I in a ring R, if √ √ ∃f1 , . . . , fr ∈ I s.t. I = (f1 , . . . , fr ),

for each i

r = ht I,

then I is called a set-theoretic complete intersection. We consider a question if every monomial curve C in affine N -space is a set-theoretic complete intersection. Namely, if we put I be the defining ideal of C, it asks whether there exist √ √ f1 , . . . , fN −1 ∈ I s.t. I = (f1 , . . . , fN −1 ). There are partrial answers for it. If N = 3, it is affirmatively answered (e.g. [6]). If N = 4 and if its defining ideal is Gorenstein or an almost complete intersection, then the monomial curve is a set-theoretic complete intersection ([1], [4]). Further, if the characteristic of the field k is positive, Cowsik and Nori prove that any affine algebraic curve is a set-theoretic complete intersection ([2]). Thus we may assume that its characterisitic is zero. Here is the main theorem: Theorem 1 ([5, Theorem 3.11]). If N = 4 and if min{n1 , n2 , n3 , n4 } ≤ 13 then the associated monomial curve with n1 , n2 , n3 and n4 is a set-theoretic complete intersection.

This follows from the two theorems: Theorem 2 ([5, Theorem 1.6]). If n1 + n4 is contained in the semigroup generated by n2 and n3 , then the associated monomial curve with n1 , n2 , n3 and n4 is a set-theoretic complete intersection. Theorem 3 ([5, Theorem 1.5]). Let I be the defining ideal of a monomial curve defined by n1 , n2 , n3 and n4 . Assume that there ∑ are vj ∈ Ker(n1 , n2 , n3 , n4 ) + and dj > 0 for j = 1, 2, 3, 4 with supp vj = {j}, j dj vj = 0 and Ker(n1 , n2 , n3 , n4 ) =

∑

Zvj ⊂ Z4 .

j

If two of dj are one, then I is a set-theoretic complete intersection. ⊕ We give some regard (n1 , n2 , n3 , n4 ) as a map Z4 = 4i=1 Zei → ∑notes. We ∑ Z which sends σi (v)ei to σi (v)ni . Then Ker(n1 , n2 , n3 , n4 ) is a submod4 4 ule in Z of rank 3. For v ∈ Z , we denote the support of v by supp v. Eliahou (1984) proved the following: Proposition 4 ([3]). If (1) n1 = 4, n2 , n3 , n4 ≥ 4, (2) n2 ≡ 1, n3 ≡ 2, n4 ≡ 3 mod 4, (3) 2n3 ≥ n2 + n4 , then I is a set-theoretic complete intersection. Applying the above, Eliahou also gives an example ([3]) in which I is a set-theoretic complete intersection if (n1 , n2 , n3 , n4 ) = (4, 5, 6, 7). But, if (n1 , n2 , n3 , n4 ) = (4, 6, 7, 9), then we cannot apply the above. We observe the case n1 = 4 again. If n2 ≡ n3 mod 4, then I is a settheoretic complete intersection, since it is reduced to the case N = 3. If n2 ≡ 1, n3 ≡ 2, n4 ≡ 3 mod 4, then v = t (a, −1, −1, 1) ∈ Ker(4, n2 , n3 , n4 ). If a ≤ 0, then n4 is contained in the semigroup generated by 4, n2 and n3 . If a > 0, then I is a set-theoretic complete intersection by Theorem 2. Hence, if n1 = 4, then I is a set-theoretic complete intersection. To generalize this argument, I is a set-theoretic complete intersection in the following cases: (1) ni ≡ nj mod n1 for any 1 < i < j,

(2) ni + nj ≡ 0 mod n1 for any 1 < i < j, (3) ni + nj ≡ nk mod n1 for any 1 < i < j and k > 0. We observe another case. Assume n1 = 7. If n2 ≡ 1, n3 ≡ 2, n4 ≡ 4 mod 7, then         a4 a3 a2 a1 −1 0 2 −1        v1 =  −1 , v2 = −1 , v3 =  2  , v4 =  0  ∈ V, 2 −1 0 −1 where V = Ker(7, n2 , n3 , n4 ). If one of a2 , a3 or a4 is non negative, then I is a set-theoretic complete intersection. If a2 , a3 and a4 are negative, then ∑ v1 , v2 , v3 and v4 satisfy the conditions of Theorem 3 (note V = Zvj and v1 + v2 + v3 + v4 = 0). In any case, I is a set-theoretic complete intersection. Definition 1. Let V = Ker(n1 , n2 , n3 , n4 ). If there are v1 , v2 , v3 , v4 ∈ V with supp vj+ ∋ j such that the existence of them implies that I(V ) is a settheoretic complete intersection, then we denote M (V ) = (σi (vj ))i>1,j≥1 . We also denote it by M (n1 ; n2 , n3 , n4 ). For example,   −1 2 0 −1 0 . M (7; 1, 2, 4) = −1 −1 2 −1 0 −1 2 Note 1. The following are valid: (1) M (n1 ; n2 , n3 , n4 ) = M (n1 ; n′2 , n′3 , n′4 ), if (n2 , n3 , n4 ) ≡ (n′2 , n′3 , n′4 ) mod n1 , (2) M (n1 ; n2 , n3 , n4 ) exists, if ni ≡ nj mod n1 for any 1 < i < j, (3) M (n1 ; n2 , n3 , n4 ) exists, if ni + nj ≡ 0 mod n1 for any 1 < i < j, (4) M (n1 ; n2 , n3 , n4 ) exists, if ni + nj ≡ nk mod n1 for any 1 < i < j and k > 0, (5) M (n1 ; n2 , n3 , n4 ) exists, if M (n1 ; n1 − n2 , n1 − n3 , n1 − n4 ) exists. For example, M (7; 3, 5, 6) exists, since M (7; 4, 2, 1) exists and (3, 5, 6) ≡ 6(4, 2, 1) mod 7. We investigate whether M (n1 ; n2 , n3 , n4 ) exists under the following assumption:

(1) 0 < n2 < n3 < n4 < n1 − n2 , (2) ni + nj ̸≡ 0 mod n1 for any 1 < i < j, (3) ni + nj ̸≡ nk mod n1 for any 1 < i < j and k > 0, (4) gcd(n1 , n3 , n4 ) = gcd(n1 , n2 , n4 ) = gcd(n1 , n2 , n3 ) = 1, (5) there are not a known triple (n′2 , n′3 , n′4 ) and d > 0 satisfying (n2 , n3 , n4 ) ≡ d(n′2 , n′3 , n′4 ) mod n1 . For example, we have (1, 2, 5) ≡ 5(2, 4, 1) mod 9, hence we have M (9; 1, 2, 5) = M (9; 2, 4, 1). If n1 is odd, then M (n1 ; 1, 2, 4) exists, since   −1 2 0 −1 M (4µ1 + 2µ2 + 1; 1, 2, 4) = −µ2 −1 2 µ2 − 1 , −µ1 0 −1 µ1 + 1 where µ1 > 0 and µ2 is 0 or 1. Note that we need not give M (n1 ; 1, 2, 4) if n1 is even, since the condition (4) above. By Mathematica, (1) if n1 ≤ 7, the only case is M (n1 ; 1, 2, 4), (2) if 11 ≤ n1 ≤ 14, the following cases of (n2 , n3 , n4 ) are the rest:

(n1 = 8) (n1 = 9) (n1 = 10) (n1 = 11) (n1 = 12)

(1, 2, 5), (1, 2, 4), (1, 2, 6), (1, 4, 7), (1, 2, 5), (1, 2, 7), (1, 3, 5), (1, 5, 8), (1, 2, 4), (1, 2, 5), (1, 2, 7), (1, 2, 8), (1, 3, 5), (1, 2, 5), (1, 2, 7), (1, 2, 9), (1, 3, 5), (1, 3, 7), (1, 3, 8), (1, 4, 7), (1, 5, 9), (1, 7, 10), (n1 = 13) (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 2, 8), (1, 2, 9), (1, 2, 10), (1, 3, 5), (1, 3, 9), (1, 4, 6), (1, 4, 11), (n1 = 14) (1, 2, 5), (1, 2, 7), (1, 2, 9), (1, 2, 11), (1, 3, 5), (1, 3, 7), (1, 3, 8), (1, 3, 10), (1, 4, 7), (1, 6, 11), (1, 7, 9), (1, 7, 10), (1, 7, 12), (1, 9, 11).

If n1 ≤ 13, the matrix M (n1 ; n2 , n3 , n4 ) exists! Here is the list of the matrices M (n1 ; n2 , n3 , n4 ) ([5]):

  −1 2 −1 0 M (8; 1, 2, 5) = −1 −1 3 −1, v1 + v2 + v3 + v4 = 0, −1 0 −1 2  −1 2 0 −1  M (9; 1, 2, 6) = −1 −1 3 −1, v1 + v2 + v3 + v4 = 0, −1 0 −1 2  −1 2 −1 −1  M (9; 1, 4, 7) = −2 −1 2 −1, 0v1 + v2 + v3 + v4 = 0, 0 −1 −1 2  −1 2 −1 0  M (10; 1, 2, 5) = −2 −1 3 0, v1 + v2 + v3 + v4 = 0, −1 0 −1 2  −1 2 −1 0 M (10; 1, 2, 7) = −1 −1 4 −2, v1 + v2 + v3 + v4 = 0, or  −1 0 −1 2 −1 2 0 −1 −1 −1 2 0 , v1 + v2 + v3 + v4 = 0. −1 0 −2 3 Note that, if π(v) = t (0, 2, −2) for v ∈ V , then supp v + = {3} or supp v − = {4}. Theabove means thatthere exists M (V ) in each case. −2 3 −1 0 M (10; 1, 3, 5) = −1 −1 2 0, v1 + v2 + v3 + v4 = 0, −1 0 −1 2 −2 3 0 −1 M (10; 1, 5, 8) =  0 −1 2 −1, v1 + v2 + v3 + v4 = 0, −1 −1 0 2  −1 2 −1 −1 M (11; 1, 2, 5) =  0 −1 3 −1, 2v1 + 2v2 + v3 + v4 = 0, −2 0 −1 5  0 2 −1 −1  M (11; 1, 2, 7) = −2 −1 4 −1, v1 + v2 + v3 + v4 = 0, −1 0 −1 2  −1 2 0 0  M (11; 1, 2, 8) = −1 −1 4 −1, 2v1 + v2 + v3 + v4 = 0, −1 0 −1 3  −1 3 −1 −1  0 −1 2 −1, v1 + v2 + v3 + v4 = 0, M (11; 1, 3, 5) = −2 0 −1 3



 0 2 −1 −1 M (12; 1, 2, 5) = −1 −1 3 −1, v1 + v2 + v3 + v4 = 0, −2 0 −1 3  −1 2 −1 0  M (12; 1, 2, 7) = −2 −1 4 −1, v1 + v2 + v3 + v4 = 0, −1 0 −1 2  −1 2 0 −1  M (12; 1, 2, 9) = −1 −1 3 −1, v1 + v2 + v3 + v4 = 0, or  −1 0 −2 3 −1 2 −1 0 −1 −1 5 −3, v1 + v2 + v3 + v4 = 0. −1  0 −1 2  −2 3 −1 0 M (12; 1, 3, 5) =  0 −1 2 −1, v1 + v2 + v3 + v4 = 0, −2 0 −1 3 0 2 0 −1 M (12; 1, 3, 7) = −1 0 4 −1, 2v1 + v2 + v3 + 2v4 = 0, or  −3 −2 0 4 −2 3 0 −2 −1 −1 4 0 , 2v1 + 2v2 + v3 + v4 = 0. −1  0 0 2  −1 3 −1 0 M (12; 1, 3, 8) = −1 −1 3 0, 2v1 + v2 + v3 + v4 = 0, −1 0 −1 3  −1 4 −1 0  M (12; 1, 4, 7) = −1 −1 2 −1, 2v1 + v2 + 2v3 + v4 = 0, −1 0 −1 4  −2 2 −1 −1  M (12; 1, 5, 9) = −2 −1 2 −1, v2 + v3 + v4 = 0, 0 −1 −1 2  −2 5 −2 −1  0 −1 2 −1, v1 + v2 + v3 + v4 = 0, or M (12; 1, 7, 10) =  −1 −1 0 2 0 2 −1 −1 −2 −2 5 −1, v1 + v2 + v3 + v4 = 0, −1 0 −1 2   −1 2 −1 0  M (13; 1, 2, 5) = −1 −1 3 −1, v1 + v2 + v3 + v4 = 0, −2 0 −1 3

  −1 2 0 −1 M (13; 1, 2, 6) =  0 −1 3 −1, 3v1 + 2v2 + v3 + v4 = 0, −2 0 −1 7  −1 2 0 −1  M (13; 1, 2, 8) = −2 −1 4 −1, v1 + v2 + v3 + v4 = 0, −1 0 −1 2  0 2 −1 −1  0 , 2v1 + v2 + v3 + v4 = 0, M (13; 1, 2, 9) = −2 −1 5 −1 0 −1 3  −1 2 0 −1  0 , 3v1 + 2v2 + v3 + v4 = 0, M (13; 1, 2, 10) = −1 −1 5  −1 0 −1 4 0 3 −1 −1 M (13; 1, 3, 5) = −1 −1 2 −1, 2v1 + v2 + 2v3 + v4 = 0, −2 0 −1 6  −1 3 0 −1 0 , 2v1 + v2 + v3 + v4 = 0, M (13; 1, 3, 9) = −1 −1 3 −1 0 −1 3  −1 6 0 −1 0 3 −1, 3v1 + v2 + v3 + 3v4 = 0 or M (13; 1, 4, 6) =  0  −2 −1 −2 3 −1 4 −1 0 −3 −1 5 −3, 2v1 + v2 + 2v3 + v4 = 0, 0  0 −1 2  0 2 −1 −1 M (13; 1, 4, 11) = −1 −1 3 −1, v1 + v2 + v3 + v4 = 0, −2 −1 −1 4 Note 2. If n1 = 14, the exsitence of M (14, 1, 9, 11) is not proved, while the others exists.

References [1] H. Bresinsky, Monomial Gorenstein curves in A4 as set-theoretic complete intersections, Manuscripta Math. 27 (1979), 353–358. [2] R. C. Cowsik and M. V. Nori, Affine curves in characteristic p are set theoretic complete intersections, Invent. Math. 45 (1978), 111-114.

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