Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 1, 195-198.

Construction of Set Theoretic Complete Intersections via Semigroup Gluing

Apostolos Thoma

Department of Mathematics, University of Ioannina, Ioannina 45110, Greece e-mail: athoma@cc.uoi.gr

Abstract. In this note we present a very simple, but powerful, technique for finding monomial varieties which are set theoretic complete intersections. The technique is based on the concept of gluing semigroups that was defined by J. C. Rosales in [6] and used by K. Fischer, W. Morris and J. Shapiro in [2] to characterize complete intersection affine semigroup rings. There are several techniques in the literature proving that certain varieties are set theoretic complete intersections but all of them preserve the dimension of the variety and are mainly results about curves, see [5, 7] and the references in there. The technique presented here does not preserve necessarily the dimension of the variety and it can combine the known results to produce set theoretic complete intersection varieties of any dimension, see Examples 4 and 5.

Let T = {α_{1}, …, α_{n}} be a set of nonzero elements in N^{m}. Let S(T) = {l_{1}α_{1} + … + l_{n}α_{n} : l_{1}, …, l_{n} ∈ N} denote the affine semigroup generated by T, G(T) = {k_{1}α_{1} + … + k_{n}α_{n} : k_{1}, …, k_{n} ∈ Z} denote the group generated by T and r denote the rank_{Z}G(T). Let ϕ be the homomorphism of semigroup algebras from K[X_{1}, …, X_{n}] to the polynomial ring K[t_{1}, …, t_{m}] induced by T, that is: ϕ(X_{i}) = t^{α_{i}}. The kernel of ϕ is the ideal of T and is denoted by I(T), while the set of zeroes V (I(T)) in K^{n} is called the monomial variety of T. K[S(T)] = ϕ(K[X_{1}, …, X_{n}]) is the affine semigroup ring of S(T).

We will write rad(I) for the radical of I. The arithmetical rank of I(T), written ara(I(T)), is the smallest integer s for which there exist elements f_{1}, f_{2}, ⋯, f_{s} in I(T), such that rad(I(T)) = rad(f_{1}, f_{2}, ⋯, f_{s}). The ideal I(S) is called set theoretic complete intersection if ara(I(T)) = n − r.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

196 A. Thoma: Construction of Set Theoretic Complete Intersections via Semigroup Gluing

Let T_{1} and T_{2} be two nonempty subsets of T such that T = T_{1} ∪ T_{2} and T_{1} ∩ T_{2} = ∅. T is said to be a gluing of T_{1} and T_{2} if there exists a nonzero a ∈ S(T_{1}) ∩ S(T_{2}) such that G(a) = G(T_{1}) ∩ G(T_{2}). In terms of the ideals this is equivalent to I(T) = I(T_{1}) + I(T_{2})+ < F_{a} >, see [6], where F_{a} is a binomial in the form M_{1} − M_{2}, M_{1} involves variables corresponding to T_{1}, M_{2} involves variables corresponding to T_{2}, and ϕ(M_{1}) = t^{a} = ϕ(M_{2}).

Lemma 1. Suppose that T is the gluing of T_{1} and T_{2}. Then

ara(I(T)) ≤ ara(I(T_{1})) + ara(I(T_{2})) + 1.

Proof. Suppose that ara(I(T_{1})) = s_{1} and ara(I(T_{2})) = s_{2}, therefore there exist f_{1}, …, f_{s_{1}}, g_{1}, …, g_{s_{2}} such that I(T_{1}) = rad(f_{1}, …, f_{s_{1}}) and I(T_{2}) = rad(g_{1}, …, g_{s_{2}}). The result follows from the observation that I(T) = rad(f_{1}, …, f_{s_{1}}, g_{1}, …, g_{s_{2}}, F_{a}), since I(T) = I(T_{1}) + I(T_{2})+ < F_{a} >.

Theorem 2. If both I(T_{1}), I(T_{2}) are set theoretic complete intersections then I(T) is set theoretic complete intersection.

Proof. Let n_{1} be the number of elements in T_{1} and n_{2} the number of elements in T_{2}, we have n = n_{1}+n_{2}. Let r_{1} be the rank of G(T_{1}) and r_{2} the rank of G(T_{2}). Since G(a) = G(T_{1})∩G(T_{2}) and rank_{Z}G(a) = 1, we have r + 1 = r_{1} + r_{2}. Then we have n − r = (n_{1} − r_{1}) + (n_{2} − r_{2}) + 1.

Now suppose that both I(T_{1}), I(T_{2}) are set theoretic complete intersections. Therefore ara(I(T_{1})) = n_{1} −r_{1} and ara(I(T_{2})) = n_{2} −r_{2}. Then, from the Lemma 1 and the previous remarks we have that n−r ≤ ara(I(T)) ≤ ara(I(T_{1}))+ara(I(T_{2}))+1 = (n_{1}−r_{1})+(n_{2}−r_{2})+1 = n − r, that means I(T) is set theoretic complete intersection.

Based on this theorem one can develop a technique producing set theoretic complete intersection monomial varieties. One can make any combination of the known results to produce new examples and iterate this procedure many times. Note also that we get explicitly the defining equations of the new variety, provided that we know the defining equations of the initial two.

Example 3. Affine monomial curves are set theoretic complete intersections in positive characteristic, while in characteristic zero the general problem is still open, even for monomial curves in A_{K}^{4}.

In [4] D. Patil proves that any affine monomial curve (t^{n_{1}}, t^{n_{2}}, …, t^{n_{s}}), for which s − 1 numbers among n_{1}, …, n_{s} form an arithmetic sequence, is set theoretic complete intersection in A_{K}^{s}, while in [3] W. Gastinger proves that an almost complete intersection monomial curve (t^{m_{1}}, t^{m_{2}}, t^{m_{3}}, t^{m_{4}}) is set theoretic complete intersection in A_{K}^{4}. Note that only for special values of the exponents m_{1}, m_{2}, m_{3}, m_{4} the curve is an almost complete intersection. Choose any positive integers m, n prime to each other such that n ∈ S(n_{1}, …, n_{s}) and m ∈ S(m_{1}, …, m_{4}), where s − 1 numbers among n_{1}, …, n_{s} form an arithmetic sequence and (t^{m_{1}}, t^{m_{2}}, t^{m_{3}}, t^{m_{4}}) is an almost complete intersection. From Theorem 2, the monomial curve (t^{mn_{1}}, …, t^{mn_{s}}, t^{nm_{1}}, …, t^{nm_{4}}) in A_{K}^{s+4} is set theoretic complete intersection.

Example 4. In [1] S. Eliahou proves that the curve (t^{4}, t^{6}, t^{7}, t^{9}) is set theoretic complete intersection in A_{C}^{4}. In [7] it was proved that the Eliahou’s curve is the set theoretic complete intersection in A_{K}^{4} of X_{2}^{3} − X_{1}X_{3}^{2}, X_{3}^{5} − 3X_{2}^{2}X_{3}^{2}X_{4} + 3X_{1}X_{2}X_{3}X_{4}^{2} − X_{1}^{2}X_{4}^{3} and X_{1}^{9} − 5X_{1}^{6}X_{2}^{2} + 10X_{1}^{4}X_{2}X_{3}^{2} − 10X_{1}^{2}X_{3}^{4} + 5X_{2}^{3}X_{4}^{2} − X_{4}^{4}. Also the curve (u^{13}, u^{5}v^{8}, u^{2}v^{11}, v^{13}) in P_{K}^{3} is arithmetically Cohen-Macaulay, therefore it is the set theoretic complete intersection of Y_{2}^{3} − Y_{1}Y_{3}Y_{4} and Y_{3}^{13} − 3Y_{2}^{2}Y_{3}^{8}Y_{4}^{3} + 3Y_{1}Y_{2}Y_{3}^{4}Y_{4}^{7} − Y_{1}^{2}Y_{4}^{11}. From Theorem 2 we have that the semigroup S((13, 0), (5, 8), (2, 11), (0, 13), (4, 4), (6, 6), (7, 7), (9, 9)) is set theoretic complete intersection, since

(13, 13) ∈ S((13, 0), (5, 8), (2, 11), (0, 13)) ∩ S((4, 4), (6, 6), (7, 7), (9, 9)) and
G((13, 13)) = G((13, 0), (5, 8), (2, 11), (0, 13)) ∩ G((4, 4), (6, 6), (7, 7), (9, 9)).

Therefore the monomial surface

V = (u^{13}, u^{5}v^{8}, u^{2}v^{11}, v^{13}, u^{4}v^{4}, u^{6}v^{6}, u^{7}v^{7}, u^{9}v^{9})

in A_{K}^{8} is set theoretic complete intersection and

I(V) = rad(X_{2}^{3} − X_{1}X_{3}^{2},
X_{3}^{5} − 3X_{2}^{2}X_{3}^{2}X_{4} + 3X_{1}X_{2}X_{3}X_{4}^{2} − X_{1}^{2}X_{4}^{3},
X_{1}^{9} − 5X_{1}^{6}X_{2}^{2} + 10X_{1}^{4}X_{2}X_{3}^{2} − 10X_{1}^{2}X_{3}^{4} + 5X_{2}^{3}X_{4}^{2} − X_{4}^{4},
X_{1}X_{4} − Y_{1}Y_{4}, Y_{2}^{3} − Y_{1}Y_{3}Y_{4},
Y_{3}^{13} − 3Y_{2}^{2}Y_{3}^{8}Y_{4}^{3} + 3Y_{1}Y_{2}Y_{3}^{4}Y_{4}^{7} − Y_{1}^{2}Y_{4}^{11}).

Example 5. One can use the same result two or more times, for example in [7] it was proved that all monomial curves in P_{K}^{4} (surfaces in A_{K}^{5}) of the form

(u^{k+7l}, u^{7l}v^{k}, u^{6l}v^{k+l}, u^{4l}v^{k+3l}, v^{k+7l})

are set theoretic complete intersections. From Theorem 2 we have that the monomial varieties of the form

(u^{k_{1}+7l_{1}}, u^{7l_{1}}v^{k_{1}}, u^{6l_{1}}v^{k_{1}+l_{1}}, u^{4l_{1}}v^{k_{1}+3l_{1}}, v^{k_{1}+7l_{1}}, v^{k_{2}+7l_{2}}, v^{7l_{2}}t^{k_{2}}, v^{6l_{2}}t^{k_{2}+l_{2}}, v^{4l_{2}}t^{k_{2}+3l_{2}}, t^{k_{2}+7l_{2}})

are set theoretic complete intersections.

References

[1] Eliahou, S.: Symbolic powers of monomial curves. J. Algebra 117 (1988), 437–456.

[2] Fischer, K.; Morris, W.; Shapiro, J.: Affine semigroup rings that are complete intersections. Proc. Amer. Math. Soc. 125 (1997), 3137–3145.

[3] Gastinger, W.: Über die Verschwindungsideale monomialer Kurven. PhD thesis, Univ. Regensburg, Landshut, 1989.

[4] Patil, D.: Certain monomial curves are set-theoretic complete intersections. Manuscripta Math. 68 (1990), 399–404.

[5] Robbiano, L.; Valla, G.: On set-theoretic complete intersections in the projective space. Rend. Sem. Mat. Fis. Milano LIII (1983), 333–346.

[6] Rosales, J.C.: On presentations of subsemigroups of N^{n}. Semigroup Forum 55(2) (1997), 152–159.

[7] Thoma, A.: On the set-theoretic complete intersection problem for monomial curves in A^{n} or P^{n}. J. Pure Appl. Algebra 104(3) (1995), 333–344.

Received March 9, 1999