Study of toric varieties in ring theory

我们考虑一个问题，是否每一个仿射单项曲线是一个集论完全相交，这是来自一个著名的问题，问Kronecker，是否每一个仿射代数曲线是一个集论完全相交。本文成功地给出了N-空间中的单项式曲线的定义理想由N-2个二项式和一个直到根的多项式生成的等价条件，其中N>2。我们考虑这个条件的原因是，大多数集合论完全相交单项曲线的早期例子都满足这个条件。应用该条件，由17、19、25和27定义的单项曲线不满足该条件。因此，它的定义理想不是由两个二项式和一个多项式生成的。另一方面，我们可以证明它是一个集合论完全交。事实上，它是由一个二项式和两个多项式生成的。进一步地，我们可以将这个结果推广如下：如果一条单项曲线与一个平衡半群相关联，那么它是一个集论完全交。平衡半群是由四个自然数定义的加法半群，比如a，B，c和d，其中a+d=B+c。此外，我们可以证明与扩张平衡半群有关的单项曲线的相同结果。

We consider a problem whether every affine monomial curve is a set-theoretic complete intersection, which is derived from a famous problem, asked by Kronecker, whether every affine algebraic curve is a set-theoretic complete intersection. We successfully give an equivalent condition in which the defining ideal of a monomial curve in affine N-space is generated by N-2 binomials and one polynomial up to radical, where N>2. The reason why we consider this condition is that most of earlier examples of set-theoretic complete intersection monomial curves satisfy this. Applying this condition, the monomial curve defined by 17, 19, 25 and 27 does not satisfy this condition. Consequently, its defining ideal is not generated by two binomials and a polynomial up to radical. On the other hand, we can prove that it is a set-theoretic complete intersection. Indeed, it is generated by one binomial and two polynomials up to radical. Further, we can extend this result as follows: If a monomial curve associated with a balanced semigroup, then it is a set-theoretic complete intersection. A balanced semigroup is an additive semigroup defined by four natural numbers, say a, b, c and d, with a+d=b+c Thus the previous example is associated with a balanced semigroup. In addition, we may prove the same result for monomial curves associated with extended balanced semigroups.