The Michigan Mathematical Journal

Infinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curves

Kazuhiko Kurano and Koji Nishida

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In this paper, we study finite generation of symbolic Rees rings of the defining ideal p of the space monomial curve (ta,tb,tc) for pairwise coprime integers a, b, c. Suppose that the base field is of characteristic 0, and the ideal p is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element ξ of the minimal degree in p is a negative curve, we determine the minimal degree of an element η such that the pair {ξ,η} satisfies Huneke’s criterion in the case where the symbolic Rees ring is Noetherian. By this result we can decide whether the symbolic Rees ring Rs(p) is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of p in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of p in which the homogeneous element of the minimal degree in p(2) is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.

Article information

Michigan Math. J., Volume 68, Issue 2 (2019), 409-445.

Received: 28 July 2017
Revised: 12 April 2018
First available in Project Euclid: 10 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Secondary: 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]


Kurano, Kazuhiko; Nishida, Koji. Infinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curves. Michigan Math. J. 68 (2019), no. 2, 409--445. doi:10.1307/mmj/1557475399.

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