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June 2019 Infinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curves
Kazuhiko Kurano, Koji Nishida
Michigan Math. J. 68(2): 409-445 (June 2019). DOI: 10.1307/mmj/1557475399


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.


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Kazuhiko Kurano. Koji Nishida. "Infinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curves." Michigan Math. J. 68 (2) 409 - 445, June 2019.


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

zbMATH: 07084769
MathSciNet: MR3961223
Digital Object Identifier: 10.1307/mmj/1557475399

Primary: 13A30
Secondary: 13F20

Rights: Copyright © 2019 The University of Michigan


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Vol.68 • No. 2 • June 2019
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