## The Michigan Mathematical Journal

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

#### Abstract

In this paper, we study finite generation of symbolic Rees rings of the defining ideal ${\mathfrak{p}}$ of the space monomial curve $(t^{a},t^{b},t^{c})$ for pairwise coprime integers $a$, $b$, $c$. Suppose that the base field is of characteristic $0$, and the ideal ${\mathfrak{p}}$ is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element $\xi$ of the minimal degree in ${\mathfrak{p}}$ is a negative curve, we determine the minimal degree of an element $\eta$ such that the pair $\{\xi ,\eta \}$ 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 $\mathcal{R}_{s}({\mathfrak{p}})$ is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of ${\mathfrak{p}}$ in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of ${\mathfrak{p}}$ in which the homogeneous element of the minimal degree in ${\mathfrak{p}}^{(2)}$ is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1557475399

Digital Object Identifier
doi:10.1307/mmj/1557475399

Mathematical Reviews number (MathSciNet)
MR3961223

Zentralblatt MATH identifier
07084769

#### Citation

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. https://projecteuclid.org/euclid.mmj/1557475399

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