## The Michigan Mathematical Journal

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

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

Kazuhiko Kurano and Koji Nishida

#### 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

**Subjects**

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]

#### 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