## On the stable Harbourne conjecture for ideals defining space monomial curves

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- by Kosuke Fukumuro and Yuki Irie PDF
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**151**(2023), 1445-1458 Request permission

## Abstract:

For the ideal $\mathfrak {p}$ in $k[x, y, z]$ defining a space monomial curve, we show that $\mathfrak {p}^{(2 n - 1)} \subseteq \mathfrak {m} \mathfrak {p}^{n}$ for some positive integer $n$, where $\mathfrak {m}$ is the maximal ideal $(x, y, z)$. Moreover, the smallest such $n$ is determined. It turns out that there is a counterexample to a claim due to Grifo, Huneke, and Mukundan, which states that $\mathfrak {p}^{(3)} \subseteq \mathfrak {m} \mathfrak {p}^2$ if $k$ is a field of characteristic not $3$; however, the stable Harbourne conjecture holds for space monomial curves as they claimed.## References

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## Additional Information

**Kosuke Fukumuro**- Affiliation: Department of Mathematics and Informatics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan
- MR Author ID: 1015594
- ORCID: 0000-0001-6446-0661
- Email: blackbox@tempo.ocn.ne.jp
**Yuki Irie**- Affiliation: Research Alliance Center for Mathematical Sciences, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-8578, Japan
- MR Author ID: 1287511
- ORCID: 0000-0002-6034-656X
- Email: yirie@tohoku.ac.jp, yuki@yirie.info
- Received by editor(s): April 22, 2022
- Received by editor(s) in revised form: July 2, 2022
- Published electronically: January 24, 2023
- Additional Notes: The second author was partially supported by JSPS KAKENHI Grant Number JP20K14277.

The second author is the corresponding author. - Communicated by: Claudia Polini
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**151**(2023), 1445-1458 - MSC (2020): Primary 13A15; Secondary 13H05
- DOI: https://doi.org/10.1090/proc/16258

Dedicated: Dedicated to Professor Koji Nishida on the occasion of his sixtieth birthday