On the stable Harbourne conjecture for ideals defining space monomial curves
HTML articles powered by AMS MathViewer
- by Kosuke Fukumuro and Yuki Irie;
- Proc. Amer. Math. Soc. 151 (2023), 1445-1458
- DOI: https://doi.org/10.1090/proc/16258
- Published electronically: January 24, 2023
- HTML | PDF | 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
- Thomas Bauer, Sandra Di Rocco, Brian Harbourne, MichałKapustka, Andreas Knutsen, Wioletta Syzdek, and Tomasz Szemberg, A primer on Seshadri constants, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 33–70. MR 2555949, DOI 10.1090/conm/496/09718
- Shiro Goto, Koji Nishida, and Yasuhiro Shimoda, Topics on symbolic Rees algebras for space monomial curves, Nagoya Math. J. 124 (1991), 99–132. MR 1142978, DOI 10.1017/S0027763000003792
- Shiro Goto, Koji Nishida, and Yasuhiro Shimoda, The Gorensteinness of the symbolic blow-ups for certain space monomial curves, Trans. Amer. Math. Soc. 340 (1993), no. 1, 323–335. MR 1124166, DOI 10.1090/S0002-9947-1993-1124166-4
- Eloísa Grifo, A stable version of Harbourne’s conjecture and the containment problem for space monomial curves, J. Pure Appl. Algebra 224 (2020), no. 12, 106435, 23. MR 4101479, DOI 10.1016/j.jpaa.2020.106435
- Eloísa Grifo, Craig Huneke, and Vivek Mukundan, Expected resurgences and symbolic powers of ideals, J. Lond. Math. Soc. (2) 102 (2020), no. 2, 453–469. MR 4171422, DOI 10.1112/jlms.12324
- Brian Harbourne and Craig Huneke, Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28A (2013), 247–266. MR 3115195
- Jürgen Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175–193. MR 269762, DOI 10.1007/BF01273309
- Melvin Hochster and Craig Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), no. 2, 349–369. MR 1881923, DOI 10.1007/s002220100176
- G. Knödel, P. Schenzel, and R. Zonsarow, Explicit computations on symbolic powers of monomial curves in affine space, Comm. Algebra 20 (1992), no. 7, 2113–2126. MR 1167091, DOI 10.1080/00927879208824449
- Koji Nishida, On the third symbolic powers of prime ideals defining space monomial curves, Symposium on Commutative Ring Theory (Japan) 2008, pp. 155–160.
- Peter Schenzel, Examples of Noetherian symbolic blow-up rings, Rev. Roumaine Math. Pures Appl. 33 (1988), no. 4, 375–383. MR 950134
Bibliographic 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
- MathSciNet review: 4550341
Dedicated: Dedicated to Professor Koji Nishida on the occasion of his sixtieth birthday