Commutative ring theory and singularity theory

交换环理论和奇点理论

• 批准号: 13440015
• 负责人: WATANABE Keiichi
• 金额: $ 4.54万
• 依托单位: Nihon University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440015/
• 关键词: multiplier ideal; lc threshold; F-pure; Hilbert-Kunz multiplicity; Frobenius endomorphism; log terminal singularity; tight closure; regular local ring; F-pure threshold; 整閉イデアル; log resolution; UFD; Seifelt手術; Chow群; 特異点; 可換環論; 全座標環; Hilb

The results are mainly concerning the followings 3 themes1.Multiplier ideals ;J.Lipman and K.Watanabe proved that every integrally closed ideal in 2 dimensional regular local rings is a multiplier ideal.N.Hara and K.Yoshida defined a generalization of "tight closures" in characteristic p>0 and by using that concept, they Succeeded to calculate multiplier ideals by purely algebraic (by commutative ring theory) method.S.Takagi and K.Watanabe established the notion of "F-pure thresholds", which corresponds to the notion of lc(=log canonical) threshold in characteristic 0, used in algebraic geometry. This concept has many interesting features in both singularity theory and commutative ring theory.2.Hilbert-Kunz multiplicity ;Hilbert-Kunz multiplicity is a kind of multiplicity defined for rings of positive characteristics. Watanabe and Yoshida proved before that a ring is regular if and only if the HK multiplicity of the ring is 1. This time we determined the rings whose HK multiplicity is smallest among non-regular rings in dimension 2 and 3.

结果主要涉及以下3个方面：1.乘子理想; J.Lipman和K.Watanabe证明了2维正则局部环中的每一个整闭理想都是乘子理想，N.Hara和K.Yoshida定义了特征p>0的“紧闭包”的推广，并利用这一概念，他们成功地计算乘子理想的纯代数S.Takagi和K.Watanabe建立了“F-纯阈值”的概念，其对应于在代数几何中使用的特征0中的LC（=对数正则）阈值的概念。这个概念在奇点理论和交换环理论中都有许多有趣的特征。2.希尔伯特-昆兹重数：希尔伯特-昆兹重数是定义在正特征环上的一种重数。Watanabe和Yoshida证明了环是正则的当且仅当环的HK重数是1。这一次我们确定了在2维和3维的非正则环中HK重数最小的环。

项目成果

When does the subadditivity theorem for multiplier ideals hold?

乘数理想的次可加性定理何时成立？

• 发表时间: 2004
• 期刊: Trans.Amer.Math.Soc. 356
• 作者: K.-i.Watanabe;S.Takagi
• 通讯作者: S.Takagi

F-regular and F-pure rings vs. log terminal and log canonical singularities

• DOI: 10.1090/s1056-3911-01-00306-x
• 发表时间: 2000-02
• 期刊: Journal of Algebraic Geometry
• 影响因子: 1.8
• 作者: Nobuo Hara;Kei-ichi Watanabe

A characterization of semi-quasihomogeneous functions in terms of Milnor numbers

用 Milnor 数表示半拟齐次函数的特征

• 发表时间: 2004
• 期刊: Proceedings of American Mathematical Society 132
• 作者: M.Furuya;M.Tomari
• 通讯作者: M.Tomari

N.Hara, K.Watanabe: "F-regular and F-pure rings vs. log terminal and log canonical singularities"J. of Algebraic Geometry. 11. 363-392 (2002)

N.Hara，K.Watanabe：“F-正则环和 F-纯环与对数终端和对数规范奇点”J。

A characterization of semi-quasihomogeneous function in terms of the Milnor number

半拟齐次函数的 Milnor 数表征

• 发表时间: 2004
• 期刊: Proc.Amer.Math.Soc. 132・7
• 作者: M.Furuya;M.Toamri
• 通讯作者: M.Toamri