Characteristic p method in singularity theory

奇点理论中的特征p法

• 批准号: 10640042
• 负责人: WATANABE Keiichi
• 金额: $ 2.18万
• 依托单位: NIHON UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640042/
• 关键词: Frobenius map; rational singularities; integrall closed ideals; Hilbert-Kunz multiplicity; F-regular ring; log terminal Singularity; log terminal singularity; integrally closed ideals; Mac Kay correspondence; multiplicity; good ideals; integral

We applied "characteristic p" methods to singularity theory and obtained the fokkowing results.1 Characterization of Singularities in Characteristic 0 via Frobenius endomorphism. We found that log-terminal singularity and F-regular rings are equivalent notions in the case the ring is Q- Gorenstein. The same is true for rational singularities and F-rational rings. Also, we found that the same is true for "singularity of pairs" (KLT, PLT etc.). The latter result is a joint work with N.Hara.2 The notion of "Hilbert-Kunz multiplicity", which is a new "multiplicity" for local rings. We characterized regular local rings, certain rational singularities in dimension 2 by this multiplicity. Also we found a very beautiful and mysterious formula for integrally closed ideals in 2-dimensional rational double points. (a joint work with K.Yoshida)3 We investigated chains of integrally closed ideals and found the existence of a composition series only by integrally closed ideals. We found that the family of integrally closed ideals of colength 1 corresponds to the closed points of the fiber cone. Also, we found a new characterization of simple integrally closed ideals in 2-dimensional regular local rings. (The last result is a joint work with S.Noh.)

我们将“特征p”方法应用于奇异理论并获得了fokking的结果。1特征性0中通过Frobenius内态的表征。我们发现对数末端的奇异性和f指标环是等效的概念，因为环为Q- Gorenstein。理性的概念和F理性环也是如此。另外，我们发现“对的奇异性”（KLT，PLT等）也是如此。后一个结果是与N.Hara.2的共同作品。2“ Hilbert-Kunz多重性”的概念，这是本地环的新“多重性”。我们表征了常规的局部环，通过这种多重性在维度2中的某些理性概念。另外，我们发现了一个非常美丽和神秘的公式，用于以二维合理的双点为综合的理想。 （与K.Yoshida的联合作品）3我们研究了整体封闭理想的链，发现仅通过封闭的理想才能存在组成序列。我们发现，Colength 1的整合封闭理想的家族对应于纤维锥的封闭点。此外，我们发现了在二维常规局部环中简单封闭理想的新特征。 （最后的结果是与s.noh的联合合作。）

项目成果

Katsura Miyazaki and Kimihiko MOTEGI: "Toroidal surgery on periodic knots"Pacific J.Math.. 193. 381-396 (2000)

Katsura Miyazaki 和 Kimihiko MOTEGI：“周期性结的环形手术”Pacific J.Math.. 193. 381-396 (2000)

K Watanabe and K.Yoshida: "Hilbert-Kunz multiplicity of two-dimensional local rings" J.of pure and applied Algebra. 未定. 未定 (1999)

K Watanabe 和 K. Yoshida：“二维局部环的 Hilbert-Kunz 重数”J.of 纯代数和应用代数 待定（1999）。

N.Hara and K.Watanale: "F-regular and F-puse rings vs log-terminal and log-canonical singularities"J.of Algelraic Geometry. (To appear).

N.Hara 和 K.Watanale：“F-正则环和 F-puse 环与对数终端和对数规范奇点”J.of Algelraic Geometry。

K.Watanabe and K.Yoshida: "Hilbert-Kunz multiplicity and an Ineguality between multiplicity and colenqth" J of Algebra. 未定. 未定 (1999)

K.Watanabe 和 K.Yoshida：“Hilbert-Kunz 重数以及重数与列数之间的不等式”《代数》杂志待定（1999 年）。

Nobuo HARA and Kei-ichi WATANABE: "F-regular and F-pure Rings vs. Log-terminal and Log-canonical Singularities"J.of Alg.Geom. (to appear).

Nobuo HARA 和 Kei-ichi WATANABE：“F-正则环和 F-纯环与对数末端和对数规范奇点”J.of Alg.Geom。