Research on singularities of a variety

品种奇异性研究

• 批准号: 14340005
• 负责人: ISHII Shihoko
• 金额: $ 8.19万
• 依托单位: Tokyo Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340005/
• 关键词: singularities; geometric genus; jet scheme; arc space; moduli spase; singularity; toric variety; graded ring; integral closed; complete intersection; multipier ideal; plurigenera; jet scheme

We gave an affirmative answer to the Nash problem for an arbitrary dimensional toric variety.We gave a negative answer to the Nash problem in general by showing a counter example of dimension 4.We studied the structure of the arc space of a toric variety and obtain that the dominating relation of the orbits is translated by the relation of valuations corresponding to the orbits.We gave an affirmative answer to the Nash problem for non-normal toric variety.We gave an affirmative answer to the local Nash problem for quasi-ordinary singularities.We defined a maximal divisorial set on the arc space of a variety and proved that any irreducible component of a contact locus is a maximal divisorial set.We proved that a maximal divisorial set is represented by the intersection of finite number of contact loci of functions.We proved that every integrally closed ideal of 2-dimensional regular local ring is a multiplier ideal.We gave a characterization of 2-dimensional Gorenstein singularities with arithmetic genus 1.

我们对任意维复曲面簇的纳什问题给出了肯定的答案。通过展示 4 维的反例，我们对一般纳什问题给出了否定答案。我们研究了复曲面簇的弧空间结构，并得到轨道的主导关系是通过与轨道相对应的估值关系来转换的。我们对非正态环面变体的纳什问题给出了肯定的答案。我们对局部纳什问题给出了肯定的答案我们在簇的弧空间上定义了一个最大除数集，并证明了接触轨迹的任何不可约分量都是一个最大除数集。我们证明了最大除数集可以用有限个函数的接触轨迹。我们证明了二维正则局部环的每一个全闭理想都是乘子理想。我们用算术亏格给出了二维Gorenstein奇点的刻画1.

项目成果

渡辺敬一: "Chains do integrally closed ideals"Contemporary Math.(AMS). 331. (2003)

Keiichi Watanabe：“链实现整体封闭的理想”当代数学。（AMS）331。（2003）。

Arcs, valuations and the Nash maps

弧线、估值和纳什图

• 发表时间: 2005
• 期刊: J.refine angew.Math. 588
• 作者: Thompson;E.R.;Kuhara;M.;都丸正;泊昌孝;S.Ishii;K.Watanabe;石井志保子
• 通讯作者: 石井志保子

Arcs, valuations and the Nash map

弧线、估值和纳什图

• 发表时间: 2005
• 期刊: J.reine angew.Math. 588
• 作者: Thompson;E.R.;Kuhara;M.;都丸正;泊昌孝;S.Ishii
• 通讯作者: S.Ishii

Multiplicity of filtered rings and simple K3 singularities of mutiplicity two,

滤波环的重数和重数二的简单 K3 奇点，

• 发表时间: 2002
• 期刊: Publ.RIMS. 38-4
• 作者: S.Ishii;P.Milman;M.Tomari
• 通讯作者: M.Tomari

Integrally closed ideals in two-dimensional regular local rings are multiplier ideals

二维正局部环中的全闭理想是乘子理想

• 发表时间: 2003
• 期刊: Math.Reasearch Letters vol.10
• 作者: J.Lipman;K.Watanabe
• 通讯作者: K.Watanabe