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.

对任意维复曲面簇的纳什问题给出了肯定的回答。通过给出4维反例，对一般纳什问题给出了否定的回答。研究了复曲面簇的弧空间的结构，得到了轨道的控制关系可以通过轨道对应的赋值关系来转化。对非复曲面簇的纳什问题给出了肯定的回答。正规复曲面簇.我们对拟常奇点的局部Nash问题给出了肯定的回答.我们在簇的弧空间上定义了极大除集，证明了切迹的任何不可约分支都是极大除集.我们证明了极大除集由有限个函数的切迹的交表示.我们证明了2-二维正则局部环是一个乘子理想，给出了算术亏格为1的二维Gorenstein奇点的一个刻划。

项目成果

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

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）。

The local Nashproblem on arc families of singularities

奇点弧族上的局部纳什问题

• 发表时间: 2006
• 期刊: Ann. Institute Fourier 56
• 作者: T.Kitano;T.Morifuji;M.Takasawa;Akito Futaki;Akihiro Tsuchiya;M.Ishikawa;Kenji Fukaya;Shihoko Ishii
• 通讯作者: Shihoko Ishii

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