Research on singularities of a variety

我们给出了任意维环面变换的纳什问题的肯定答案。我们通过一个四维的反例给出了纳什问题的否定答案。研究了一个环变弧空间的结构，得到了轨道的支配关系由轨道对应的值关系来转换。给出了非正态环型纳什问题的肯定答案。对拟平凡奇点的局部纳什问题给出了肯定的答案。定义了一簇弧空间上的极大分集，证明了接触轨迹的任何不可约分量都是极大分集。证明了一个极大分集是由有限个函数的接触轨迹的交表示的。证明了二维正则局部环的每一个整闭理想都是乘子理想。我们给出了算术属为1的二维Gorenstein奇点的一个刻划。

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.