Research on singularities of a variety

我们给出了任意维环面变分的Nash问题的一个肯定的回答。我们通过给出一个4维的反例，给出了一般情况下Nash问题的否定回答。研究了环面变元的弧空间的结构，得到了轨道的支配关系被对应于轨道的赋值关系所转换。我们给出了非正规环面变分的Nash问题的肯定回答。我们给出了准普通奇点的局部Nash问题的肯定回答。我们在环面变元的弧空间上定义了一个极大除集，并证明了任何不可约分支证明了函数的有限个接触轨迹的交集表示极大除集，证明了2维正则局部环的每个积分闭理想都是乘子理想，给出了亏格为1的2维Gorenstein奇点的刻画.

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.