Research on the singularities of a variety

我们阐明了奇异点的一些性质与弧空间上的一些性质以及奇异点上的射流格式之间的关系。我们用射流格式的截断态射的平整度给出了基的变化规律的一个表征。代数曲线退化族与二维奇异点的关系。利用正特征法引入了交换环上理想对的f阈值，并推广了关于多重性的公式。

We clarified the relationship between some properties of singularities on a variety and the properties on the arc space and the jet schemes over the variety. We gave a characterization of the regularity of the base variety by the flatness of a truncation morphism of jet schemes. The relationship between degenerating families of algebraic curves and two dimensional singularities. We introduced F-threshold for pair of ideals in commutative rings by means of positive characteristic method and generalized the formula on multiplicity.