Research on the singularities of a variety

阐明了簇上奇点的某些性质与弧空间上的性质及簇上的喷流格式之间的关系。我们给出了一个表征的正则性的基品种的平坦性的截断态射喷射计划。退化代数曲线族与二维奇点的关系。利用正特征线方法引入了交换环中理想对的F-阈值，推广了关于重数的公式。

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.