Research on the singularities of a variety

我们阐明了簇上奇点的一些性质与弧空间上的性质以及簇上的喷射格式之间的关系。我们用JET格式的截断态射的平坦性刻画了基簇的正则性。退化代数曲线族与二维奇点之间的关系。利用正特征线法引入交换环中理想对的F-门限，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.