Commutative Ring Theory of Singularities of Algebraic Varieties

本研究的目的是将交换环理论应用于代数簇的奇点的研究，反过来，用奇点的几何语言来表达交换环的性质，并对其进行分析。利用维数和嵌入维数得到了f-纯环重数的上界。利用正特征理想的紧闭包理论刻画了参数理想的拟脚柱理想，并得到了一个界。得到了二维有理奇点中Ulrich理想的完全分类。用自由分解刻画了仿射3-空间中的几乎Gorenstein半群环，5.得到了齐次Toric环的a-不变量与F-纯阈值之间的不等式。确定超曲面的喷流格式何时具有有理奇点。

The aim of this research is to apply commutative ring theory to the study of singular points of algebraic varieties and, conversely, to express properties of commutative rings in terms of geometric language of singularities and analyze it.Main results of this research are the followings; 1. Obtained upper bound of multiplicities of f-pure rings in terms of dimension and embedding dimension, 2. Described the quasi-socle ideals of parameter ideals using the tight closure theory of ideals of positive characteristic and got a bound, 3. Obtained the complete classification of Ulrich ideals in 2-dimensional rational singularities, 4. Characterized almost Gorenstein semigroup rings in affine 3-space in terms of its free resolution, 5. Obtained inequalities between a-invariant and F-pure threshold of homogeneous toric rings, 6. Determined when the jet schemes of a hypersurface have rational singularities.