Commutative Ring Theory of Singularities of Algebraic Varieties

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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.