Applications of Frobenius map to Commutative Ring Theory and Algebraic Geometry

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1 Characterization of Singularities in Characteristic 0 via Frobenius endomorphism. We found that log-terminal singularity and F-regular rings are equivalent notions in the case the ring is Q-Gorenstein. The same is true for rational singularities and F-rational rings.2 By definig F-terminal rings, the terminal singularities are characterized in 3-dimensional case. F-terminal and Q-Gorenstein imply terminal singularity in any dimension. But in dimension 4, unfortunately, there is a counterexample to the converse and F-rational ring is characterized by the property that its general hyperplane section is F-rational in Gorenstein case.3 The characterization of regular local rings by Hilbert-Kunz multiplicity=1.Namely, an unmixed local ring of characteristic p is regular if and only if its H-K multiplicity is 1.Also we succeeded to classify the 2-dimensional rings with Hibert-Kunz multiplicity less than 9/4.