Filtered blowing-up of local rings and algebraic geometric classification of singularities

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On our research, the following results are given. Some of these were already published, and the rest will be published soon.(1)The head investigator Tomari studied the normal graded rings whose Veronese subring is a polynomial rings. As an extension of the result in the case of 2-dimensional U.F.D., he obtain the classification of such cases by using famous results of Orlik-Wagreich type. In Tomari's paper, a geometric characterization of complex normal 2-dimensional Gorenstein elliptic singularities was shown by using the recent work of Okuma. Also, in the case of positive characteristic, similar problem was studied and turned out that the techniques in p_g-formula and elliptic singularities can be applied. The lower best bound of the geometric genus of normal 2-dimensional singularities were studied by comparing the topological bound the arithmetic genus. In the graded case, by using non-effective Pinkham-Demazure divisor, the bound for star-shaped singularity is very near the arithm … More etic genus. The studies are still in progress. (2)Hayakawa classified the divisorial contractions to 3-dimensional Gorenstein terminal singularity with small discrepancy. Here he used his special filtered blowing-ups which were introduced in previous his works. In the case of the index with greater than 1, he constructed the terminalization where the exceptional divisor appear with 1/index discrepancy by a composition of irreducible blowing-ups. (3)Watanabe gave a characterization of F-rational ring by using the theory of F-threshold which is a generalization of F-pure threshold. He also gave some interesting inequality of multiplicity in terms of F-threshold. This also induces new results of the case of characteristic zero. (4)Okuma has shown the equisingularity of the universal abelian covers of 2-dimensional rational singularity or minimal elliptic singularity to the complete intersection singularity. This gave a partial answer to the Neuman-Wahl conjecture. He also showed the relation between singularity and its UAC in terms of geometric genus. It turned out that, in the case UAC is of splice type, that the relation is topological. (5)Fukuda obtained the criterion for the existence of solutions for implicit differential systems. He also characterized the singularities which are integrable in the case of the generalized Hamiltonian system, which is important object in the mathematical physics. Less