Algebro-geometric studies of rational singularities and related singularities by blowing-ups

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On the main theme of this project :(1)In 1997, M.Tomari found more 3 examples of simple K3 singularities which do not belong to the famous 95 classess. It was a natural continuation of studies of previous year. Tomari also found a. counter example to an analogus conjecture of M.Reid about 4-dimensional terminal singularities in terms of Newton boundary. In the both studies, the theory of filtered blowing-up by Tomari-Watanabe plays an essential role. In 1998, Tomari succeeded to prove the criterion about the rational singularities and isolated singularities about the Segre product of two normal graded rings. The criterions are natural generalizations to those for the normal graded rings in terms of Pinkham-Demazure's construction.(2)T.Hayakawa studied several partial resolutions of 3-dimensional terminal singularities by weighted blowing-ups. In particular he succeded to show a special corespondence between the set of divisorial blowing-ups with minimal discrepancy and the set of the maximal blowing-ups with "big weight". He classified the elementary contraction with the minimal discrepancy in his situation.(3)M.Takamura gave a very good estiamte about the arithemetic genus of normal two-dimensional singularities of multiplicity two in terms of the Horikawa canonical resolution. Combined with the previous result of Tomari, he obtained the complete classification of the case of p_<alpha> = 2.As related works on complex analysis :(4)K.Morita studied the special log forms which gives a-basis of higher dimensional de Rham cohomology which is related to the arrangements of hyperfurface on the complex affine space. The work is aimed to give application to integral representaion of hypergeomeric functions of several variables and a natural generalization of Aomoto-Kita's theory.