Filtered blowing-up of singularities and algebraic geometric properties of tangent cone

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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 showed; (a) By the theory of multiplicy of filtered rings, a new proof of characterization of 2-dim, regular local ring was given. (b) There are new progress for the study of lower bound of L^2 plurigenus from the point of views concerning the tangent cone of the filtration. A characterization of the equality in the above had done. (c) Seeking for a future progress to extend the results of (b), he introduced L^2 plurigenus for 1-dim singularities. Using a technique of filtered blowing-up, he gave a classification of 1-dim, singularities by the asymptotic behavior of this L^2 genus as in the same way as higher dim, cases. This is an evidence to extend the studies of (b) in more general situations. (d) Also he made a progress concerning linear complementary inequality about the order functions on normal 2-dim, singularities of multip … More licity two. (e) Combining recent results by T. Okuma, Tomari gave a geometric characterization of normal 2-dim. Gorenstein elliptic singularities, which was conjectured about 20 years ago by Tomari. Through these studies, all the investigators of this project contributed in several forms. Among others, Hayakawa and Iwase had been contacting to Tomari and giving many contributions in these periods. Further, as related subjects of the project, there are several individual results as follows; (2) In the minimal model theory, Hayakawa classified all the 3-dim. extremal contractions to one point, where the discrepancy is less than one. (3) Iwase continued the studies of a kind of Gluck surgery of 4-dim, manifold, and showed the all cases become the Dehn surgery along torus as same as 3-dim, studies. (4) Matsuura gave a criterion of affine properties of algebraic stacks ; which has been interested by algebraic geometers in many years. (5) Watanabe made several essential progress about the relations between integral closed ideals and multiplier ideals. Less