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

在我们的研究中，给出了以下结果。其中一些已经出版，其余的将很快出版。（1）主要研究者托马里证明了：（a）利用滤环的重数理论，给出了2-dim正则局部环的一个刻画的新证明。(b)从滤子切锥的角度研究L^2 plurigenus的下界有了新的进展。对上述平等性做了定性分析。(c)为了进一步扩展（B）的结果，他引入了一维奇点的L^2 plurigenus。利用过滤爆破技术，他通过L^2亏格的渐近行为给出了1维奇异性的分类，这与更高维情形的分类方法相同。这是在更一般的情况下扩展（B）的研究的证据。(d)在正规2维、多重奇点上的序函数的线性互补不等式方面也取得了进展。 关于我们 licity二(e)结合T. Okuma，托马里给出了正规2维Gorenstein椭圆奇点的一个几何刻画，这个刻画是托马里在20年前提出的.通过这些研究，该项目的所有研究人员都以多种形式做出了贡献。其中，早川、岩濑等人与托马里有着密切的联系，并在这一时期做出了不少贡献。（2）在最小模型理论中，Hayakawa把所有的三维极值收缩都归为一个点，在这个点上的偏差小于1。(3)Iwase继续研究了一种4维、多径的Gluck手术，并显示所有病例都成为与3维研究相同的沿着环面的Dehn手术。(4)Matsuura给出了代数栈仿射性质的一个判别准则，多年来一直是代数几何学家们感兴趣的问题。(5)Watanabe在积分闭理想与乘子理想的关系方面取得了一些重要的进展。少

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