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。其中一些已经出版了，其余的很快就会出版。(1)首席研究员Tomari证明了：(A)利用滤环的乘法理论，给出了2维正则局部环的一个新的刻划。(2)从滤子的切锥角度研究L^2的下界有了新的进展。对上述平等的特征进行了描述。(C)为了寻求进一步的进展来推广(B)的结果，他引入了L^2关于一维奇点的多项式。利用过滤爆破技巧，他利用L^2亏格的渐近行为给出了一维奇点的分类，其方法与高维情形相同。这是将(B)的研究扩展到更一般情况的证据。(D)关于多维…的正规二维奇点上的序函数的线性互补不等式的研究也取得了进展再来点二度的。(E)结合T.Okuma的最新结果，Tomari给出了正规2-dim的几何刻画。戈伦斯坦椭圆奇点，这是大约20年前由Tomari猜想的。通过这些研究，该项目的所有调查人员都以几种形式做出了贡献。其中，早川和岩泽一直与Tomari保持联系，并在这两个时期做出了许多贡献。此外，作为项目的相关课题，有几个单独的结果如下：(2)在最小模型理论中，早川对所有的3维空间进行了分类。极值收缩到一点，其中的差异小于1。(3)Iwase继续了一种四维流形的Gluck手术的研究，发现所有病例都成为环面上的Dehn手术，与三维研究一样。(4)Matsuura给出了代数堆栈仿射性质的一个判据，该判据多年来一直受到代数几何学者的关注。(5)渡边在积分闭理想与乘子理想之间的关系方面取得了一些实质性进展。较少

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