Classification of higher dimensional hypersurface singularities in terms of non-degenerate complete intersections

关于本课题的主题：(1)2000年，M. Tomari研究了相关联的梯度环是具有孤立奇点的积分域的终端奇点上理想的同化问题。作为一个特例，他展示了它们在三维正则局部环上相关联的梯度环的规律性。2001年，托马里用定义方程f的加权泰勒展开式给出了关于超曲面孤立奇点u的米尔诺数μ(f)的不等式。这里，只有当初始形式定义了孤立奇点时，等式才成立。给出了半拟齐次条件用μ表示的刻画。利用托马里关于过滤环多重性的一个结果进行证明。(2) T. Hayakawa研究了指数不小于2的三维终端奇点的几种部分分辨率。2000年，他构造了一个有趣的例子，在最坏的情况下，戈伦斯坦终端奇点仍然存在，这个例子承认部分解决。这很自然地通过对有理双点的特殊部分解析的研究得到了理解，这些点是奇点的反规范线性系统的一般成员。2001年，他还研究了这类部分分辨率的不可约分量。(3) H. Fujimoto成功地构造了n维复保护空间中2^n次双曲超曲面的一系列新例子。在n = 2的情况下，这个例子给出了这种情况下环境空间的最小可能度的世界记录。(4) A. Kodama从韦伯斯特度规的角度研究了具有非光滑边界的一般椭球体。

On the main theme of this project :(1) In 2000, M. Tomari studied the nitration of ideals on terminal singularities where the associated graded rings are integral domains with isolated singularity. As a special case, he showed the regularity of their associated graded rings of 3-dimensional regular local ring. In 1 2001, Tomari gave an inequality about Milnor number μ(f) of a hypersurface isolated singularity Uin terms of weighted Taylor expansion of the defining equation f. Here the equality holds iiand only if the initial form defines an isolated singularity. This gives a characterization of a semiquasi-homogeneous condition in terms of μ. The proof uses a result of Tomari on multiplicity of filtered ring.(2) T. Hayakawa studied several partial resolutions of 3-dimensipnal terminal singularities with index is not less than two. In 2000, he constructed an interesting example which admits a partial resolution where at worst Gorenstein terminal singularities remain. This was understood naturally by the' studies of the special partial resolution of rational double points which appears as the general members of anti-canonical linear system of the singularity. In 2001 he also studied the irreducible components of this type of partial resolution.As related-works on complex analysis :(3) H. Fujimoto had succeeded to construct a new series of examples of hyperbolic hypersurfaces of degree 2^n in n-dimensional complex protective spaces. In the case of n = 2, this example gives the world record of the minimal possible degree of the ambient spaces for such situation.(4) A. Kodama studied the general ellipsoids with not necessary smooth boundaries from the points of view of the Webster metric.