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

关于这个项目的主题：(1)1997年，M.Tomari发现了3个不属于著名的95类的简单K3奇点的例子。这是前一年研究的自然延续。Tomari还发现了一个与M.Reid关于牛顿边界下的四维终端奇点的类似猜想的反例。在这两个研究中，Tomari-Watanabe的过滤爆破理论起着至关重要的作用。1998年，Tomari成功地证明了关于两个正规分次环的Segre积的有理奇点和孤立奇点的判据。根据Pinkham-Demazure的构造，这些判据是正规分次环上的判据的自然推广。(2)T.Hayakawa通过加权爆破研究了三维终端奇点的几种部分分辨。特别地，他成功地证明了具有最小偏差的除数爆破集与具有“大权”的最大爆破集之间的特殊对应。(3)M.Takamura根据Horikawa正则分解给出了二重正规二维奇点算术亏格的一个很好的估计。结合Tomari的结果，他得到了p_&lt；α&gt；=2情形的完全分类。作为复分析的相关工作：(4)K.Morita研究了特殊的对数形式，它给出了高维De Rham上同调的基础，它与复仿射空间上的超曲面的排列有关。工作的目的是应用于多元超几何函数的积分表示和Aomoto-Kita理论的自然推广。

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.