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

通过爆炸对有理奇点和相关奇点进行代数几何研究

• 批准号: 09640021
• 负责人: TOMARI Masataka
• 金额: $ 1.86万
• 依托单位: Kanazawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640021/
• 关键词: the normal graded rings; rational singularities; weighted blowing-ups; conjecture of M.Reid; the Segre product; terminal singularities; the arithemetic genus; the special log forms; ネバンリンナ理論; 単純K3特異点; 因子的ブロ-イングアップ; 正則曲線の一意性定理; 多様体のホモトピー同値; 正則; the normal graded rings; rational singularities; weighted blowing-ups; conjecture of M.Reid; the Segre product; terminal singularities; the arithemetic genus; the special log forms; ネバンリンナ理論; 単純K3特異点; 因子的ブロ-イングアップ; 正則曲線の一意性定理; 多様体のホモトピー同値; 正則

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.

在该项目的主题上：（1）在1997年，M.Tomari找到了更多的3个简单K3奇点的例子，这些例子不属于著名的95级课。这是上一年研究的自然延续。托马里也发现了一个。在牛顿边界方面，M.Reid的类似物猜想的示例。在这两项研究中，托马里·沃塔纳布（Tomari-Watanabe）过滤吹传的理论都起着至关重要的作用。 1998年，托马里（Tomari）成功地证明了有关两个正常分级环的segre产物的理性概念和孤立的奇异性的标准。这些标准是对普通分级环的结构来对正常分级环的自然概括。（2）T.Hayakawa通过加权吹式爆破研究了几种三维终端奇异性的部分分辨率。特别是，他成功地展示了与最小差异的一组分区爆破与以“大重量”的最大爆炸量之间的特殊对比。 （3）M.Takamura将基本收缩归类为最小的差异。结合了托马里（Tomari）的先前结果，他获得了p_ <alpha> = 2的完整分类。这项工作的目的是将几个变量的高晶函数的整体代表和对aomoto-kita理论的自然概括进行应用。