STUDY ON SINGULARITIES OF VARIETIES

品种奇异性研究

基本信息

批准号：
11640015
负责人：
KUROKAWA Nobushige
金额：
$ 2.3万
依托单位：
TOKYO INSTITUTE OF TECHNOLOGY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640015/
关键词：
Zeta function log-canonical singularity index of singularity exceptional singularity modular L-function Singularities toric variety index Zariski decomposition birational geometry

项目摘要

We gave an estimation of the multiplicity of the principal series. We obtained basic properties of the spectra of categories and studied examples.It was proved by Chen-Ishii that the set of the values of -K^2 for normal surface singularities has no accumulation points from above and has many accumulation points from below. We checked the closedness of this set in the real number field. The closedness is equivalent to the fact that every accumulation point is a value of -K^2 of a singular point. We proved that every accumulation point is the sum of finite number of the value of -K^2.And non-closedness of the set was proved.We proved the boundedness of the indices of isolated strictly log-canonical singularities of dimension 3 and also obtained all possible values of the indices.We constructed counter examples of Reid's conjecture : hypersurface rational singularities are characterized by weights. As one of the consequences, we obtained simple K3 singularities which is not in the category of simple K3 singularities of famous 95 types.We proved that for every hypersurface canonical singularity defined by a non-degenerate function there is a weight such that the singularity defined by the leading tern with respect to this weight is exceptional if and only if the original singularity is exceptional. So we can reduce the problem of exceptionality of the singularity into the weighted homogeneous case. We also proved that the number of the weights whose singularities are exceptional is finite.We studied the order of zeros at the center of equations of modular L-functions. In particular we studied Rankin zta function corresponding to the pair of elliptic modular forms.

我们估计了主系列的多样性。我们获得了类别光谱的基本特性和研究的示例。陈式证明，正常表面奇点的-k^2值集的集合没有上面的积累点，并且从下方有许多积累点。我们在实际数字字段中检查了该集合的闭合度。闭合度等同于以下事实：每个积累点都是单个点的-k^2值。我们证明，每个积累点都是-k^2的有限数量的总和。作为后果之一，我们获得了简单的K3奇异性，而不是著名的95种类型的简单K3奇异性类别。我们证明，对于每个由非分类功能定义的高度表面典型的奇异性，都有一个重量，使得由领先的奇异性与仅在原始singuniqu的范围中所定义的奇异之处。因此，我们可以将奇异性异常的问题减少到加权均匀的情况下。我们还证明，奇异性的权重数量是有限的。我们研究了模块化L功能方程中心的零命令。特别是我们研究了与一对椭圆形模块形式相对应的Rankin ZTA函数。