The research of 2-dimensional complex singularities associated to degenerations of closed Riemann surfaces

闭黎曼曲面退化相关的二维复奇点研究

• 批准号: 16540052
• 负责人: TOMARU Tadashi
• 金额: $ 0.77万
• 依托单位: Gunma University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540052/
• 关键词: singularity; degeneration of Riemann surfaces; pencil genus; singularity with C^*-action; degeneration with C^*-action; rational triple point; Kodaira singularity; 閉Riemann面; 閉Riemann面の退化族; 複素乗法群; 局所モノドロミー群; Milnorモノドロミー群

In this research, we investigated the following and obtained following results.(1) We have been investigated the structure of degenerations of closed Riemann surfaces with C^*-action. Four years ago, Tomaru proved that there is a very natural construction of degenerations of closed Riemann surfaces from complex surface singularities and holomorphic functions on the singularities. We prove similar result for normal surface singularities with C^*-action.(2) Let (X,o) be a normal surface singularity obtained by the contraction of the zero-section of a line bundle on a curve. We gave a necessary and sufficient condition for (X,o) to be Kodaira (or Kulikov) singularity. Using this, we gave an example which is a Kodaira singularity but not a Kulikov singularity.(3) We determined the value of pencil genus of rational triple points by using Artin's classification of rational triple points and Kodaira's classification of elliptic degenerations.(4) We prove some results on some relation between quasi-rational singularities and cyclic coverings.

在本研究中，我们进行了以下研究，并获得了以下结果。(1)研究了具有C^* 作用的闭Riemann曲面退化的结构。四年前，Tomaru证明了有一个非常自然的结构退化的封闭黎曼曲面从复杂的表面奇点和全纯函数的奇点。对于具有C^*-作用的法向曲面奇点，我们证明了类似的结果。(2)设（X，o）是由线丛的零截面在曲线上的收缩而得到的法向曲面奇点。给出了（X，o）为科代拉（或库利科夫）奇点的一个充要条件.利用这一点，我们给出了一个例子，这是一个科代拉奇点，但不是一个库利科夫奇点。(3)利用Artin的有理三重点分类和科代拉的椭圆退化分类，确定了有理三重点的束亏格的值. (4)证明了拟有理奇点与循环覆盖之间的一些关系。

项目成果

Universal abelian covers of certain surface singularities

• DOI: 10.1007/s00208-005-0693-8
• 发表时间: 2005-03
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Tomohiro Okuma

Pencil genus for normal surface singularities

法线表面奇点的铅笔亏格

• 发表时间: 2007
• 期刊: J.Math.Soc.Japan 59
• 作者: 仲村龍介;松林玄;土谷博昭;藤本慎司;中嶋英雄;T.Tomaru
• 通讯作者: T.Tomaru

On ($-Pcdot P$)-constant deformations of Gorenstein surface singularities

关于 Gorenstein 表面奇点的 ($-Pcdot P$) 恒定变形

• 期刊: Commentarli Mathematici Helvetici (印刷中)
• 作者: S.Goto;W.Heinzer;Mee-Kyoung Kim;K.Kurano;都丸 正;奥間智弘
• 通讯作者: 奥間智弘

On some classification of weakly Kodaira singularities

关于弱小平奇点的某种分类

• 发表时间: 2005
• 期刊: Seminaires & Congres 10
• 作者: S.Goto;S.lai;T.Tomaru
• 通讯作者: T.Tomaru

On some classes of weakly Kodaira singularities

关于某些类的弱小平奇点

• 发表时间: 2005
• 期刊: Seminaires & Congres 10
• 作者: Fukuda;S.Janeczko;都丸 正
• 通讯作者: 都丸 正