Filtered blowing-up of singularities and algebraic geometric properties of tangent cone

奇点的过滤吹胀和正切锥体的代数几何性质

• 批准号: 14540017
• 负责人: TOMARI Masataka
• 金额: $ 2.3万
• 依托单位: Nihon University (2003)Kanazawa University (2002)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540017/
• 关键词: resolution of singularities; filtered blowing-up; plurigenus; 3-dimensional terminal singularities; Gluck surgery; algebraic stacks; order function of local rings; associated graded rings; フィルター付ブロウイングアップ; フリップの存存定理; 双正則同形問題; 4次元多様体; Dehn手術

On our research, the following results are given. Some of these were already-published, and the rest will be published soon. (1)The head investigator Tomari showed; (a) By the theory of multiplicy of filtered rings, a new proof of characterization of 2-dim, regular local ring was given. (b) There are new progress for the study of lower bound of L^2 plurigenus from the point of views concerning the tangent cone of the filtration. A characterization of the equality in the above had done. (c) Seeking for a future progress to extend the results of (b), he introduced L^2 plurigenus for 1-dim singularities. Using a technique of filtered blowing-up, he gave a classification of 1-dim, singularities by the asymptotic behavior of this L^2 genus as in the same way as higher dim, cases. This is an evidence to extend the studies of (b) in more general situations. (d) Also he made a progress concerning linear complementary inequality about the order functions on normal 2-dim, singularities of multip … More licity two. (e) Combining recent results by T. Okuma, Tomari gave a geometric characterization of normal 2-dim. Gorenstein elliptic singularities, which was conjectured about 20 years ago by Tomari. Through these studies, all the investigators of this project contributed in several forms. Among others, Hayakawa and Iwase had been contacting to Tomari and giving many contributions in these periods. Further, as related subjects of the project, there are several individual results as follows; (2) In the minimal model theory, Hayakawa classified all the 3-dim. extremal contractions to one point, where the discrepancy is less than one. (3) Iwase continued the studies of a kind of Gluck surgery of 4-dim, manifold, and showed the all cases become the Dehn surgery along torus as same as 3-dim, studies. (4) Matsuura gave a criterion of affine properties of algebraic stacks ; which has been interested by algebraic geometers in many years. (5) Watanabe made several essential progress about the relations between integral closed ideals and multiplier ideals. Less

研究结果如下：其中一些已经出版，其余的也将很快出版。(1)首席调查员托马里表明；(a)利用滤波环的乘法理论，给出了2-dim正则局部环的表征的一个新的证明。(b)从滤波切线锥的角度对L^2 plurigenus下界的研究有了新的进展。前面已经对等式进行了描述。(c)为了进一步扩展(b)的结果，他引入了1暗淡奇点的L^2 plurigenus。利用一种过滤放大技术，他根据L^2属的渐近行为，给出了1-dim奇点的分类，与高dim情况的分类方法相同。这是在更一般的情况下扩展(b)研究的证据。(d)关于正规2-dim上的阶函数的线性互补不等式，复数的奇异性，也取得了一些进展。(e)结合T. Okuma最近的结果，Tomari给出了正规2-dim的几何表征。戈伦斯坦椭圆奇点，这是托马里在20年前推测出来的。通过这些研究，该项目的所有研究者都以多种形式做出了贡献。在这些时期，早川和岩濑一直与托玛莉保持联系，并做出了许多贡献。此外，作为项目的相关主题，有以下几个单独的结果；(2)在最小模型理论中，Hayakawa对所有3-dim进行了分类。极端收缩到一点，差值小于1。(3)我继续了一种4-dim， manifold的Gluck手术的研究，并表明所有病例都成为沿环面Dehn手术与3-dim，研究相同。(4) Matsuura给出了代数叠的仿射性质的判据；代数几何学者多年来一直对此感兴趣。(5)渡边在积分封闭理想与乘数理想的关系上取得了若干重要进展。少

项目成果

M.TOMARI: "Multiplicity of filtered rings and simple K3 singulairties of multiplicity two"Publ.Res.Inst.Math.Sci.(Kyoto Univ.). vol.38-4. 693-727 (2002)

M.TOMARI：“滤波环的重数和重数二的简单 K3 奇点”Publ.Res.Inst.Math.Sci.（京都大学）。

T.HAYAKAWA: "Flips in dimension three via crepant descent method"Proceedings of the Japan Acad.Ser.A. vol.79-2. 46-51 (2003)

T.HAYAKAWA：“通过绉纱下降法进行第三维翻转”日本学术院学报.Ser.A.

Takayuki HAYAKAWA: "Flips in dimension three via crepant descent method"Proceedings of the Japan Acad.Ser.A. 79-2. 46-51 (2003)

Takayuki HAYAKAWA：“通过绉纱下降法在第三维度中翻转”日本 Acad.Ser.A 论文集。

Masako FURUYA, Masataka TOMARI: "A characterization of semi-quasihomogeneous function in terms of the Milnor number"Proc.Of Amer.Math.Soc.. (冊子体は印刷中)(電子雑誌としては、2004年一月に発行済み). (2004)

Masako FURUYA、Masataka TOMARI：“A Character of semi-quasihomous function in terms of the Milnor number”Proc.Of Amer.Math.Soc..（目前正在印刷的小册子）（2004 年 1 月出版的电子杂志）（2004 年）

Masako FURUYA, Masataka TOMARI: "A characterization of semi-quasihomogeneous function in terms of the Milnor number"Proc.Amer.Math.Soc.. (to appear).

Masako FURUYA、Masataka TOMARI：“用 Milnor 数描述半拟齐次函数”Proc.Amer.Math.Soc..（待发表）。