Filtered blowing-up of local rings and algebraic geometric classification of singularities

局部环的滤波放大和奇点的代数几何分类

• 批准号: 16540043
• 负责人: TOMARI Masataka
• 金额: $ 1.92万
• 依托单位: Nihon 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-16540043/
• 关键词: resolution of singularities; filtered blowing-up; elliptic singularity; geometric genus; 3-dimensional terminal singularity; F-threshold; 2-dimensional singularity; universal abelican covering; 2次元特異点; ベロネーゼ部分環; 完全交叉性; 2次元楕円型特異点; 整閉イデアルの対数的特異点

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 studied the normal graded rings whose Veronese subring is a polynomial rings. As an extension of the result in the case of 2-dimensional U.F.D., he obtain the classification of such cases by using famous results of Orlik-Wagreich type. In Tomari's paper, a geometric characterization of complex normal 2-dimensional Gorenstein elliptic singularities was shown by using the recent work of Okuma. Also, in the case of positive characteristic, similar problem was studied and turned out that the techniques in p_g-formula and elliptic singularities can be applied. The lower best bound of the geometric genus of normal 2-dimensional singularities were studied by comparing the topological bound the arithmetic genus. In the graded case, by using non-effective Pinkham-Demazure divisor, the bound for star-shaped singularity is very near the arithm … More etic genus. The studies are still in progress. (2)Hayakawa classified the divisorial contractions to 3-dimensional Gorenstein terminal singularity with small discrepancy. Here he used his special filtered blowing-ups which were introduced in previous his works. In the case of the index with greater than 1, he constructed the terminalization where the exceptional divisor appear with 1/index discrepancy by a composition of irreducible blowing-ups. (3)Watanabe gave a characterization of F-rational ring by using the theory of F-threshold which is a generalization of F-pure threshold. He also gave some interesting inequality of multiplicity in terms of F-threshold. This also induces new results of the case of characteristic zero. (4)Okuma has shown the equisingularity of the universal abelian covers of 2-dimensional rational singularity or minimal elliptic singularity to the complete intersection singularity. This gave a partial answer to the Neuman-Wahl conjecture. He also showed the relation between singularity and its UAC in terms of geometric genus. It turned out that, in the case UAC is of splice type, that the relation is topological. (5)Fukuda obtained the criterion for the existence of solutions for implicit differential systems. He also characterized the singularities which are integrable in the case of the generalized Hamiltonian system, which is important object in the mathematical physics. Less

在我们的研究中，给出了以下结果。其中一些已经出版，其余的将很快出版。（1）托马里研究了Veronese子环是多项式环的正规分次环。作为2维U.F.D.情况下结果的推广，他利用Orlik-Wagreich型的著名结果得到了这类情况的分类。在托马里的论文中，利用Okuma最近的工作给出了复正规二维Gorenstein椭圆奇点的一个几何特征.在正特征的情形下，研究了类似的问题，证明了p_g公式和椭圆奇点中的技巧是可以应用的。通过比较拓扑亏格和算术亏格，研究了二维正规奇点的几何亏格的最佳下界。在分次情形下，通过引入非有效Pinkham-Demazure因子，得到了星型奇异性的上界，其上界非常接近于算术上界 ...更多信息 客位属有关研究仍在进行中。（2）Hayakawa将分裂收缩分类为具有小偏差的三维Gorenstein终端奇点。在这里，他使用他的特殊过滤吹了这是介绍了他以前的作品。在指数大于1的情况下，他通过不可约爆破的合成构造了例外因子出现1/指数差异的终止化。（3）Watanabe利用F-门限理论给出了F-有理环的一个刻划，F-门限理论是F-纯门限理论的推广。他还给出了一些有趣的不等式的多重性方面的F-阈值。这也导出了特征零点情形的新结果。（4）Okuma证明了2维有理奇点或极小椭圆奇点的泛阿贝尔覆盖对完全交奇点的等奇异性。这部分回答了诺伊曼-瓦尔猜想。他还表明了奇异性和UAC之间的关系的几何亏格。事实证明，在UAC是拼接类型的情况下，该关系是拓扑的。（5）Fukuda得到了隐式微分方程组解的存在性判据。他还刻画了广义哈密顿系统中可积的奇点，这是数学物理学中的重要对象。少

项目成果

Gorenstein resolutions of 3-dimensional tereminal singularities

3 维终端奇点的 Gorenstein 分辨率

• 发表时间: 2005
• 期刊: Nagoya Math.J. 178(未定)
• 作者: K.-i.Watanabe;K.-i.Yoshida;T.Hayakawa
• 通讯作者: T.Hayakawa

Numerical Gorenstein elliptic singularities

数值 Gorenstein 椭圆奇点

• 发表时间: 2005
• 期刊: Math.Z. 249
• 作者: Z.;Liu;T.Okuma
• 通讯作者: T.Okuma

A geometric characterization of normal two-dimensional Gorenstein singularities with $p_a=1$

$p_a=1$ 的正常二维 Gorenstein 奇点的几何特征

• 发表时间: 2005
• 期刊: Proc.The Inst.of Natural Scie.Nihon Univ. vol.40
• 作者: Nakashima;Toshiki;T.Nakashima;T.Nakashima;中島 俊樹;諏訪紀幸;Noriyuki SUWA;諏訪紀幸;Fumiyuki Momose;Tomohiro OKUMA;T.Okuma;Takayuki HAYAKAWA;M.Tomari
• 通讯作者: M.Tomari

Singularities of implicit differential systems and their integrability

隐式微分系统的奇异性及其可积性

• 发表时间: 2004
• 期刊: Banach Center Publications vol.65
• 作者: Fukuda;S.Janeczko
• 通讯作者: S.Janeczko

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