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

在我们的研究中，给出了以下结果。主要研究人员Tomari研究了Verones子环为多项式环的正规分次环。作为对二维UF.D.情形的推广，他利用Orlik-Wgreich类型的著名结果得到了这类情形的分类。在Tomari的论文中，利用Okuma最近的工作，证明了复正则二维Gorenstein椭圆奇点的几何特征。在特征为正的情况下，研究了类似的问题，证明了p_g公式和椭圆奇点的技巧是可以应用的。通过比较拓扑界和算术亏格，研究了正规二维奇点几何亏格的最佳下界。在分级情况下，通过使用无效的Pinkham-Demazure因子，星形奇点的界非常接近算术…更有精神的属。研究仍在进行中。(2)Hayakawa将除法收缩归结为具有小偏差的三维Gorenstein末端奇点。在这里，他使用了他以前的作品中介绍的特殊的过滤爆炸。在指数大于1的情况下，他构造了一个终结化，其中例外因子通过不可约爆破的组合出现，具有1/指数的偏差。(3)Watanabe利用F-门限理论刻画了F-有理环，F-门限理论是F-纯门限的推广。他还给出了一些关于F-门限的有趣的重数不等式。这也给出了特征为零的情形的新结果。(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到完全交奇点的等价性。这给出了纽曼-瓦尔猜想的部分答案。他还用几何亏格证明了奇点与它的UAC之间的关系。事实证明，在UAC是拼接型的情况下，关系是拓扑的。(5)Fukuda得到了隐式微分方程解存在的判据。他还刻画了在广义哈密顿系统的情况下可积的奇点，这是数学物理中的重要对象。较少

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