Project/Area Number  01460002 
Research Category 
GrantinAid for Scientific Research (B).

Research Field 
代数学・幾何学

Research Institution  University of Tokyo 
Principal Investigator 
KAWAMATA Yujiro University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (90126037)

CoInvestigator(Kenkyūbuntansha) 
IHARA Yasutaka Kyoto University, Research Institute for Mathematical Sciences, 数理解析研究所, 教授 (70011484)
NAKAMURA Hiroaki University of Tokyo, Faculty of Science, Assistant, 理学部, 助手 (60217883)
織田 孝幸 東京大学, 理学部, 助教授 (10109415)
MATSUMOTO Yukio University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (20011637)
服部 晶夫 東京大学, 理学部, 教授 (80011469)
TERADA Itaru University of Tokyo, Faculty of Science, Assistant, 理学部, 助手 (70180081)
TSUBOI Takashi University of Tokyo, Faculty of Science, Associate Professor, 理学部, 助教授 (40114566)

Project Fiscal Year 
1989 – 1990

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥6,900,000 (Direct Cost : ¥6,900,000)
Fiscal Year 1990 : ¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 1989 : ¥3,800,000 (Direct Cost : ¥3,800,000)

Keywords  GaIois group / Monodromy group / Index / ガロア群 / モノドロミ群 / インデックス / 基本群 / 表現 / 代数曲線 / リマン面 / 代表曲面 / 組み糸群 / モノドロミ表現 
Research Abstract 
In the case of dimension 3, a minimal model has singular points and each singular point has the the invariant called the index. In spite of disappearance in a nonsingular model, it is known that the whole of indices = the basket is a birational invariant and also a deformation invariant. By the way, in the calssification of 3dimensional varieties, the boundedness of indices for some classes of varieties is needed. I have shown before that the indices are bounded when the canonical divisor k is numerically equivalent to zero. As a continuation of this, I have shown that both indices and K^3 are bounded when K is negative in the paper "Boundedness of QFano threefolds". By the totally different method, I have also shown the boundesness of indices for degenerations of elliptic surfaces in the paper "Moderate degenerations of algebraic surfaces". Moreover, in the same paper, I have shown that the topology of minimal models of degenerations of surfaces are very similar to that of semistable degenerations. If X is an algebraic variety defined over an algebraic number field k, then the fundamental group pi, (X) can be considered as a group extension of the absolute Galois group Gal (k/k). Nakamura studied conditions under which X can be characterized by the group theoretical properties of pi, (X), and obtained several results. Firstly, he showed that when X is a certain hyperbolic curve, pi, (X) as a group extension of Gal (k/k) determines X uniquely up to isomorphisms. He also showed that, when X is P'minus three points, every automorphism of pi, (X) as a group extension must be induced from an automorphim of X itself. Matsumoto studied topological classification of singular fiders in degenerating families of Riemann surfaces ; he proved that topological types of singular fivers are determined by nonadelian monodromy and conversely that, for any monodromy of algebraically finite type, there exists a degenerating family of Riemann surfaces which realizes the monodromy.
