Project/Area Number  10640042 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  NIHON UNIVERSITY 
Principal Investigator 
WATANABE Keiichi Nihon Univ., College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)

CoInvestigator(Kenkyūbuntansha) 
MOTEGI Kimihiko Nihon Univ., College of Humanities and Sciences, Associate Professor, 文理学部, 助教授 (40219978)
黒田 耕嗣 日本大学, 文理学部, 教授 (50153416)
SUZUKI Masahiko Nihon Univ., College of Humanities and Sciences, Professor, 文理学部, 教授 (00171249)
MATSUURA Yutaka Nihon Univ., College of Humanities and Sciences, Associate Professor, 文理学部, 助教授 (50096905)
MORI Makoto Nihon Univ., College of Humanities and Sciences, Professor, 文理学部, 教授 (60092532)
鈴木 理 日本大学, 文理学部, 教授 (10096844)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥3,400,000 (Direct Cost : ¥3,400,000)
Fiscal Year 2000 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1999 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1998 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  Frobenius map / rational singularities / integrall closed ideals / HilbertKunz multiplicity / Fregular ring / log terminal Singularity / Frobenius写像 / rational singularity / 整閉イデアル / logterminal singularity / log terminal singularity / integrally closed ideals / Mac Kay correspondence / multiplicity / good ideals / integral closure / tight closure (of ideals) / Veronese sub rings / HilbertKung multiplicity 
Research Abstract 
We applied "characteristic p" methods to singularity theory and obtained the fokkowing results. 1 Characterization of Singularities in Characteristic 0 via Frobenius endomorphism. We found that logterminal singularity and Fregular rings are equivalent notions in the case the ring is Q Gorenstein. The same is true for rational singularities and Frational rings. Also, we found that the same is true for "singularity of pairs" (KLT, PLT etc.). The latter result is a joint work with N.Hara. 2 The notion of "HilbertKunz multiplicity", which is a new "multiplicity" for local rings. We characterized regular local rings, certain rational singularities in dimension 2 by this multiplicity. Also we found a very beautiful and mysterious formula for integrally closed ideals in 2dimensional rational double points. (a joint work with K.Yoshida) 3 We investigated chains of integrally closed ideals and found the existence of a composition series only by integrally closed ideals. We found that the family of integrally closed ideals of colength 1 corresponds to the closed points of the fiber cone. Also, we found a new characterization of simple integrally closed ideals in 2dimensional regular local rings. (The last result is a joint work with S.Noh.) ・整閉イデアルの族に関し、整閉イデアルのみによる組成列の存在、colength1の整閉イデアルの族が"fiber cone"と1対1対応が付くなどの大変興味深い事実を発見した,また,これにより,2次元正則局所環での単純イデアルの新しい特徴付けを発見した(S.Noh氏との共同の結果).
