| Project/Area Number |
10640042
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | NIHON UNIVERSITY |
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Principal Investigator |
WATANABE Keiichi Nihon Univ., College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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)
鈴木 理 日本大学, 文理学部, 教授 (10096844)
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| Project Period (FY) |
1998 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| 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)
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| Keywords | Frobenius map / rational singularities / integrall closed ideals / Hilbert-Kunz multiplicity / F-regular ring / log terminal Singularity / log terminal singularity / integrally closed ideals / Mac Kay correspondence / multiplicity / good ideals / integral closure / tight closure (of ideals) / Veronese sub rings / Hilbert-Kung multiplicity |
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| 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 log-terminal singularity and F-regular rings are equivalent notions in the case the ring is Q- Gorenstein. The same is true for rational singularities and F-rational 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 "Hilbert-Kunz 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 2-dimensional 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 2-dimensional regular local rings. (The last result is a joint work with S.Noh.)
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