2005 Fiscal Year Final Research Report Summary
Applications of tight closure and F-singularity to algebraic geometry
| Project/Area Number |
16540005
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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 | Tohoku University |
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Principal Investigator |
HARA Nobuo Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90298167)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIDA Masanori Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (30124548)
KAJIWARA Takeshi Tohoku University, Graduate School of Science, Research Associate, 大学院・理学研究科, 助手 (00250663)
WATANABE Kei-ichi Nihon Univ., College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)
YOSHIDA Ken-ichi Nagoya Univ., Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (80240802)
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| Project Period (FY) |
2004 – 2005
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| Keywords | tight closure / F-singularity / algebraic geometry / F-pure threshold / toric variety / tropical geometry / multiplier ideal / multiplicity |
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| Research Abstract |
Given a pair of a variety of characteristic p and an effective divisor on it, one can associate a real number called the F-pure threshold. Since this invariant is defined as a characteristic p analog of the log canonical threshold in characteristic 0, it is desirable that F-pure thresholds are rational numbers similarly as log canonical thresholds. N.Hara studied F-pure thresholds of pairs of a nonsingular surface and an effective divisor, and proved based on Monsky's idea of p-fractals that the F-pure thresholds are rational provided that the base field is finite. When the divisor is defined by a homogeneous polynomial f (x, y), the F-pure threshold c(f) can be estimated more precisely, and we can obtain a finite list of possible value of c(f) for a fixed degree d=deg f and characteristic p. We also proved that the Monsky's function ψ_f(t) has a piecewise quadratic limit as p→∞.
M.Ishida studied real fans from a viewpoint of toric geometry, as well as moduli parameter of Catanese-Ciliberto-Ishida surface. T.Kajiwara studied the theory of logarithmic abelian varieties, the relationship of tropical hypersurfaces and degeneration of projective toric varieties, and the theory of tropical toric varieties. K.-i.Watanabe studied geometric interpretation of integrally closed monomial ideals in 3 variables, multiplier ideals, and F-thresholds. K.Yoshida gave estimates of multiplicities of Stanley-Reisner rings and Buchsbaum homogeneous algebras, and studied the structure of these rings when they have minimal multiplicities.
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