2003 Fiscal Year Final Research Report Summary
Algebro-Geometric Method in Commutative Algebra
| Project/Area Number |
13640005
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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) |
YOSHIDA Ken-ichi Nagoya University, Graduate School of Mathematics, Assistant, 大学院・多元数理科学研究科, 助手 (80240802)
WATANABE Kei-ichi Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Tight closure / F-singularity / test ideal / multiplier ideal / positive characteristic / commutative algebra / algebraic geometry / Rees algebra |
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| Research Abstract |
We reinterpreted the theory of tight closure in prime characteristic commutative algebra from the viewpoint of singularity theory and birational geometry. Namely, we generalized the concepts of tight closure and F-singularities, gave foundation to the theory thereof, and applied it to problems in commutative algebra and al-gebraic geometry. Our results are summarized as follows:
1.Study of F-singularities of Rees algebras : There have been few researches on Rees algebras from a geometric viewpoint, although a Rees algebra is a geometric object in the sense that its "Proj" gives a blow-up. Taking this into account, we studied ring-theoretical and geometric aspects of Rees algebras R(1) associated to an in-primary ideal I of a normal local ring (R,m) in terms of miscellaneous methods such as F-singularities, blow-up and desingularization.
2.A generalization of tight closure : We generalized the notion of the tight closure of an ideal in a ring R of characteristic p to those of "D-tight clo
… More
sure" associated to an effective Q-divisor D on Spec R and of "I-tight closure" associated to an ideal I of R. We established foundation of the theory of I-tight closure and the ideal r(I) defined via I-tight closure, and proved various properties of the ideal -r(I) such as Skoda's theorem, restriction theorem and subadditivity.
3.Applications of I-tight closure : We considered the global generation of adjoint bundles K_X+nL of a polarized variety (X, L), as an application of a variant of Skoda's theorem in the canonical module of the section ring of (X,L). In particular, we obtained an alternative proof of K.E.Smith's result on a special case of Fujita's conjecture in characteristic p, assuming that Litself is spanned.
We also constructed a characteristic p analog T(‖I.‖) of the asymptotic multiplier ideal associated to a filtration of ideals I.={I_n|n= 1,2,...}. As an application, we reinterpret the result on the uniform behavior of symbolic powers due to Ein-Lazarsfeld-Smith and Hochster-Huneke. Less
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