2004 Fiscal Year Final Research Report Summary
Free resolutions of the coordinate rings of projective varieties
| Project/Area Number |
14540036
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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 | University of the Ryukyus |
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Principal Investigator |
MIYAZAKI Chikashi University of the Ryukyus, Faculty of Science, Associate Professor, 理学部, 助教授 (90229831)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Takashi University of the Ryukyus, Faculty of Science, Professor, 理学部, 教授 (30229306)
AMASAKI Mutsumi Hiroshima University, Graduate School of Education, Associate Professor, 大学院・教育学研究科, 助教授 (10243536)
NOMA Atsushi Yokohama National University, Faculty of Education and Human Sciences, Associate Professor, 教育人間科学部, 助教授 (90262401)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Castelnuovo-Mumford regularity / syzygy / free resolution / projective curve / rational normal scroll |
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| Research Abstract |
My research has been devoted to the study of minimal free resolutions for the coordinate rings of projective varieties. In particular, the Castelnuovo-Mumford regularity is the main theme of this research project, and I have studied the upper bounds of the Castelnuovo-Mumford regularity in terms of the other invariants of give varieties. The Castelnuovo-Mumford regularity is the maximal integer m such that the ideal sheaf of the varietiy is m-regular in Murnford sense, and it regulates the degree of the defining equations of the variety and the syzygies of the defining ideal of the variety. Le Tuan Hoa and I had given a conjecture on an upper bound of the regularity by using so-called the Castelnuovo bound. Also we had conjectured the extremal case happens only in the case of a divisor on a rational normal scroll for its degree large enough. For the curve case, I have obtained an extremal bound by showing that the generic hyperplane section of an extremal curve is contained in a rational normal curve. The classical Caltelnuovo method plays an important role for the proof in positive characteristic case. Moreover, I have classified all the extremal varieties for the bound among the divisors on rational normal scrolls. These facts should guess the conjecture. I would like to extend them to the corresponding results for weighted projective space.
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