2006 Fiscal Year Final Research Report Summary
Explicit studies on singularities which appear on 3-dimensional algebraic varieties
| Project/Area Number |
17540019
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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 | Kanazawa University |
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Principal Investigator |
HAYAKAWA Takayuki Graduate School of Natural Science & Technology, Lecturer, 自然科学研究科, 講師 (20198823)
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| Co-Investigator(Kenkyū-buntansha) |
SUGANO Takashi Graduate School of Natural Science & Technology, Professor, 自然科学研究科, 教授 (30183841)
MORISHITA Masanori Kyushu University, Faculty of Mathematics, Professor, 数理学研究院, 教授 (40242515)
IWASE Zjunici Graduate School of Natural Science & Technology, Assistant, 自然科学研究科, 助手 (70183746)
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| Project Period (FY) |
2005 – 2006
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| Keywords | terminal singularity / divisorial contraction / resolution of singularity |
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| Research Abstract |
We studied on resolutions of 3-dimensional terminal singularities and 3-dimensional divisorial contractions and get the results below.
(1) We construct birational morphisms from 3-folds with only terminal singularities with index 1 to 3-dimensional terminal singularities with index m greater than one, by repeating divisorial contractions with minimal discrepancy. As a result we determined all prime divisors over 3-dimensional terminal singularities with index greater than or equal to 2 which have discrepancy not greater than 1.
(2) We showed that there does not exist economic resolutions due to M.Reid in its original form, which states that if there are partial resolutions of 3-dimensional terminal singularities with all prime divisors with discrepancy not greater than one as its exceptional divisors. We also showed that there are such economic resolutions after changing conditions on singularities.
(3) We gave an explicit description of divisorial contractions which contract divisors to points with discrepancy one, for 3-dimensional terminal singularities with index greater than one. Such divisorial contractions are obtained by embedding singularities into four or five dimensional spaces and by making suitable weighted blowing ups.
(4) We gave an explicit description of divisorial contractions of Gorenstein terminal singularities of type cD and cE with discrepancy one.
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