2001 Fiscal Year Final Research Report Summary
CLASSIFICATION OF WEAK FANO MANIFOLDS WHICH ARE DIVISORS IN THE WEIGHTED PROJECTIVE SPACE BUNDLES OVER THE PROJECTIVE LINE
| Project/Area Number |
12640048
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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 | Gifu Shotoku Gakuen University |
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Principal Investigator |
TAKEUCHI Kiyohiko GIFU SHOTOKU GAKUEN UNIVERSITY, FACULTY OF ECONOMICS AND INFORMATION, ASSOCIATED PROFESSOR, 経済情報学部, 助教授 (30236418)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Fano variety / algebraic variety / the classification theory / anti-canonical model / projective space bundle |
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| Research Abstract |
The research for several years about weak Fano threefolds tells us the following classification (I am in preparation for the paper titled "Weak Fano threefolds with del Pezzo fibrations"):
Let V be a smooth weak Fano threefold whose anti-canonical model has only terminal singularities. Assume that V has an extremal ray of type D (the type having del Pezzo fibration) of degree d. Denote the number of deformation classes for degree d by N(d). Then, we have N(1)=2, N(2)=4, N(3)=7, N(4)=11, N(5)=11, N(8)=9, and N(9)=3, and can determine the structure of the general weak Fano threefold belonging to each deformation class.
The problem for the case d=6 is open. The weak Fano threefold for the case d=2 can be regarded as a divisor in weighted projective space bundles. This point of view leads to the same classification as the previous one. See the paper "Weak Fano threefolds with del Pezzo fibration of degree two," Economics and Information Studies (working paper series in our faculty), 2001.
This research studied the smooth weak Fano fourfold which is a divisors in weighted projective space bundles as the case of threefolds above. We obtained several examples in weak Fano fourfolds of this type. Although the result is not yet sufficient from the viewpoint of classification theory, a classification was obtained for the case that the fourfold is a divisor in projective space bundles. See the paper "Weak Fano fourfolds in the projective space bundle over the projective line", Economics and Information Studies, 2002.
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