2006 Fiscal Year Final Research Report Summary
Study on integral convex polytopes and toric varieties
| Project/Area Number |
16540004
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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 |
OGATA Shoetsu Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (90177113)
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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)
HARA Nobuo Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (90298167)
KAJIWARA Takeshi Yokohama National University, Faculty of Enginieering, Associate Professor, 大学院工学研究院, 助教授 (00250663)
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| Project Period (FY) |
2004 – 2006
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| Keywords | convex polytope / toric variety / algebraic geometry |
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| Research Abstract |
Ogata studied about ideals defining projective toric varieties. We say the ideal satisfies the property (Np) if its free resolution is the simplest up to the degree p part. We obtained a best possible estimate of this p with respect to the dimension of the variety and the times of tensor product of the same ample line bundle. We also obtained an algebro-geometric proof of the Theorem of Fakhruddin, which states that any global section of Ample bundle multiplied by a nef bundle is a nultiple of sections of the ample bundle and the nef bundle. Next we proved that an ample line bundle on a nonsingular toric 3-fold is normally generated if the bundle after added with the twice of the canonical bundle has no global sections. As a consequence, we showed that the anti-canonical bundle on nonsingular toric Fano 4-fold is normally generated.
Ishida studied about completion of real fans by using the notion of Zariski-Riemann spaces. We also determined the moduli number of CCI surfaces which are defined by Hirotaka Ishida.
Hara studied about isolated singularities of dimension two in positive characteristic. We define the F-pure threshold ftp(X, D) for a pair of a surface and an effective divisor which is an analog of the logarithmic canonical threshold. And we showed that ftp(X, D) has values in rational numbers by using the method of p-fractal.
Kajiwara studied abuot the relation between tropical geometry and toric geometry. We characterized degenerations of projective toric varieties defined from polyhedral decompositions which are determined naturally by tropical hypersurfaces.We also reconstructed tropical toric varieties in terms of algebraic geometry. As a consequence, we obtained
We also reconstructed tropical toric varieties in terms of algebraic geometry. As a consequence, we obtained an intersection theory on tropical nonsingular toric surfaces.
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