| Project/Area Number |
16K05092
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tokyo University of Agriculture and Technology |
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Principal Investigator |
Hara Nobuo 東京農工大学, 工学(系)研究科(研究院), 教授 (90298167)
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| Project Period (FY) |
2016-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| Keywords | 正標数 / フロベニウス直像 / 有限F表現型（FFRT） / 2次元正規次数環 / ファノ多様体 / 大域的F正則 / デルペッツォ曲面 / 代数幾何 / 有限F-表現型（FFRT） / del Pezzo 曲面 / フロベニウス写像 / 反標準環 / del Pezzo曲面 / 有限F表現型(FFRT) / Frobenius直像 / 5次del Pezzo曲面 / ベクトル束 / F-符号数 / 有限F表現型 / 2次元特異点 / 代数的スタック / 特異点 |
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| Outline of Final Research Achievements |
We studied the structure of iterated Frobenius direct images on algebraic varieties and their singularities in positive characteristic, from the viewpoint of the finite F-representation type (FFRT). Our results are as follows.
1. (joint work with Ryo Ohkawa) We studied the FFRT property of 2-dimensional normal graded rings (quasi-homogeneous singularities) in positive characteristic p using the method of algebraic stacks. We proved that a 2-dimensional normal graded ring has FFRT if it has a log terminal singularity; but it does not have FFRT otherwise, except for some exceptional cases depending on p.
2. We studied the FFRT property of the anti-canonical ring of a quintic del Pezzo surface X in positive characteristic. We constructed a self-dual indecomposable vector bundle of rank 3 that appears as a direct summand of self-dual Frobenius direct images on X. We have also shown that the anti-canonical ring has FFRT in characteristics 2 and 3.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究は，正標数p，すなわち素数pについて，任意の数をp回足すと0になってしまう世界で，多項式系の零点集合として定義される図形（代数多様体）の大域的および局所的な性質を研究するものです．正標数の数学は暗号符号などへの応用もありますが，本研究はこれらの応用と直接的には関係せず，正標数特有の時として奇妙にも映る現象の中に，純粋数学的な意義と美しさを見出して，これを探求するものです．
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