Semi-orthogonal decomposition of derived categories of rationally connected varieties and vector bundles

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04810
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Electro-Communications
• Principal Investigator: Ohno Masahiro 電気通信大学, 大学院情報理工学研究科, 准教授 (70277820)
• Co-Investigator(Kenkyū-buntansha): 寺川 宏之 都留文科大学, 教養学部, 教授 (80277863) ; 山口 耕平 電気通信大学, 大学院情報理工学研究科, 教授 (00175655)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: ベクトル束 / ネフ / 大域生成

Outline of Final Research Achievements

We work over an algebraically closed field of characteristic zero. First we classified nef vector bundles on a smooth projective quadric surface with first Chern class (2,1). They consist of 5 types of vector bundles and they are all globally generated. Second we classified nef vector bundles on a projective space with first Chern class three. Note here that in this case the second Chern class is non-negative and less than or equal to nine. We showed in particular that if the second Chern class is less than or equal to seven, then they are all globally generated, and that if the second Chern class is eight, then they are not globally generated and they exist only on a projective plane.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

ベクトル束の性質に，大域生成と呼ばれる幾何学的な性質と，ネフ(nef, 数値的半正)と呼ばれる数値的な性質がある．一般に，大域生成ならばネフだが，この逆は成り立たない．一方，不変量等の数値的な条件から幾何学的状況をどれだけ回復できるか？という観点から，どのような条件のもとでネフならば大域生成がわかるか？は興味の持たれるところである．本研究の研究成果は特にこの観点からの学術的意義があると考えている．