Brill-Noeter theory for semi stable bundles on curves which are contained in a K3 surface and around the fields

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05101
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nihon University
• Principal Investigator: WATANABE Kenta 日本大学, 理工学部, 助教 (70582683)
• Research Collaborator: KOMEDA Jiryo
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 安定 ACM 束 / Lazarsfeld-Mukai 束 / Mercat 予想 / Weierstrass 半群

Outline of Final Research Achievements

In our research, the author obtained several results on the stability and the splitting of Lazarsfeld-Mukai bundles of rank two associated with smooth curves on K3 surfaces and base point free pencils on them in the point of view of the classification of indecomposable ACM bundles on polarized K3 surfaces. On the other hand, the author have studied linear systems on curves on a K3 surface which is given by a double covering of a Hirzebruch surface to construct a certain semistable bundles of rank two on curves which contribute to the second Clifford indices of them.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

代数曲面上の与えられた偏極に関するベクトル束の安定性や分解問題に付随した偏極代数曲面の表現型の決定に関する研究は環論・代数幾何学における興味深い話題である。ところが、K3 曲面をはじめとする多くの対象に対しそれらの問題は難しく、解決されていない部分が多い。しかしながら、偏極 K3 曲面上の階数 2 ACM 束は大域切断で生成されていれば Lazarsfeld-Mukai 束である為、本研究では問題をそのようなベクトル束に帰着させることで新しい着眼点を得ることができた。