Study on Durfee-type inequalities of complete intersection singularities

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K23407
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Tokyo University of Science
• Principal Investigator: Enokizono Makoto 東京理科大学, 理工学部数学科, 助教 (30843461)
• Project Period (FY): 2019-08-30 – 2023-03-31
• Keywords: 特異点 / ファイバー多様体 / 代数曲面 / 消滅定理

Outline of Final Research Achievements

In this study, we aimed to derive inequalities and equations between the Milnor number, the geometric genus, the embedding dimension, and the self-intersection numbers of canonical cycles for isolated complete intersection singularities. By embedding the resolution space of the singularities into certain types of fibered varieties, we reduced the problem to numerical invariants of fibered varieties. We first examined this problem in the case where the fiber is two dimensional, and also addressed the problem of factorization of divisors on surfaces and derived the cohomology vanishing theorem for surfaces in positive characteristic.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究は特異点の問題とファイバー多様体の問題を結びつけるものである。本研究の意義の一つは、代数多様体やファイバー多様体の数値的不変量の地誌学的研究を特異点の研究に応用できることである。また、本研究で行った代数曲面の因子の分解やコホモロジーの消滅定理に関する研究は、平面曲線の有理点の特徴付けなど整数論的な応用もあり、応用数学など他の分野への意外な応用なども将来的に期待される。