Toward the fundamental inequality for normal surface singularities

• Project/Area Number: 23654008
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: KONNO Kazuhiro 大阪大学, 理学(系)研究科(研究院), 教授 (10186869)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 特異点 / 標準サイクル / 種数

Research Abstract

We considered normal surface singularities in order to obtain an inequality between the self-intersection number of the canonical cycle and the geometric genus. Our inequality is modeled on the slope inequality for fibred algebraic surfaces, and thus the ratio shuld be expressed as a function in the fundamental genus. We could establish the desired inequalities for double and triple points of cyclic type, by using the canonical resolution. They lead us to another proof for Durfee's conjecture in such special cases.

Research Products

• [Journal Article] On the Yau cycle of a normal surface singularity (2012)
  • Author(s): Kazuhiro Konno
  • Journal Title: Asian J. of Math Volume: 16 Pages: 279-298
  • Peer Reviewed
• [Journal Article] Maximal ideal cycles over normal surface singularities of Brieskorn type (2012)
  • Author(s): Kazuhiro Konno and Daisuke Nagashima
  • Journal Title: Osaka J. Math Volume: 49 Pages: 225-245
  • NAID: 120004838543
  • Peer Reviewed
• [Journal Article] On the Yau cycle of a normal surface singularity (2012)
  • Author(s): Kazuhiro KONNO
  • Journal Title: Asian J. of Math. Volume: 16 Pages: 279-298
  • Peer Reviewed
• [Journal Article] Maximal ideal cycles over normal surface singularities of Brieskorn type (2012)
  • Author(s): Kazuhiro KONNO and Daisuke NAGASHIMA
  • Journal Title: Osaka J. Math. Volume: 49 Pages: 225-245
  • NAID: 120004838543
  • Peer Reviewed
• [Presentation] On the fixed part of a non-special linear system over a normal surface singularity (2013)
  • Author(s): Kazuhiro Konno
  • Organizer: Classification of Algebraic Varieties and Related Topics
  • Place of Presentation: Cetraro (Italy)
  • Year and Date: 2013-09-09
• [Presentation] Normal sextics which are canonical (2012)
  • Author(s): 今野一宏
  • Organizer: 高次元代数多様体とベクトル束の代数幾何
  • Place of Presentation: 九州大学
  • Year and Date: 2012-03-16
• [Presentation] Numerical invariants of fibred algebraic surfaces (2012)
  • Author(s): 今野一宏
  • Organizer: 特異点と多様体の幾何学2012
  • Place of Presentation: 山形大学理学部
• [Presentation] Normal sextic surfaces which are canonical (2012)
  • Author(s): Kazuhiro Konno
  • Organizer: WCU workshop on classification and construction of algebraic varieties
  • Place of Presentation: Busan (Korea)
• [Presentation] Normal sextic surfaces which are canonical (2012)
  • Author(s): 今野一宏
  • Organizer: 高次元代数多様体とベクトル束の代数幾何学(招待講演)
  • Place of Presentation: 九州大学
• [Presentation] Regular canonical surfaces of degree 3n-3 in P^n, Parts I and II (2011)
  • Author(s): Kazuhiro Konno
  • Organizer: Two Weeks of Classical Algebraic Geometry
  • Place of Presentation: Univ. Padova (Italy)
• [Book] 代数曲線束の地誌学 (2013)
  • Author(s): 今野一宏
  • Publisher: 内田老鶴圃