整閉イデアルの族の研究

• Project/Area Number: 13874006
• Research Category: Grant-in-Aid for Exploratory Research
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Nihon University
• Principal Investigator: 渡辺 敬一 日本大学, 文理学部, 教授 (10087083)
• Co-Investigator(Kenkyū-buntansha): 後藤 四郎 明治大学, 理工学部, 教授 (50060091)
• Project Period (FY): 2001 – 2003
• Project Status: Completed (Fiscal Year 2003)
• Budget Amount: ¥2,000,000 (Direct Cost: ¥2,000,000); Fiscal Year 2003: ¥600,000 (Direct Cost: ¥600,000); Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000); Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
• Keywords: 整閉イデアル / 重複度 / multiplier ideal / Riemann-Rochの公式 / 正則局所環 / 有理特異点 / toric ring / Hilbert-Kunz multiplicity / multiplicr ideal

Research Abstract

本年度得られた主な成果は以下の通りである.
・2次元の正則局所環において任意の整閉イデアルが有理数係数の整閉イデアルのmultiplier idealとして得られることをJ.Lipman氏との共同研究で示した.
・3次元正則局所環のtoric整閉イデアルの分解の様子をanti-nef divisorの言葉で与え,C.Huneke, J.Lipmanの例の幾何的解釈に成功した.また,mIが整閉でないtoric整閉イデアルの例を構成した.
・3次元のRiemann-Rochの定理を整閉イデアルに応用し,colength,最小生成系の個数などに関する公式を得た.
・整閉イデアルに対するlc(log canonical) thresholdに対応するF-pure thresholdの概念を標数pの手法を用いて定義し,lc thresholdに関する様々な理論の簡易化に成功した(高木俊輔氏との共同研究成果).
・後藤はI=Q : m(Qは局所環(A, m)のパラメーターイデアル)に関する等式I^2=QIの重要性に注目し,これが成り立つための条件,整閉性との関係などにおいて大変興味ある結果を得た.
このように多くの興味深い結果を得たが,3次元以上における整閉イデアルの分解,整閉イデアルの列の挙動など多くの重要な問題が今後の課題として残っている.

Research Products

• [Publications] K.Watanabe: "Chains of Integrally Closed Ideals"Contemporary Mathematics (AMS). 331. 353-358 (2003)
• [Publications] J.Lipman, K.Watanabe: "Integrally closed ideals\\ in two-dimensional regular local rings are multiplier ideals"Math.Reasearch Letters. 10. 423-434 (2003)
• [Publications] S.takagi, K.Watanabe: "When does the subadditivity theorem for multiplier ideals hold?"Trans.Amer.Math.Soc.. To appear.
• [Publications] J.Elizondo, K.Kurano, K.Watanabe: "The total coordinate ring of a normal projective variety"J.of Algebra. To appear.
• [Publications] S.Goto, H.Sakurai: "The equality I^2=QI in Buchsbaum rings"Rend.Sem.Mat.Univ.Padova. 110. 25-56 (2003)
• [Publications] S.Goto, F.Hayasaka, S.Iai: "The a-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings"Proc.Amer.Math.Soc. 131. 87-94 (2003)
• [Publications] K.Watanabe: "Chains of Integrally closed ideals"contemporary math. (Amer. Math. Soc.). (To appear (accepted)).
• [Publications] J.Lipman, K.Watanabe: "Integrally closed ideals in two-dimensional regular local rings are multiplier ideals"Mathematical Research Letters. (To appear (accepted)).
• [Publications] N.Hara, K.Watanabe, K.Yoshida: "F-rationality of Rees algebras"J. of Algebra. 247. 153-190 (2002)
• [Publications] S.Goto, F.Hayasaka: "Towards a theory of Gorenstein m-primary integrally closed ideals"Proc. NATO advanced workshop. (To appear).
• [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals"Proc. Amer. Math. Soc.. 130. 3159-3164 (2002)
• [Publications] K. Watanabe, K. Yoshida: "Hilbert-Kunz multiplicity of two-dimensional local rings"Nagoya Math. J.. 162. 87-110 (2001)
• [Publications] S. Goto, S. Iai, K. Watanabe: "Good ideals in Gorenstein local rings"Trans. American Math. Soc.. 353. 2309-2346 (2001)
• [Publications] K. Watanabe, K. Yoshida: "Hilbert-Kunz multiplicity, McKay correspondence and Good ideals in 2-dimensional Rational Singularities"Manuscripta Math.. 104. 275-294 (2001)