Applications of tight closure and F-singularity to algebraic geometry

紧闭包和F-奇异性在代数几何中的应用

• 批准号: 16540005
• 负责人: HARA Nobuo
• 金额: $ 2.18万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540005/
• 关键词: tight closure; F-singularity; algebraic geometry; F-pure threshold; toric variety; tropical geometry; multiplier ideal; multiplicity; 正標数; 対数的標準閾値; CCI曲面; トロピカル・トーリック多様体; Buchsbaum斉次代数; 判定イデアル; P-フラクタル; tight closure; F-singularity; algebraic geometry; F-pure threshold; toric variety; tropical geometry; multiplier ideal; multiplicity; 正標数; 対数的標準閾値; CCI曲面; トロピカル・トーリック多様体; Buchsbaum斉次代数; 判定イデアル; P-フラクタル

Given a pair of a variety of characteristic p and an effective divisor on it, one can associate a real number called the F-pure threshold. Since this invariant is defined as a characteristic p analog of the log canonical threshold in characteristic 0, it is desirable that F-pure thresholds are rational numbers similarly as log canonical thresholds. N.Hara studied F-pure thresholds of pairs of a nonsingular surface and an effective divisor, and proved based on Monsky's idea of p-fractals that the F-pure thresholds are rational provided that the base field is finite. When the divisor is defined by a homogeneous polynomial f (x, y), the F-pure threshold c(f) can be estimated more precisely, and we can obtain a finite list of possible value of c(f) for a fixed degree d=deg f and characteristic p. We also proved that the Monsky's function ψ_f(t) has a piecewise quadratic limit as p→∞.M.Ishida studied real fans from a viewpoint of toric geometry, as well as moduli parameter of Catanese-Ciliberto-Ishida surface. T.Kajiwara studied the theory of logarithmic abelian varieties, the relationship of tropical hypersurfaces and degeneration of projective toric varieties, and the theory of tropical toric varieties. K.-i.Watanabe studied geometric interpretation of integrally closed monomial ideals in 3 variables, multiplier ideals, and F-thresholds. K.Yoshida gave estimates of multiplicities of Stanley-Reisner rings and Buchsbaum homogeneous algebras, and studied the structure of these rings when they have minimal multiplicities.

给定一对多种特征P和有效的除数，可以将一个称为F-Pure阈值的实际数字关联。由于此不变性被定义为特征0中对数规范阈值的特征p类似物，因此希望F-Pure阈值与对数规范阈值类似。 N.hara研究了一个非词性表面和有效除数对的F-Pure阈值，并基于Monsky的P-fractals的观念证明了F-Pure阈值是理性的，只要基地是有限的。当除数通过均匀的多项式f（x，y）定义时，可以更精确地估计f-pure阈值c（f），我们可以获得固定度d = deg f和特征p的可能值的有限列表。我们还证明，蒙斯基的函数ψ_f（t）从曲折的几何学的角度以及catanese-ciliberto-ishida表面的模量参数。 T.Kajiwara研究了对数的Abelian品种的理论，热带超曲面的关系和投射福利品种的变性以及热带感谢您的品种的理论。 K.-i.Watanabe研究了3个变量，乘数思想和F阈值的整体封闭单体思想的几何解释。 K.Yoshida估计了Stanley-Reisner Rings和Buchsbaum均匀代数的多重性，并在这些环的结构最少时研究了这些环的结构。