Arcs, Valuations, and Multiplier Ideals on Algebraic Varieties

代数簇上的弧线、估值和乘数理想

基本信息

  • 批准号:
    1402907
  • 负责人:
  • 金额:
    $ 13.1万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2014
  • 资助国家:
    美国
  • 起止时间:
    2014-09-01 至 2018-08-31
  • 项目状态:
    已结题

项目摘要

This research project is in the field of algebraic geometry. At the core of algebraic geometry is the quest for classification of algebraic varieties, which are sets of solutions of systems of polynomial equations (or various generalizations of this concept). Classification roughly splits into two parts: birational classification, in which varieties that look alike almost everywhere are organized in the same class, and moduli theory, where continuous deformations of varieties are studied. The minimal model program is the major tool for the birational classification of algebraic varieties, and it has important applications to the theory of moduli. Designed over the model of classification of surfaces, the minimal model program has become one of the major trends in algebraic geometry, earning S. Mori the Fields Medal for work on three dimensional varieties. Recently, there has been groundbreaking progress which brings the subject very close to a complete program in all dimensions. There are however many fundamental questions that still remain unanswered, and some of these constitute the main objectives of this research project. The local study of algebraic varieties and their singularities is a natural component of this study.The project has two main objectives. The first objective concerns a conjecture of Shokurov from 1988 which predicts that a certain local invariant of singularities, known as the minimal log discrepancy, varies semicontinuously. This property is closely connected to a conjecture on termination of flips which has implications to the minimal model program, and relates to a series of specific questions concerning the geometry of spaces of arcs and their connections to valuation spaces and Berkovich analytic spaces. The second objective deals with multiplier ideals on algebraic varieties. There are various notions of multiplier ideals on singular varieties, and the PI will explore a unifying approach which incorporates all theories and explains their connections. This part of the project contains various applications, from a Reider-type theorem on the global generation of twists of the dualizing sheaf of a normal surface, to the comparisons between certain notions of singularities defined in characteristic zero and in positive characteristics. The project also includes educational activities and other broader impacts. These include the participation in outreach programs, various forms of dissemination including the writing of a book on the birational geometry of algebraic varieties, and the organization of several activities at various levels, ranging from undergraduate summer schools to international conferences.
该研究项目属于代数几何领域。代数几何的核心是寻求代数簇的分类,代数簇是多项式方程组的解的集合(或这个概念的各种推广)。分类大致分为两个部分:双理性分类,其中几乎在任何地方看起来都相似的品种被组织在同一类中,以及模理论,其中研究品种的连续变形。极小模型程序是代数簇的双有理分类的主要工具,它在模理论中有重要的应用。基于曲面分类模型的最小模型程序设计已成为代数几何的主要发展趋势之一。森喜朗因其在三维变量方面的工作而获得菲尔兹奖。最近,已经取得了突破性的进展,使该主题在各个方面都非常接近一个完整的程序。然而,仍有许多基本问题尚未得到解答,其中一些问题构成了本研究项目的主要目标。代数簇及其奇点的局部研究是本研究的一个自然组成部分。第一个目标涉及一个猜想Shokurov从1988年预测,一定的本地不变量的奇异性,称为最小的日志差异,不断变化。这一性质与一个关于翻转终止的猜想密切相关,该猜想对最小模型程序有影响,并涉及一系列关于弧空间几何及其与赋值空间和伯科维奇解析空间的联系的具体问题。第二个目标是代数簇上的乘子理想。奇异变种上有各种乘数理想的概念,PI将探索一种统一的方法,该方法包含所有理论并解释它们的联系。该项目的这一部分包含各种应用,从一个雷德型定理的全球生成扭曲的对偶层的正常表面,以比较某些概念的奇点定义的特征零和积极的特点。该项目还包括教育活动和其他更广泛的影响。其中包括参与外联方案,各种形式的传播,包括写一本关于代数簇的双有理几何的书,以及在各级组织一些活动,从本科生暑期学校到国际会议。

项目成果

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Tommaso de Fernex其他文献

Divisorial valuations viaarcs
通过arcs进行除数估值
  • DOI:
  • 发表时间:
    2008
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Tommaso de Fernex;Lawrence Ein and Shihoko Ishii
  • 通讯作者:
    Lawrence Ein and Shihoko Ishii
On planar Cremona maps of prime order
在素数平面克雷莫纳地图上
  • DOI:
    10.1017/s0027763000008795
  • 发表时间:
    2004
  • 期刊:
  • 影响因子:
    0.8
  • 作者:
    Tommaso de Fernex
  • 通讯作者:
    Tommaso de Fernex
行列式イデアルとその仲間たちの低次のシジジ
行列式理想的低阶 syjiji 及其同伴
  • DOI:
  • 发表时间:
    2016
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Tommaso de Fernex;Roi Docampo;Shunsuke Takagi and Kevin Tucker;橋本光靖
  • 通讯作者:
    橋本光靖
Birationally rigid hypersurfaces
  • DOI:
    10.1007/s00222-012-0417-0
  • 发表时间:
    2012-08
  • 期刊:
  • 影响因子:
    3.1
  • 作者:
    Tommaso de Fernex
  • 通讯作者:
    Tommaso de Fernex
Grothendieck–Lefschetz for ample subvarieties
  • DOI:
    10.1007/s00209-020-02693-4
  • 发表时间:
    2021-01-18
  • 期刊:
  • 影响因子:
    1.000
  • 作者:
    Tommaso de Fernex;Chung Ching Lau
  • 通讯作者:
    Chung Ching Lau

Tommaso de Fernex的其他文献

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{{ truncateString('Tommaso de Fernex', 18)}}的其他基金

Arc Spaces, Singularities, and Motivic Integration
弧空间、奇点和动机整合
  • 批准号:
    2001254
  • 财政年份:
    2020
  • 资助金额:
    $ 13.1万
  • 项目类别:
    Standard Grant
Algebraic Varieties and Valuation Theory
代数簇和估价理论
  • 批准号:
    1700769
  • 财政年份:
    2017
  • 资助金额:
    $ 13.1万
  • 项目类别:
    Standard Grant
FRG: Collaborative research: Birational geometry and singularities in zero and positive characteristic
FRG:合作研究:双有理几何以及零和正特征中的奇点
  • 批准号:
    1265285
  • 财政年份:
    2013
  • 资助金额:
    $ 13.1万
  • 项目类别:
    Continuing Grant
CAREER: Singularities in the Minimal Model Program and Birational Geometry
职业:最小模型程序和双有理几何中的奇点
  • 批准号:
    0847059
  • 财政年份:
    2009
  • 资助金额:
    $ 13.1万
  • 项目类别:
    Continuing Grant
Topological Invariants and Singularities in Birational Geometry
双有理几何中的拓扑不变量和奇点
  • 批准号:
    0456990
  • 财政年份:
    2005
  • 资助金额:
    $ 13.1万
  • 项目类别:
    Standard Grant
Topological Invariants and Singularities in Birational Geometry
双有理几何中的拓扑不变量和奇点
  • 批准号:
    0548325
  • 财政年份:
    2005
  • 资助金额:
    $ 13.1万
  • 项目类别:
    Standard Grant

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