Studies in Moduli Theory and Birational Geometry

模理论与双有理几何研究

• 批准号: 0070970
• 负责人: Dan Abramovich
• 金额: $ 9.78万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070970&HistoricalAwards=false
• 关键词: Studies; Moduli; Theory; Birational; Geometry

The investigator and his colleagues study problems in moduli theory and birational geometry. In moduli theory they study questions related to the moduli stacks of twisted stable maps into a tame Deligne-Mumford stack. Questions addressed include constructions towards Gromov-Witten invariants of stacks, generalizations of twisted stable maps to higher dimensions and to wild stacks, and applications to the theory of admissible covers, level structures, spin structures and higher dimensional varieties. The investigator and colleagues will also study problems in birational geometry: the strong factorization conjecture, which says that a birational map of smooth projective varieties should factor as a sequence of smooth blowings up followed by a sequence of smooth blowings down; and the toroidalization conjecture, which says that, given a surjective morphism of complex projective manifolds, there is a way to blow up both source and target along smooth centers so that the resulting map is a toroidal morphism.The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. While algebraic geometry has contributed applications in coding, industrial control, and computation, the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as components of the fine structure of our universe. This is especially true with the first topic, moduli theory. This theory studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is often manifested as an algebraic variety, called a moduli space, in its own right. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not a just a metaphor but a rigorous and quite useful fact. Sometimes a collection of algebraic varieties manifests itself as a slightly more general object, called a stack, rather than a variety. Such stacks are a central object of study of this project. The other topic studied in this project is birational geometry, which is devoted to a certain abstract relationship, called birational equivalence, among algebraic varieties, which lies at the foundation of algebraic geometry. A goal of this project is to understand the connection between this abstract notion, and a much more concrete relationship given by an algebraic "surgery" procedure called "blowing up".

调查员和他的同事研究问题的模理论和双有理几何。 在模理论中，他们研究了与扭曲稳定映射的模堆栈成驯服的Deligne-Mumford堆栈有关的问题。 解决的问题包括建设Gromov-Witten不变量的堆栈，广义扭曲稳定映射到更高的维度和野生堆栈，和应用理论的容许覆盖，水平结构，自旋结构和高维品种。 研究者和他的同事们还将研究双有理几何中的问题：强因式分解猜想，它说一个光滑射影簇的双有理映射应该分解为一个光滑爆破序列，然后是一个光滑爆破序列;以及环面化猜想，它说，给定复射影流形的满射态射，有一种方法可以沿着沿着光滑的中心将源和目标都放大，这样得到的映射是一个环形的态射。2这个项目的研究领域是代数几何，是数学的分支，专门研究被称为代数簇的几何形状，由多项式方程定义。虽然代数几何在编码，工业控制和计算中的应用做出了贡献，但这个项目的主题与理论物理学中的应用更密切相关，物理学家认为代数簇是我们宇宙精细结构的组成部分。第一个主题，模量理论，尤其如此。这个理论研究了一个显著的现象，其中所有相同类型的代数簇的集合通常表现为一个代数簇，称为模空间。因此，在代数几何学中，把一个“有机体”群落看作是一个“有机体”的隐喻，不仅是一个隐喻，而且是一个严格而非常有用的事实。有时候代数簇的集合表现为一个稍微更一般的对象，称为栈，而不是一个簇。这种书库是本项目研究的中心对象。在这个项目中研究的另一个主题是双有理几何，它致力于代数簇之间的某种抽象关系，称为双有理等价，这是代数几何的基础。这个项目的一个目标是理解这个抽象概念之间的联系，以及一个更具体的关系，这个关系是由一个叫做“爆炸”的代数“手术”过程给出的。