Geometry of Compact Moduli Spaces

紧模空间的几何

• 批准号: 0701191
• 负责人: Evgueni Tevelev
• 金额: --
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701191&HistoricalAwards=false
• 关键词: Geometry; Compact; Moduli; Spaces

The principal investigator plans to study geometry of moduli spaces of stable curves of Deligne, Knudsen, and Mumford and its higher dimensional analogues: moduli spaces of stable pairs introduced by Kollar, Shepherd-Barron, and Alexeev. The deepest results are expected for moduli of Del Pezzo surfaces and certain classes of surfaces of general type, for moduli of stable rational curves, and for their higher-dimensional analogues: compact moduli spaces of hyperplane arrangements. New methods are based on the interaction between Mori theory and tropical algebraic geometry, which assigns to any algebraic variety an object of combinatorial nature, basically a polytope. This polytope (a tropical variety) surprisingly encodes a lot of geometric information about the variety and most importantly about its compactifications. Tropical varieties were originally introduced in the community of computational algebraic geometers who study how to read invariants of algebraic varieties from their equations.It turns out that proposed methods can also be used backwards to advance computational algebraic geometry. In particular, they can be used to find implicit equations of algebraic varieties given parametrically.Algebraic geometry studies algebraic varieties shapes defined by systems of polynomial equations. Algebraic varieties have discrete characteristics that allow to classify their species: rational curves, Del Pezzo surfaces, Abelian varieties, Calabi-Yau varieties, varieties of general type, etc. Varieties of each type depend on certain continuous parameters (called moduli) and the set of all parameters has a rich structure of the so-called moduli space. One is particularly interested in compact moduli spaces that parametrize varieties with allowed mild degenerations.For example, a hyperbola xy=C on the plane can degenerate to the union of two lines xy=0 when C goes to 0. The principal investigator will study these compact moduli spaces and related problems in algebraic geometry. This centuries-old concept of pure mathematics has rich relationship with physics, and applications to computational algebraic geometry will lead to new algorithms useful in algebraic statistics and mathematical biology.

主要研究者计划研究Deligne，Knudsen和Mumford稳定曲线的模量空间的几何形状及其更高的类似物：Kollar，Shepherd-Barron和Alexeev引入的稳定对的模量稳定对。预计Del Pezzo表面的模量和一般类型的某些表面，稳定理性曲线的模量以及它们的较高维度类似物的最深结果是：超平面布置的紧凑型模量空间。新方法基于MORI理论与热带代数几何形状之间的相互作用，该几何形状分配给任何代数变体是组合性质的对象，基本上是多层。这种多层（热带品种）令人惊讶地编码了许多有关该品种的几何信息，最重要的是关于其压缩。热带品种最初是在计算代数几何学社区中引入的，他们研究了如何从其方程式中读取代数品种的不变性。事实证明，提议的方法也可以向后使用以推进计算代数几何形状。特别是，它们可用于找到给定参数的代数品种的隐式方程。代数几何研究代数品种由多项式方程系统定义的代数品种形状。代数品种具有允许对其物种进行分类的离散特征：理性曲线，del Pezzo表面，Abelian品种，Calabi-Yau品种，一般类型的品种等。每种类型的品种都取决于某些连续参数（称为Moduli）以及所有参数的集合。一个人特别感兴趣的是与允许的轻度变性一起参数化品种的紧凑型模量空间。这个数百年历史的纯数学概念与物理学具有丰富的关系，并且在计算代数几何形状上的应用将导致新算法在代数统计和数学生物学中有用。