Higher Dimensional Algebraic Geometry

高维代数几何

• 批准号: 1001336
• 负责人: Lawrence Ein
• 金额: $ 43.7万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-05-15 至 2016-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001336&HistoricalAwards=false
• 关键词: Higher; Dimensional; Algebraic; Geometry

This is a project on algebraic geometry. Algebraic geometry studies properties of algebraic varieties, which are geometric objects defined by algebraic equations. Classically, algebraic geometers understood the geometry of algebraic curves and surfaces. But the geometry of varieties of dimension three or higher remains rather mysterious. The goals of this project are to study the properties of the equations defining these varieties and investigate the properties of the spaces of cycles. Another main difficulty of studying higher dimensional varieties is that it seems singularities are unavoidable when one studies birational geometry of varieties of dimension three or higher. Ein proposes to use various new technical tools to study the invariants that measure the complexity of these singularities. The new techniques involve the geometry of the arc spaces, generic limits and multiplier ideals from complex analysis. These numerical invariants also occur naturally in questions on birational rigidity, the theory of D-modules and positive characteristic commutative algebra. The appearance of the same invariants in so many different areas of mathematics is surprising. One of the goals of this proposal is to understand the links of these different aspects better.Algebraic geometry is one of the oldest disciplines in mathematics. In recent years, mathematicians have found that there are many important applications of algebraic geometry to mathematical physics, number theory topology and cryptography. The intellectual impacts of the proposal are on finding new scientific results and gaining a deeper understanding of the geometry of higher dimensional algebraic varieties. In particular, Ein plans to study syzygies, spaces of higher co-dimensional cycles and the singularities that occur naturally in studying higher dimensional birational geometry.

这是一个关于代数几何的项目。代数几何研究代数簇的性质，代数簇是由代数方程定义的几何对象。传统上，代数几何学家了解代数曲线和曲面的几何。但三维或更高维度的各种几何形状仍然相当神秘。这个项目的目标是研究定义这些簇的方程的性质，并调查圈空间的性质。研究高维簇的另一个主要困难是，当研究三维或更高维簇的二次几何时，奇点似乎是不可避免的。Ein建议使用各种新的技术工具来研究衡量这些奇点复杂性的不变量。新技术涉及弧空间的几何、一般极限和复数分析中的乘子理想。这些数值不变量也自然地出现在关于二元刚性、D-模理论和正特征交换代数的问题中。同样的不变量出现在这么多不同的数学领域令人惊讶。这项建议的目的之一是更好地理解这些不同方面的联系。代数几何是数学中最古老的学科之一。近年来，数学家们发现，代数几何在数学物理、数论拓扑和密码学中有许多重要的应用。这一提议的智力影响在于发现新的科学结果，并加深对高维代数簇几何的理解。尤其是，Ein计划研究合子，即具有更高共维循环的空间，以及在研究更高维双曲面几何时自然出现的奇点。