Singularities of Pairs and Linear Systems

偶对和线性系统的奇异性

• 批准号: 0700774
• 负责人: Lawrence Ein
• 金额: $ 25.91万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700774&HistoricalAwards=false
• 关键词: Singularities; Pairs; Linear; Systems

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 algebraic surfaces. But the geometry of algebraic varieties of dimension three or higher remains rather mysterious. One of main difficulties of the problem 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 construct numerical invariants that measure the complexity of these singularities. He also plans to find new applications for these invariants to commutative algebra and birational geometry. In particular, the numerical invariants he studies will play important roles in the study of the Minimal Model Program, which is one the central problems in higher dimensional algebraic geometry.The techniques involved in his investigations include non-classical methods such as the geometry of the arc spaces 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. While the connections of some these areas are well understood. the appearance of the same invariants in so many different areas is surprising. One of the goals of this proposal is to understand the links of these different aspects better.Algebraic geometry is one of 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 proposals 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 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。古典代数几何理解几何的代数曲线和代数曲面。但三维或更高维的代数簇的几何学仍然是相当神秘的。这个问题的主要困难之一是，在研究三维或更高维的双有理几何时，奇点似乎是不可避免的。Ein建议使用各种新的技术工具来构造数值不变量来衡量这些奇异点的复杂性。他还计划为这些不变量在交换代数和双有理几何中找到新的应用。特别是，他研究的数值不变量将发挥重要作用的最小模型程序，这是一个中心问题，在高维代数几何的研究。在他的调查涉及的技术包括非经典的方法，如几何的弧空间和乘子理想从复杂的分析。这些数值不变量也自然地出现在双有理刚性、D-模理论和正特征交换代数的问题中。虽然这些领域的一些联系是很好理解的。同样的不变量出现在如此多的不同领域是令人惊讶的。这个建议的目标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、拓扑学和密码学中有许多重要的应用。这些建议的智力影响在于发现新的科学成果，并更深入地了解高维代数簇的几何。 特别是，艾因计划研究奇点自然发生在研究高维双有理几何。