Singularities of Pairs and Linear Systems

这是一个关于代数几何的课题。代数几何研究代数变量的性质，代数变量是由代数方程定义的几何对象。经典代数几何学者了解代数曲线和代数曲面的几何。但是三维或更高维度的代数变换的几何仍然相当神秘。问题的主要困难之一是，当研究三维或更高维的变化的双几何时，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建测量这些奇异性复杂性的数值不变量。他还计划寻找这些不变量在交换代数和双对数几何中的新应用。特别是，他所研究的数值不变量将在高维代数几何的核心问题之一——最小模型程序的研究中发挥重要作用。在他的研究中涉及的技术包括非经典方法，如弧空间的几何和复数分析的乘法器理想。这些数值不变量也自然地出现在关于双空间刚性、d模理论和正特征交换代数的问题中。虽然这些区域之间的联系已经被很好地理解。在这么多不同的领域出现相同的不变量是令人惊讶的。本提案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结果，并对高维代数变体的几何有更深的理解。特别地，Ein计划研究在高维几何中自然出现的奇点。

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.