Higher Dimensional Algebraic Geometry

这是一个关于代数几何的项目。代数簇是由代数方程定义的几何对象。经典的代数几何学家理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变量的方程的性质，并研究循环空间的性质。另一个主要的困难，研究高维品种是，它似乎奇点是不可避免的，当一个研究双有理几何品种的三维或更高。艾因建议使用各种新的技术工具来研究测量这些奇点复杂性的不变量。新的技术涉及几何的弧空间，一般的限制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。同样的不变量出现在如此多的不同数学领域是令人惊讶的。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。该建议的智力影响是寻找新的科学成果，并获得更深层次的理解几何的高维代数簇。特别是，艾因计划研究syzygies，空间的更高的co-dimensional周期和奇点自然发生在研究高维birational几何。

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.