Topics in Algebraic Geometry

这是一个关于代数几何的项目，代数几何是数学中最古老的学科之一。它研究由代数方程定义的几何对象。代数几何与数学物理和编码理论有着重要的联系。该研究将提高对代数曲线几何与高维簇之间的关系以及定义簇的方程之间的一组更高关系的理解。这些关系被称为合朔的品种。研究者和合作者将研究代数曲线和高维簇的合点的性质。他们将研究高维多样性的奇异性及其与最小模型程序和其他数学分支的关系。该项目还包括广泛的培训内容。PI已经培养了几个成功的年轻代数几何学家，目前正在指导两名研究生，他们正在研究与该项目有关的各种问题。PI和合著者正在写一本关于乘子理想、扩张定理和弧空间几何的书。这本书的目的是帮助年轻的研究人员了解如何这些技术可以应用到理解高维代数几何。研究项目涉及研究syzygies通过研究性质的希尔伯特计划的点上的曲线。它还计划调查的渐近行为syzygies的代数簇和它们的关系的内在几何的品种。该项目将扩展的经典结果以前只知道在代数曲线的情况下，以更高的维品种。乘子理想及其消失定理在高维极小模型程序的发展中起着重要的作用;这些理想也是奇点复杂性的重要度量。Mather-Jacobian理想比经典的乘子理想有优势，因为这些理想是针对非Q-Gorenstein奇点定义的。高余维有效圈对于理解高维簇的几何结构是非常有用的。研究者和合作者正计划研究数值不变量，如对数规范阈值和最小对数差异附加到代数簇的奇点。这些不变量在极小模型规划的研究中起着核心作用。它们也自然地出现在D-模理论、正特征交换代数和弧空间几何中。

This is a project on algebraic geometry, which is one of the oldest disciplines in mathematics. It studies geometric objects defined by algebraic equations. Algebraic geometry has important connections to mathematical physics and coding theory. The research will improve understanding of the relations between the geometry of algebraic curves and higher dimensional varieties and the set of higher relations among the equations defining the varieties. These relations are called the syzygies of the varieties. The investigator and collaborators will investigate the properties of the syzygies of algebraic curves and higher dimensional varieties. They will study singularities of higher dimensional varieties and their relations to the minimal model program and other branches of mathematics. The project has extensive training components as well. The PI has already trained several successful young algebraic geometers and is currently supervising two graduate students who are working on various problems related to the project. The PI and co-authors are writing a book on multiplier ideals, extension theorems, and geometry of arc spaces. The book aims to help young researchers understand how these techniques can be applied to understanding higher dimensional algebraic geometry.The research project involves study of syzygies by studying properties of the Hilbert schemes of points on the curve. It is also planned to investigate the asymptotic behavior of syzygies of an algebraic variety and their relations to the intrinsic geometry of the variety. The project would extend the classical results previously only known in the case of algebraic curves to higher dimensional varieties. Multiplier ideals together with their vanishing theorems play an important role in the development of the higher dimensional minimal model program; these ideals are also an important measure of the complexity of singularities. Mather-Jacobian ideals have an advantage over the classical multiplier ideals because these ideals are defined for non-Q-Gorenstein singularities. Higher co-dimension effective cycles should be very useful in understanding the geometry of higher dimensional varieties. The investigator and collaborators are planning to study numerical invariants such as log-canonical thresholds and minimal log-discrepancies attached to the singularities of algebraic varieties. These invariants play central roles in the study of minimal model program. They also show up naturally in the theory of D-modules, positive characteristic commutative algebra, and the geometry of arc spaces.