Commutative Algebra: Set-Theoretic Complete Intersections, Local Cohomology, Free Resolutions, and Rees Rings

这项研究项目涉及交换代数，着眼于代数几何。通常，解决问题最有效的方法是建立一个数学模型。通常，这样的模型涉及未知参数，这些未知参数由几个方程联系在一起，这些方程往往不可能精确求解。交换代数是对这类多项式方程组的定性研究。它的应用是深远的，包括计算机科学、密码学、编码理论、机器人学、模式识别和理论物理等多个领域。人们可以用几何或代数的方法研究这些系统的解集。这个项目是关于代数方法的。该项目的目标之一是了解描述曲线或曲面等几何对象所需的最小数量的方程。另一种方法是构造定义给定几何对象的方程组。第三个目标是研究给定的一组多项式方程之间的关系，这些关系之间的关系，等等。该项目涉及本科生、研究生和博士后研究员。这项研究项目有三个主要主题。第一个是给出射影空间中的一个簇是集合论完全交的判据。解决这一问题的一个基本工具是局部上同调模理论。局部上同调模编码代数簇的代数结构和拓扑结构。作为环上的模，局部上同调模是巨大的(既不是有限生成的，也不是Artin的)，因此很难处理。然而，作为Weil代数上的模块，它们可以被简单的对象过滤并变得可管理。因此，了解局部上同调模的D-模结构是一个重要的任务。第二个主题是使用距离的概念来研究局部环。这一概念被引入到研究人员和合作者最近的工作中，以理解理想的整体闭包。其思想是用距离代替齐次分辨率中的位移和分次模的Castelnuovo-Mumford正则性。主要目标是证明受评分案例中的陈述启发的一般结果。第四个主题是研究定义射影空间之间有理映射的图和象的隐式方程。这是消去论、交换代数和代数几何中的经典问题及其应用，例如在几何建模中。

This research project concerns commutative algebra with a view towards algebraic geometry. Often, the most effective method to solve a problem is to create a mathematical model. Frequently, such models involve unknown parameters related by several equations that are often impossible to solve exactly. Commutative algebra is the qualitative study of such systems of polynomial equations. Its applications are far-reaching and include diverse fields such as computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. One can study the sets of solutions of these systems either geometrically or algebraically. This project deals with the algebraic approach. One of the goals of the project is to understand the smallest number of equations needed to describe a geometric object like a curve or surface. Another is to construct the system of equations defining a given geometric object. A third goal is the study of the relations among a given set of polynomial equations, the relations among the relations, and so on. The project involves undergraduate students, graduate students, and postdoctoral fellows in the research. This research project has three main themes. The first is to develop criteria for a variety in projective space to be a set-theoretic complete intersection. A fundamental tool to solve this problem is the theory of local cohomology modules. Local cohomology modules encode the algebraic and topological structure of an algebraic variety. As modules over the ring, local cohomology modules are huge (neither finitely generated nor Artinian), hence intractable. However, as modules over the Weil algebra they can be filtered by simple objects and become manageable. Hence an important task is to understand the D-module structure of local cohomology modules. The second theme is the study of local rings using the notion of distance. This notion was introduced in recent work of the investigator and collaborators to understand the integral closure of ideals. The idea is to use distance as a substitute for shifts in homogeneous resolutions and for the Castelnuovo-Mumford regularity of graded modules. The main goal is to prove general results that are inspired by statements in the graded case. The last theme is to study the implicit equations defining the graph and the image of rational maps between projective spaces. This is a classical problem in elimination theory, commutative algebra, and algebraic geometry with applications, for instance, in geometric modeling.