Problems in Commutative Algebra: Free Resolutions, Multiplicities, and Blowup Rings

这个项目是交换代数，一个与代数几何密切相关的纯数学领域。交换代数，在更广泛的意义上，处理在几个未知数多项式方程组的定性研究。这些系统的解在许多科学和工程领域发挥着重要的作用，因此所提出的研究也具有应用方面。它的目标之一是为不同方程组的解集何时“相似”设计数值标准，另一个目标是构建具有给定几何对象（如曲线或曲面）作为解集的方程组。第二个问题也与几何建模和计算机辅助设计的应用有关。该项目的第三个目标是进一步开发代数的基本工具，称为“自由分辨率”。自由解析度是一种研究复杂代数结构的方法，它通过一个可能无限的简单对象序列，即矩阵来研究。在更专业的术语中，这个项目的主题是等奇异性理论，Rees代数的隐化问题，乘子理想的代数性质，结果的局部环的扩展和梯度设置固有的技术。更具体地说，等奇异性理论的一般目标是设计出一组解析空间在拓扑平凡、惠特尼等奇异或满足其他等奇异条件时的精确数值准则。与Kleiman和Validashti一起，PI证明了一个孤立奇点族是惠特尼等奇异的，因此拓扑平凡，如果一个新定义的广义多重性，即多重性，在整个族中是常数。现在，PI计划研究多重性的恒常性意味着哪些更强的等奇异性条件。一个相关的，基本的问题是如何理解其雅可比模的多重为零的奇点。等奇异性条件与模间积分依赖的数值判据密切相关，PI希望设计一个基于交数的等奇异性判据。消去理论中的一个经典问题是确定Rees代数的定义理想，特别是给出了投影空间间有理映射的图和象的隐式方程。PI计划对曲面参数化的正则映射，或者更一般地说，对具有余维三维基轨迹的有理映射，研究这个问题。在最近与Corso， Huneke和Polini的合作中，PI引入了局部环上自由分辨率的距离概念，并用它来研究理想的积分闭包。现在PI提出了这个概念作为梯度自由分辨率的位移的替代品，以便将已知的结果从梯度扩展到局部情况，例如相关的梯度环的Cohen-Macaulay和Gorenstein性质的准则以及具有有限射影维的模块的Loewy长度的界。PI的另一个目标是研究奇异变量上（Mather-Jacobian）乘子理想的代数性质，并确定哪些理想可以实现为乘子理想。

This project is in commutative algebra, an area of pure mathematics with close ties to algebraic geometry. Commutative algebra deals, in a wider sense, with the qualitative study of systems of polynomial equations in several unknowns. Solutions of such systems play an important role in many areas of science and engineering, hence the proposed research has applied aspects as well. One of its objectives is to devise numerical criteria for when the solution sets of different systems of equations are "alike," another one aims at constructing systems of equations that have a given geometric object, like a curve or a surface, as its solution set. This second problem is also relevant for applications in geometric modeling and computer-aided design. A third goal of the project is to further develop a fundamental tool in algebra, called "free resolutions." Free resolutions are a method to study complex algebraic structures by means of a possibly infinite sequence of simpler objects, namely, matrices.In more technical terms, the topics of this project are equisingularity theory, the implicitization problem for Rees algebras, algebraic properties of multiplier ideals, the extension to local rings of results and techniques that are inherent to the graded setting. To be more specific, a general goal in equisingularity theory is to devise fiberwise numerical criteria for when a family of analytic spaces is topologically trivial, is Whitney equisingular, or satisfies other equisingularity conditions. With Kleiman and Validashti the PI proved that a family of isolated singularities is Whitney equisingular, and hence topologically trivial, if a newly defined generalized multiplicity, the epsilon multiplicity, is constant across the family. Now the PI plans to examine which stronger equisingularity conditions the constancy of the epsilon multiplicity implies. A related, fundamental problem is to understand singularities whose Jacobian module has epsilon multiplicity zero. Equisingularity conditions are closely related to numerical criteria for integral dependence of modules, and the PI wishes to devise such a criterion based on intersection numbers. A classical problem in elimination theory is to determine the defining ideals of Rees algebras, which gives, in particular, the implicit equations of graphs and images of rational maps between projective spaces. The PI plans to work on this problem for regular maps parametrizing a surface or, more generally, for rational maps with codimension three base locus. In recent joint work with Corso, Huneke, and Polini, the PI introduced the notion of distance in free resolutions over local rings and used it to study integral closures of ideals. Now the PI proposes this notion as a substitute for the shifts in graded free resolutions, in order to extend known results from the graded to the local case -- such as criteria for the Cohen-Macaulay and Gorenstein properties of associated graded rings and bounds for the Loewy length of modules having finite projective dimension. Another of the PI's goals is tostudy algebraic properties of (Mather-Jacobian) multiplier ideals on singular varieties and to determine which ideals can be realized as multiplier ideals.