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

这个项目是在交换代数中进行的，这是一个与代数几何有着密切联系的纯数学领域。在更广泛的意义上，交换代数研究的是几个未知数的多项式方程组的定性研究。这类系统的解在许多科学和工程领域中都具有重要的作用，因此所提出的研究也具有广泛的应用前景。它的目标之一是为不同方程系统的解集何时“相似”设计数字标准，另一个目标是构建以给定几何对象(如曲线或曲面)作为其解集的方程系统。第二个问题也与几何建模和计算机辅助设计中的应用有关。该项目的第三个目标是进一步开发一个基本的代数工具，称为“自由分解”。自由归结是一种通过简单对象的可能无限序列来研究复杂代数结构的方法，也就是矩阵。用更专业的术语来说，这个项目的主题是均衡性理论、Rees代数的隐含化问题、乘子理想的代数性质、结果的局部环的推广以及分次设置所固有的技巧。更具体地说，均衡性理论的一个一般目标是为一族解析空间何时拓扑平凡、惠特尼等奇性或满足其他等奇性条件设计纤维状的数值准则。PI用Kleiman和Validashti证明了一族孤立奇点是Whitney等分的，因此如果一个新定义的广义重数--epsilon重数在整个族中是恒定的，那么它在拓扑平凡上是平凡的。现在，PI计划检查epsilon多重数的不变性意味着哪些更强的均衡性条件。一个相关的基本问题是理解雅可比模为epsilon重数为零的奇点。均衡性条件与模的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判据。消去论中的一个经典问题是确定Rees代数的定义理想，它特别给出了射影空间之间有理映射的图和映象的隐式方程。PI计划解决这个问题，用于将曲面参数化的规则映射，或者更一般地，用于具有余维三个基轨迹的有理映射。在最近与Corso、Huneke和Polini的联合工作中，PI在局部环上的自由分解中引入了距离的概念，并用它来研究理想的积分闭包。现在PI提出了这个概念来替代分次自由分解中的位移，以便将分次环的已知结果推广到局部情况--例如相关分次环的Cohen-Macaulay和Gorenstein性质的准则以及具有有限投射维度的模的Loewy长度的界。PI的另一个目标是研究奇异簇上(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.