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Implicitization, Residual Intersections, and Differential Methods in Commutative Algebra

交换代数中的隐式化、残差交点和微分方法

基本信息

批准号：
1802383
负责人：
Bernd Ulrich
金额：
$ 32.15万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802383&HistoricalAwards=false
关键词：
Implicitization Residual Intersections Differential Methods

项目摘要

This award funds research in Commutative Algebra, the study of systems of polynomial equations in several unknowns. To this end, one considers the collection of all solutions of a system of equations as a geometric object and investigates the functions defined on this object. Reversing the perspective leads to the implicitization problem, which the investigator plans to work on: Given a geometric object, such as a curve or a surface, one wishes to construct a system of polynomial equations that has the geometric object as its solution set. Once the system of polynomial equations is known, it becomes much easier, for instance, to decide whether a specific point lies on the surface or whether the surface is smooth at one of its points. The implicitization problem is difficult and requires advanced techniques from pure mathematics, but its solution even in particular cases has numerous applications, for instance in computer aided design, robotics, and other areas of engineering. This project addresses several topics in Commutative Algebra that have close connections with Algebraic and Analytic Geometry and with Elimination Theory. They include the implicitization problem for Rees algebras and rational maps, equisingularity theory, the Poincare problem for plane foliations, and residual intersection theory. Determining the implicit equations of graphs and images of rational maps is a classical, but open problem in elimination theory, which amounts to finding defining ideals of Rees algebras. Previously, the PI and his collaborators solved this problem for Rees algebras of codimension three Gorenstein ideals, under the additional assumption that the entries of a syzygy matrix of the ideal generate a complete intersection. Now the PI intends to remove this crucial hypothesis. The PI also plans to investigate Rees algebras of more general ideals, with the aim to obtain at least qualitative statements and bounds for the implicit equations. A goal in equisingularity theory is to devise fiberwise numerical criteria for when a family of analytic spaces is topologically trivial. An important intermediate step are numerical characterizations of integral dependence of modules. The PI intends to prove such a characterization using a notion of multiplicity that is inspired by intersection theory. Poincare had asked how to decide whether a singular algebraic foliation of the complex plane has an algebraic curve as a leaf. In more recent times, this question has often been treated as a problem about relating invariants of a vector field to invariants of curves or varieties that are left invariant by the vector field. The PI will investigate this problem, using his expertise from prior work on algebraic differentials and Castelnuovo-Mumford regularity. The notion of residual intersection, a generalization of linkage or liaison, is ubiquitous and appears naturally in intersection theory and in the study of Rees algebras, for instance. Of central importance are the Cohen-Macaulayness and duality properties of residual intersections. Based on partial results and experimental evidence, David Eisenbud and the PI have observed that, unexpectedly, many residual intersections, even when they fail to be Cohen-Macaulay, admit maximal Cohen-Macaulay modules of rank one that are self-dual. The PI and his collaborators intend to give a proof of this unusual phenomenon.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项资助交换代数的研究，即研究几个未知数的多项式方程组。为此，我们将一个方程组的所有解的集合视为一个几何对象，并研究在该对象上定义的函数。相反的观点导致了隐含化问题，这是研究者计划研究的问题：给定一个几何对象，如一条曲线或曲面，人们希望构造一个以该几何对象作为其解集的多项式方程组。一旦知道了多项式方程组，就变得容易得多，例如，确定一个特定点是否位于曲面上，或者曲面在其某个点处是否光滑。隐含问题是困难的，需要纯数学的高级技术，但它的解决方案即使在特定情况下也有许多应用，例如在计算机辅助设计、机器人学和其他工程领域。这个项目涉及交换代数中与代数、解析几何和消元理论有密切联系的几个主题。它们包括Rees代数和有理映射的隐含化问题、均衡性理论、平面层的Poincare问题和剩余交理论。确定有理映射图和映象的隐方程是消去论中一个经典而又开放的问题，其实质就是寻找Rees代数的定义理想。以前，PI和他的合作者解决了余维三个Gorenstein理想的Rees代数的这个问题，另外假设理想的合矩阵的项生成完全交。现在，PI打算去掉这一关键假设。PI还计划研究更一般理想的Rees代数，目的是至少得到隐式方程的定性陈述和界。等价性理论的一个目标是为一族解析空间何时是拓扑平凡的设计纤维状的数值准则。一个重要的中间步骤是模的积分依赖性的数值刻画。PI打算使用受交集理论启发的多重性概念来证明这样的特征。庞加莱曾问过，如何确定复平面的奇异代数叶层是否具有作为叶的代数曲线。最近，这个问题经常被看作是一个关于矢量场的不变量与矢量场留下的不变量的曲线或簇的不变量之间的关系的问题。PI将利用他之前在代数微分和Castelnuovo-Mumford正则性方面的专业知识来调查这个问题。剩余交的概念是链接或联络的推广，它是普遍存在的，并自然地出现在交理论和Rees代数的研究中，例如。最重要的是剩余交的Cohen-Macaulay性和对偶性质。基于部分结果和实验证据，David Eisenbud和PI已经观察到，出乎意料的是，许多剩余交，即使它们不是Cohen-Macaulay，也允许一阶的极大Cohen-Macaulay模是自对偶的。PI和他的合作者打算证明这一不寻常的现象。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。