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.

该奖项资助了交换代数的研究，这是多个未知数中多项式方程系统的研究。为此，人们认为方程系统的所有解决方案的收集是一个几何对象，并研究了该对象上定义的函数。逆转透视图导致了隐式问题，研究者计划处理的问题：给定一个几何对象，例如曲线或表面，人们希望构建具有将几何对象作为其解决方案集的多项式方程系统。一旦知道多项式方程的系统，例如，决定特定点是否位于表面或表面是否在其一个点之一上平滑变得更容易。隐式问题很困难，需要纯数学的高级技术，但是它的解决方案即使在特定情况下也具有许多应用程序，例如在计算机辅助设计，机器人技术和其他工程领域中。该项目介绍了与代数和分析几何以及消除理论有密切联系的交换代数中的几个主题。它们包括REES代数和理性图的隐式问题，平等性理论，平面叶子的庞加罗问题以及残留的相交理论。确定图形的隐式方程和理性图的图像是消除理论中的经典但开放的问题，它等于找到定义REES代数的理想。以前，PI和他的合作者解决了Codimension三戈伦斯坦理想的REES代数的问题，这是在理想的Syzygy矩阵的条目产生完整的交叉点的其他假设下。现在，PI打算去除这一关键假设。 PI还计划调查更一般理想的REES代数，目的是为隐式方程提供至少定性陈述和界限。方程性理论的目标是设计何时拓扑空间的何时拓扑。一个重要的中间步骤是模块的积分依赖性的数值特征。 PI打算使用受相交理论启发的多重性概念来证明这种表征。 Poincare询问了如何确定复杂平面的奇异代数叶面是否具有代数曲线为叶子。在最近的时候，这个问题通常被视为将向量场的不变性与矢量场留下的曲线或品种不变的变种的不变性。 PI将利用他从代数差异和Castelnuovo-Mumford的规律性方面进行的专业知识来调查这个问题。剩余交叉的概念是无处不在的连锁或联络的概括，在相交理论和REES代数的研究中自然而然。至关重要的是残留相交的Cohen-Macaulays和二元性能。基于部分结果和实验证据，David Eisenbud和PI观察到，即使许多残留的交叉路口，即使它们无法成为Cohen-Macaulay，也承认最大的Cohen-Macaulay模块是自我双重的。 PI及其合作者打算证明这种不寻常的现象。该奖项反映了NSF的法定使命，并被认为是使用基金会的知识分子优点和更广泛的影响评估标准的评估值得支持的。