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Studies on Local Cohomology, Derivations, Integral Dependence, and Blowup Algebras

局部上同调、导数、积分相关性和爆炸代数的研究

基本信息

批准号：
1902033
负责人：
Claudia Polini
金额：
$ 15万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902033&HistoricalAwards=false
关键词：
Studies Local Cohomology Derivations Integral

项目摘要

This project is in commutative algebra, but has also been inspired by algebraic geometry. Systems of polynomial equations in several variables occur in many applications in science and technology, for instance in engineering, computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. Commutative algebra and algebraic geometry are concerned with the qualitative study of such systems of polynomial equations. This project focuses on the algebraic approach. One of the goals of the project is to understand the vectors that are tangent to solution sets of systems of polynomial equations. Another objective is to find systems of polynomial equations describing a given geometric object. The latter has applications to computer graphics, algebraic statistics, and rigidity of structures. The PI intends to involve undergraduate students, graduate students, and postdoctoral fellows in her research. She will continue to organize national and international meetings. Throughout her activities, the PI will continue to promote underrepresented groups in mathematics. The first objective of this research is to relate degrees of vector fields on projective space to invariants of curves, or varieties, that they leave invariant. This is a difficult problem that has been studied for over a century, mainly from the point of view of complex analysis, dynamical systems, algebraic and differential geometry. The PI will investigate this question using tools from commutative algebra. The second objective is to find criteria for a variety in projective space to be a set-theoretic complete intersection. When there is only one non vanishing local cohomology module the PI believes that the property of being a set-theoretic complete intersection is encoded in the structure of this module. The third objective is to prove a numerical characterization of integral dependence of modules using a notion of multiplicity that arises in intersection theory. Theorems of this kind have a bearing on equisingularity theory, in fact they lead to fiber-wise numerical conditions for a family of analytic spaces to be Whitney equisingular, hence topologically trivial. The last objective is to study the implicit equations defining the graph and the image of rational maps between projective spaces.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.

这个项目是在交换代数，但也受到了代数几何的启发。多变量多项式方程组出现在科学和技术的许多应用中，例如工程学、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学。交换代数和代数几何关注的是这种多项式方程组的定性研究。这个项目的重点是代数方法。该项目的目标之一是了解与多项式方程组的解集相切的向量。另一个目标是找到描述给定几何对象的多项式方程组。后者应用于计算机图形学、代数统计和结构的刚性。PI打算让本科生、研究生和博士后研究员参与她的研究。她将继续组织国家和国际会议。在她的活动中，PI将继续促进数学代表性不足的群体。本研究的第一个目的是将射影空间上向量场的度与它们保持不变的曲线或变种的不变量联系起来。这是一个困难的问题，已经研究了超过世纪，主要是从复分析，动力系统，代数和微分几何的观点。PI将使用交换代数的工具来研究这个问题。第二个目标是找到射影空间中的簇是集合论完全交的准则。当只有一个非零局部上同调模时，PI认为集合论完全交的性质编码在这个模的结构中。第三个目标是证明一个数值表征的积分依赖模块使用的概念，出现在交叉理论的多重性。这类定理与等奇异性理论有关系，事实上，它们导致一个解析空间族成为惠特尼等奇异的纤维数值条件，因此拓扑平凡。最后一个目标是研究定义投影空间之间的有理映射的图形和图像的隐式方程。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Multiplicity sequence and integral dependence

多重序列和积分依赖性

DOI：
10.1007/s00208-020-02059-5
发表时间：
2020
期刊：
Mathematische Annalen
影响因子：
1.4
作者：
Polini, Claudia;Trung, Ngo Viet;Ulrich, Bernd;Validashti, Javid
通讯作者：
Validashti, Javid

Degree bounds for local cohomology

局部上同调的度界

DOI：
10.1112/plms.12364
发表时间：
2020
期刊：
Proceedings of the London Mathematical Society
影响因子：
1.8
作者：
Kustin, Andrew R.;Polini, Claudia;Ulrich, Bernd
通讯作者：
Ulrich, Bernd

Quasi-cyclic modules and coregular sequences

准循环模和共正则序列

DOI：
10.1007/s00209-020-02676-5
发表时间：
2021
期刊：
Mathematische Zeitschrift
影响因子：
0.8
作者：
Hartshorne, Robin;Polini, Claudia
通讯作者：
Polini, Claudia

Simple D -module components of local cohomology modules

局部上同调模的简单 D 模分量

DOI：
10.1016/j.jalgebra.2018.09.005
发表时间：
2021
期刊：
Journal of Algebra
影响因子：
0.9
作者：
Hartshorne, Robin;Polini, Claudia
通讯作者：
Polini, Claudia

Relations between the 2 × 2 minors of a generic matrix

泛型矩阵的 2××2 个次数之间的关系

DOI：
10.1016/j.aim.2021.107807
发表时间：
2021
期刊：
Advances in Mathematics
影响因子：
1.7
作者：
Huang, Hang;Perlman, Michael;Polini, Claudia;Raicu, Claudiu;Sammartano, Alessio
通讯作者：
Sammartano, Alessio