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Singularities and Multiplicities in Commutative Algebra

交换代数中的奇异性和多重性

基本信息

批准号：
1901672
负责人：
Linquan Ma
金额：
$ 21.11万
依托单位：
Purdue University
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=1901672&HistoricalAwards=false
关键词：
Singularities Multiplicities Commutative Algebra

项目摘要

The PI plans to study questions in commutative algebra. This is a field that deals with the study of algebraic varieties, that is, the solution set of a system of polynomial equations in several variables. For example, the parabola defined by the equation y=x^2 in the plane is smooth, meaning that locally, at any point, it looks like a line, while the curve defined by the equation y^2=x^3 is not smooth at the origin (it looks like a cusp near the origin). The singular or non-smooth points of an algebraic variety have very rich algebraic and geometric structures, and investigating their properties has many applications in other sciences and engineering. The projects that will be explored are the study of singular points of algebraic varieties (i.e., singularities), with a focus on measuring the singularities (e.g., assigning some invariant to compare them with each other) using the theory multiplier/test ideals and multiplicities. The PI plans to develop a mixed characteristic singularity theory that is parallel to singularities in the minimal model program and tight closure theory from equal characteristic. This is motivated by the recent breakthrough of Yves Andre that solved Hochster's long-standing direct summand conjecture and the existence of big Cohen-Macaulay algebras. Together with Karl Schwede, the PI utilizes Andre`'s techniques from perfectoid algebras and spaces to start building such a theory, introducing a perfectoid version of multiplier/test ideals. There will be applications on the study of symbolic powers in mixed characteristic, as well as the study of family of singularities when the characteristic varies. Another research direction is the Hilbert-Samuel multiplicities attached to singularities, with a focus on Lech's conjecture, Lech's inequality, and the Stuckrad--Vogel invariant. The PI has settled the dimension three equal characteristic case of the longstanding Lech's conjecture, and has recently settled the Stuckrad--Vogel conjecture (together with Klein, Pham, Smirnov and Yao). The proposed projects include attacking the higher dimensional case of Lech's conjecture by studying and improving the classical Lech's inequality. A related project is to investigate the relationship between multiplicity and colength for sufficiently deep ideals.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.

PI计划学习交换代数中的问题。这是一个研究代数簇的领域，也就是研究多变量多项式方程组的解集。例如，由方程y=x^2定义的抛物线在平面上是光滑的，这意味着在任何局部，它看起来像一条直线，而由方程y^2=x^3定义的曲线在原点处不光滑(它看起来像原点附近的尖点)。代数族的奇点或非光滑点具有非常丰富的代数和几何结构，研究它们的性质在其他科学和工程中有着广泛的应用。将探索的项目是研究代数簇(即奇点)的奇点，重点是使用理论乘子/检验理想和重数来测量奇点(例如，指定一些不变量来相互比较)。PI计划发展一种与最小模型程序中的奇点平行的混合特征奇点理论和由特征相等的紧闭包理论。这是由于Yves Andre最近的突破解决了Hochster长期存在的直接求和猜想和大Cohen-Macaulay代数的存在。与卡尔·施韦德一起，PI利用安德鲁的S技巧从完美拟态代数和空间开始建立这样的理论，引入了乘子/检验理想的完美拟态版本。这将在混合特征中的符号幂的研究以及特征变化时奇异族的研究中得到应用。另一个研究方向是与奇点有关的希尔伯特-塞缪尔多重性，重点是Lech猜想、Lech不等式和Stuckrad-Vogel不变量。PI解决了长期存在的Lech猜想的三维等特征情形，最近又解决了Stuckrad-Vogel猜想(与Klein、Pham、Smirnov和姚一起)。建议的项目包括通过研究和改进经典的Lech不等式来攻击Lech猜想的高维情况。一个相关的项目是调查多样性和足够深的理想的长度之间的关系。这个奖项反映了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。

项目成果

期刊论文数量（15）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Symbolic power containments in singular rings in positive characteristic

奇异环中的象征性权力遏制具有积极特征

DOI：
10.1007/s00229-021-01359-7
发表时间：
2023
期刊：
manuscripta mathematica
影响因子：
0.6
作者：
Grifo, Eloísa;Ma, Linquan;Schwede, Karl
通讯作者：
Schwede, Karl

Koszul and local cohomology, and a question of Dutta

Koszul 和局部上同调，以及 Dutta 问题

DOI：
10.1007/s00209-020-02619-0
发表时间：
2021
期刊：
Mathematische Zeitschrift
影响因子：
0.8
作者：
Ma, Linquan;Singh, Anurag K.;Walther, Uli
通讯作者：
Walther, Uli

Covers of rational double points in mixed characteristic

混合特征中有理双点的覆盖

DOI：
10.5427/jsing.2021.23h
发表时间：
2021
期刊：
Journal of Singularities
影响因子：
0.4
作者：
Carvajal-Rojas, Javier;Ma, Linquan;Polstra, Thomas;Schwede, Karl;Tucker, Kevin
通讯作者：
Tucker, Kevin

Maximal Cohen-Macaulay complexes and their uses: A partial survey

最大科恩-麦考利复合体及其用途：部分调查