Local Cohomology and Singularities

局部上同调和奇点

• 批准号: 1502282
• 负责人: Craig Huneke
• 金额: $ 8.8万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502282&HistoricalAwards=false
• 关键词: Local; Cohomology; Singularities

The Principal Investigator plans to study geometric problems using different techniques in commutative algebra. This research focuses on spaces given by the set of points that satisfy certain polynomial equations in many variables. Since many phenomena can be described in terms of polynomial equations, these spaces appear in many fields of science and its applications. In such spaces most points are what is called "smooth", which, roughly speaking, means that after zooming in, their vicinity looks like a linear space. For instance, in a sphere every point is smooth and, just as the Earth, from a very close view its neighborhood looks like a plane. Then, those points that are not smooth present a particular behavior and, for that reason, are called "singular points". For instance, a cone has exactly one singular point at its vertex. The set of singular points can be described in terms of the derivatives of the polynomial equations that the points in the space satisfy. For many purposes, detecting singularities is not enough, as some are worse than others. For instance, the sharper vertices of cones are considered worse. To distinguish different singularities, one needs to use more sophisticated algebraic techniques. This research project seeks to study singularities using local cohomology modules, which can be seen as algebraic objects associated to a point. This has already proven to be a powerful tool to detect different kinds of singularities. The Principal Investigator plans to use local cohomology to study measurements of how bad a singular point is. The research includes long-standing problems in the study of singularities as well as new conjectures that could have theoretical and computational consequences. The project involves graduate students in the research. The Principal Investigator seeks to study the structure of local cohomology modules and singularities in positive and mixed characteristic. One of the main problems that one encounters while working with local cohomology modules is that they are usually very large and difficult to handle. However, these modules behave as if they were finitely generated over regular local rings that contain a field. An example of a regular ring in mixed characteristic for which injective dimension behaves differently from equal characteristic was recently found. Motivated by this result, the Principal Investigator intends to explore potential counter-examples for the properties regarding associated primes and Bass numbers of local cohomology modules over regular local rings of mixed characteristic. In addition, the Principal Investigator plans to work on the following related conjecture: the support of a local cohomology module is a Zariski closed set in the spectrum of the ring. Using local cohomology over rings containing a field, Lyubeznik introduced a family of invariants now called Lyubeznik numbers. These invariants have shown several connections with the algebraic and geometric properties of a ring. This inspired an analogous definition of these numbers in mixed characteristic. The project aims to compare the Lyubeznik numbers of rings that contain fields with those that do not. In particular, the research seeks a topological or arithmetic criterion that relates both notions of Lyubeznik numbers. In addition, the project seeks to find geometric properties encoded by the Lyubeznik numbers in mixed characteristic. Lastly, the Principal Investigator plans to work on singularities in positive characteristic via the Frobenius map. In particular, he is planning to work on the ACC conjecture for F-pure thresholds and its corollaries. In addition, the Principal Investigator and a collaborator will investigate a conjectured inequality that relates the F-pure thresholds with the Hilbert-Kunz multiplicities. If this project succeeds, the conjectured relation could have several computational and geometric consequences.

主要研究者计划在交换代数中使用不同的技术研究几何问题。这项研究的重点是满足许多变量中某些多项式方程的点的空间。由于可以用多项式方程来描述许多现象，因此这些空间出现在许多科学领域及其应用中。在这样的空间中，大多数要点就是所谓的“平滑”，这大概是意味着在放大后，它们的附近看起来像是线性空间。例如，在一个球中，每个点都很光滑，就像地球一样，从非常近的角度看，它的邻居看起来像飞机。然后，那些不平滑的点提出了特定的行为，因此称为“单数点”。例如，一个圆锥在其顶点上完全具有一个单数点。可以用空间中点的多项式方程的衍生物来描述奇异点的集合。出于许多目的，检测奇异性是不够的，因为有些人比其他差异更糟。例如，锥体的尖锐顶点被认为更糟。为了区分不同的奇异性，需要使用更复杂的代数技术。该研究项目旨在使用局部的共同体学模块研究奇异性，该模块可以看作是与某个点相关的代数对象。这已经被证明是检测不同种类奇异性的强大工具。首席研究人员计划使用当地的共同体来研究一个奇异点有多糟糕的测量。这项研究包括长期存在的奇点研究以及可能带来理论和计算后果的新猜想。该项目涉及研究生。首席研究者试图研究正面和混合特征中局部的共同体学模块和奇异性的结构。使用当地的共同体学模块时遇到的主要问题之一是它们通常很大且难以处理。但是，这些模块的行为似乎是在包含一个字段的常规本地环上有限生成的。最近发现了混合特征中的常规环的示例与同等特征的行为不同。受此结果的促进，首席研究人员打算探索有关相关素数和低音的局部共同体学模块数量的潜在的反审查，这些属性与常规混合特征的局部局部环相比。此外，主要研究人员计划研究以下相关的猜想：当地共同体学模块的支持是环形谱系中的Zariski封闭设置。 Lyubeznik利用局部的共同体学上的圈子，介绍了一个名为Lyubeznik数字的一群不变的家族。这些不变性显示了与环的代数和几何特性的多个连接。这启发了混合特征中对这些数字的类似定义。该项目旨在比较包含字段的环数与不包含字段的环数。特别是，该研究寻求拓扑或算术标准，该标准与Lyubeznik数字的两个概念有关。此外，该项目旨在找到由Lyubeznik数字编码的混合特征中编码的几何特性。最后，主要研究人员计划通过Frobenius Map来研究积极特征的奇异性。特别是，他正计划为F-Pure阈值及其推论的ACC猜想进行工作。此外，首席研究员和合作者将调查将F-Pure阈值与Hilbert-Kunz多重性相关的猜想不平等。如果这个项目成功，猜想的关系可能会带来几种计算和几何后果。