Local Cohomology and Singularities

首席研究员计划使用交换代数中的不同技术来研究几何问题。本研究的重点是空间的一组点，满足某些多项式方程在许多变量。由于许多现象可以用多项式方程来描述，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，大多数点都是所谓的“平滑”，粗略地说，这意味着放大后，它们的附近看起来像线性空间。例如，在一个球体中，每个点都是光滑的，就像地球一样，从非常近的角度看，它的邻居看起来像一个平面。然后，那些不光滑的点呈现出特定的行为，因此被称为“奇点”。例如，一个圆锥在其顶点上只有一个奇点。奇异点的集合可以用空间中的点所满足的多项式方程的导数来描述。对于许多目的，检测奇点是不够的，因为有些比其他更糟糕。例如，圆锥体的较尖锐顶点被认为是更差的。为了区分不同的奇点，需要使用更复杂的代数技巧。这个研究项目试图使用局部上同调模来研究奇点，局部上同调模可以被看作是与一个点相关联的代数对象。这已经被证明是一个强大的工具来检测不同种类的奇点。首席研究员计划使用局部上同调来研究奇点有多坏的测量。这项研究包括奇点研究中长期存在的问题，以及可能产生理论和计算后果的新发现。该项目涉及研究生的研究。主要研究局部上同调模的结构和正特征及混合特征的奇点。在处理局部上同调模时遇到的主要问题之一是它们通常非常大并且难以处理。然而，这些模块的行为就好像它们是在包含字段的规则局部环上生成的。最近发现了一个混合特征正则环的内射维数与相等特征维数不同的例子，基于这个结果，本文主要研究了混合特征正则局部环上局部上同调模的相关素数和Bass数的性质.此外，主要研究者计划致力于以下相关猜想：局部上同调模的支撑是环的谱中的Zebrski闭集。利用包含域的环上的局部上同调，Lyubeznik引入了一族不变量，现在称为Lyubeznik数。这些不变量与环的代数和几何性质有着密切的联系。这启发了一个类似的定义，这些数字的混合特征。该项目旨在比较包含字段的环与不包含字段的环的Lyubeznik数。特别是，研究寻求一个拓扑或算术标准，涉及这两个概念的Lyubeznik数。此外，该项目还试图找到由Lyubeznik数编码的混合特征几何属性。最后，首席研究员计划通过Frobenius映射研究正特性的奇点。特别是，他正计划研究ACC猜想的F-纯阈值及其推论。此外，主要研究者和合作者将研究一个将F-纯阈值与希尔伯特-昆兹多重性联系起来的固定不等式。如果这个项目成功，这个被约束的关系可能会产生一些计算和几何上的后果。

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.