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。本文主要研究由满足多变量多项式方程的点集所给出的空间。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“光滑的”，粗略地说，这意味着在放大后，它们的附近看起来像一个线性空间。例如，在一个球体中，每个点都是光滑的，就像地球一样，从非常近的角度来看，它的周围看起来像一个平面。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，一个锥体的顶点恰好有一个奇点。奇异点的集合可以用空间中点所满足的多项式方程的导数来描述。对于许多目的来说，检测到奇点是不够的，因为有些奇点比其他奇点更糟糕。例如，锥体的尖角被认为是较差的。为了区分不同的奇点，需要使用更复杂的代数技术。本研究项目旨在利用局部上同调模来研究奇点，局部上同调模可以看作是与点相关的代数对象。这已经被证明是一种检测不同奇异点的强大工具。首席研究员计划使用局部上同学来研究奇点有多糟糕的测量。该研究包括奇点研究中长期存在的问题，以及可能具有理论和计算后果的新猜想。这个项目有研究生参与研究。主要研究正、混合特性下局部上同模和奇异的结构。使用局部上同调模块时遇到的主要问题之一是它们通常非常大且难以处理。然而，这些模块的行为就好像它们是在包含字段的正则局部环上有限地生成的。本文给出了一个正则环混合特征的例子，其内射维与等特征不同。在此结果的激励下，本课题拟探索混合特征正则局部环上局部上同模的相关素数和Bass数性质的潜在反例。此外，项目负责人还计划对以下相关猜想进行研究：环谱上局部上同模的支持是Zariski闭集。利用包含域的环上的局部上同，Lyubeznik引入了一组不变量，现在称为Lyubeznik数。这些不变量与环的代数和几何性质有若干联系。这启发了在混合特性中对这些数的类似定义。该项目旨在比较包含油田和不包含油田的环的柳别兹尼克数。特别是，该研究寻求一种拓扑或算术准则，将吕别兹尼克数的两个概念联系起来。此外，该项目寻求寻找混合特征中由柳别兹尼克数编码的几何性质。最后，首席研究员计划通过Frobenius地图研究正特征的奇点。特别是，他计划研究f纯阈值的ACC猜想及其推论。此外，首席研究员和合作者将研究一个推测的不等式，该不等式将f纯阈值与Hilbert-Kunz多重性联系起来。如果这个项目成功，推测的关系可能会有几个计算和几何结果。

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.