Computing and Interpreting Frobenius Invariants

计算和解释弗罗贝尼乌斯不变量

• 批准号: 1602070
• 负责人: Kevin Tucker
• 金额: $ 20.48万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1602070&HistoricalAwards=false
• 关键词: Computing; Interpreting; Frobenius; Invariants

Commutative algebra and algebraic geometry are among the oldest and yet most active disciplines in mathematics. The fields have strong ties to diverse areas of mathematics, and are also used in a wide variety of applied settings. This research project will explore important open questions in these fields, aiming for a deeper understanding of the singularities of varieties over positive characteristic number systems. These systems, those where a prime number vanishes, include the finite fields at the heart of essentially all electronic computation. While carrying out this research program, the investigator will also organize research seminars, support and supervise graduate student research, and conduct activities, including an annual symposium, to promote undergraduate research. Explicitly, the investigator plans to continue study of singularities of and invariants defined via the Frobenius map in positive characteristic commutative algebra. In particular, the research will focus on the circle of ideas surrounding F-signature, Hilbert-Kunz multiplicity, and test ideals, in two very different directions. First, in a fixed characteristic, the project aims to expand upon recent progress in computing these invariants, and to approach some long standing important open questions about them, such as the equivalence of weak and strong F-regularity. Second, the research aims to show these invariants have limiting values under reduction to positive characteristic. Here, it is hoped that the limits as the characteristic tends towards infinity have simpler interpretations and values, and can be related to geometric measures of singularities for complex algebraic varieties. The interaction with geometric methods in characteristic zero stemming from complex algebraic geometry is central to the project, and one of the main objectives of the work is to better describe the geometry and broader connections of F-invariants.

交换代数和代数几何是数学中最古老也是最活跃的学科。这些领域与数学的各个领域有着密切的联系，也被广泛应用于各种应用环境中。本研究项目将探索这些领域的重要开放问题，旨在更深入地了解正特征数系统上的品种奇点。这些系统，那些质数消失的系统，包括在本质上所有电子计算核心的有限域。在开展本研究计划的同时，研究者还将组织研究研讨会，支持和监督研究生的研究，并开展包括年度研讨会在内的活动，以促进本科生的研究。明确地，研究者计划继续研究正特征交换代数中由Frobenius映射定义的和不变量的奇异性。特别是，研究将集中在围绕f签名，Hilbert-Kunz多重性和测试理想的两个截然不同的方向上。首先，在一个固定的特征中，该项目旨在扩展计算这些不变量的最新进展，并接近一些长期存在的重要开放问题，例如弱和强f正则性的等价。其次，研究旨在证明这些不变量在约简为正特征时具有极限值。在这里，我们希望当特征趋于无穷时的极限有更简单的解释和值，并且可以与复杂代数变量的奇点的几何度量有关。复杂代数几何中特征零与几何方法的相互作用是该项目的核心，工作的主要目标之一是更好地描述f不变量的几何和更广泛的联系。

项目成果

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