FRG: Collaborative Research: Birational Geometry and Singularities in Zero and Positive Characteristic

FRG：协作研究：双有理几何和零特征和正特征中的奇点

• 批准号: 1265261
• 负责人: Karl Schwede
• 金额: $ 22.91万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2014-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265261&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Birational; Geometry

The goal of this project is to bring together a group of researchers with experience in a broad array of topics in higher dimensional algebraic geometry, in order to make progress in two closely related areas: birational geometry in positive characteristics and the theory of singularities and linear series arising in the minimal model program. While there has been a lot of recent progress in our understanding of the geometry of higher-dimensional varieties, almost all of this body of work is restricted to characteristic zero, due to the use of vanishing theorems that can fail in positive characteristic. The first goal of this collaborative research is to build tools and a framework that would allow the main results in birational geometry to be extended to positive characteristics. This would make systematic use of the recent techniques that have been devised to exploit the Frobenius morphism. A second goal of the project is to further develop the study of invariants of singularities and of linear series, with an eye toward the remaining problems in the minimal model program. There have been many recent advances in this area. In particular, a conjecture of Shokurov asserting that certain invariants of singularities (the log canonical thresholds) satisfy the so-called ACC property has been solved by some of the PIs. This suggests that other related but harder questions might be within reach, questions whose importance comes from the connection with one of the remaining conjectures in the minimal model program, the termination of flips. The PIs propose to attack one of these problems, predicting the ACC property of another invariant, the minimal log discrepancy. In a separate direction, the PIs plan to undertake a systematic study of examples of linear systems on algebraic varieties that exhibit a pathological behavior from the point of view of various positivity invariants.The last ten years have seen major breakthroughs in the study of higher-dimensional algebraic varieties, but several important problems are still open. A central such problem is intimately related to the study of singularities and one of the goals of this project is to make progress on understanding the properties of the invariants of singularities that appear in this setting. Another general goal of the PIs is to develop systematically the study of algebraic varieties of dimension at least 3 in positive characteristic. A lot less is known in this setting, where new phenomena (sometimes considered pathological) arise. The PIs expect that cross-pollination of ideas with other areas, in particular with commutative algebra, will play an important role in making progress in this direction. Moreover, it is likely that techniques and results in this context would have many applications to other fields (for example, in arithmetic geometry). As part of the collaborative effort, the PIs plan several events that will bring together members of the mathematical community working on related problems and also help disseminate the results and the techniques developed as part of this project.

该项目的目的是将一群具有在较高维代数几何形状的广泛主题中经验的研究人员组合在一起，以便在两个紧密相关的领域中取得进展：积极特征和奇异性和线性序列中产生的奇异性和线性序列。 尽管我们对高维品种的几何形状的理解取得了很大的进步，但由于使用消失的定理可能会在积极特征上失败，因此几乎所有这些作品都限于特征零。 这项协作研究的第一个目标是建立工具和框架，该框架将使赋予生育几何形状的主要结果扩展到积极的特征。 这将系统地使用用于利用Frobenius形态的最新技术。 该项目的第二个目标是进一步发展奇异性和线性系列的不变性研究，并着眼于最小模型计划中的剩余问题。 该领域最近有许多进步。特别是，shokurov的猜想断言某些不变性（原木规范阈值）满足所谓的ACC特性，已由某些PI解决了。 这表明其他相关但更困难的问题可能涉及到涉及的问题，其重要性来自与最小模型计划中其余的猜想之一的联系，即翻转的终止。 PI提议攻击这些问题之一，预测另一个不变的ACC特性，即最小的日志差异。 在一个单独的方向上，PIS计划对代数品种的线性系统实例进行系统的研究，这些示例从各种阳性不变的角度表现出病理行为。过去十年来，在高维代数品种的研究中，有一些重大突破，但是几个重要问题仍然开放。 一个核心的问题与研究奇异性的研究密切相关，该项目的目标之一是在理解这种环境中出现的奇异事物的属性方面取得进展。 PI的另一个一般目标是系统地开发对尺寸的代数品种至少3种阳性特征的研究。 在这种情况下，新现象（有时被认为是病理）的情况少得多。 PI期望与其他领域的思想交叉授粉，尤其是与换向代数的交叉授粉，将在朝这个方向取得进步方面发挥重要作用。 此外，在这种情况下，技术和结果很可能会在其他字段（例如，在算术几何形状中）具有许多应用程序。 作为协作努力的一部分，PIS计划了几项活动，这些活动将汇集数学社区的成员，从事相关问题，并有助于传播结果和作为该项目的一部分开发的技术。