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: 1523233
• 负责人: James McKernan
• 金额: $ 10.64万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1523233&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.

该项目的目标是汇集一组在高维代数几何领域具有广泛主题经验的研究人员，以便在两个密切相关的领域取得进展：正特性的双有理几何以及奇点和线性理论系列出现在最小模型程序中。 虽然我们对高维簇几何的理解最近取得了很多进展，但由于使用了可能在正特征上失败的消失定理，几乎所有这些工作都仅限于特征零。 这项合作研究的首要目标是构建工具和框架，使双有理几何的主要结果能够扩展到积极的特征。 这将系统地利用最近为利用弗罗贝尼乌斯态射而设计的技术。 该项目的第二个目标是进一步发展奇点和线性级数不变量的研究，着眼于最小模型程序中的剩余问题。 该领域最近取得了许多进展。特别是 Shokurov 的一个猜想，即某些奇点不变量（对数正则阈值）满足所谓的 ACC 性质，已被一些 PI 解决。 这表明其他相关但更难的问题可能是可以解决的，这些问题的重要性来自于与最小模型程序中剩余猜想之一（翻转的终止）的联系。 PI 建议解决其中一个问题，预测另一个不变量的 ACC 属性，即最小对数差异。 在一个单独的方向上，PI 计划对代数簇上的线性系统的例子进行系统的研究，这些例子从各种正不变量的角度表现出病态行为。在过去的十年里，高等代数簇的研究取得了重大突破。维代数簇，但几个重要问题仍然悬而未决。 这样的一个核心问题与奇点的研究密切相关，该项目的目标之一是在理解这种情况下出现的奇点不变量的性质方面取得进展。 PI 的另一个总体目标是系统地开展对正特征至少为 3 维的代数簇的研究。 在这种新现象（有时被认为是病态的）出现的环境中，人们所知甚少。 PI 预计，思想与其他领域（尤其是交换代数）的交叉传播将在这一方向取得进展中发挥重要作用。 此外，这种情况下的技术和结果很可能在其他领域（例如算术几何）有许多应用。 作为协作努力的一部分，PI 计划了几项活动，将数学界的成员聚集在一起解决相关问题，并帮助传播作为该项目一部分开发的结果和技术。