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: 1265263
• 负责人: James McKernan
• 金额: $ 16.08万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2015-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265263&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.

这个项目的目标是汇集一组研究人员的经验，在广泛的主题在高维代数几何，以取得进展，在两个密切相关的领域：双有理几何的积极特点和理论的奇点和线性系列中产生的最小模型程序。 虽然最近在我们对高维簇的几何的理解上有了很多进展，但由于使用了消失定理，几乎所有的工作都局限于特征零，这可能会在正特征中失败。 这项合作研究的第一个目标是建立工具和框架，使双有理几何的主要成果扩展到积极的特点。 这将系统地利用最近的技术，已被设计来利用弗罗贝纽斯态射。 该项目的第二个目标是进一步发展奇点和线性级数的不变量的研究，着眼于最小模型程序中的剩余问题。 最近在这一领域取得了许多进展。特别是，Shokurov的一个猜想声称，某些不变量的奇点（对数正则阈值）满足所谓的ACC属性已经解决了一些PI。 这表明，其他相关的但更难的问题可能是触手可及的，这些问题的重要性来自于与最小模型程序中剩余的一个问题的联系，翻转的终止。PI提出攻击这些问题之一，预测ACC属性的另一个不变，最小的日志差异。 在一个单独的方向，PI计划进行一个系统的研究的例子，线性系统的代数簇，表现出病态的行为，从角度来看，各种积极的不变量。在过去的十年里，已经看到了重大突破，在研究高维代数簇，但一些重要的问题仍然是开放的。 这样的一个中心问题是密切相关的奇点的研究和这个项目的目标之一是取得进展，了解在这种情况下出现的奇点的不变量的属性。 PI的另一个总体目标是系统地发展至少3维正特征的代数簇的研究。 在这种情况下，人们对新现象（有时被认为是病态的）的了解要少得多。 PI希望与其他领域的思想交叉授粉，特别是与交换代数，将在这一方向取得进展方面发挥重要作用。 此外，这方面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。 作为合作努力的一部分，PI计划举办几次活动，将数学界的成员聚集在一起研究相关问题，并帮助传播作为该项目一部分开发的成果和技术。