Boundedness and termination

有界性和终止性

• 批准号: 1200656
• 负责人: James McKernan
• 金额: $ 73.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200656&HistoricalAwards=false
• 关键词: Boundedness; termination

The PI will work on the birational classification of algebraic varieties. The minimal model program is an ambitious program to classify varieties up to birational equivalence. To finish the program and to establish existence of minimal models, it suffices to show termination of flips. The PI will use recent results concerning ACC for the log canonical threshold to attack termination. A related project is to study birational boundedness of Fano varieties, especially the conjecture of Borisov-Alexeev-Borisov. Birational geometry also offers a potential way to study vector bundles on projective space, to be used as a means to attack Hartshorne's conjectures. Finally, the PI also plans to study the connection between birational geometry, foliations and the abundance conjecture.Algebraic Geometry is one of the oldest and most challenging of areas of research in mathematics, which combines some very classicial geometry, for example that of conic sections and the more modern techniques of algebra. There has been a lot exciting recent work in higher dimensional geometry. The PI will write a survey article for the Proceedings of the Royal Society on this work which is intended for a general scientific literate audience and the PI will also try to impart some of the exciting research in algebraic geometry to undergraduate and graduate students in his teaching.

PI 将致力于代数簇的双有理分类。 最小模型程序是一个雄心勃勃的程序，用于将品种分类为双有理等价。 为了完成程序并建立最小模型的存在性，只需显示翻转的终止即可。 PI 将使用有关 ACC 的最新结果作为终止攻击的​​日志规范阈值。 一个相关的项目是研究 Fano 簇的双有理有界性，特别是 Borisov-Alexeev-Borisov 猜想。 双有理几何还提供了一种研究射影空间上矢量丛的潜在方法，可用作攻击哈特肖恩猜想的手段。 最后，PI还计划研究双有理几何、叶状结构和丰度猜想之间的联系。代数几何是数学研究中最古老和最具挑战性的领域之一，它结合了一些非常经典的几何学，例如圆锥曲线的几何学以及更现代的代数技术。 最近在高维几何领域出现了许多令人兴奋的工作。 PI 将在《英国皇家学会学报》上撰写一篇关于这项工作的调查文章，该文章面向一般具有科学素养的读者，同时 PI 还将尝试在教学中向本科生和研究生传授一些令人兴奋的代数几何研究成果。