CAREER: Singularities in the Minimal Model Program and Birational Geometry

职业：最小模型程序和双有理几何中的奇点

• 批准号: 0847059
• 负责人: Tommaso de Fernex
• 金额: $ 40万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0847059&HistoricalAwards=false
• 关键词: CAREER; Singularities; Minimal; Model; Program

The proposed research is in the field of algebraic geometry. It focuses on a series of problems in classification theory that are connected by the use of similar tools coming from singularity theory and the minimal model program. The project has three main objectives:(1) Investigate abstract properties of numerical invariants of singularities and their impact in birational geometry, and thus address longstanding open problems of primary importance in the field such as Shokurov's ACC Conjectures and the Termination of Flips Conjecture. (2) Determine rigidity properties of the birational geometric structure of certain class of projective Fano hypersurfaces, and use variations of these properties to introduce a new point of view in measuring various degrees of non-rationality. (3) Study different aspects concerning the behavior of curves on Fano varieties, especially in connection to the deformations of the variety and within the context of the minimal model program: expected deformation properties, if satisfied, would bring to light a rather surprising rigidity nature of Fano varieties. Other more speculative projects, notably one regarding a possible connections between Shokurov's ACC Conjecture and resolution of singularities via normalized Nash blow-ups, and one on the behavior of the Cox ring in families of Fano varieties, are outlined throughout the proposal.The minimal model program, the main program geared towards the birational classification of algebraic varieties, is the general framework of the proposed research. Designed over the model of classification of surfaces (one of the main achievements of the Italian school at the beginning of the 20th century), the minimal model program has become one of the major trends in algebraic geometry, earning S. Mori the Fields Medal for is work on three dimensional varieties. Several leading mathematicians have deeply contributed to the program throughout the years, and very recently there has been some spectacular progress which is bringing us very close to a complete program in all dimensions. There are however many fundamental questions that still remain open, and some of these constitute the main objectives of the proposed research. A series of educational activities are also proposed: the range of such activities covers a large spectrum, from of high school students to peer researchers, passing through undergraduate students, graduate students, and young researchers.

拟议的研究是在代数几何领域。它集中讨论了分类理论中的一系列问题，这些问题通过使用奇点理论和最小模型程序中的相似工具而联系在一起。该项目有三个主要目标：(1)研究奇点数值不变量的抽象性质及其在二次几何中的影响，从而解决该领域长期存在的主要开放问题，如Shokurov的ACC猜想和Flips猜想的终止。(2)确定了一类射影Fano超曲面的双曲面几何结构的刚性性质，并利用这些性质的变化为度量不同程度的非理性提供了一种新的观点。(3)研究Fano变种上曲线行为的不同方面，特别是与变形性有关的Fano变种，并在最小模型程序的背景下：如果满足预期变形性质，将揭示Fano变种相当惊人的刚性性质。其他更具投机性的项目，特别是关于Shokurov的ACC猜想和通过归一化Nash爆破分解奇点之间的可能联系的项目，以及关于Fano簇族中Cox环的行为的项目，在整个提案中都被概述。最小模型程序是主要程序，主要用于代数簇的二元性分类，是所提议的研究的总体框架。在曲面分类模型(20世纪初意大利学派的主要成就之一)的基础上设计的最小模型程序已成为代数几何的主要趋势之一，为S.Mori赢得了菲尔兹奖，以表彰其在三维变种方面的工作。多年来，几位顶尖的数学家为该计划做出了巨大贡献，最近取得了一些令人瞩目的进展，使我们非常接近于一个全面的计划。然而，仍有许多基本问题仍然悬而未决，其中一些构成了拟议研究的主要目标。还提出了一系列的教育活动：这些活动的范围很广，从高中生到同行研究人员，通过本科生、研究生和青年研究人员。