Birational geometry of algebraic varieties

代数簇的双有理几何

• 批准号: 1300750
• 负责人: Christopher Hacon
• 金额: $ 47.77万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300750&HistoricalAwards=false
• 关键词: Birational; geometry; algebraic; varieties

The proposed research deals with topics in Higher dimensional Algebraic Geometry. It is mainly focused on natural questions in the birational geometry of projective varieties such as the study of questions related to the minimal model program. The PI will investigate the existence of minimal models for varieties not of general type, abundance and the termination of flips, the existence of flips for 3-folds in positive characteristic, the geometry of Fano varieties (in particular the boundedness of log-Fanos), the singularities of normal varieties and multiplier ideal sheaves, the pluricanonical maps of algebraic varieties and the birational geometry of irregular varieties.The birational classification of surfaces was understood by the Italian school at the beginning of the twentieth century. If a surface is not covered by rational curves then there is a natural choice of a birational surface (known as the minimal model) which has many useful properties. The Minimal Model Program aims to generalize these results to higher dimensions. This program is complete in dimension 3 (by work of Mori, Kawamata, Kollar, Reid, Shokurov and others) and is known to work for complex projective varieties of general type (by work of Birkar, Cascini, Hacon, McKernan, Siu and others). This project hopes to answer some of the remaining questions and conjectures that naturally arise in this context.

提出的研究涉及高维代数几何的主题。它主要集中于射影变量的自然几何问题，如与最小模型规划有关的问题的研究。PI将研究非一般型品种的极小模型的存在性，翻转的丰度和终止性，正特征中3倍翻转的存在性，Fano品种的几何（特别是log-Fanos的有界性），正规品种和乘子理想束的奇异性，代数品种的多分形映射和不规则品种的双分形几何。二十世纪初，意大利学派理解了曲面的出生地分类。如果一个曲面没有被有理曲线覆盖，那么自然会选择一个双曲面（称为最小模型），它有许多有用的性质。最小模型程序旨在将这些结果推广到更高的维度。该程序在3维中完成（由Mori， Kawamata, Kollar, Reid， Shokurov等人完成），并且已知可用于一般类型的复杂投影变体（由Birkar， Cascini, Hacon, McKernan， Siu等人完成）。这个项目希望回答在这种情况下自然产生的一些剩余的问题和猜想。