Singularities in Characteristic Zero and Singularities in Positive Characteristic

特征零奇点和正特征奇点

• 批准号: 1064485
• 负责人: Karl Schwede
• 金额: $ 10.54万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1064485&HistoricalAwards=false
• 关键词: Singularities; Characteristic; Zero; Positive

The program proposed by Karl Schwede aims to study relations between higher dimensional algebraic geometry and positive characteristic commutative algebra. Over the past 30 years, a dictionary has been developed linking, by reduction to characteristic p, seemingly unrelated concepts coming from these two distinct areas of mathematics. However, the dictionary as it exists seems incomplete. For example, while there is a good positive characteristic analog of ``log terminal singularities'', there is no known analog of ``terminal singularities''. Schwede will attempt to fill in these gaps. Furthermore, inspired by this dictionary, Schwede will explore both the local and global geometry of varieties in positive characteristic and characteristic zero. One problem that Schwede plans to explore is how the test ideal (a positive characteristic analog of the multiplier ideal) behaves under birational maps. Another problem that Schwede plans to study is whether Du Bois singularities deform.Algebraic geometry is a centrally important and very active field of mathematics with strong ties to many other areas including fields as disparate as string theory and coding theory. Explicitly, algebraic geometry is the study of geometric objects (called algebraic varieties) made up of the solutions to polynomial equations (such as y = x^2). At the most basic level, Schwede plans to study relations between the geometry of these algebraic varieties, with algebraic properties of the equations themselves. In the past, this interplay has led to new insights in both the geometric and algebraic theories. In algebraic geometry, one of the major areas of research in the last 30 years has been the classification of algebraic varieties -- the "minimal model program". In order to accomplish this, one must study singular varieties (an example of a singular variety is the solution set to the quadric cone, z^2 = x^2 + y^2). The particular questions that Schwede proposes to study will hopefully lead to a deeper understanding of the varieties and the singularities that appear in this classification.

Karl Schwede提出的计划旨在研究更高维代数几何形状和积极特征交换代数之间的关系。 在过去的30年中，通过简化特征P的链接开发了词典，该词典似乎来自这两个不同的数学领域。 但是，其存在的字典似乎不完整。 例如，虽然``对数末端奇点''有一个很好的积极特征类似物，但尚无``终端奇点''的类似物。 Schwede将尝试填补这些空白。 此外，灵感来自该字典的启发，Schwede将探索积极特征和特征为零的品种的局部和全球几何形状。 Schwede计划探索的一个问题是测试理想（对乘数理想的积极特征类似物）如何在Birational图下行为。 Schwede计划研究的另一个问题是Du Bois的奇异性是否变形。代数几何学是一个非常重要且非常活跃的数学领域，与许多其他领域有着牢固的联系，包括诸如弦乐理论和编码理论等领域。 明确的，代数几何形状是由多项式方程（例如y = x^2）组成的几何对象（称为代数品种）的研究。 在最基本的层面上，Schwede计划研究这些代数品种的几何形状与方程本身的代数特性之间的关系。 过去，这种相互作用导致了几何和代数理论的新见解。 在代数几何形状中，过去30年中的主要研究领域之一是代数品种的分类 - “最小模型计划”。 为了实现这一目标，必须研究奇异品种（单数品种的示例是将溶液设置为四锥，z^2 = x^2 + y^2）。 Schwede提议研究的特定问题希望将对这种分类中出现的品种和奇异性有更深入的了解。