Singularities in Characteristic Zero and Singularities in Positive Characteristic

特征零奇点和正特征奇点

• 批准号: 0969145
• 负责人: Karl Schwede
• 金额: $ 10.54万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 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 2010-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0969145&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高维代数几何与正特征交换代数之间的关系。 在过去的 30 年里，一本字典被开发出来，通过简化为特征 p，将来自这两个不同数学领域的看似不相关的概念联系起来。 然而，现有的词典似乎并不完整。 例如，虽然“对数终端奇点”有一个良好的正特性类似物，但没有已知的“终端奇点”类似物。 Schwede 将尝试填补这些空白。 此外，受这本词典的启发，Schwede 将探索正特征和特征零的品种的局部和全局几何形状。 Schwede 计划探索的一个问题是测试理想（乘数理想的正特征模拟）在双有理映射下的表现。 施韦德计划研究的另一个问题是杜波依斯奇点是否变形。代数几何是一个非常重要且非常活跃的数学领域，与许多其他领域（包括弦理论和编码理论等不同领域）有着密切的联系。 明确地说，代数几何是对由多项式方程（例如 y = x^2）的解组成的几何对象（称为代数簇）的研究。 在最基本的层面上，施韦德计划研究这些代数簇的几何形状与方程本身的代数性质之间的关系。 过去，这种相互作用导致了几何和代数理论的新见解。 在代数几何中，近30年的主要研究领域之一是代数簇的分类——“最小模型程序”。 为了实现这一目标，必须研究奇异簇（奇异簇的一个例子是二次锥的解集，z^2 = x^2 + y^2）。 施韦德建议研究的特定问题有望使人们更深入地了解该分类中出现的品种和奇点。