CAREER: Test Ideals and the Geometry of Projective Varieties in Positive Characteristic

职业：检验理想和正特征中射影多样性的几何

• 批准号: 1252860
• 负责人: Karl Schwede
• 金额: $ 44.26万
• 依托单位: Pennsylvania State Univ University Park
• 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: 2013-09-01 至 2014-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1252860&HistoricalAwards=false
• 关键词: CAREER; Test; Ideals; Geometry; Projective

The Principal Investigator will explore questions in characteristic p 0 algebraic geometry and commutative algebra. Explicitly, he proposes to apply methods of commutative algebra to study adjoint line bundles in characteristic p 0, especially to produce global sections. Such line bundles are crucial when classifying the central objects of algebraic geometry. Over the complex numbers, the standard techniques to study adjoint line bundles include Kodaira-type vanishing theorems. These theorems are unavailable in characteristic p 0, thus methods of the Frobenius morphism and test ideals from commutative algebra will be employed instead. The PI also proposes to use methods of algebraic geometry to study questions related to singularities in commutative algebra. The Principal Investigator will also run several mathematics camps for high school students. He will employ students to help develop software related to test ideals and singularities in characteristic p 0. Finally he will continue to organize conferences.Algebraic geometry is a very active field of mathematics. Explicitly, it is the study of geometric objects (called algebraic varieties) made up of the solutions to polynomial equations (for example, the parabola is the solution to y = x^2). Characteristic p 0 algebraic geometry is the study of these equations and their solutions when the underlying rules of arithmetic have changed. For example, instead of simply 4 + 4 = 8, in characteristic p = 5, once we get to 5 we start counting at 1 again so that 4 + 4 = 3 (like counting hours of a clock). This type of arithmetic and related algebra and geometry is heavily employed in modern cryptography and coding theory. The running of camps, conferences and the development of software will also help train the next generation of researchers.

主要研究者将探索特征P 0代数几何和交换代数中的问题。 明确地，他建议采用交换代数的方法来研究特征p 0中的伴随线束，尤其是为了产生全球部分。 当对代数几何形状的中心对象分类时，此类捆绑包至关重要。 在复数上，研究伴随线束的标准技术包括Kodaira型消失定理。 这些定理在特征P 0中不可用，因此将采用Frobenius形态的方法和来自交换代数的理想的方法。 PI还建议使用代数几何方法来研究与奇异性代数相关的问题。 首席调查员还将为高中生开设几个数学营地。 他将雇用学生来开发与特征p 0中的理想和奇异性有关的软件。最后，他将继续组织会议。代理几何是一个非常活跃的数学领域。 明确的是，它是由多项式方程的解决方案组成的几何对象（称为代数品种）（例如，抛物线是y = x^2的解决方案）。特征P 0代数几何是对这些方程式及其解决方案的研究，当时算术的基本规则发生了变化。 例如，在特征p = 5中而不是简单的4 + 4 = 8，一旦到达5，我们就开始以1个开始计数，以便4 + 4 = 3（如时钟的计数小时）。这种类型的算术和相关代数和几何形状在现代密码学和编码理论中大量采用。 营地，会议和软件的开发也将有助于培训下一代研究人员。