Singularities in Arbitrary Characteristic and Positivity

任意特性和积极性中的奇点

• 批准号: 2201251
• 负责人: Takumi Murayama
• 金额: $ 18.34万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201251&HistoricalAwards=false
• 关键词: Singularities; Arbitrary; Characteristic; Positivity

The project concerns questions in algebraic geometry and commutative algebra. Algebraic geometry is the study of algebraic varieties, which are solution sets for systems of polynomial equations. For example, in the xy-plane, the solutions for y=0 consist of all points along the x-axis, while the solutions for xy=0 consist of all points along both coordinate axes. Since the tangent line at the origin (0,0) is not defined for the algebraic variety defined by xy=0, we say that this variety has a singularity at the origin (0,0). The first goal of the present project is to further develop the theory of singularities of algebraic varieties and more general objects. The second goal is to build the tools and techniques necessary for this research program, which would have applications to other fundamental open questions in algebraic geometry and commutative algebra.The research will investigate singularities of algebraic varieties over arbitrary fields, or more generally of arbitrary Noetherian rings and schemes. Even when the primary interest is in non-singular complex algebraic varieties, working in this much more general context is often unavoidable. A major focus in the present project is to build the foundations necessary for birational geometry and the minimal model program for schemes of arbitrary characteristic. In previous work, the PI proved that Kodaira-type vanishing theorems hold for schemes in equal characteristic zero of arbitrary dimension and developed new methods to replace Kodaira-type vanishing theorems in positive characteristic. In the present project, the PI will apply the techniques from this previous work to study the behavior of singularities under flat morphisms. The PI will also investigate inversion of adjunction-type results for various classes of singularities. Finally, the PI plans to extend techniques in positive characteristic to study positivity of line bundles on algebraic varieties over arbitrary fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目涉及代数几何和交换代数的问题。代数几何是研究代数簇的学科，代数簇是多项式方程组的解集。例如，在xy平面中，y=0的解由沿x轴沿着的所有点组成，而xy=0的解由沿两个坐标轴沿着的所有点组成。由于在原点（0，0）处的切线对于xy=0定义的代数簇没有定义，我们说这个簇在原点（0，0）处有一个奇点。本项目的第一个目标是进一步发展代数簇和更一般对象的奇点理论。第二个目标是建立必要的工具和技术，这个研究计划，这将有应用到其他基本的开放问题在代数几何和交换代数。研究将调查奇异的代数簇在任意领域，或更一般的任意Noether环和计划。即使主要的兴趣是在非奇异复代数簇，在这个更一般的背景下工作往往是不可避免的。本项目的一个主要重点是建立必要的基础，双有理几何和最小模型计划的任意特性。在以前的工作中，PI证明了Kodaira型消失定理对任意维数的等特征零点格式成立，并发展了新的方法来取代Kodaira型消失定理的正特征。在本项目中，PI将应用以前工作中的技术来研究平坦态射下奇点的行为。PI还将研究各类奇异点的附加类型结果的反演。最后，PI计划扩展技术的积极特征，以研究积极的线丛代数簇在任意field.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。