Birational Geometry: Invariants, Reconstruction, and Deformation Problems

双有理几何：不变量、重构和变形问题

• 批准号: 2201195
• 负责人: Alena Pirutka
• 金额: $ 28万
• 依托单位: New York 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: 2022-04-15 至 2025-03-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201195&HistoricalAwards=false
• 关键词: Birational; Geometry; Invariants; Reconstruction; Deformation

This project is in algebraic geometry, an active branch of modern mathematics, originating in the theory of diophantine equations. The main objects of study are algebraic varieties, which are solution sets of systems of polynomial equations in several variables. The goal of the project is to develop methods to measure how unconstrained the solutions of these equations are. More precisely, it aims at understanding several invariants of algebraic varieties, constructed by modern techniques using methods from algebraic geometry, arithmetic geometry, and number theory. The project includes concrete problems for graduate student training and incorporates the use of computational resources and algebraic software packages. In more detail, the PI will use methods from Galois cohomology of function fields, properties of algebraic cycles over non-algebraically closed fields, K-theory, and degeneration techniques, to prove results in the following directions: (1) rationality and stable rationality for families of algebraic varieties, in particular for fourfolds fibered by del Pezzo surfaces; (2) equivariant birational invariants; (3) variation of Chow groups of algebraic cycles, in particular over number fields; and (4) reconstruction of birational classes from set-theoretical data.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.

该项目位于代数几何形状，这是现代数学的活跃分支，起源于二磷理论。研究的主要对象是代数品种，它们是多个变量中多项式方程系统的解决方案集。该项目的目的是开发方法来衡量这些方程式的不受约束的解决方案。更确切地说，它旨在理解几种代数品种的不变性，这些品种是由现代技术使用代数几何，算术几何形状和数字理论的方法构建的。该项目包括用于研究生培训的具体问题，并结合了计算资源和代数软件包的使用。 更详细地，PI将使用功能领域的GALOIS共同学，代数周期的属性，非代数封闭的领域的特性，K理论和退化技术在以下方向上证明：（1）代数的合理性和稳定的合理性，尤其是Pebirs for for for for fy fy for pefipers fyfiper fy fy fy fyf peef fy fyf peef fyf peelf。 （2）模棱两可的男性不变； （3）代数循环的食物组的变化，尤其是数量场； （4）从集合理论数据中重建了男性班级。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来获得支持。