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.

这个项目是在代数几何，一个活跃的分支现代数学，起源于理论的丢番图方程。主要研究对象是代数簇，它是多元多项式方程组的解集。该项目的目标是开发方法来衡量这些方程的解有多不受约束。更确切地说，它旨在理解代数簇的几个不变量，这些不变量是通过使用代数几何、算术几何和数论的方法的现代技术构造的。该项目包括研究生培训的具体问题，并结合使用计算资源和代数软件包。 更详细地说，PI将使用函数域的Galois上同调，非代数闭域上代数圈的性质，K-理论和退化技术的方法来证明以下方向的结果：（1）代数簇族的合理性和稳定合理性，特别是四重纤维化的del Pezzo曲面;（2）等变双有理不变量;（3）代数圈的Chow群的变化，特别是在数域上;（4）从集合理论数据重建双有理类。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。