Birational Geometry: Invariants, Reconstruction, and Deformation Problems

这个项目是在代数几何，现代数学的一个活跃分支，起源于丢番图方程理论。主要研究对象是代数变量，即多变量多项式方程组的解集。该项目的目标是开发方法来测量这些方程的解的无约束程度。更准确地说，它旨在理解代数变量的几个不变量，这些不变量是用代数几何、算术几何和数论的现代技术构造的。该项目包括研究生培训的具体问题，并结合了计算资源和代数软件包的使用。更详细地说，PI将使用函数场的伽罗瓦上同调方法，非代数闭域上代数循环的性质，k理论和退化技术来证明以下方向的结果：(1)代数变量族的合理性和稳定合理性，特别是对于del Pezzo曲面的四倍纤维；(2)等变双分不变量；(3)代数循环的Chow群的变分，特别是在数域上；(4)从集合理论数据重构民族类。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.