Rationality and Stable Rationality of Algebraic Varieties

本课题主要研究多元非线性代数方程组解的性质。这一领域的一个主要开放问题是确定这些解空间何时可以由独立变量参数化；这可以用于线性或二次方程，但在三次方程中已经未知。这样的代数问题具有广泛的应用，例如，在数据科学、优化或机器人中。将这一问题转化为几何问题，通过关注由方程系统定义的形状的几何不变量，打开了大量更直观的技术的大门：研究小变形或极限形状。该研究项目的主要目的是在2维和3维上理解这种参数化性质，即研究非闭域上Del Pezzo曲面和Fano三重曲面的合理性和稳定合理性。需要解决的具体问题包括：理性和稳定理性的障碍，如未分枝上同调、对角线和Burnside群的积分分解；代数变种家族中理性的变化和专门化；等变二元几何中的不变量，特别是等变Burnside群，以及它在Cremona群研究中的应用。该奖项反映了NSF的法定使命，并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，被认为值得支持。

This project is concerned with the study of properties of solutions of systems of nonlinear algebraic equations in many variables. A major open problem in this field is to determine when these solution spaces can be parametrized by independent variables; it can be done for linear or quadratic equations, but is already unknown in degree three. Such algebraic questions have wide-ranging applications, e.g., in data science, optimization, or robotics. Translating this problem into geometry, by focusing on geometric invariants of shapes defined by the systems of equations, opens the door to a plethora of more intuitive techniques: the study of small deformations, or limit shapes. The project also provides research training opportunities for graduate students.The main goal of the research project is to understand this parametrization property in dimensions 2 and 3, i.e., to study rationality and stable rationality of Del Pezzo surfaces and Fano threefolds over nonclosed fields. Among concrete problems to be addressed are: Obstructions to rationality and stable rationality, such as unramified cohomology, integral decomposition of the diagonal, and Burnside groups; Variation of rationality in families of algebraic varieties, and specialization; Invariants in equivariant birational geometry, in particular, the equivariant Burnside group, as well as its applications to the study of the Cremona group.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.