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Rationality and Stable Rationality of Algebraic Varieties

代数簇的有理性和稳定有理性

基本信息

批准号：
2000099
负责人：
Yuri Tschinkel
金额：
$ 31.5万
依托单位：
New York University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000099&HistoricalAwards=false
关键词：
Rationality Stable Algebraic Varieties

项目摘要

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.

本课题研究多元非线性代数方程组解的性质。这个领域的一个主要问题是确定这些解空间何时可以由独立变量参数化;它可以用于线性或二次方程，但在三度上已经是未知的。这样的代数问题有广泛的应用，例如，数据科学、优化或机器人技术。将这个问题转化为几何，通过关注方程组定义的形状的几何不变量，为大量更直观的技术打开了大门：研究小变形或极限形状。该项目还为研究生提供了研究培训机会。该研究项目的主要目标是了解二维和三维中的这种参数化属性，即，研究非闭域上DelPezzo曲面和Fano三重曲面的合理性和稳定合理性。其中要解决的具体问题是：合理性和稳定的合理性，如非分歧上同调，积分分解的对角线，和伯恩赛德群的障碍;变化的合理性在家庭的代数簇，和专业化;等变双有理几何中的不变量，特别是等变伯恩赛德群，该奖项反映了NSF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。