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.

该项目涉及许多变量中非线性代数方程系统系统解决方案的性质。该领域的一个主要开放问题是确定何时可以通过自变量来参数化这些解决方案空间。它可以用于线性或二次方程式，但在第三级中已经未知。此类代数问题具有广泛的应用程序，例如在数据科学，优化或机器人技术中。将这个问题转化为几何形状，通过关注由方程式系统定义的形状的几何不变，为多种更直观的技术打开了大门：小变形或限制形状的研究。该项目还为研究生提供了研究培训的机会。研究项目的主要目标是了解第2和3的参数化属性，即，在未公开的领域中研究理性和稳定的合理性和稳定的合理性。要解决的具体问题包括：理性和稳定理性的障碍，例如未经塑造的共同体，对角线的整体分解和伯恩赛德群体；代数品种家庭中理性的变化和专业化；尤其是伯恩赛德集团（Eprivariant Burnside Group）及其对Cremona Group的研究的应用，特别是在等效的birational几何形状中不变。审查标准。