Moduli and Arithmetic of Higher Dimensional Varieties

高维簇的模和算术

• 批准号: 2302550
• 负责人: Kenneth Ascher
• 金额: $ 18.06万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302550&HistoricalAwards=false
• 关键词: Moduli; Arithmetic; Higher; Dimensional; Varieties

Algebraic geometry is the study of geometric shapes which arise as solutions to polynomial equations. Some fundamental examples of these shapes are circles or hyperboloids, and as such, algebraic geometry is largely intertwined with many other fields of mathematics. One of the guiding research directions in algebraic geometry as well as the overall focus of this research project is understanding the classification of these shapes, a subfield known as the study of moduli spaces. Algebraic geometry as well as the study of moduli spaces have numerous applications, for example within cryptography as well as in understanding the structure underlying our universe via string theory and mathematical physics. This project includes training opportunities for both undergraduate and graduate students, as well as outreach efforts involving high school students from the local community.While we have an extensive understanding of moduli spaces of algebraic curves, we understand far less about moduli spaces of higher dimensional algebraic varieties (i.e. varieties of complex dimension at least two). In short, the main goal of this project is to further our understanding of higher dimensional moduli by leveraging many recent results in moduli theory and birational geometry, such as wall-crossing results that the PI has obtained in joint work with collaborators. The first project aims to use wall-crossing techniques to understand the underlying geometric structure and geometric properties of moduli spaces of higher dimensional algebraic varieties. Additionally, as most known results regarding moduli spaces of higher dimensional varieties focus on two cases, namely (log) general type pairs and (log) Fano pairs, the goal of the second project is to construct and study moduli spaces of (log) Calabi-Yau pairs using tools from the minimal model program and K-stability. Finally, the third project uses techniques from birational geometry, moduli spaces, and the minimal model program to understand various notions of hyperbolicity on varieties of (log) general type, including the distribution of rational and integral points.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.

代数几何是研究几何形状的多项式方程的解决方案。这些形状的一些基本例子是圆或双曲面，因此，代数几何在很大程度上与许多其他数学领域交织在一起。代数几何的指导研究方向之一以及该研究项目的总体重点是了解这些形状的分类，这是一个被称为模空间研究的子领域。代数几何和模空间的研究有许多应用，例如在密码学中，以及通过弦理论和数学物理来理解我们宇宙的结构。该项目包括为本科生和研究生提供培训机会，以及与当地社区高中生的外联活动。虽然我们对代数曲线的模空间有广泛的了解，但对高维代数簇（即复维数至少为2的簇）的模空间的了解要少得多。简而言之，该项目的主要目标是通过利用模量理论和双有理几何中的许多最新结果，例如PI与合作者联合工作获得的跨壁结果，进一步了解高维模量。第一个项目的目的是使用跨壁技术来理解基本的几何结构和几何性质的模空间的高维代数簇。此外，由于大多数已知的结果，关于模空间的高维品种集中在两种情况下，即（日志）一般类型对和（日志）Fano对，第二个项目的目标是构建和研究（日志）Calabi-Yau对的模空间使用工具从最小模型程序和K-稳定性。最后，第三个项目使用的技术，从双有理几何，模空间，和最小模型程序，以了解各种（日志）一般类型的双曲线的各种概念，包括合理和积分点的分布。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。