Studies in Moduli Theory and Birational Geometry

模理论与双有理几何研究

• 批准号: 2100548
• 负责人: Dan Abramovich
• 金额: $ 33.5万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100548&HistoricalAwards=false
• 关键词: Studies; Moduli; Theory; Birational; Geometry

The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. Algebraic geometry has significant applications in coding, industrial control, computation, and theoretical physics, where physicists consider algebraic varieties as a piece of the fine structure of our universe. One focus in this project is Moduli theory, which studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is often manifested as an algebraic variety in its own right, called a moduli space. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not just a metaphor but a rigorous and quite useful fact. The other focus in this project is birational geometry, focusing here on resolution of singularities. Resolution of singularities is a fundamental procedure where "bad" points of an algebraic variety are removed and replaced by "good" points. This project includes research opportunities for undergraduate and graduate students.In more detail, the PI will continue studying problems in birational geometry, focusing on resolution of singularities, semistable reduction, and the geometry of stack theoretic birational modifications. The PI and collaborators will build on recently completed work to functorially resolve singularities of proper families of varieties; to study the geometry of weighted blowings up and more general stack-theoretic procedures; and to study subtle phenomena in positive characteristics. In addition, the PI will continue to study moduli spaces. The main foci are Moduli and arithmetic of subvarieties of families of abelian varieties, aiming to extend recent non-degeneracy and uniformity results of rational points on curves to symmetric squares of curves and related constructions.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.

该项目的研究领域在于代数几何，数学的分支致力于几何形状称为代数簇，由多项式方程定义。代数几何在编码、工业控制、计算和理论物理学中有着重要的应用，物理学家认为代数簇是我们宇宙精细结构的一部分。这个项目的一个重点是模理论，它研究一个显着的现象，其中所有相同类型的代数簇的集合通常表现为一个代数簇本身，称为模空间。因此，在代数几何学中，把一个“有机体”群落看作是一个“有机体”的隐喻不仅是一个隐喻，而且是一个严格而非常有用的事实。这个项目的另一个重点是双有理几何，这里侧重于奇点的解决。奇点的解析是一个基本的过程，其中代数簇的“坏”点被删除并被“好”点取代。该项目包括本科生和研究生的研究机会。更详细地说，PI将继续研究双有理几何中的问题，重点是奇异点的解决，半稳定的减少，以及堆栈理论双有理修改的几何。PI和合作者将在最近完成的工作的基础上，泛函地解决适当的品种族的奇点;研究加权爆破的几何和更一般的堆栈理论程序;并研究正特征中的微妙现象。此外，PI将继续研究模空间。主要研究方向是阿贝尔簇族的子簇的模和算术，旨在将最近关于曲线上有理点的非退化性和均匀性的结果扩展到曲线的对称平方和相关构造。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而被认为值得支持。