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Syzygies, Moduli Spaces, and Brill-Noether Theory

Syzygies、模空间和布里尔-诺特理论

基本信息

批准号：
2013730
负责人：
Michael Kemeny
金额：
$ 2.92万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2013730&HistoricalAwards=false
关键词：
Syzygies Moduli Spaces Brill Noether

项目摘要

This project concerns research in algebraic geometry, the study of polynomial equations. The investigator works on the connections between the algebraic properties of equations and the geometric properties of the spaces they define. In particular, this research project uses syzygies to study fundamental questions regarding Riemann surfaces, one of the most important classes of geometric objects. The investigation of syzygies, or the relations amongst equations, has long played a central role in algebra, with a history ranging from 19th-century invariant theory to 21st-century theoretical physics. Applications of this research include moduli spaces, matrix factorizations, string theory, enumerative geometry, and mirror symmetry.In more detail, the investigator is working on relating the algebraic invariants associated to the extrinsic geometry of a Riemann surface embedded in projective space to its intrinsic geometry. The relevant algebraic invariants are the Betti numbers of the minimal free resolution of the coordinate ring, as defined by Hilbert, whereas the intrinsic geometry is encoded in the invariants of Brill-Noether theory. The investigator will explore longstanding fundamental conjectures predicting precise relationships between these invariants. The project investigates these conjectures and generalizations of them using several new techniques such as intersection theory, the moduli space of curves, Hurwitz space, and vector bundle techniques.

这个项目涉及代数几何的研究，即多项式方程的研究。研究人员致力于研究方程的代数性质和它们所定义的空间的几何性质之间的联系。特别是，这项研究项目使用合集来研究关于黎曼曲面的基本问题，黎曼曲面是最重要的几何对象之一。从19世纪的不变理论到21世纪的理论物理学，合子或方程之间的关系的研究长期以来一直在代数中扮演着核心角色。这项研究的应用包括模空间、矩阵分解、弦理论、计数几何和镜像对称。更详细地，研究人员致力于将嵌入在射影空间中的Riemann曲面的外几何的代数不变量与其内在几何联系起来。相应的代数不变量是希尔伯特定义的坐标环最小自由分辨的Betti数，而固有几何则编码在Brill-Noether理论的不变量中。研究人员将探索长期存在的预测这些不变量之间精确关系的基本猜想。该项目使用几种新的技术来研究这些猜想和它们的推广，例如交集理论、曲线的模空间、Hurwitz空间和向量丛技术。