Syzygies, Moduli Spaces, and Brill-Noether Theory

Syzygies、模空间和布里尔-诺特理论

• 批准号: 1701245
• 负责人: Michael Kemeny
• 金额: $ 14.07万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701245&HistoricalAwards=false
• 关键词: Syzygies; Moduli; Spaces; Brill; Noether

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.

该项目涉及代数几何学的研究，即多项式方程的研究。研究者致力于方程的代数特性与它们定义的空间的几何特性之间的连接。特别是，该研究项目使用Syzygies来研究有关Riemann表面的基本问题，这是最重要的几何对象之一。对Syzygies或方程之间的关系的调查长期以来在代数中发挥了核心作用，历史范围从19世纪不变理论到21世纪的理论物理学。 这项研究的应用包括模量空间，矩阵因素化，弦理论，枚举几何形状和镜像对称性。在更详细的范围内，研究人员正在努力将与嵌入在投影空间中嵌入的riemann表面的外部几何形状相关的代数不变性与其内在的几何形状相关。相关的代数不变性是希尔伯特（Hilbert）定义的坐标环的最小自由分辨率的betti数字，而固有的几何形状是在Brill-Noether理论的不变式中编码的。研究人员将探索长期以来的基本猜想，以预测这些不变性之间的精确关系。该项目使用几种新技术（例如交叉路口理论，曲线的模量空间，Hurwitz空间和矢量束技术）研究了这些猜想和概括。