Geometry of Moduli Spaces and Applications

模空间几何及其应用

• 批准号: 1101153
• 负责人: Dawei Chen
• 金额: $ 12.6万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2011-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101153&HistoricalAwards=false
• 关键词: Geometry; Moduli; Spaces; Applications

This proposal aims to study the geometry of moduli spaces, such as their dimensions, irreducible components, birational types, and geometric invariants. The principal investigator plans to carry out the study in several directions. Firstly, he would like to study Teichmueller curves, which are rigid geodesics in the moduli space of curves, and hence can provide crucial information for the geometry of the moduli space. Secondly, he wants to explore moduli spaces parameterizing curves in an ambient space, focusing on the comparison between the moduli space of stable maps, the moduli space of semi-stable sheaves, Hilbert scheme and Chow variety. Finally, he plans to study the Jacobian variety of line bundles on a non-reduced curve, which may reveal geometric properties of smooth curves by deformation and degeneration techniques. This project belongs to the subject of algebraic geometry, whose main objects are algebraic varieties defined by the solution sets of polynomial equations. Moduli spaces parameterize varieties of a given type. For instance, a donut and a car tire are of the same type, because they both have one hole, but a pretzel with three holes is different. An attractive aspect is that a moduli space for its objects tends itself to be a variety. Therefore, studying moduli spaces can help us understand the classification of algebraic varieties. In addition, moduli spaces have been extensively used in many other fields. The principal investigator expects that the outcome of this project can enrich the studies of other subjects, including combinatorics, dynamics, enumerative geometry and mathematical physics.

该提案旨在研究模量空间的几何形状，例如其尺寸，不可还原成分，异常类型和几何不变性。主要研究人员计划向多个方向进行研究。首先，他想研究Teichmueller曲线，这些曲线是曲线模量空间中的刚性大地测量学，因此可以为模量空间的几何形状提供关键信息。其次，他想在环境空间中探索模量空间参数化曲线，重点是稳定地图的模量空间之间的比较，半稳定滑轮的模量空间，希尔伯特方案和chow综艺。最后，他计划在非还原曲线上研究雅各布的各种线束，这可能会揭示通过变形和变性技术平滑曲线的几何特性。 该项目属于代数几何学的主题，其主要对象是由多项式方程的解决方案集定义的代数品种。模量空间使给定类型的种族变种。例如，甜甜圈和汽车轮胎的类型相同，因为它们都有一个孔，但是带有三个孔的椒盐脆饼是不同的。一个有吸引力的方面是，其物体的模量空间往往是一种多样性。因此，研究模量空间可以帮助我们了解代数品种的分类。另外，模量空间已在许多其他领域广泛使用。主要研究人员预计，该项目的结果可以丰富其他受试者的研究，包括组合，动力学，枚举几何和数学物理学。