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.

这一建议旨在研究模空间的几何，如它们的维度、不可约分支、双胞型和几何不变量。首席调查员计划从几个方向开展这项研究。首先，他想研究Teichmueller曲线，它是曲线的模空间中的刚性测地线，因此可以为模空间的几何提供关键信息。其次，他研究了环境空间中参数曲线的模空间，重点比较了稳定映射的模空间、半稳定层的模空间、Hilbert格式和Chow簇。最后，他计划研究非约化曲线上的线丛的雅可比变化，这可能通过变形和退化技术来揭示光滑曲线的几何性质。本课题属于代数几何学科，其主要研究对象是由多项式方程解集定义的代数簇。模空间将给定类型的簇参数化。例如，甜甜圈和汽车轮胎是同一类型的，因为它们都有一个洞，但有三个洞的椒盐卷饼是不同的。一个吸引人的方面是，对象的模空间本身往往是多种多样的。因此，研究模空间可以帮助我们理解代数簇的分类。此外，模空间在许多其他领域也有广泛的应用。首席研究人员期望该项目的成果可以丰富其他学科的研究，包括组合学、动力学、计数几何和数学物理。