Geometry of Moduli Spaces and Applications

这个建议的目的是研究模空间的几何，例如它们的维数、不可约分量、双有理类型和几何不变量。主要研究者计划从多个方向开展研究。首先，他想研究Teichmueller曲线，这是刚性测地线的模空间的曲线，因此可以提供至关重要的信息几何的模空间。其次，他想探索模空间参数化曲线在周围的空间，侧重于稳定映射的模空间，半稳定层，希尔伯特计划和周品种之间的比较。最后，他计划研究非约化曲线上的线丛的雅可比簇，这可能通过变形和退化技术揭示光滑曲线的几何性质。 本项目属于代数几何学科，其主要对象是由多项式方程组的解集所定义的代数簇。模空间是给定类型的参变量簇。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。一个有吸引力的方面是，其对象的模空间本身往往是一个品种。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着广泛的应用。主要研究者期望该项目的成果可以丰富其他学科的研究，包括组合数学，动力学，枚举几何和数学物理。

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.