Geometry of Moduli Spaces, Geometric Invariant Theory, and Deformations of Singularities

这一建议涉及到代数几何中模空间的研究。通过使用几何不变理论、形变理论和计数几何的方法，研究者将继续进行几个项目，以阐明研究得很好的模空间的几何，并对较少研究的模空间采用新的方法。在第一个项目中，研究者将继续研究嵌入曲线的有限Hilbert稳定性，目的是提出稳定曲线的Deligne-Mumford模空间的对数最小模型程序。在第二个项目中，研究人员将探讨关于这个模空间及其变体上的有效和充裕因子的基本开放问题。奇点的研究是这两个项目的重要组成部分。在第三个项目中，研究者将曲线理论的最新结果推广到某些高维簇(例如，K3曲面)及其模空间。代数簇是一组多项式方程组的解的集合。代数簇是数学的基本研究对象，特别是在本提议所属的代数几何领域。多项式系数的变化给出了一类给定变元的模空间。对模空间的研究对于理解代数簇本身以及最终求解多项式方程组是必不可少的。作者建议用经典和现代的方法研究依赖于一个或两个自由参数的代数簇的模空间。该提案的更广泛影响包括推进一项积极的研究计划，并共同组织关于模空间和相关问题的研讨会。

This proposal is concerned with the study of moduli spaces in algebraic geometry. By using methods of Geometric Invariant Theory, deformation theory, and enumerative geometry, the investigator will pursue several projects in order to elucidate the geometry of well-studied moduli spaces and to adopt new approaches to less studied ones. In the first project, the investigator will continue the study of finite Hilbert stability of embedded curves with the goal of advancing the log minimal model program for the Deligne-Mumford moduli space of stable curves. In the second project, the investigator will approach fundamental open questions about effective and ample divisors on this moduli space and its variants. The study of singularities is an important ingredient of these two projects. In the third project, the investigator will extend recent results in the theory of curves to certain classes of higher-dimensional varieties (e.g., K3 surfaces) and their moduli spaces.An algebraic variety is a collection of solutions to a system of polynomial equations. Algebraic varieties are fundamental objects of study in mathematics and, in particular, in the field of algebraic geometry, to which this proposal belongs. Variation of the polynomials' coefficients gives rise to a moduli space for a given class of varieties. The study of moduli spaces is essential to understanding algebraic varieties themselves and, ultimately, to solving systems of polynomial equations. The investigator proposes to study moduli spaces of algebraic varieties depending on one or two free parameters using both classical and modern techniques. The broader impacts of the proposal include advancing an active research program and co-organizing workshops on moduli spaces and related problems.