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

模空间几何、几何不变量理论和奇点变形

• 批准号: 1201286
• 负责人: Maksym Fedorchuk
• 金额: $ 11.8万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2012-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201286&HistoricalAwards=false
• 关键词: Geometry; Moduli; Spaces; Geometric; Invariant

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.

这个建议是关于代数几何中模空间的研究。通过使用几何不变理论，变形理论和枚举几何的方法，研究人员将进行几个项目，以阐明研究充分的模空间的几何，并采用新的方法来研究较少。在第一个项目中，研究人员将继续研究嵌入曲线的有限Hilbert稳定性，目标是推进稳定曲线的Deligne-Mumford模量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模空间及其变体上的有效和充足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，研究人员将把曲线理论的最新结果扩展到某些高维变量（例如，代数簇是多项式方程组的解的集合。代数簇是数学研究的基本对象，特别是在代数几何领域，这一建议属于。多项式的系数的变化产生了一个模空间为给定类的品种。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。研究者建议使用古典和现代技术来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积极的研究计划，并共同组织关于模数空间和相关问题的研讨会。