Geometry of Linear Systems on Curves: Birational Geometry of Moduli Spaces

曲线上线性系统的几何：模空间的双有理几何

• 批准号: 1001926
• 负责人: Joseph Harris
• 金额: $ 31.15万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001926&HistoricalAwards=false
• 关键词: Geometry; Linear; Systems; Curves; Birational

The Principal Investigator proposes to work on birational models of moduli spaces of curves. Specifically, he will continue his investigation of the relationships among the compactifications of the space of smooth abstract curves arising from the Minimal Model Program, Geometric Invariant Theory, and the deformation theory of curve singularities. In addition, he proposes to study moduli of curves together with maps to projective space; specifically, birational models of the Kontsevich space of stable maps arising from the Minimal Model Program, and also generalizations of Viscardi's construction of moduli spaces of stable maps with domains having singularities other than nodes.An algebraic curve is, simply put, a polynomial in two variables; and the theory of algebraic curves is largely concerned with the relationship between the solutions of such a polynomial and the geometry of the set of solutions. To fully understand this relationship, we need to study the space of all possible curves 'the moduli space of curves' and its geometry. This in turn requires that we study different models of this space; and this is what the Principal Investigator proposes to do.

首席研究员建议研究曲线的模空间的双态模型。具体地说，他将继续研究由最小模型程序、几何不变理论和曲线奇点形变理论产生的光滑抽象曲线空间的紧致化之间的关系。此外，他还建议研究曲线的模和映射到射影空间的映射；具体地说，由最小模型程序产生的稳定映射的Kontsevich空间的双调模型，以及Viscardi构造具有除节点之外的域的奇点的稳定映射的模空间的推广。简单地说，代数曲线是两个变量的多项式；而代数曲线理论主要涉及这样的多项式的解和解集的几何之间的关系。为了充分理解这种关系，我们需要研究所有可能的曲线的空间‘曲线的模空间’及其几何。这反过来要求我们研究这个空间的不同模型；这就是首席调查员建议做的事情。