Spaces of Rational Curves in Projective Varieties

射影簇中的有理曲线空间

• 批准号: 1204567
• 负责人: Roya Beheshti Zavareh
• 金额: $ 14.29万
• 依托单位: Washington 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: 2012-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1204567&HistoricalAwards=false
• 关键词: Spaces; Rational; Curves; Projective; Varieties

The projects described in this proposal aim to contribute towards understanding moduli spaces of rational curves on complete intersections. The study of rational curves on smooth complete intersections is fundamental to a broad spectrum of important problems about Fano varieties, rationally connected varieties, and diophantine geometry. Despite some progress over the past few years, some of the basic properties of these spaces are still unknown. In the proposed research, some open questions on the dimension, irreducibility, Kodaira dimension, and several other aspects of the geometry of spaces of rational curves on complete intersections in projective space and other homogeneous varieties are investigated. Algebraic varieties are common zeros of collections of polynomial equations. An important approach to study the geometry of algebraic varieties is to study parameter spaces of rational curves contained in them. These parameter spaces are themselves varieties with rich geometry, and in the case of hypersurfaces, the study of them has broad applications in higher dimensional algebraic geometry, modern enumerative geometry, and questions inspired by mirror symmetry.

该提案中描述的项目旨在为理解完整交叉点的理性曲线模量空间做出贡献。对平滑完整交叉点上的理性曲线的研究是关于Fano品种，合理联系品种和Diophantine几何形状的广泛重要问题的基础。尽管在过去几年中取得了一些进展，但这些空间的一些基本特性仍然未知。在拟议的研究中，研究了关于射影空间和其他均匀品种的完整交集的理性曲线空间几何几何几何几何几何几何形状的一些开放问题。代数品种是多项式方程集合的常见零。研究代数品种的几何形状的一种重要方法是研究其中包含的理性曲线的参数空间。这些参数空间本身就是具有丰富几何形状的品种，在高度曲面的情况下，对它们的研究在较高维数的几何形状，现代枚举几何形状以及受镜像对称性启发的问题中具有广泛的应用。