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.

本提案中描述的项目旨在帮助理解完整交叉点上有理曲线的模空间。光滑完全交线上的有理曲线的研究是关于Fano簇、有理连通簇和丢番图几何等一系列重要问题的基础。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性质仍然是未知的。在所提出的研究中，一些开放的问题上的维数，不可约，科代拉维，和其他几个方面的空间几何的有理曲线在完全相交的射影空间和其他齐次品种进行了调查。代数簇是多项式方程集合的公共零点。研究代数簇几何的一个重要途径是研究其中包含的有理曲线的参数空间。这些参数空间本身就是具有丰富几何的变种，在超曲面的情况下，它们的研究在高维代数几何，现代枚举几何和镜像对称启发的问题中有广泛的应用。