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有助于理解完全交点上有理曲线的模空间。光滑完全相交上有理曲线的研究是有关法诺簇、有理连通簇和丢番图几何等一系列重要问题的基础。尽管过去几年取得了一些进展，但这些空间的一些基本属性仍然未知。在所提出的研究中，研究了射影空间和其他齐次簇中完全交集上有理曲线空间几何的维数、不可约性、小平维数以及其他几个方面的一些悬而未决的问题。代数簇是多项式方程组的公共零点。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形状的变体，就超曲面而言，它们的研究在高维代数几何、现代枚举几何以及受镜像对称启发的问题中具有广泛的应用。