Rational Curves on Fano Varieties

Fano 品种的有理曲线

• 批准号: 2101935
• 负责人: Roya Beheshti Zavareh
• 金额: $ 20.39万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101935&HistoricalAwards=false
• 关键词: Rational; Curves; Fano; Varieties

Algebraic varieties are common zeros of collections of polynomial equations. Rational curves are the simplest algebraic varieties, and 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 their study has broad applications in higher dimensional algebraic geometry, enumerative geometry, arithmetic geometry, and questions inspired by mathematical physics. In this project, several open questions on various aspects of the geometry of spaces of rational curves on algebraic varieties are investigated. The project provides training opportunities for graduate students. The focus of the first part of the project is the study of spaces of rational curves (as well as rational surfaces and linear subvarieties) contained in hypersurfaces. Hypersurfaces of low degree in projective space form an important testing ground for the study of rationally connected and Fano varieties as well as several other questions in birational geometry. Despite some progress over the past few years, some of the basic properties of these spaces are still unknown. A major guiding question for the study of rational curves on hypersurfaces is which Fano hypersurfaces are rational or unirational. The second part of the project is on the study of rational curves on varieties from the perspective of Geometric Maninís conjecture which predicts the growth rate of a counting function associated to the irreducible components of moduli spaces of rational curves on a variety. In this part, several questions on the geometry of spaces of rational curves on Fano threefolds in characteristic zero and on del Pezzo surfaces over fields of finite characteristic are investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数簇是多项式方程组的公共零点，有理曲线是最简单的代数簇，研究代数簇几何的一个重要途径就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丰富的簇。几何，他们的研究在高维代数几何、枚举几何、算术几何以及受数学物理启发的问题中具有广泛的应用。在这个项目中，有几个关于几何各个方面的开放问题。该项目为研究生提供了培训机会，该项目的第一部分的重点是研究超曲面中包含的有理曲线空间（以及有理曲面和线性子簇）。射影空间中的低次超曲面构成了研究有理连通簇和法诺簇以及双有理几何中的其他几个问题的重要试验场，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属性仍然存在。超曲面有理引导曲线研究的一个主要问题是法诺超曲面是有理还是非有理，该项目的第二部分是从预测增长率的几何马尼尼斯猜想的角度研究有理曲线。与有理曲线模空间不可约分量相关的计数函数的研究在这一部分中，关于特征零和德尔佩佐的法诺三重有理曲线空间几何的几个问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。