Rational Curves on Fano Varieties

代数族是多项式方程集合的公共零点。有理曲线是最简单的代数簇，研究其包含的有理曲线的参数空间是研究代数簇几何的一个重要途径。这些参数空间本身就是丰富的几何变种，它们的研究在高维代数几何、计数几何、算术几何以及数学物理启发的问题中有着广泛的应用。在这个项目中，研究了代数簇上有理曲线空间几何的各个方面的几个公开问题。该项目为研究生提供了培训机会。该项目第一部分的重点是研究超曲面中包含的有理曲线(以及有理曲面和线性子簇)的空间。射影空间中的低次超曲面是研究有理连通和Fano簇以及双曲几何中的其他几个问题的重要试验场。尽管在过去的几年里取得了一些进展，但这些空间的一些基本性质仍然未知。研究超曲面上的有理曲线的一个主要指导问题是Fano超曲面是有理的还是单圆的。本项目的第二部分是从几何ManiníS猜想的角度研究簇上的有理曲线，该猜想预测了与簇上有理曲线的模空间的不可约分支相关的计数函数的增长率。在这一部分中，研究了特征为零的Fano三重有理曲线和有限特征域上Del Pezzo曲面上的有理曲线空间的几何问题。这一裁决反映了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了评估，被认为值得支持。

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.