CAREER: Hyperbolicity Properties of Hypersurfaces

与平面上的直线相比，平面上的高阶曲线显示出许多特殊的性质。例如，高阶曲线只有有限个有理点，而直线有无限多个。高次曲线的这些性质一般称为双曲性。人们已经投入了大量的精力来理解这些双曲性质的类似物在高维中应该是什么，并且证明哪些变体以哪种方式是双曲的仍然是代数几何中的一个基本问题。这个项目将进一步阐明这些问题，特别关注哪些超曲面满足各种类型的双曲性质。此外，该项目将通过一项针对K-12学生的强大教育计划，帮助培养下一代科学家和数学家，该计划将看到本科生和教师的参与。该计划包括扩大将本科生送到南本德学校的辅导项目，试点帮助南本德学生为圣母大学科学博览会制作项目的项目，以及培训研究生管理数学圈。PI还将通过指导和组织讲习班和暑期学校，在接近该项目的研究领域培训研究生。更具体地说，本项目的研究将研究规范束如何控制品种的品种的双曲性和其他正性。这是代数、算术和复杂几何中的一个基本驱动问题。研究将集中在三个主要问题上。首先，PI将研究投影空间中一般完全交的双曲性。考虑到最近对这些问题的一系列研究，包括对小林猜想和德巴雷关于完全交点余切束充裕性的猜想的研究，这是及时的。其次，PI将研究在射影空间中非常一般的Fano超曲面上有理曲线的模空间的正性质，着眼于找到第一个理性连接但非非酉的变种的例子。最后，PI将研究源自Manin猜想的问题，研究Fano三重的几何Manin猜想，并对a值大于预期的超曲面的子变种进行分类。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

High-degree curves in the plane have been shown to have many special properties as compared to lines in the plane. For instance, high degree curves have only finitely many rational points, while lines have infinitely many. These properties of high degree curves are loosely referred to as hyperbolicity properties. A huge amount of effort has been devoted to understanding what the analogues of these hyperbolicity properties should be in higher dimension, and proving which varieties are hyperbolic in which ways remains a fundamental question in algebraic geometry. This project will shed further light on these questions, focusing particularly on which hypersurfaces satisfy various types of hyperbolicity properties. Furthermore, this project will help train the next generation of scientists and mathematicians through a strong educational plan aimed to K-12 students, that sees the involvement of undergraduate students and faculty. The plan includes the expanding of a tutoring program that sends undergraduate students to a South Bend school, the piloting of a program to help South Bend students make projects for Notre Dame science fair, and the training graduate students to run math circles. The PI will also train graduate students in the area of research close to this project, through mentoring and the organizing of workshops and summer schools.More specifically, the research for this project will study how the canonical bundle controls the hyperbolicity and other positivity properties of varieties of varieties. This is a fundamental driving question in algebraic, arithmetic and complex geometry. The research will focus on three principal problems. First, the PI will study the hyperbolicity of general complete intersections in projective space. This is timely given the flurry of recent activity on these questions, including work on the Kobayashi Conjecture and Debarre's Conjecture on the ampleness of the cotangent bundle of complete intersections. Second, the PI will investigate positivity properties of the moduli spaces of rational curves on very general Fano hypersurfaces in projective space, with an eye toward finding the first examples of varieties that are rationally connected but not unirational. Finally, the PI will investigate questions originating from Manin's Conjecture, studying Geometric Manin's Conjecture for Fano threefolds and classifying subvarieties of hypersurfaces with larger-than-expected a-value.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.