CAREER: Hyperbolicity Properties of Hypersurfaces

职业：超曲面的双曲性质

• 批准号: 1945144
• 负责人: Eric Riedl
• 金额: $ 40万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945144&HistoricalAwards=false
• 关键词: CAREER; Hyperbolicity; Properties; Hypersurfaces

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.

与平面中的直线相比，平面中的高次曲线具有许多特殊的性质。例如，高次曲线只有200个有理点，而直线有无穷多个有理点。高次曲线的这些性质被松散地称为双曲性性质。大量的努力一直致力于了解这些双曲性质的类似物应该在更高的维度，并证明哪些品种是双曲的方式仍然是一个基本的问题，在代数几何。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性质。此外，该项目将通过针对K-12学生的强有力的教育计划帮助培养下一代科学家和数学家，该计划将吸引本科生和教师的参与。该计划包括扩大一项将本科生送往南本德学校的辅导计划，试点一项帮助南本德学生为圣母大学科学博览会制作项目的计划，以及培训研究生管理数学圈。PI还将通过指导和组织研讨会和暑期学校来培训与该项目相关的研究领域的研究生。更具体地说，该项目的研究将研究正则束如何控制各种品种的双曲性和其他正性属性。这是一个基本的驱动问题，在代数，算术和复杂的几何。研究将集中在三个主要问题上。首先，PI将研究射影空间中一般完全交的双曲性。这是及时考虑到最近的活动对这些问题，包括工作的小林猜想和德巴尔的猜想的余切丛的丰富性完全交叉。其次，PI将研究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性质，着眼于找到有理连通但非单有理的簇的第一个例子。最后，PI将调查源于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簇进行分类。该奖项反映了NSF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。

项目成果

Rational curves on del Pezzo surfaces in positive characteristic

• DOI: 10.1090/btran/138
• 发表时间: 2021-10
• 期刊: Transactions of the American Mathematical Society, Series B
• 作者: Roya Beheshti;Brian Lehmann;Eric Riedl;Sho Tanimoto

Clustered families and applications to Lang-type conjectures

聚类族及其在 Lang 型猜想中的应用

• 发表时间: 2022
• 期刊: Proceedings of the London Mathematical Society
• 影响因子: 1.8
• 作者: Coskun, Izzet;Riedl, Eric
• 通讯作者: Riedl, Eric

Moduli spaces of rational curves on Fano threefolds

Fano 三重上有理曲线的模空间

• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Beheshti, Roya;Lehmann, Brian;Riedl, Eric;Tanimoto, Sho
• 通讯作者: Tanimoto, Sho

Restricted tangent bundles for general free rational curves

一般自由有理曲线的限制切丛

• 发表时间: 2023
• 期刊: International mathematics research notices
• 影响因子: 1
• 作者: Lehmann, Brian;Riedl, Eric
• 通讯作者: Riedl, Eric

Restrictions on rational surfaces lying in very general hypersurfaces

对非常一般的超曲面中的有理曲面的限制

• 发表时间: 2022
• 期刊: Forum of mathematics Sigma
• 影响因子: 1.7
• 作者: Beheshti, Roya;Riedl, Eric
• 通讯作者: Riedl, Eric