CAREER: Hyperbolicity Properties of Hypersurfaces

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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.