LEAPS-MPS: Moduli of Stable Surfaces and Beyond

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Algebraic geometry is rooted in the relationship between polynomial equations and the geometric objects they define, such as curves, surfaces, and threefolds. Within algebraic geometry, parameter spaces known as moduli spaces form one of the most central objects of study; understanding their structure can have extensive applications throughout mathematics, physics, and statistics.

This project explores fundamental questions in algebraic geometry related to compactifications of moduli spaces of surfaces and threefolds. The project has a significant educational and outreach component: the PI will design and implement multiple activities, each of which shares the objective of recruiting and retaining underrepresented students in math, particularly students identifying as Black, African American, Latinx, and Indigenous.

This project investigates the moduli space of surfaces of general type and its compactifications, as well as connections to related moduli problems for threefolds. The methods follow the spirit of the renowned Hassett-Keel program, the overarching goal of which is to completely understand compactifications of the moduli spaces of algebraic curves. In the case of surfaces, the jump in dimension adds an additional challenge to understanding this moduli problem, and requires developing new techniques to build and study singular objects.

To this end, the PI will work on three interrelated projects. The first is a study of T singularities, an important type of singularity that frequently appears on stable surfaces. The second focuses on the relationship between connected and irreducible components of the moduli space of surfaces of general type and its compactification.

The third project branches out from the moduli of surfaces into Noether-Lefschetz problems and moduli of threefold-surface pairs. In addition to dissemination of work through talks and publications, the PI will conduct outreach activities in local schools, develop an upper-level undergraduate course focusing on work of modern BIPOC mathematicians, lead two undergraduate research projects, and organize readings and discussions focused on retaining BIPOC students in math.

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.