Moduli of higher dimensional varieties and families of hypersurfaces

Algebraic geometry is the study of objects defined by polynomial equations, called varieties. These equations can be studied algebraically (e.g. solving the equation to find all solutions) or geometrically (e.g. graphing the shape defined by the equation) and algebraic geometry uses the tools from both perspectives to analyze varieties. An overarching goal of the field to be explored in this project is to classify all possible varieties, which is done through the construction of moduli spaces, or parameter spaces for varieties of a given type. The study of moduli spaces has a rich history and these spaces arise naturally in algebraic geometry, symplectic geometry, differential geometry, enumerative geometry and combinatorics, mirror symmetry, number theory, and physics. The work involved with this project has connections to each of these fields. The PI will mentor both undergraduate and graduate students and continue with a variety of activities that encourage the participation of women in mathematics. The main objective of the PI is to research moduli spaces of higher dimensional algebraic varieties, specifically to study degenerations of hypersurfaces (varieties defined by a single polynomial equation). The PI will approach two related questions: studying smooth limits of hypersurfaces from a moduli-theoretic perspective, focusing on when such limits are again hypersurfaces, and also an explicit classification of singular varieties appearing in these moduli spaces, focusing on degenerations of projective space, Fano varieties, and log Calabi-Yau pairs. The main tools used to accomplish these goals will be wall crossing in K-stability and KSBA moduli, the minimal model program, geometric invariant theory, and interpolation between these different perspectives on moduli spaces. The projected outputs of this project include several theoretical results, such as (non-)existence of particular degenerations of projective space, techniques for constructing moduli of log Calabi-Yau varieties, and some classification of smooth limits of hypersurfaces. The outputs will also include several explicit examples of moduli spaces of log canonically polarized, log Calabi-Yau, and log Fano pairs. 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.