The Geometry of Hyperkähler Varieties and Rationality

Algebraic geometry is the study of geometric objects, called varieties, which are defined by the solution sets of systems of polynomial equations. It is a far-reaching branch of mathematics, making connections with many other research areas such as commutative algebra, number theory, differential and complex geometry, representation theory, and mathematical physics. In this project the PI will study certain families of varieties that play an important role in the classification of all varieties, namely hyperkaehler varieties and rational varieties. This project focuses on arithmetic questions about these two families. The project includes research training opportunities for undergraduate and graduate students, as well as outreach activities to strengthen the community of individuals in algebraic geometry from underrepresented backgrounds. This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research. This research program is centered around three projects. In the first, birational transformations of hyperkaehler varieties will be used to study Brauer classes on K3 surfaces in order to identify which Brauer classes can arise as exceptional loci in hyperkaehler contractions. This makes connections to questions about the rationality of families of cubic fourfolds. The second is to study the behavior of rationality of fourfolds in arithmetic families, giving an analogue to previous results in families over the complex numbers. The third project is centered around the intermediate Jacobian torsor obstruction to rationality for geometrically rational threefolds, with the goal of characterizing rationality for a certain family of conic bundle threefolds. 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.