Birational classification of varieties: connections to arithmetic and algebra

An algebraic variety is a geometric object defined as the set of points at which a set of multivariate polynomial equations vanishes, with the most elementary examples being the conic sections. While algebraic varieties are the fundamental objects of algebraic geometry, they also appear in many other diverse fields, such as number theory and computer vision.

Birational geometry aims to classify higher dimensional varieties in terms of their intrinsic properties such as curvature. Over the past few decades, there has been tremendous progress in the classification of varieties whose points take values in the complex numbers, and many of the major conjectures are now known in that context. However, there are many other situations in which the theory could apply, and much less is known about these.

For example, number theorists are mainly interested in varieties whose points take values in finite fields or the integers, since the (non-)existence of these points appears in statements such as Fermat's last theorem. In this project, the PI will investigate birational geometry in these other settings. In addition, the project will provide research training opportunities for students and will support various initiatives promoting broadening participation in mathematics, with particular focus on first-generation college students.

In more detail, the project has three main research goals. The first is to investigate the moduli theory and boundedness of Fano varieties over the integers, particularly in dimension two and three. In the process, the PI will explore connection with commutative algebra and new techniques from arithmetic geometry.

The second objective is to investigate the birational classification of non-commutative surfaces from the point of view of Mori theory. Finally, the PI will apply new techniques from derived algebraic geometry to investigate problems involving the purely inseparable covers which lie behind many positive characteristic pathologies.

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.