Geometry, rationality, and arithmetic of Fano varieties

Algebraic varieties are geometric objects defined as the solution sets of systems of polynomial equations in several variables. Classifying algebraic varieties is a fundamental problem, and from this perspective, there are three basic building blocks of varieties. One of these building blocks consists of Fano varieties, which are a class of varieties with many important applications in geometry and number theory. In the case when the variety is defined by a single polynomial equation, the Fano condition states that the degree of the polynomial is low compared to the number of variables. This project will study Fano varieties, their geometric properties (over the complex numbers), and their arithmetic properties (such as what types of numbers arise as solutions to the system of polynomials). The project will also include the training of graduate and undergraduate students. Specifically, the project will study the following three directions. The first direction will be the study of rationality questions for varieties over non-closed fields. This will focus on varieties that admit Fano fibration structures, in particular for a class of such threefolds, by pursuing generalizations of the intermediate Jacobian torsor obstruction. The second direction will focus on the relationship between rationality and group actions. Thirdly, this project will also study geometric and arithmetic aspects of pencils of quadrics, which are an important class of Fano varieties with numerous applications to the existence of rational points and to moduli spaces. 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.