Birational Geometry: Invariants, Reconstruction, and Deformation Problems

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This project is in algebraic geometry, an active branch of modern mathematics, originating in the theory of diophantine equations. The main objects of study are algebraic varieties, which are solution sets of systems of polynomial equations in several variables. The goal of the project is to develop methods to measure how unconstrained the solutions of these equations are. More precisely, it aims at understanding several invariants of algebraic varieties, constructed by modern techniques using methods from algebraic geometry, arithmetic geometry, and number theory. The project includes concrete problems for graduate student training and incorporates the use of computational resources and algebraic software packages. In more detail, the PI will use methods from Galois cohomology of function fields, properties of algebraic cycles over non-algebraically closed fields, K-theory, and degeneration techniques, to prove results in the following directions: (1) rationality and stable rationality for families of algebraic varieties, in particular for fourfolds fibered by del Pezzo surfaces; (2) equivariant birational invariants; (3) variation of Chow groups of algebraic cycles, in particular over number fields; and (4) reconstruction of birational classes from set-theoretical data.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.