Speaker: Ignacio Barros
Date: 10:30 10/06/2022
Title: On the irrationality of moduli spaces.

Abstract: I will talk about the problem of determining the birational complexity of moduli spaces of curves and 
K3 surfaces. I will recall some recently introduced invariants that measure irrationality and talk about what is 
known for these moduli spaces. In the second half, I will report on joint work with D. Agostini and K.-W. Lai, 
where we study how the degrees of irrationality of the moduli spaces of polarized K3 surfaces grow with respect 
to the genus g. We provide polynomial bounds. The proof relies on Kudla's modularity conjecture for Shimura 
varieties of orthogonal type. For special genera, we exploit the deep Hodge theoretic relation between K3 surfaces 
and special hyperkähler fourfolds to obtain much sharper bounds.