Verra fourfolds: lattice theory, derived categories and birational geometry

A Verra fourfold is a double cover of $\mathbb P^2 \times \mathbb P^2$, branched along a divisor of bidegree (2,2). Its two natural quadric surface fibrations give rise to twisted K3 surfaces, providing a connection between the geometry and derived categories.

In this talk, I will present a joint work with Augustinas Jacovskis and Reinder Meinsm. I will discuss how lattice-theoretic methods can be used to study these fourfolds and construct examples with interesting geometric and categorical properties. In particular, I will focus on the count of their Fourier-Mukai partners and construct arbitrarily large families of pairwise non-isomorphic Verra fourfolds with equivalent Kuznetsov components.