Hypersurfaces in products of projective spaces

Hypersurfaces in multiprojective spaces are defined by a multigraded polynomial. Despite their apparent simplicity, their birational geometry can be quite complicated. For example, they can have infinite birational automorphism groups. We give a complete picture of their birational geometry and compute their effective cones and nef cones. As an application, we give a negative answer to a question of Cascini and Gongyo.