Birational geometry of hypersurfaces in products of weighted projective spaces

Abstract

We study the birational geometry of hypersurfaces in products of weighted projective spaces, extending results previously established by J. C. Ottem. For most cases where these hypersurfaces are Mori dream spaces, we determine all relevant cones and characterise their birational models, along with the small $\mathbf{Q}$-factorial modifications to them. We also provide a presentation of their Cox ring. Finally, we establish the birational form of the Kawamata-Morrison cone conjecture for terminal Calabi-Yau hypersurfaces in Gorenstein products of weighted projective spaces.