Birational geometry of hypersurfaces in products of weighted projective spaces

TL;DR

This paper investigates the birational geometry of hypersurfaces in products of weighted projective spaces, identifying their cones, models, Cox rings, and confirming the Kawamata-Morrison cone conjecture for certain Calabi-Yau hypersurfaces.

Contribution

It extends previous work by explicitly describing the birational models, cones, and Cox rings of these hypersurfaces, and proves the cone conjecture in specific cases.

Findings

Determined all relevant cones for hypersurfaces that are Mori dream spaces.

Characterized birational models and small $f{Q}$-factorial modifications.

Established the birational form of the Kawamata-Morrison cone conjecture for certain Calabi-Yau hypersurfaces.

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.