Remarks on hypersurfaces in ${\mathbf{P}}^1\times Z$

We study the birational geometry of hypersurfaces in projective varieties of the form ${\mathbf{P}}^1\times Z$, where $Z$ satisfies mild assumptions. Building on recent results of Herrera–Laface–Ugaglia, we study their Cox rings (when finitely generated) in terms of the Cox ring of $Z$. In particular, we obtain a complete picture when $Z$ is a Fano variety of dimension at least 3, and with class group of rank 1.