Hypersurfaces in Mori dream spaces

Let X be a hypersurface of a Mori dream space Z. We provide necessary and sufficient conditions for the Cox ring R(X) of X to be isomorphic to R(Z)/(f), where R(Z) is the Cox ring of Z and f is a defining section for X. We apply our results to Calabi–Yau hypersurfaces of toric Fano fourfolds. Our second application is to general degree d hypersurfaces in Pn containing a linear subspace of dimension n−2.