The classification of  Mori dream spaces - equivalently, normal projective varieties with finitely generated Cox ring - is closely tied to positivity properties of the anticanonical class. In particular, toric and Fano varieties provide fundamental classes of examples with finitely generated Cox rings. In contrast, Calabi-Yau varieties lie at the boundary of positivity: there is no general criterion deciding when they are Mori dream spaces, and their birational geometry (and birational automorphism groups) can be remarkably rich.

In this talk we focus on Calabi--Yau varieties $X$ arising as  general anticanonical hypersurfaces of smooth toric Fano varieties $Z$. Our results are formulated in terms of primitive pairs of the anticanonical polytope of $Z$, which is a smooth reflexive polytope. We present two complementary theorems. The first result provides sufficient combinatorial conditions on primitive pairs ensuring that $X$ is a Mori dream space, and it yields an explicit presentation of the Cox ring $R(X)$ in terms of $R(Z)$  and the defining equation of $X$. The second result goes in the opposite direction: the existence of certain relations among primitive pairs forces $\mathrm{Bir}(X)$ to be infinite, and hence $X$ cannot be a Mori dream space. The proof of the first theorem builds on the approach of Herrera-Laface-Ugaglia on Cox rings of embedded varieties, while the second generalizes ideas of Kawamata and Ottem for anticanonical hypersurfaces in products of projective spaces. As an application, we obtain a complete classification of Mori dream Calabi-Yau hypersurfaces in dimensions $2$ and $3$. In particular, for these hypersurfaces there is a sharp dichotomy:  either $R(X)$ is finitely generated  or $\mathrm{Bir}(X)$ is infinite.

This is joint work with Antonio Laface and Luca Ugaglia.