Remarks on nef and movable cones of hypersurfaces in Mori dream spaces

We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let Z be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let X be a smooth ample divisor in Z, then X is a Mori dream space as well.The second result is: Let Z be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let X be a smooth anti-canonical hypersurface in Z, which is a smooth Calabi–Yau variety, then the unique minimal model of X up to isomorphism is X itself, and moreover, the movable cone conjecture holds for X, namely, there exists a rational polyhedral cone which is a fundamental domain for the action of birational automorphisms on the effective movable cone of X.The third result is: Let $P:={\mathbb{P}}^n\times ?\times {\mathbb{P}}^n$ be the N-fold self-product of the n-dimensional projective space. Let X be a general complete intersection of $n+1$ hypersurfaces of multidegree $(1,\dots ,1)$ in P with $\dim ?X\ge 3$. Then X has only finitely many minimal models up to isomorphism, and moreover, the movable cone conjecture holds for X.