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

Abstract

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 $P:= \mathbb{P}^n \times \cdots \times \mathbb{P}^n$ 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 $\dim X \geq 3$. Then $X$ has only finitely many minimal models up to isomorphism, and moreover, the movable cone conjecture holds for $X$.