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

TL;DR

This paper studies the structure of nef and movable cones of hypersurfaces in Mori dream spaces, establishing conditions under which these hypersurfaces are Mori dream spaces and verifying the movable cone conjecture in various geometric contexts.

Contribution

It proves that smooth ample divisors in certain Mori dream spaces are themselves Mori dream spaces, and confirms the movable cone conjecture for specific classes of hypersurfaces.

Findings

Smooth ample divisors in Mori dream spaces are Mori dream spaces.

The movable cone conjecture holds for certain Calabi-Yau hypersurfaces.

Complete intersections in product of projective spaces have finitely many minimal models.

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 $X$.