On the relative version of Mori dream spaces

Abstract

This paper is devoted to a study of the relative version of a Mori dream space (MDS for short), which was first introduced by Andreatta and Wi\'{s}newski and will be called Mori dream morphism (MDM) in this paper. An MDM is defined to be an algebraic fiber space between normal quasi-projective varieties $ X \to U$ such that $\operatorname{Pic}(X/U)_{\mathbb{Q}}=\operatorname{N} ^1(X/U)_{\mathbb{Q}}$ and the (relative) movable cone is decomposed into the semi-ample cones of finitely many small $\mathbb{Q}$-factorial modifications, each of which is assumed to be rational polyhedral. An MDS is an MDM where $U$ is a point. We prove that the relative $D$-MMP runs and terminates in either a good $D$-minimal model or a $D$-Mori fiber space for an arbitrary divisor $D$ on an MDM, and that an algebraic fiber space satisfying $\operatorname{Pic}(X/U)_{\mathbb{Q}}= \operatorname{N} ^1(X/U)_{\mathbb{Q}}$ is an MDM if and only if a Cox sheaf is finitely generated over $U$. These are generalizations to an MDM of the fundamental results by Hu and Keel for an MDS. We also show that if the composition of two algebraic fiber spaces $f$ and $g$ is an MDM, then so are $f$ and $g$. In the end we investigate base changes of MDMs. We prove that a base change of an MDM by a proper flat morphism is again an MDM, provided that the base change is normal $\mathbb{Q}$-factorial and that any $\mathbb{Q}$-line bundle on the base change descends to the original MDM.