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On the relative version of Mori dream spaces

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 Wiśnewski 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.