On the relative version of Mori dream spaces

This paper is devoted to a study of the relative version Mori dream space (MDS for short), which was first introduced by Andreatta and Wiśnewski will be called morphism (MDM) in this paper. For MDMs we prove that MMP an arbitrary divisor D runs terminates either good minimal model or fiber space, algebraic $$X \rightarrow U$$ satisfying $${\mathrm{Pic}}(X/U)_{{\mathbb {Q}}} \simeq {\text {N}}^1(X/U)_{{\mathbb {Q}}}$$ MDM if only Cox sheaf finitely generated over U. We also show composition two spaces f g MDM, then so are g. give sufficient condition property being preserved under base change.