论文标题
在莫里梦的相对版本上
On the relative version of Mori dream spaces
论文作者
论文摘要
本文致力于研究Mori Dream Space（简称MDS）的相对版本，该文章最初是由Andreatta和Wiśnewski引入的，并将在本文中称为Mori Dream Morphism（MDM）。 MDM被定义为正常的准标准品种$ x \ to u $之间的代数纤维空间，使得$ \ operatoTorName {pic}（x/u）_ {\ mathbb {q}}}}分解为有限的许多小$ \ mathbb {q} $ - 阶乘修改的半样品锥，每个阶段都被认为是理性的多面体。 MDS是MDM，其中$ U $是一个点。我们证明，相对$ d $ -mmp在良好的$ d $ - 米数模型或$ d $ -mori纤维的空间中用于MDM上的任意分隔线$ d $，以及满足$ \ operatorname {pic}（x/u）_ {x {n n的代数光纤空间， ^1（x/u）_ {\ Mathbb {q}} $是MDM，并且仅当Cox Sheaf在$ U $上有限生成。这些是HU和Keel的MDS基本结果的MDM的概括。我们还表明，如果两个代数纤维空间$ f $和$ g $的组成是MDM，那么$ f $和$ g $也是如此。最后，我们研究了MDM的基本变化。我们证明，如果基本变化为正常的$ \ m athbb {q} $ - fortorial，则通过适当的平坦形态的基本变化是MDM，并且任何$ \ Mathbb {Q} $ - 基本变化上的任何$ \ Mathbb {q} $ - 线捆绑包下降到原始MDM。
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.
