Mathematics – Algebraic Geometry
Scientific paper
2011-04-07
Mathematics
Algebraic Geometry
34pages, 2 figures. Comments and corrections are welcome
Scientific paper
The purpose of this paper is to study the geometry of the images of morphisms from Mori dream spaces. Firstly we prove that a variety which admits a surjective morphism from a Mori dream space again is a MDS. Secondly we introduce a fan structure on the effective cone of a MDS and show that under a surjective morphism between MDSs the fan of the target space coincides with the restriction of the fan of the source. We see that this fan encodes some information of the Zariski decompositions, which in turn is equivalent to the information of the GIT equivalence of VGIT of the Cox ring. Corresponding to these two interpretations, two different proofs are given to the second theorem. Generalizations to non-\mathbb{Q}-factorial cases and to Mori dream re- gions are also treated.
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