[Paper Review] On images of Mori dream spaces

Shinnosuke Okawa|arXiv (Cornell University)|Apr 7, 2011

Algebraic Geometry and Number Theory20 references4 citations

TL;DR

This paper establishes that the image of a Mori dream space under a surjective morphism is again a Mori dream space, proving finite generation of the target's Cox ring via the source's structure. It introduces a fan structure on the effective cone that encodes Zariski decompositions and GIT equivalence, showing that the fan of the target is the restriction of the source's fan to the effective cone of the target.

ABSTRACT

The purpose of this paper is to study the geometry of images of morphisms from Mori dream spaces. First we prove that a variety which admits a surjective morphism from a Mori dream space is again a Mori dream space. Secondly we introduce a natural fan structure on the effective cone of Mori dream spaces. We show that it encodes the information of Zariski decompositions, which in turn is equivalent to the information of the variation of GIT quotients of their Cox rings. Finally we show that under a surjective morphism between Mori dream spaces, the fan of the target space coincides with the restriction of the fan of the source.

Motivation & Objective

• To establish that the image of a Mori dream space under a surjective morphism is again a Mori dream space.
• To define and study a canonical fan structure on the effective cone of a Mori dream space.
• To relate this fan structure to Zariski decompositions and variation of GIT quotients of the Cox ring.
• To show that the fan of the target space under a surjective morphism is the restriction of the source's fan to the target's effective cone.
• To extend the theory to Mori dream regions and examine the limits of fan compatibility in such cases.

Proposed method

• Prove finite generation of the Cox ring of the target variety by using the finite generation of the source's Cox ring and the Stein factorization of the morphism.
• Decompose finite morphisms into separable and purely inseparable parts, treating each case separately using properties of multi-section rings.
• Define a fan on the effective cone of a Mori dream space via strong Mori equivalence, where two line bundles are equivalent if their Zariski decomposition negative parts have the same support and their positive parts define the same Iitaka fibration.
• Establish equivalence between strong Mori equivalence and GIT equivalence for line bundles on Mori dream spaces via the action of the dual torus on the Cox ring.
• Use the pullback map on rational Picard groups to restrict the fan of the source to the effective cone of the target, and show this restriction equals the fan of the target.
• Apply the results to Mori dream regions by reducing to the Q-factorial case via small modifications and verifying that the fan structure and finite generation properties descend.

Experimental results

Research questions

• RQ1 Is the image of a Mori dream space under a surjective morphism necessarily a Mori dream space?
• RQ2 Can a canonical fan structure be defined on the effective cone of a Mori dream space that encodes Zariski decomposition data?
• RQ3 Is the fan structure on the effective cone of a Mori dream space equivalent to the variation of GIT quotients of its Cox ring?
• RQ4 Does the fan of the target space under a surjective morphism between Mori dream spaces coincide with the restriction of the source's fan to the target's effective cone?
• RQ5 To what extent do these fan-theoretic and GIT-theoretic structures extend to Mori dream regions, and what are the limitations?

Key findings

• The image of a Mori dream space under a surjective morphism between normal Q-factorial projective varieties is again a Mori dream space.
• A canonical fan structure on the effective cone of a Mori dream space exists, where each cone corresponds to a class of line bundles that are strongly Mori equivalent.
• Strong Mori equivalence—defined by matching support of negative parts and Iitaka fibration of positive parts in Zariski decomposition—is equivalent to GIT equivalence of the corresponding characters in the Cox ring.
• For a surjective morphism f: X → Y between Mori dream spaces, the fan of Y is equal to the restriction of the fan of X to the effective cone of Y.
• The results extend to Mori dream regions, where the restriction of a Mori dream region in the source to the effective cone of the target yields a Mori dream region in the target.
• The fan structure does not generally extend to arbitrary Mori dream regions due to issues with unstable loci and the failure of GIT equivalence to match strong Mori equivalence in the absence of ample divisors.