On images of Mori dream spaces

TL;DR

This paper investigates the geometric properties of images of Mori dream spaces, establishing their stability under surjective morphisms, and introduces a fan structure on their effective cones that encodes key geometric and algebraic information.

Contribution

It proves that images of Mori dream spaces under surjective morphisms are Mori dream spaces and introduces a fan structure on their effective cones that captures Zariski decompositions and GIT variation.

Findings

Images of Mori dream spaces under surjective morphisms are Mori dream spaces.

A natural fan structure on the effective cone encodes Zariski decompositions.

The fan of the target space coincides with the restriction of the source's fan under surjective morphisms.

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.