Maps of Mori dream spaces

Abstract

Given a map $\varphi :X\to Y$ of $\mathbb{Q}$-factorial Mori dream spaces, one can ask whether this map is induced by a homogeneous homomorphism $R(Y)\to R(X)$ of Cox rings. As soon as $Y$ is singular, such a homomorphism needs not to exist, as pulling back Weil divisors is not well-defined. In this article, we prove that there is a unique Cox lift $\Phi :\mathcal{X}\to \mathcal{Y}$ of Mori dream stacks coming from a homogeneous homomorphism $R(Y)=R(\mathcal{Y})\to R(\mathcal{X})$, where $\mathcal{Y}$ is a canonical stack to $Y$ and $\mathcal{X}$ is obtained from $X$ by root constructions. Moreover, $\varphi$ is induced from $\Phi$ by passing to coarse moduli spaces.

20 pages, content identical to published version

References in corpus (3)

• Smooth toric DM stacks
• Functorial destackification of tame stacks with abelian stabilisers
• Mori Dream Stacks

Cited by in corpus (3)

• Cox rings of algebraic stacks
• Selected topics on Toric Varieties
• Cox Rings and Algebraic Maps