Maps of Mori dream spaces

TL;DR

This paper establishes a unique Cox lift for maps between Mori dream spaces, linking them to homogeneous homomorphisms of Cox rings via stack constructions, even when the target space is singular.

Contribution

It proves the existence and uniqueness of a Cox lift for maps between Mori dream spaces using stack and root constructions, extending the understanding of Cox ring homomorphisms.

Findings

Existence of a unique Cox lift for maps between Mori dream spaces.

Construction of the Cox lift via root constructions on stacks.

Connection between the Cox lift and the original map through coarse moduli spaces.

Abstract

Given a map $\phi: X \to Y$ 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, $\phi$ is induced from $\Phi$ by passing to coarse moduli spaces.