Maps of Mori Dream Spaces in Cox coordinates. Part I: existence of descriptions

TL;DR

This paper extends the concept of describing rational maps via Cox coordinates from classical varieties to Mori Dream Spaces, demonstrating the existence of finite Cox ring extensions that facilitate such descriptions.

Contribution

It proves the existence of finite Cox ring extensions for Mori Dream Spaces that enable the lifting of regular maps, involving only roots of homogeneous elements.

Findings

Existence of finite Cox ring extensions for Mori Dream Spaces.

Lifting of regular maps to Cox rings using roots of homogeneous elements.

Practical applicability in computations of maps between Mori Dream Spaces.

Abstract

Any rational map between affine spaces, projective spaces or toric varieties can be described in terms of their affine, homogeneous, or Cox coordinates. We show an analogous statement in the setting of Mori Dream Spaces. More precisely (in the case of regular maps) we show there exists a finite extension of the Cox ring of the source, such that the regular map lifts to a morphism from the Cox ring of the target to the finite extension. Moreover the extension only involves roots of homogeneous elements. Such a description of the map can be applied in practical computations.