The Cox ring of an embedded variety

TL;DR

This paper presents a method to compute the Cox ring of an embedded variety within a Mori dream space, providing an algorithm for finite generation and applications to hypersurfaces in toric varieties.

Contribution

It introduces a new approach to determine the Cox ring of embedded varieties, generalizing previous results and offering an explicit algorithm for finite generation.

Findings

Cox ring of X is the intersection of finitely many localizations of a quotient of Z's Cox ring.

The algorithm terminates if and only if the Cox ring of X is finitely generated.

Applied to compute Cox rings of hypersurfaces in smooth projective toric varieties.

Abstract

We compute the Cox ring of an embedded variety $X \subseteq Z$ within a Mori dream space, under the assumption that the pullback map induces an isomorphism at the level of divisor class groups. We show that the Cox ring of $X$ is the intersection of finitely many localizations of a quotient image of the Cox ring of $Z$. As a consequence, we provide an algorithm that terminates if and only if the Cox ring of $X$ is finitely generated, thereby generalizing previous works on the subject. We apply these results to compute the Cox ring of hypersurfaces in smooth projective toric varieties.