The Cox ring of an embedded variety
Crist\'obal Herrera, Antonio Laface, Luca Ugaglia

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 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 is the intersection of finitely many localizations of a quotient image of the Cox ring of . As a consequence, we provide an algorithm that terminates if and only if the Cox ring of 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.
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