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We highlight a relation between the existence of Sarkisov links and thefinite generation of (certain) Cox rings. We introduce explicit methods to usethis relation in order to prove birational rigidity statements. To illustrate,we complete the birational rigidity results of Okada for Fano completeintersection 3-folds in singular weighted projective spaces.

Mori dream spaces and birational rigidity of Fano 3-folds