Mori dream spaces and birational rigidity of Fano 3-folds

TL;DR

This paper explores the connection between Sarkisov links and Cox rings to establish birational rigidity of Fano 3-folds, providing explicit methods and completing previous rigidity results.

Contribution

It introduces new methods linking Sarkisov links and Cox rings to prove birational rigidity of Fano 3-folds, extending prior results.

Findings

Established a relation between Sarkisov links and Cox rings.

Developed explicit techniques for birational rigidity proofs.

Completed rigidity results for Fano complete intersections in weighted spaces.

Abstract

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