Birationally rigid Pfaffian Fano 3-folds

We classify birationally rigid orbifold Fano 3-folds of index one delined by 5 x 5 Pfaf- ffans. We give a sharp criterion for birational rigidity of these families based on the type of singularities that the varieties admit. Various conjectures are born out of our study, highlighting a possible approach to the classification of terminal Fano 3-folds. The birationally rigid cases are the first known rigid examples of Fanos that are not (weighted) complete intersection.