We classify birationally rigid orbifold Fano 3-folds of index 1 defined by 5 × 5 Pfaffian varieties. We give a sharp criterion for the 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 Fano varieties that are not (weighted) complete intersections.

In this paper, we consider a double-struckQ‐Fano 3‐fold weighted complete intersection of codimension 2 in the 85 families listed in Iano‐Fletcher's list and determine which cycle is a maximal center or not. For each maximal center, we construct either a birational involution which untwists the maximal singularity or a Sarkisov link centered at the cycle to another explicitly described Mori fiber space. As a consequence, nineteen families are proved to be birationally rigid and the remaining 66 families are proved to be birationally non‐rigid.

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We determine the rationality of very general quasismooth Fano 3fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted hypersurfaces of index 1 or 2 are not stably rational except possibly for the cubic threefolds, (ii) among the 27 families of Fano 3-fold weighted hypersurfaces of index greater than 2, very general members of specific 7 families are not stably rational and the remaining 20 families consists of rational varieties.

We prove a relation between birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas [RT07].