STABLE RATIONALITY OF ORBIFOLD FANO 3-FOLD HYPERSURFACES

We determine the rationality of very general quasi-smooth Fano 3-fold 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 3-folds, (ii) among the 27 families of Fano 3-fold weighted hypersurfaces of index greater than 2, very general members of 7 specific families are not stably rational, and the remaining 20 families consist of rational varieties.