Explicit study of rationality of Q-Fano varieties and its related topics

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We studied 85 families of Q-Fano 3-fold weighted complete intersections of codimension 2 with the aim to determine their birational Mori fiber structures. As a result, we proved that a general member belonging to 19 families is birationally rigid, and every (quasismooth) member belonging to 14 families has exactly two birational Mori fiber structures. This in particular implies nonrationality of general members belonging to 33 = 19 + 14 families. We also studied 35 families of Q-Fano 3-fold weighted hypersurfaces and determines (non)rationality of generic members belonging to 34 families.