On Q-Fano 3-folds of Fano index 2
Let X be a Q-Fano 3-fold of rank 1 with qQ(X)=qF(X)=2 and KX not Cartier. Let A be a Weil divisor on X such that −KX = 2A. Then dim |A| ≤ 4. Moreover, if dim |A| = 4, then X belongs to the single irreducible family.