On Classification of Q-fano 3-folds with Gorenstein Index 2 and Fano Index

We generalize the theory developed by K. Takeuchi in [T1] and restrict the birational type of a Q-factorial Q-Fano 3-fold X with the following properties: (1) the Picard number of X is 1; (2) the Gorenstein index of X is 2; (3) the Fano index of X is 1 2 ; (4) h(−KX) ≥ 4; (5) there exists an index 2 point P such that (X, P ) ≃ ({xy + f(z, u) = 0}/Z2(1, 1, 1, 0), o) with ordf(Z, U) = 1. This gives an effective bound of the value of (−KX ) 3 and h(−KX ) for a Qfactorial Q-Fano 3-fold X with (1)∼(4) by a deformation theoretic result of T. Minagawa in [Mi2]. Furthermore based on the main result, we prove that if X is a Q-factorial Q-Fano 3-fold with (1)∼(4) and with only 1 2 (1, 1, 1)-singularities as its non Gorenstein points, then (1) | − KX | has a member with only canonical singularities; (2) for any 1 2 (1, 1, 1)-singularity, there is a smooth rational curve l through it such that −KX .l = 1 2 ; (3) by a blow up at a 1 2 (1, 1, 1)-singularity, a flopping contraction and a smoothing of Gorenstein singularities, X can be transformed to a Q-Fano 3-fold with (1)∼(4) and with only 1 2 (1, 1, 1)-singularities as its singularities; (4) X can be embedded into a weighted projective space P(1, 2 ), where h := h(−KX) and N is the number of 1 2 (1, 1, 1)-singularities on X.