On the Anti-canonical Geometry of Weak Q-fano Threefolds, Ii

By a canonical (resp. terminal) weak Q-Fano 3-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is QCartier, nef and big. For a canonical weak Q-Fano 3-fold V , we show that there exists a terminal weak Q-Fano 3-fold X , being birational to V , such that the m-th anti-canonical map defined by | −mKX | is birational for all m ≥ 52. As an intermediate result, we show that for any K-Mori fiber space Y of a canonical weak Q-Fano 3-fold, the m-th anti-canonical map defined by | −mKY | is birational for all m ≥ 52.