On the anti-canonical geometry of Q-Fano threefolds

For a Q-Fano 3-fold X on which KX is a canonical divisor, we investigate the geometry induced from the linear system -mKX and prove that the anti-m-canonical map φ - m is birational onto its image for all m ≥ 39. By a weak Q-Fano 3-fold X we mean a projective one with at worst terminal singularities on which -KX is Q-Cartier, nef and big. For weak Q-Fano 3-folds, we prove that φ - m is birational onto its image for all m ≥ 97.