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Let X be a projective 3-fold with at most Q-factorial terminal singularitieson which K_X is nef and big. Suppose the canonical index r(X)>1. For anypositive integer m, it is interesting to consider the base point freeness andbirationality of the divisor mK_X. For example, we know the following results:  (1) the system |5rK_X| is base point free (Ein-Lazarsfeld-Lee);  (2) |mK_X| gives a birational map for all m>4r+2 (M. Hanamura).  This article aims to present a better result in direction (2). As far as ourmethod can tell here, |mK_X| gives a birational map for all m>2r+5. (Q-divisormethod + patient calculation)Comment: 27 pages, The final version, Accepted for publication in Math. Proc.  Camb. Phil. So

Canonical stability in terms of singularity index for algebraic threefolds