Canonical stability in terms of singularity index for algebraic   threefolds
  Let X be a projective 3-fold with at most Q-factorial terminal singularities on which K_X is nef and big. Suppose the canonical index r(X)>1. For any positive integer m, it is interesting to consider the base point freeness and birationality 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 our method can tell here, |mK_X| gives a birational map for all m>2r+5. (Q-divisor method + patient calculation) 
Meng Chen