On generic finiteness of pluricanonical maps of threefolds of general type

We prove that $\mathrm{\mid }6K_X\mathrm{\mid }$ defines a generically finite map for all minimal 3-folds $X$ of general type with $P_2(X)\ge 2$, which is optimal. We also prove that $\mathrm{\mid }n{K}_X\mathrm{\mid }$ defines a generically finite map for all minimal 3-folds $X$ of general type when $n\ge 38$. The essential technical ingredients of this paper are a new generic finiteness criterion for surfaces and an effective comparison inequality under a special resolution.