The third and fourth pluricanonical maps of threefolds of general type

Abstract

For a nonsingular projective threefold of general type X over the field of complex numbers, we show that the fourth pluricanonical map ϕ 4 is not birational onto its image if and only if X is birationally fibred by (1, 2)-surfaces, provided that vol(X ) 30 3 . We also have similar characterization of birationality of ϕ 3 .