Explicit birational geometry of 3-folds and 4-folds of general type, III

Abstract

Nonsingular projective 3-folds V of general type can be naturally classified into 18 families according to the pluricanonical section index δ(V ) := min{m | Pm ≥ 2} since 1 ≤ δ(V ) ≤ 18 due to our previous series (I, II). Based on our further classification to 3-folds with δ(V ) ≥ 13 and an intensive geometrical investigation to those with δ(V ) ≤ 12, we prove that Vol(V ) ≥ 1/1680 and that the pluricanonical map Φm is birational for all m ≥ 61, which greatly improves known results. An optimal birationality of Φm for the case δ(V ) = 2 is obtained. As an effective application, we study projective 4-folds of general type with pg ≥ 2 in the last section.