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 {\it pluricanonical section index} $\delta(V):=\text{min}\{m|P_m\geq 2\}$ since $1\leq \delta(V)\leq 18$ due to our previous series (I, II). Based on our further classification to 3-folds with $\delta(V)\geq 13$ and an intensive geometrical investigation to those with $\delta(V)\leq 12$, we prove that $\text{Vol}(V) \geq \frac{1}{1680}$ and that the pluricanonical map $\Phi_{m}$ is birational for all $m \geq 61$, which greatly improves known results. An optimal birationality of $\Phi_m$ for the case $\delta(V)=2$ is obtained. As an effective application, we study projective 4-folds of general type with $p_g\geq 2$ in the last section.