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} $\delta (V):=\text{min}\{m|P_m\ge 2\}$ since $1\le \delta (V)\le 18$ due to our previous series (I, II). Based on our further classification to 3-folds with $\delta (V)\ge 13$ and an intensive geometrical investigation to those with $\delta (V)\le 12$, we prove that $\text{Vol}(V)\ge \frac{1}{1680}$ and that the pluricanonical map ${\Phi }_m$ is birational for all $m\ge 61$, 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 $p_g\ge 2$ in the last section.

Final version, 44 pages. Compositio Math (to appear)

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