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

TL;DR

This paper classifies nonsingular projective 3-folds of general type into 18 families based on the pluricanonical section index, establishing volume bounds and birationality results, and applies findings to 4-folds with geometric genus at least 2.

Contribution

It provides a detailed classification of 3-folds by pluricanonical section index and proves new bounds on volume and birationality, improving previous results.

Findings

Volume of 3-folds is at least 1/1680.

Pluricanonical map $ ext{Phi}_m$ is birational for all $m \,\geq\, 61$.

Optimal birationality achieved for $ ext{delta}(V)=2$.

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.