Explicit birational geometry of 3-folds of general type, I

TL;DR

This paper establishes positivity results for pluricanonical sections of complex 3-folds of general type, proving birationality of pluricanonical maps for large m, and providing a universal lower bound for volume.

Contribution

It proves the existence of nontrivial pluricanonical sections at specific levels and shows the birationality of pluricanonical maps for all m ≥ 126, advancing the understanding of 3-folds of general type.

Findings

P_{12}(V) > 0, indicating nontrivial sections at level 12

P_{m_0}(V) > 1 for some m_0 ≤ 24

Birationality of al_m for all m  126

Abstract

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=\text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0\leq 24$. A direct consequence is the birationality of the pluricanonical map $\varphi_m$ for all $m\geq 126$. Besides, the canonical volume $\text{Vol}(V)$ has a universal lower bound $\nu(3)\geq \frac{1}{63\cdot 126^2}$.