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

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 $m_0\le 24$. A direct consequence is the birationality of the pluricanonical map ${\phi }_m$ for all $m\ge 126$. Besides, the canonical volume $\text{Vol}(V)$ has a universal lower bound $\nu (3)\ge \frac{1}{63\cdot 126^2}$.

29 pages, Ann Sci Ecole Norm Sup (to appear)