Explicit birational geometry of threefolds of general type

Abstract

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar and S. Mori). We also prove that the canonical volume has an universal lower bound $\text{Vol}(V)\ge 1/2660$ and that the pluri-canonical map ${\Phi }_m$ is birational onto its image for all $m\ge 77$. As an application of our method, we prove Fletcher's conjecture on weighted hyper-surface 3-folds with terminal quotient singularities. Another featured result is the optimal lower bound $\text{Vol}(V)\ge 1/420$ among all those 3-folds $V$ with $\chi ({\mathcal{O}}_V)\le 1$.

(updated version on October 15, 2007) 55 pages, a couple of missing P_2 terms in Section 5 added and slight rearrangements to the context

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